NCAR/TN-331+STRNCAR TECHNICAL NOTE

May 1989

Signal Processing

for Atmospheric Radars

R. Jeffrey KeelerRichard E. Passarelli

ATMOSPHERIC TECHNOLOGY DIVISION

NATIONAL CENTER FOR ATMOSPHERIC RESEARCHBOULDER, COLORADO

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iI

TBSIE OF COTENTS

TABLE OF CONTENTS ..................... . iii

LIST OF FIGURES ......................... v

LIST OF TABLES ................... . .. .vii

PREFACE. .. .. i.......................

1. Purpose and scope ................. 1

2. General characteristics of atmospheric radars. 32.1 Characteristics of processing .......... 32.1.1 Sampling ................... 32.1.2 Noise ...... ............... 42.1.3 Scattering ............... 52.1.4 Signal to noise ratio (SNR) .......... 62.2 Types of atmospheric radars .......... 62.2.1 Microwave radars . ........... 72.2.2 ST/MST radars or wind profilers ....... 82.2.3 FM-CW radars .. .............. 82.2.4 Mobile radars ............ .. 92.2.5 Lidar ............ ....... 102.2.6 Acoustic sounders . .......... 11

3. Doppler power spectrum moment estimation . .... 133.1 General features of the Doppler power spectrum. 143.2 Frequency domain spectral moment estimation . 183.2.1 Fast Fourier transform techniques . .... 183.2.2 Maximum entropy techniques .......... 203.2.3 Maximum likelihood techniques . ....... 233.2.4 Classical spectral moment computation ..... 253.3 Time domain spectral moment estimation. ..... 273.3.1 Geometric interpretations ........... 273.3.2 "Pulse pair" estimators ........... 283.3.3 Circular spectral moment computation for

sampled data. . ............. 313.3.4 Poly pulse pair techniques ..... 333.4 Uncertainties in spectrum moment estimators . . 353.4.1 Reflectivity. ... ............ 353.4.2 Velocity. . . ..... .. ..... 363.4.3 Velocity spectrum width ........... 37

4. Signal processing to eliminate bias and artifacts. 434.1 Doppler techniques for ground clutter suppression 434.1.1 Antenna and analog signal considerations. ... 444.1.2 Frequency domain filtering. ......... 454.1.3 Time domain filtering ............. 464.2 Range/velocity ambiguity resolution ....... 504.2.1 Resolution of velocity ambiguities ...... 51

iii

4.2.2 Resolution of range ambiguities ....... 554.3 Polarization switching consequences ....... 56

5. Exploratory signal processing techniques . .... 575.1 Pulse compression .... .......... 575.1.1 Advantages of pulse compression . ...... 585.1.2 Disadvantages of pulse compression. ...... 595.1.3 Ambiguity function. .. 615.1.4 Comparison with multiple frequency scheme . 635.2 Adaptive filtering algorithms ......... 635.2.1 Adaptive filtering applications ....... 645.2.2 Adaptive antenna applications .. ..... 685.3 Multi-channel processing. ............ 695.4 A priori information. ............. 70

6. Signal processor implementation ......... 716.1 Signal processing control functions ..... 716.2 Signal Z?D conversion and calibration ...... 746.3 Reflectivity processing ... .......... 766.4 Thresholding for data quality ......... 78

7. Trends in signal processing. ............ 817.1 Realization factors ............... 817.1.1 Digital signal processor chips ....... 817.1.2 Storage media ................. 827.1.3 Display technology . .............. 837.1.4 Commercial radar processors .......... 837.2 Trends in programmability of DSP. ........ 847.3 Short term expectations .......... .... 857.3.1 Range/velocity ambiguities ......... 857.3.2 Ground clutter filtering .......... 867.3.3 Waveforms for fast scanning radars ...... 867.3.4 Data compression. ............. 877.3.5 Artificial intelligence based feature extraction 877.3.6 Real time 3D weather image processing .. ... 877.4 Long term expectations . ............ 877.4.1 Advanced hardware ...... . .. 887.4.2 Optical interconnects and processing ..... 887.4.3 Communications . . . . . . . . . . . . . . . . 887.4.4 Electronically scanned array antennas ..... 887.4.5 Adaptive systems ............... 89

8. Conclusions. . ................... 918.1 Assessment of our past. ............. 918.2 Recommendations for our future . ........ 928.3 Acceptance of new techniques ........... 938.4 Acknowledgements. .............. 93

ACRONYM LIST ........................... 95

BIBLIOGRAPHY ....... ........... .... . . 97

iv

TIST OF FJIGRES

Fig 3.1 Doppler power spectrum (128 point periodogram) of 15typical weather echo in white noise. Estimatedparameters are velocity ~ 0.4 Vax velocity spectrumwidth ~ .04 Vmax, and SNR 10 dB.

Fig 3.2 Three dimensional representation of the complex 29autocorrelation function as a helix. Radius of helixRs(0) is proportional to total signal power, Ps;rotation rate of helix is proportional to velocity, V;width of envelope is inversely proportional to velocityspectrum width, W. Delta function Rn(0) representsnoise power.

Fig 3.3 Periodogram power spectrum plotted on unit circle in the 32z-plane. Note velocity aliasing point, the Nyquistvelocity, at z=-l.

Fig 3.4 Comparison of classical and circular (pulse pair) first 34moment estimators. Classical estimate is determined bylinear weighting of spectrum estimate and circularestimate, by sinusoidal weighting.

Fig 3.5 Velocity error as function of spectrum width and SNR. 39Spectrum width is normalized to Nyquist interval,vn=W/2Vmx=2WTs/X. M is number of sample pairs and

error is normalized to Nyquist velocity interval, 2va =2Vmax. Small circles represent simulation values(Doviak and Zrnic, 1984).

Fig 3.6 Width error as a function of spectrum width and SNR. 42Spectrum width is normalized to Nyquist interval,

vn=W/2Vmax=2Wrs/X. M is number of sample pairs anderror is normalized to Nyquist interval, 2Vmax. Smallcircles represent simulation values (Doviak and Zrnic,1984).

Fig 4.la Clutter filter frequency response for a 3 pole infinite 47impulse response (IIR) high pass elliptic filter. Forground clutter width of 0.6 ms- 1 and scan rate of 5 rpmthis filter gives about 40 dB suppression. V = stopband. Vp = pass band cutoff, Vmax = 16 ms- (Hamidiand Zrnic, 1981).

Fig 4.lb Implementation of 3rd order IIR clutter suppression 48filter; z-1 is 1 PRT delay. K1 - K4 are filter coeffi-cients (Hamidi and Zrnic, 1981).

v

Fig 5.1 Ambiguity diagram for single FM chirped pulse waveform 62with TB=10. T is range dimension. 0 is velocitydimension. Targets distributed in (r,q) space contributeto the filter output proportional to the ambiguityfunction. For atmospheric targets, Doppler shiftfrequencies are typically very small relative to pulsebandwidth (Rihaczek, 1969).

Fig 5.2 Prediction error surface for 2 weight adaptive filter. 65The LMS algorithm estimates the negative gradient of thequadratic error and steps toward the minimum mean squareerror (mse). The optimum weight vector is W* = (0.65,-2.10). If the input statistics change so that theerror surface varies with time, the adaptive weightswill track this change (Widrow and Stearns, 1985).

Fig 5.3 Adaptive filter structure. The desired response (dk) is 66determined by the application. The adaptive filter.coefficients (Wk) and/or the output signal (Yk) are theparameters used for spectrum moment estimation (Widrowand Stearns, 1985).

Fig 6.1 Block diagram of a typical signal processor. 26

vi

LSTr OF TAHBI

Table 1 Comparison of remote sensor sampling schemes and rates. 7

Table 2 Characteristics of several popular windows when applied 20to time series data analysis (Marple, 1987).

Table 3 Expressions for variance of velocity estimators at high 38SNR. Assumes Gaussian spectra in white noise, lownormalized velocity width (Wn=W/2Vmx) and large M.Expressions apply to both pulse pair and Fouriertransform estimators.

Table 4 Expressions for variance of width estimators at high 41SNR. Assumes Gaussian spectra in white noise, lownormalized velocity width (Wn=W/2Vmax) and large M.Expressions apply to both pulse pair and Fouriertransform estimators.

vii

PiRFACE

This review of signal processing for atmospheric radars was originally

written as Chapter 20 of the book Radar in Meteorology, edited by Dave Atlas

(1989) for the Proceedings of the 40th Anniversary and Louis Battan Memorial

Radar Meteorology Conference. We have attempted to give the reader an

overview of signal processing techniques and the technology that are

applicable to the atmospheric remote sensing tools of weather radar, lidar,

ST/MST radars and wind profilers.

This NCAR Technical Note includes the signal processing chapter and the

relevant references in a single document. The text has had minor editing

and the references have been slightly expanded over the version published in

Radar in Meteorology.

We hope that this Technical Note will assist the many individuals who want a

better understanding of signal processing to achieve that goal.

R. Jeffrey Keeler

Richard E. Passarelli

March 1989

ix

1. PURPOSE AND SODFE

Signal processing is perhaps the area of atmospheric remote sensing where

science and engineering make their point of closest contact. Signal

processing offers challenges to engineers who enjoy developing state-of-the-

art systems and to scientists who enjoy being at the crest of the wave in

observing atmospheric phenomena in unique ways.

The primary function of radar signal processing is the accurate, efficient

extraction of information from radar echoes. A typical pulsed Doppler radarsystem samples data at 1000 range bins at 1 kilohertz pulse repetition

frequency (PRF), generating approximately 3 million samples per second

(typically in-phase (I) and quadrature phase (Q) components from a linear

channel and often a log receiver). These "time series", in their raw form,

convey little information that is of direct use in determining the state of

the atmosphere. The volume of time series data is sufficiently large that

storage for later analysis is impractical except for limited regions of time

and space. The data must be processed in real time to reduce its volume and

to convert it to more useful form.

In this paper the current state of signal processing for atmospheric radars

(weather radars, ST/MST radars or wind profilers, and lidars) shall be

discussed along with how signal processing is currently optimized for

various applications and remote sensors. The focus shall be on signal

processing for weather radar systems but the techniques and conclusions

apply equally well to ST/MST radars and lidars. Zrnic (1979a) has given an

excellent review of spectral moment estimation for weather radars and

Woodman (1985) has done the same for MST radars. Problem areas and

promising avenues for future research shall be identified. Finally, we

shall discuss the scientific and technological forces that are likely to

shape the future of atmospheric radar signal processing.

We will differentiate between "signal processing" (the topic of this review)

and "data processing" in the following way. "Signal processing" is that set

1

of operations performed on the analog or digital signals for efficiently

extracting desired information or measuring some attribute of the signal.

For atmospheric radars this information is often referred to as the "base

parameter estimates". Fundamental base parameters are:

Radar reflectivity factor Z dBZRadial velocity V ms- 1

Velocity spectrum width1 W ms-l

In the course of extracting these estimates, signal processing algorithms

will improve the signal to noise ratio (SNR) through filtering or averaging,

mitigate the effects of interfering echoes such as ground clutter, remove

ambiguities such as range or velocity aliasing, and reduce the input data

rate by a significant factor. The end result of an effective signal

processing scheme is to provide minimum mean squared error estimates of the

base parameters along with the expected error or a measure of the degree of

confidence that can be placed on the estimates (e.g., the SNR). Note that

signal processing is primarily used in atmospheric remote sensing as an

estimation procedure as well as a detection process as in some aviation

applications. The emphasis is on making estimates of atmospheric parameters

or meteorological events.

"Data processing", on the other hand, takes up where signal processing

leaves off -- although the line of demarcation is not razor sharp. Data

processing algorithms take the base parameter estimates and further process

them so that they convey information that is of direct use to the radar

user. For example, data processing techniques imply display generation,

data navigation to a desired coordinate system, wind profile analyses, data

syntheses from several Doppler radars or other sensors, applying physical

constraints to the measured data, and forecasts or "nowcasts" of severe

weather hazards. Many aspects of data processing are covered in other

chapters.

1 The width is defined as the square root of the secondcentral moment of the spectral power distribution.

2

2. GENRA CIRACJERSLR'LCS OF AICMY4SERIC RADARS

There are two main classes of "radar" -- electromagnetic and acoustic.

Electromagnetic radars include microwave, UHF, VHF, infrared and optical

systems. Acoustic radars are only briefly described here. The signal

processing techniques employed for all these systems are similar (Serafin

and Strauch, 1978).

2.1 ClARACTERISTICS OF IRDCESSING

Although the processing techniques are nearly identical for the various

atmospheric radars, the way in which this backscattered or partially

reflected radiation is sampled, the principle noise sources, and the nature

of the scattering mechanisms are different.

2.1.1 Sampling

Because electromagnetic radars employ wavelengths from several meters to

less than 1 im, they must use different sampling techniques. There are two

constraints on the sample time spacing (Ts) of the backscattered signal.

The first is that the backscattered signal should be coherent from sample-

to-sample, i.e., the motion among the scatterers should be small compared to

the wavelength so that their relative positions produce highly correlated

echoes from sample-to-sample. The nominal duration of this correlation is

called the coherence time (Nathanson, 1969), i.e.,

Ts < tcoh = /4rW (2.1)

where the true velocity spectrum width W in ms-1 is a direct measure of the

relative motions of the scatterers. The coherence time is a measure of the

maximum time between successive samples for coherent phase measurements.

Thus, for short wavelength systems, such as a lidar, the backscattered

signal must be sampled much more rapidly than for a longer wavelength

microwave system. The autocorrelation function (defined later) can provide

a direct measure of the coherence time of a fluctuating target echo.

3

The second constraint on sampling is that for regularly spaced pulses, the

sampling frequency must be at least twice the maximum desired Doppler shift

frequency which reduces the occurrence of velocity aliasing. In this case

the time between samples is governed by,

Ts < tNyq = 4V' (2.2)

where tNyq is the minimum time between samples such that the desired

velocity V' is at least the so-called Nyquist velocity. Since V' is

typically much larger than W, the latter constraint usually dominates the

sampling requirement. In fact, if we assume the desired maximum velocity is

+ 25 ms-1 , then Ts V/100 or PRF = 100/ \ is a useful rule of thumb.

2.1.2 Noise

One of the goals of signal processing is to suppress the effects of noise.

The main source of noise in microwave radar is thermal in nature. This noise

power is simply

Pn = k Tsys Bsys (2.3)

where k is Boltzman's constant (1.38 x 10-23 W/Hz/°K), Tsys is the total

system temperature, and Bsys is the total system bandwidth including effects

of preselector filters, IF filters, and all other amplifiers in the signal

path (Skolnik, 1970, 1980; Paczowski and Whelehan, 1988). With recent

improvements in low noise amplifiers (INA's), little room is left for

sensitivity improvement in conventional radar receivers. Presently, most

microwave radar systems are sufficiently sensitive that thermal radiation

from the earth makes a strong contribution to the receiver input at low

elevation angles.

ST/MST radar noise, because of its lower frequency, has a large contribution

from environmental, cosmic and atmospheric sources, and is not easily

quantified (Rottger and Larsen, Chap 21A). Therefore, antenna design and

the specific radar location and frequency band of operation define the

system noise.

4

Coherent lidar systems utilize detection schemes using optical heterodyning

onto cryogenic detectors with a local oscillator laser having relatively

high power mixing with the weak atmospheric return (Jelalian, 1980,

1981a,b). Because of the small wavelengths, quantum effects dominate the

detection process associated with random photon arrivals impacting the LD

laser. This "shot noise" contribution is a fundamental physical limitation

of lidar sensitivity.

2.1.3 Scattering

Atmospheric radars respond to a variety of scattering targets--

precipitation, cloud particles, aerosols, refractive index variations,

chaff, insects, birds, and ground targets. Probert-Jones (1962) derived the

familiar radar equation most often used by radar meteorologists for

precipitation scattering. A detailed derivation can be found in Doviak and

Zrnic (1984), Battan (1973), or Atlas (1964). The received power is

Pt G2 02 cTr 3 1k12 Ze L (2.4)Pr=

1024 ln2 X2 R2

This equation includes L, the product of several small but significant loss

terms which are necessary to accurately estimate radar reflectivity factor,

e.g. receiver filter loss, propagation loss, blockage loss, and processing

bias. Zric (1978) defines the receiver filter loss as that portion of the

input signal frequencies not passed by the finite receiver bandwidth,

typically 1-3 dB. The other losses depend on atmospheric conditions and

antenna pointing and are enumerated in Skolnik (1980). This equation is

correct for Rayleigh scattering of a distributed target that completely

fills the resolution volume. Non-Rayleigh targets or partially filled

resolution volumes will give received power estimates that cannot accurately

be related to precipitation rate. Rottger and Larsen (Chap. 21A) and

Huffaker, et al. (1976, 1984) give similar received power expressions for

returns from refractive index variations and from lidar aerosol returns,

respectively.

5

The required dynamic range for measuring the backscattered power from

atmospheric targets is very large because:

1. The effective backscatter cross-sections of atmospheric scatterers span

dynamic ranges of approximately 60 dB for precipitation but much larger

if cloud particle, "clear air", and ground target returns are included.

2. The R 2 dependence of the received power for distributed targets spans

a range of 50 dB between 1 and 300 km.

Microwave systems should accommodate the sum of these two effects and

typically can achieve a dynamic range of order 100 dB for power measurements

using either a log receiver, linear receiver with AGC, or some combination

of these.

2.1.4 Signal to noise ratio (SNR)

The ratio of the received signal power to the measured noise power is

defined to be the signal to noise ratio (SNR):

SNR = Pr/Pn (2.5)

The SNR is extremely important for analyzing tradeoffs in signal processing.

It is a key term along with spectrum width and integration time in analytic

evaluation of spectrum moment errors.

2.2 YPES OF AM[SHFERIC RAIDRS

A summary of the characteristics of the different types of electromagnetic

radars in use today for atmospheric research is discussed below. Table 1

assembles these differences.

6

Table 1

Remote Sensor Sampling Comparison

Pulse SampleSensor Wavelength Scatterers Beamwidth Duration Rate

(deg) (~sec) (Hz)RadarS-band 10cm Precipitation 0.5-3 0.25-4 103Ka-band 1 cm Precipitation 0.5-2 0.25-1 104mm-band 1 mm Cloud 0.2-1 0.25-1 105

ST/MST (profilers)UHF 75 cm Refractive 3-10 0.2-5 104->102VHF 6 m index 3-10 0.2-5 103->10

LidarIR 10 gmi Aerosols 0.01 0.1-3 107Optical <1 im Molecules (near field) <1

2.2.1 Microwave radars

Microwave pulsed radars radiate fields with wavelengths between 20 cm and 1

mm and are commonly used as "weather radars" (Smith, et al., 1974; Doviak,

et al., 1979). Depending on the wavelength, primary scattering is from

precipitation, insects (Vaughan,1985), refractive index fluctuations, and

cloud particles. Beams are typically circular in cross section with widths

0.5 to 3 degrees and the maximum usable ranges for storm observation is 200-

500 km. After a few kilometers range, the pulse volume is "pancake" shaped,

i.e., the pulse depth in range is small compared to the distance across the

beam. Attenuation effects range from severe for millimeter wavelength

systems, to nearly insignificant for 10 cm S-band systems.

Most centimeter wavelength microwave systems collect coherent samples over

several milliseconds. Millimeter wavelength radars can make use of the

double pulsing technique (Campbell and Strauch, 1976) to assure coherence

and to reduce an otherwise intolerable range ambiguity problem. Doviak and

Zrnic (1984) and Strauch (1988) have shown that since only the second pulse

of a double pulsing radar may be contaminated by overlaid echo from the

first pulse of the pair, only random errors occur in the pulse to pulse

correlations. These random errors may change very slowly with time so they

would appear to be systematic (bias) errors at a given time.

7

2.2.2 ST/ST radars or wind profilers

VHF and UHF radars which probe the mesosphere, stratosphere and/or the

troposphere are called ST/MST radars and sometimes known as wind profilers,

observe radial winds at wavelengths between 30 cm and 6 m at near vertical

incidence (Gage and Balsley, 1978; Rottger, et al., 1978). Scattering is

from atmospheric refractive index fluctuations in space, analogous to Bragg

scattering. Beamwidths may be as large as several degrees for tropospheric

sounding, but much narrower beams are used for longer stratospheric and

mesospheric ranges (Rottger and Larsen, Chap. 21A; Gage, Chap. 28A).

For a nominal 1 m wavelength, the atmospheric coherence time is typically

large fractions of a second. Consequently, the sampling rate to achieve

coherence is of order 10 Hz. Because of this and the typically weak clear

air returns, it is advantageous to perform time domain averaging of the

samples from pulse-to-pulse, e.g., at a given range, N successive complex

samples are averaged to yield a single complex pair. This operation

effectively reduces the sampling frequency and the unambiguous velocity

interval by a factor of N, but the fundamental interval is usually so large

that this reduction is of little consequence. The main feature is that the

data rate is reduced by a factor of N while the SNR is improved N times

compared to the SNR of a data set sampled N times slower. The reduced data

rate permits computationally intensive processing such as FFT analysis so

that artifacts can be more easily eliminated. Doviak, et al. (1983) and

Smith (1987) describe the optimum number of samples to average given the

expected radial velocities and dispersions. Otherwise, the processing is

similar to microwave radars following conventional techniques. Rottger and

Larsen (Chap. 21A) describe the details of ST/MST radar processing

techniques.

2.2.3 FM-CW radars

FM-CW (frequency modulated continuous wave) radars have also played an

important role in boundary layer remote sensing (Richter, 1969; Chadwick, et

al., 1976; Ligthart, et al., 1984). Using an FM chirp waveform to obtain

range resolution of order 1 m and a continuous wave (CW) to achieve

8

sensitivity 30 dB greater than a comparably chirped pulse system having thesame peak power, this system has given high resolution information on thedetailed structure of the boundary layer. Individual insects are apparentlydiscernable, and can be differentiated from atmospheric refractive indexvariations. Strauch, et al. (1975) and Chadwick and Strauch (1979) havedemonstrated both theoretically and experimentally that Doppler, as well asreflectivity, information can be extracted from a distributed target usingthis pulse compression waveform at microwave wavelengths. Any pulsecompression waveform with range-time sidelobes limits the radar's

performance in strong reflectivity gradients. Alternatively, one can use

continuous, periodic, pseudo-random phase coding in a bistatic configuration

with similar advantages as Woodman (1980b) describes for the Arecibo S-band

planetary radar.

2.2.4 Mobile radars

Airborne and spaceborne radars are an important class of atmospheric remotesensors covered by Hildebrand and Moore (Chap. 22A). Special problems areevident when a moving platform supports the remote sensor. Many of thesignal processing problems have well known solutions but have not been field

tested. The basic processing algorithms are similar to those employed withground based sensors, but special processing techniques must be employed to

suppress moving ground clutter and to obtain adequate resolution andsensitivity from spaceborne instruments.

Synthetic aperture radar (SAR) techniques can be used only if the platformmoves rapidly so that atmospheric targets remain coherent during a "dwell

time", thereby giving a synthetic aperture yielding the desired along-track

resolution. SAR mapping of precipitation is possible from space vehiclesbecause of the great distance traversed by the antenna during the coherency

time of the targets (Atlas and Moore, 1987). Quantitative measurements ofprecipitation from space involve a broad range of signal processing problemsto achieve both maximum sensitivity and a sufficiently large number ofindependent samples. Obtaining reliable average echo power from individualstorm cells while covering a large cross-track swath in the short times

available to traverse a typical along-track beam width requires extremely

9

high processing rates. Research concerning atmospheric target measurements

is just beginning in this important field (Li, et al., 1987).

2.2.5 Lidar

Optical or infrared radars, cammonly known as lidars, scatter from

atmospheric aerosols at wavelengths between 10 and 0.3 microns (Huffaker,

1974-75; Huffaker, et al., 1976; Jelalian, 1977; Bilbro, et al., 1984 and

1986; and McCaul, et al., 1986). This makes them most useful in the lower

regions of the atmosphere where aerosol concentrations are the highest.

Molecular scattering dominates at the shorter wavelengths. Lidar is

severely attenuated by cloud and precipitation so it is most useful in

"clear air" applications (Lawrence, et al., 1972; McWhirter and Pike, 1978).

Lidar requires a receiving aperture several thousand wavelengths in diameter

to achieve the necessary gain and sensitivity. Consequently, many

atmospheric lidars, both ground based and airborne, operate within the

antenna (or telescope) "near field" range. A distinct advantage of this

near field operation is the collimation of the optical energy into the "near

field tube" with minimal "sidelobe" radiation. When in the far field, the

beamwidths are measured in milliradians. Maximum ranges are a few tens of

kilometers, and pulse volumes are usually elongated.

The expected Doppler shifts and coherence times require sampling at rates of

10 - 100 MHz. This means that all the information necessary for complete

spectral processing is acquired from a single pulse. This makes lidar, by

its very nature, a "fast scanning" atmospheric remote sensor. Current laser

duty cycle constraints limit PRF's to about 100 Hz, which produces data

rates that can easily be processed and recorded (Hardesty, et al., 1988;

Alldritt, et al., 1978).

An important characteristic of acquiring the data in a single pulse is the

degraded range resolution that results when the pulse propagates outward

during the data collection interval. During the sampling interval, "new"

particles are appearing at the leading edge of the illuminated volume, while

"old" particles are disappearing at the trailing edge. This creates an

10

additional contribution to the spectrum width similar to that caused by

antenna scanning for microwave radars.

2.2.6 Acoustic sounders

Acoustic radars, also known as echosondes, sodars, or acdars, are important

sensors for the boundary layer (Little, 1969). Acoustic waves are

longitudinal in nature and propagate at about 340 ms -1 . Scattering is from

temperature and velocity fluctuations caused by turbulent motion in the

atmosphere. The processing techniques, while at audio frequencies, are

similar to those employed by lidar since spectral data representative of the

scattering medium are obtained from a single pulse rather than pulse-to-

pulse sampling. Because of the slow propagation speed and small Doppler

shifts, sampling the echoes obtained from a real (single channel) data

source is possible. Thus, complex (dual channel) data processing is

avoided. Moreover, the real echoes are sampled at a rate substantially less

than the carrier frequency of the sodar so that zero Doppler shift is offset

from zero frequency. In this manner unambiguous and signed velocity

estimates can be made.

11

3. DOPLER PE SPBCRlM MMENT ErAMHATICN

It is well established that the first three moments of the Doppler power

spectral density or the "power spectrum" (incorrectly termed the "Doppler

spectrum" in the community) are directly related to the desired atmospheric

base parameters: radar reflectivity, radial velocity, and velocity spectrum

width (Rogers and Chimera, 1960; Groginsky, 1966). Before we discuss the

power spectrum and moment estimation, we shall find it useful to define the

input waveform.

Since the return from individual range cells typically is generated by

scattering from a large number of randomly distributed particles and/or

refractive index inhomogeneities, the received signal process is (by the

central limit theorem) a very good approximation to a Gaussian random

process (Parzen, 1957; Swerling, 1960; Mitchell, 1976). Thus, signal

processing techniques should be assessed in the context of a statistical

estimation theory framework wherein one seeks to make the best estimate of

the ensemble parameters given a particular sample function (Wiener, 1949;

Davenport and Root, 1958). This statistical estimation framework becomes of

particular importance when one wishes to scan a phenomenon quickly since the

random process nature of the weather signal will necessitate a certain

amount of averaging if the desired accuracies are to be achieved.

A single stationary point target at range R reproduces the transmitted

waveform after it has been filtered by the receiver

z(t,R) = A exp[j2rf(t-2R/c) W(t-2t-2R/c) (3.1)

where A is the complex voltage amplitude and W(t)is a range weighting

function that depends on the transmit pulse length and the receiver

bandwidth (Doviak and Zrnic, 1984).

Actual targets in the atmosphere are composed of many individual scatterers,

distributed over range, radar cross section, and velocity. The received

13

waveform for a particular distributed target then is a sample function of

the random process which produces the atmospheric return. We desire to

estimate the mean characteristics of the random target over an ensemble of

sample functions. The vector sum of the return complex voltage from the

individual scatterers is

z(t,R) = Z Ai exp[j2fi(t-2Ri/c)] W(t-t2Ri/c) (3.2)i

where the subscript i represents the individual particle. Each particle has

a complex voltage return (Ai), a Doppler shifted frequency (fi), and a range

(Ri). At any given sampling instant for the kth pulse the received waveform

can be represented in the complex signal plane by a vector (or "phasor")

which has an instantaneous amplitude or voltage IVk(R) and phase Ek(R)

determined by the instantaneous vector sum of the individual scatterers.

The complex signal is then

Zk(R) = Ik(R) + j Qk(R) (3.3)

where Ik(R)=IVk(R) Icos ek(R) is the in-phase and Qk(R)=|Vk(R) Isin Ek(R) is

the quadrature phase component (Rader, 1984). These expressions illustrate

that (for a specific received polarization) only two quantities are

measurable, the complex amplitude and phase. All other quantities are

derived from these based on physical models.

3.1 GENERAL FEURES OF THE DOFPPER POWER SECRM

The concept of the Doppler power spectrum is fundamental in radar signal

processing (Haykin, 1985b). A typical power spectrum, shown in Figure 3.1,is a plot of the returned power as a function of the Doppler shifted

frequency components in the target resolution volume. The usual sign

convention (taken from spherical coordinates) is that a positive Doppler

velocity corresponds to a velocity away from the radar; the rate of change

in range is positive. This corresponds to a negative Doppler frequency

shift. The velocity limits ±Vmax are determined by the Nyquist constraint

that two samples per wavelength or period are required to unambiguously

measure a frequency (Whittaker, 1915; Nyquist, 1928; Shannon, 1949). For a

14

0

-10

dB -20

-30

-40-0.5 -0.4 -0.3 -0.2 -0.1

VELO

Fig 3.1 Doppler power spectrumecho in white noise.Vmax, velocity spectrum

0 0.1 0.2 0.3 0.4 0.5)CITY/2 Vmax

(128 point periodogram) of typical weatherEstimated parameters are velocity 0.4width ~ .04 Vax, and SNR ~ 10 dB.

15

uniform pulse repetition time Ts (equally spaced samples) the so called

"Nyquist velocity" is

Vmax = \/4Ts . (3.4)

The interval [-Vmax, +Vma] is called the "unambiguous velocity interval" or

commonly the "Nyquist velocity interval" and all possible velocities are

measured within this interval. The reality of sampling theory dictates that

sampled Doppler spectra exist on a circular frequency domain rather than a

frequency line extending both directions from zero (Gold and Rader, 1969).

Thus, as a target velocity increases beyond Vmax, it aliases or "folds" onto

the negative velocity region of the Nyquist velocity interval (Passarelli,

et al., 1984).

The signal power spectrum rests on a platform of "white noise", so called

because the noise power spectral density is independent of frequency. White

noise is caused by several factors including thermal noise from the

receiver, phase noise from the transmitter/receiver system, artifacts from

the spectrum estimation algorithm, artifacts from receiver non-linearities,

and quantization noise from the A/D converters.

It is convenient to approximate the signal portion of the power spectrum

with a Gaussian shape having some mean velocity and width. The area under

the signal portion of the spectrum, not including the contribution of white

noise, is the returned power. Depending on the distribution of velocities

in the pulse volume and the scattering mechanism, asymmetric spectra and/or

multi-modal spectra may occur. Second trip echoes are a common cause of

bimodal spectra in klystron systems. Janssen and Van der Spek (1985) found

that only about 75% of observed precipitation spectra had the assumed

Gaussian shape.

For ST/MST radars the spectrum is often assumed to be Gaussian, but spectra

measured at near vertical antenna beam directions (zenith angles less than

about 10°) very regularly show one or more strong spectral spikes superposed

on a Gaussian shaped base. The spikes result from a corresponding number of

16

quasi-horizontal laminar refractive index structures producing partial

reflections while the Gaussian floor results from scattering by turbulent

refractive index structures. Moreover, the aspect sensitivity due to the

quasi-horizontal laminar structures may produce strongly asymmetric mean

power spectra if several single power spectra are averaged for oblique

antenna beam directions.

The width of the velocity spectrum has a number of contributions including

wind shear, turbulence, particle fallspeed dispersion, antenna rotation

(Nathanson, 1969) and, in the case of lidar, range propagation of the pulse

during sarpling. It is difficult to separate instrumental effects from the

desired signal contributions.

The goal of signal processing is to deduce the characteristics of the signal

portion of the spectrum. This means that the other contributions from

clutter, noise, and artifacts must be either minimized or removed by the

various steps of processing. There are two basic approaches: frequency

domain processing using the power spectrum, and time domain processing using

the autocorrelation function. Each approach has its advantages and

disadvantages but the essential information available from each is identical

since the power spectrum of the sampled signal and its autocorrelation

function comprise a Discrete Fourier Transform (DFT) pair, (Oppenheim &

Schafer, 1975; Tretter, 1976):

N-1S(nfo) = Z R(mTs) exp [-j2mnm/N] (3.5a)

m=0

N-lR(miTs) = N- 1 Z S(nfo) exp [+j27rmn/N] (3.5b)

n=0

where S(nfo) is the Doppler spectrum in multiples of the fundamental

frequency shift fo=l/NTs and R(rTs) is the autocorrelation function in

multiples of the sample time Ts. This is the discrete version of the

celebrated Wiener-Khinchine theorem (Wiener, 1930; Khinchine, 1934). The

information content is identical in the two approaches. The primary

difference between time and frequency domain processing is that the

17

information concerning the lower spectral moments is distributed over

several frequencies of the power spectrum, while it is concentrated in the

small lags of the autocorrelation function.

It is important to realize that sampling theory dictates that both S(nfo)

and R(mrs) be periodic. That is, the spectrum repeats at multiples of the

sampling frequency and the correlation function repeats at multiples of N

times the sampling period (NTS). When highly coherent spectral components

(e.g. clutter) are present, the correlation usually will not decay to zero

within the N/2 samples. Thus, the periodicity requirement of R(mTs) will

produce a biased spectrum estimate. Care must be exercised in these cases.

3.2 FREQUENCY DOMAIN SPECTRAL MEMENT ESTIMATION

Estimating the Doppler power spectrum and its moments directly are

straightforward techniques (Haykin and Cadzow, 1982). However, some basic

questions must be answered first. We implicitly assume a data model for

weather and clutter spectra when we choose a spectrum estimation technique.

A specific data model such as a sum of sinusoids or white noise passed

through a narrowband filter is best analyzed by a spectrum analysis

technique compatible with that data model. Robinson (1982) emphasizes this

point in his historical review of spectrum estimation. Marple (1987)

stresses the importance of using an appropriate model fitting analysis and

gives a very well organized discussion of classical and modern spectral

estimates using digital techniques.

3.2.1 Fast Fburier transform techniques

The Doppler power spectrum may be estimated from the Discrete Fourier

Transform (DFT) of the complex signal. The DFT decomposes the observed data

into a sum of sinusoids having amplitude and phase that will exactly

reproduce the observed data. It is easy to show that these N discrete

components are adequate to reconstruct the entire continuous spectrum so

long as the complex data samples {zk} are taken at a rate equal to or

greater than the bandwidth of the signal. The advantage of measuring the

full Doppler spectrum is that spectral impurities such as ground clutter,

18

bi-modal spectra or artifacts can be suppressed by intuitive (if non-

optimal) algorithms.

The so called "periodogram", a frequently used estimator in weather radar as

well as many other fields, is an N point spectrum estimator in which the

standard deviation of each spectral value equals its mean value. Usually

one averages several spectra from a divided time series or smooths over

several points in the periodogram to improve the accuracy. The periodogram

is defined as the squared magnitude of the transformed data sequence {Zk}

(Blackman and Tukey, 1958; Cooley and Tukey, 1965; Oppenheim and Schafer,

1975),

N-1P(f) = N-11 Z hkzk exp [-j27fk]1 2 (3.6)

k=-O

where the """ denotes an estimate. The hk term is the "window" which

modifies the waveform being transformed.

In general, window functions have a maximum value centered on the time

series and are tapered near zero at the ends. This tapering reduces the

spectrum smearing, a "leakage" of spectral energy introduced by the

discontinuity imposed by sampling when the end points are joined. Windowing

also effectively reduces the number of points in the time series. The

simplest window is hk = 1 (or no windowing). For this window, the

periodogram of a single point target has the first side lobe only 13 dB down

from the peak. This is not a problem for estimating the mean and variance

of the designed signal, but if strong clutter is present, then the sidelobe

power from the clutter that leaks throughout the Nyquist interval can mask

weaker weather echoes. Table 2 shows characteristics of several common

windows. Harris (1978) and Marple (1987) both give an extraordinary

description of window functions. In general, the lower the sidelobes

offered by a window, the broader its main lobe response. This broadening

degrades the spectral moment estimates.

19

Table 2

Characteristics of time series data windows (Marple, 1987).

Equivalent 1/2 PowerWindow Highest Sidelobe Bandwidth Bandwidth

Name Sidelobe Decay Rate (Bins) (Bins)

Rectangle -13.3 dB -6 dB/octave 1.00 0.89Triangle -26.5 dB -12 dB/octave 1.33 1.28Hann -31.5 dB -18 dB/octave 1.50 1.44Hamming -43 dB -6 dB/octave 1.36 1.30Gaussian -42 dB -6 dB/octave 1.39 1.33Equiripple -50 dB 0 dB/octave 1.39 1.33

The windowed periodogram P(f) can be evaluated at any frequency f in the

Nyquist interval. The Fast Fourier Transform (FFT) is simply a highly

efficient technique for evaluating the DFT at N equally spaced discrete

frequencies (Welch, 1967). Although the FFT algorithm is attributed to

Cooley and Tukey (1965), a recent historical investigation into the history

of the Fast Fourier Transform by Heideman, et al. (1984) attributes an

algorithm very similar to the FFT for computation of the coefficients of a

finite Fourier series to Gauss, the German mathematician. Apparently the

first implementation of the FFT on a weather radar was in December 1970 at

the CHILL radar (Mueller and Silha, 1978).

3.2.2 Maximum entrpy techniques

The aforementioned Fourier transform techniques have been understood since

the time of Fourier and Gauss and are well documented by Jenkins and Watts

(1968). Only recently have techniques based on covariance estimates and

probabilistic concepts been explored. Kay and Marple (1981) and Childers

(1978) have termed these parametric techniques "modern spectrum analysis".

Marple (1987) points out that maximum entropy, maximum likelihood and other

techniques are "modern" in the sense that short data sequences produce

spectral resolutions better than the inverse duration of the data sequence,

which is characteristic of classical spectrum estimators. Furthermore, fast

digital algorithms have been developed which allow computing hardware to

perform the computations in the required time frames. This interest in

20

alternative spectrum estimators can be explained by categorizing expected

performance improvements as increased resolution or increased detectability.Both Jaynes (1982) and Makhoul (1986) attempt to clarify some confusion and

misleading notions related to the maximum entropy techniques.

Maximum entropy (ME) spectrum analysis estimates the spectrum using

parametric techniques to define the spectrum. The parameters are typically

derived from the data samples or some estimated autocorrelation sequence.

The ME technique was developed by J.P. Burg (1967, 1968, 1975) as a

geophysical prospecting technique for high resolution measurement of sonic

wave reflections and velocities. Makhoul (1975) shows that the all pole ME

spectrum model can approximate any spectrum arbitrarily closely by

increasing its order L. He shows that the ME spectra minimizes the log

ratio of the estimated spectrum to the true spectrum integrated over the

Nyquist interval. The MST radar community (Klostermeyer, 1986) and the

lidar community (Keeler and Lee, 1978) have used the maximum entropy method

for characterizing atmospheric targets. Sweezy (1978) and Mahapatra and

Zrnic (1983) have computed maximum entropy spectrum estimates on simulated

weather radar data and compared them with Fourier transform and pulse pair

estimators. Haykin, et al. (1982) describe how maximum entropy techniques

can be applied to Doppler processing of radar "clutter" including weather

and birds for aviation hazard identification.

Atmospheric echoes, whether from precipitation, aerosols, or turbulence, can

be modeled by "autoregressive" (AR) techniques as narrow band filtered

noise. These AR and the standard Fourier technique appear to represent the

essential spectral features well although little quantitative work is

available for comparison in the atmospheric echo application. Van den Bos

(1971) and Ulrych and Bishop (1975) show that maximum entropy spectrum

analysis is equivalent to least squares fitting of a discrete time all pole

model to the observed data. As noise is added to the observations the

autoregressive moving average (ARMA) model is more appropriate (Cadzow,

1980; Marple, 1987).

21

The justification for studying maximum entropy spectra is its ability to

estimate complete spectra from the first few lags of the autocorrelation

function rather than from all the autocorrelation lags that are required by

the Fourier transform technique (Radoski, et al., 1975). Since only the

first few autocorrelation values are known with any confidence, this

property may be critically important when the sampled data sets are very

short. Baggeroer (1976) computes confidence limits for ME spectra which are

applicable to atmospheric echoes.

The "order" of the maximum entropy spectra defines the number of lags, or

equivalently the number of poles in the filter through which white noise is

passed in modeling the data. A larger order allows non-Gaussian spectral

detail to be more accurately represented, e.g. a weak atmospheric echo in

the presence of a much stronger ground clutter. However, a larger order

requires a longer data sample to obtain accurate estimates.

The basic technique uses the sampled input data to compute R(0), R(1),...

R(L) for the Lth order estimator. Additional lags are realized by requiring

that the entropy (in an information theoretic sense) of the probability

density function having the extended autocorrelation function be maximized.

This extended autocorrelation function allows computation of coefficients

for a whitening or linear prediction filter. The ME spectrum is computed

from these filter coefficients which are defined by the matrix equation

A = R-1P (3.7)

where A is the filter coefficient vector, R is the autocorrelation matrix

and P is the autocorrelation vector (Ulrych and Bishop, 1975). The

coefficient estimates can be rapidly computed using the Levinson algorithm

(Makhoul, 1975; Anderson, 1978).

This filter removes the predictable components from the input data and the

optimum filter of order L minimizes the prediction error. The Lth order ME

spectrum estimate can then be computed

22

^2 (L)

SME (f) (3.8)L

I1 -e am exp[-j2rfm] 12

m=l

where am are the elements of A and ao2(L) is final prediction error. Burg

(1967) gives the "forward-backward" technique of estimating the linear

prediction coefficients directly from the data which frequently permits more

detail to be shown in the spectrum. Smylie, et al. (1973) and Haykin and

Kesler (1976) give the complex form of the ME spectrum estimator.

Friedlander (1982) and Makhoul (1977) describe lattice structures for ME

spectrum estimates which are computationally more efficient and identical toBurg's method. Papoulis (1981) attempts to interrelate the various aspects

of maximum entropy and spectrum estimation in his mathematical review paper.

Marple (1987) presents a more readable exposition. Cadzow (1980, 1982)

extends the ME concept to rational models.

Keeler and Lee (1978) and Mahapatra and Zrnic (1983) have shown that the

pulse pair frequency estimator is identically the mean (or the peak, in this

special case) of the first order maximum entropy spectrum. The atmospheric

remote sensing community has been using the simplest form of ME for almost

two decades! Its relevance to accurate parameter estimation for weather

radars, ST/MST profilers and lidar signals is an active research area

(Haykin, 1982).

3.2.3 Maximum likelihood techniques

Maximum likelihood (ML) estimation is a statistical concept that gives the

most likely outcome or minimum variance estimate of an experiment based on a

set of known probabilities. ML estimates of spectral parameters are

"efficient", i.e. there is no other unbiased estimator having a lower

variance. It is well suited for estimating parameters of a spectrum whose

shape is known or assumed when neither a priori knowledge nor a valid cost

function associated with moment estimator error is known (Van Trees, 1968).

Zrnic (1979a) uses ML techniques to derive the minimum variance (Cramer-Rao)

23

bounds of spectral moment estimators for application to atmospheric radar

data. He compares present estimators to these bounds and interprets Levin's

(1965) results in a modern framework. Moreover, he shows that the pulse

pair estimator is ML for a Markov process.

In general closed form solutions for ML estimates of spectrum moments are

quite complicated and difficult to compute. The optimum (ML) processor

depends on the underlying signal statistics which in turn depend on the

spectrum shape and SNR. Shirakawa and Zrnic (1983) evaluate the ML

estimator for sinusoids in noise and find a slight improvement over the

pulse pair estimator at low SNR's. Novak and Lindgren (1982) derive the

exact ML mean velocity estimator for Gaussian shaped spectra using more than

one autocorrelation lag. Their technique is similar to Lee and Lee's (1980)

poly pulse pair velocity estimator. Miller and Rochwarger (1972) show that

for independent pairs, the pulse pair estimator of mean frequency is ML for

an arbitrarily shaped spectrum so long as the normalized width is small.

Sato and Woodman (1982) use a least square fit algorithm to estimate

spectral parameters, including noise and clutter parameters, by assuming

prior knowledge of the spectral shapes. Woodman (1985) shows that this

technique is a ML estimator of the spectral characteristics. It is

gratifying that the simple pulse pair estimators approach the minimum

variance bound over a wide range of SNR's.

If the spectrum shape is completely unknown, the ML spectrum gives the most

probable estimate which concentrates the spectral energy at the input signal

frequencies while minimizing other spectral energy in a statistically

optimum sense (Capon, 1969; Lacoss, 1971). The statistical rationale for

using ML estimation is that the ML spectrum estimate provides a minimum

variance, unbiased estimate of the power at a given frequency. Burg (1972)

has shown that in the mean the Lth order ML spectrum is just the following

combination of ME spectra up to order L:

L[SML(f) - 1 = L1 Z [SME,m(f)- 1 (3.9)

m=l

24

Thus, the mean ML spectrum is a smoothed version of mean ME spectra. It has

many of the same properties as ME spectra but the details are obscured by

combining all order ME spectra. There have been theoretical studies of ML

spectra but little application to atmospheric data. Klostermeyer (1986) has

computed ML spectra for VHF radar data.

3.2.4 Classical spectral moment computation

The spectrum moments can be directly related to the reflectivity, velocity,

and dispersion parameters desired for further analysis. Computing these

moments has historically been performed using classical moment calculations

based on techniques from probability theory when considering the power

spectrum as a density function of frequency or velocity components of the

desired signal (Denenberg, 1971, 1976). For sampled data systems the

"sampling theorem" imposes certain requirements on moment and transform

computations that cannot be ignored -- namely replication in the frequency

domain and circular convolution (Oppenheim and Schafer, 1975).

Let the power spectrum of the received signal be denoted by S(f). Then the

classical spectral moments are given by

Mn = x fn S(f)df . (3.10)

The zeroth moment (MO) is the area under S(f) and represents total signal

clutter, and noise power. Of course, we are usually interested only in the

signal power, so the clutter and noise powers must be estimated and removed.

Noise power is generally easy to remove, but clutter removal causes

difficulties to the parameter estimation process.

The classical normalized first moment represents mean velocity and is given

by the linear weighting of S(f) over the Nyquist interval

fc = f S(f)df / Mo (3.11a)

V = (X/2) fc (3. 1Ib)

25

Note that white noise biases the velocity towards zero and for a pure noise

spectrum the mean velocity is identically zero. Various techniques have

been described for mitigating this bias, most of them requiring manipulation

of the power spectra. Thresholding the spectrum points with some value near

the noise spectral density is common, but some sensitivity is lost

(Hildebrand and Sekhon, 1974; Sirmans and Bumgarner, 1975a; Klostermeyer,

1986).

The "spectral balancing technique" rotates S(f) until the signal spectrum is

near zero so that the signal and the noise share the same zero mean

velocity. The amount of rotation represents the mean velocity of the signal

component and removes errors due to aliased spectra. The same effect is

obtained by computing the offset first moment

4 (f-fc) S(f)df = 0 (3.12)

where fc is varied to obtain equality.

The normalized second central moment represents the velocity dispersion

within the pulse resolution volume. Shear, turbulence and precipitation

motion (fallspeed oscillations, etc.) contribute to a distribution of radial

velocities (Nathanson and Reilly, 1968). A contribution from antenna

scanning during the finite dwell time may also be significant (Nathanson,

1969). The velocity dispersion (width) is the square root of the second

central moment of the spectrum estimate:

af2 = I (f-fc)2 S(f)df / M0 (3.13a)

W = (>/2) of . (3.13b)

Spectrum estimation algorithms are fairly time consuming to invoke, and once

the frequency domain is entered, there is still substantial computation to

accurately extract the meteorological moments. The main reason for entering

the frequency domain lies in the ability to more easily filter spectral

26

artifacts or identify multi-modal spectra. In cases where spectra are

unimodal and generally free from artifacts, more efficient time domain

processing is typically used.

3.3 TIME DIXMAIN SPECTRAL MO'ENT ESTIMATIC

The basis for time domain moment estimation is the transform relationship of

the autocorrelation function of the complex signal to the power spectrum.

An estimate of the autocorrelation can be easily calculated from the complex

input time series {Zk),

N-m-1R(m) = (N-m) 1 2 Zk* Zk+m (3.14)

k=0

where m is the lag between the two data series. For uncontaminated spectra,

usually only two or three lags are necessary to obtain the moments of

interest. This represents a substantial savings in computation over the

spectrum domain approach. The general relationship between the complex

autocorrelation function and the nth classical spectral moment is

Mn = R[n](0)/(j2w)n (3.15)

where R[n] (0) is the nth derivative of the autocorrelation function

evaluated at lag = 0 (Papoulis, 1962; Bracewell, 1965). The first three

spectral moments are used to estimate the reflectivity, radial velocity, and

velocity dispersion or width respectively (Miller, 1970; Miller, 1972).

3.3.1 Gecmetric interpretatins

The complex autocorrelation function, which is the basis for time domain

moment estimation, is often depicted as its real and imaginary components,

but an alternative 3 dimensional representation allows a better

understanding of the covariance, or pulse pair, mean frequency estimator.

Consider the complex R(m) to be a 3D helix that is wide at the center and

tapered toward zero radius at the ends having a Gaussian shaped envelope.

27

Figure 3.2 shows a drawing of this continuous autocorrelation helix. A

sampled autocorrelation helix will consist of points on this helix spaced at

the PRT. Note that zero lag, R(0), is at the center and has no imaginary

component. The radius at lag 0 represents the signal power and the real

delta function at lag 0 represents the noise power. The width of the

Gaussian envelope of the helix represents the inverse velocity spectrum

width or dispersion. The rotation rate of the helix defines the mean

velocity of the signal. For a given spacing of autocorrelation function

samples the angular rotation between a pair of samples is a measure of mean

velocity. Thus, the angle of the complex estimate R(1) gives the mean

velocity of the received signal expressed as a fraction of the Nyquist

interval which is the "pulse pair estimator" used almost universally for

mean velocity in weather radar and lidar processors.

A useful geometric interpretation of the relationship between classical

spectral moments and the autocorrelation function can be found in Passarelli

and Siggia (1983). This interpretation illustrates many of the properties

of pulse pair estimators.

3.3.2 "Pulse pair" estimators

The advent of the so-called pulse pair, double pulse, or complex covariance

technique (Rummler, 1968a; Woodman and Hagfors, 1969; Miller and Rochwarger,

1972; Berger and Groginsky, 1973; Woodman and Guillen, 1974) for mean

velocity estimation was revolutionary since the algorithm arose at about the

same time that it could be implemented in hardware for a significant number

of range bins. Lhermitte (1972) and Groginsky (1972) reported the first use

of hardware signal processors and weather radars using this technique.

However, covariance processing for velocity measurements apparently was

first used in March of 1968 for ionospheric velocity measurements (Woodman

and Hagfors, 1969). Woodman and Guillen (1974) also reported covariance

based velocity measurements in the mesosphere at the Jicamarca MST radar in

1970. This algorithm development in the MST community was independent of

Rummler's work. The pulse pair algorithm led to an exciting growth in the

use of Doppler radar by the scientific community (Groginsky, et al., 1972;

Ihennitte, 1972; Sirmans, 1975; Ihermitte and Serafin, 1984).

28

Real axisx

Pn

R(I)

Imaginaryaxis

v1J

Fig 3.2 Three dimensional representation of the complex autocorrelationfunction as a helix. Radius of helix Rs(O) is proportional tototal signal power, Ps; rotation rate of helix is proportional tovelocity, V; width of envelope! is inversely proportional tovelocity spectrum width, W. Delta function Rn(0) represents noisepower.

29

R(O)

PS

lagm

Other time domain algorithms such as the "vector phase change" (Hyde and

Perry, 1958) and the "scalar phase change" (Sirmans and Doviak, 1973) are

closely related to the pulse pair estimator, but their performance is

inferior. Sirmans and Bumgarner (1975b) capare these and other mean

frequency estimators.

It is well known that the first few lags of the autocorrelation function are

sufficient to deduce spectrum parameters of interest. Papoulis (1965 and

1984), Bracewell (1965), Woodman and Guillen (1974), and Passarelli and

Siggia (1983) show that the autocorrelation function can be represented by a

Taylor series expansion in terms of the central moments of the Doppler

spectrum with the low order moments being the leading terms. In other

words, the first few lags of the autocorrelation function contain the moment

information of interest. For an arbitrary spectrum, these expansions have

the form

R(mits) = A(mrTs) exp[-j0(mrs)] . (3.16)

The even function A(mrTs) is determined primarily by the even central moments

(e.g., power, variance and kurtosis), while the odd function 0(mTs) is

determined primarily by the mean velocity and the odd central moments (e.g.,

skewness).

Estimators can be generated for any moment, provided that a sufficient

number of autocorrelation lags are measured. White noise power Pn biases

the magnitude for lag zero. Therefore, the total received power must be

corrected for noise,

Pr = R(0) - Pn (3.17)

The pulse pair mean velocity estimator is not biased by white noise and is

obtained by taking the argument of the first autocorrelation lag,

V = ( /2) (2Ts)-1 tan[Im R(T)/Re R(Ts)] . (3.18)

30

The pulse pair spectrum width is given by

W = ( X/2) (2fTs)- 1 [1 - p(Ts) (1 + SNR-1)] (3.19)

where p(Ts) =|R(Ts) |/R(O) is the normalized first lag and the noise power

must be determined independently.

3.3.3 Circular spectral rmment computation for sampled data

Sampled data systems utilize the complex plane and z-transform theory to

formally express the relationships between the time and frequency domains

(Oppenheim and Schafer, 1975). For example, the DFT of the autocorrelation

function is formally the z-transform of the sampled autocorrelation function

evaluated on the unit circle in the z plane, i.e. |z|=1 or z = exp[-j27f]:

N-1S(f) = z R(mTs) z-m (3.20)

m=O I z=exp [-j 27rf]

The unit circle on the complex z plane is important in understanding

concepts of sampled or discrete data systems, specifically concepts of

digital signal processing. Figure 3.3 shows the z plane and the frequencies

associated with various points on the unit circle. Zero frequency, where

ground clutter usually appears, corresponds to z=l and the Nyquist frequency

(where velocity spectra alias into the next Nyquist velocity interval)

corresponds to z=-l. Thus, the z plane representation of spectral space

allows an immediate and simple geometric interpretation of velocity aliasing

and the velocity ambiguity arising from sampling too slowly. Analysis and

synthesis of digital filters requires heavy application of z transform

theory, thus easily allowing visualizing the effect of various types of

ground clutter filters, for example.

It is natural to compute spectral moments on the unit circle rather than

along the frequency line in the Nyquist velocity interval. The zeroth

moment or total receiver power, is still that area under the spectrum

31

Imag

f = (2Ts)-'

Real

I0

f = -(4Ts )-I Z plane

Fig 3.3 Periodogram power spectrum plotted on unit circle in the z-plane.Note velocity aliasing point, the Nyquist velocity, at z=-l.

32

whether on a line or on a circle. However, higher order moments can be

different for the two cases (Passarelli, et al., 1984).

A simple geometric derivation shows that the first circular moment estimate,

fc, of the estimated spectrum, S(f), is the normalized frequency at which

the center of mass on the circle is located,

S (27m/N) sin(27m/N)fc = (27)- 1 tan-1 (3.21)

Z S(27m/N) cos(2nm/N)

where the summations run over 0 to N-l. Trigonometric manipulation converts

this equation to

N-1Z S(27m/N) sin[27(n/N - fc)] = 0 . (3.22)

n=0

Thus, fc is the sinusoidal weighted mean of S(f) (Zrnic, 1979a). Further,

we see that the numerator and denominator of (3.21) are the imaginary and

real parts of R(mrTs) and that the circular first moment is identically the

pulse pair frequency or velocity estimator.

Two points are clear from this discussion: 1) white noise does not bias the

pulse pair frequency estimate because the noise does not weight any

particular frequencies on the circle, and 2) symmetric spectra have

identical first moments using either the classical (linear weighting) or the

circular (sinusoidal weighting) computations. Asymmetric spectra produce

different first moment estimators but there are no compelling reasons to

prefer linear weighting over the more common sinusoidal weighting (the pulse

pair estimator). Indeed, for sampled data systems the circular moment

computation is more natural than classical moment computation.

3.3.4 Pbly pulse pair techniques

If we accept the premise that knowing lags of the autocorrelation function

past the first allows a processor to extract additional information about

the received signal, then one should expect to reduce the variance of

velocity estimates by using, not only R(1), but R(2), R(3), etc. The

33

Classical:

2(f - f,,i) S(f)= o

2Ts freq - Y 2Ts

Circular:

Zsin[27r (f fci)]S(f) 0-t~~~~~i,,]scr, ·o~~~~~~~~~~~~

fcir +2T5

Fig 3.4 Comparison of classical and circular (pulse pair) first momentestimators. Classical estimate is determined by linear weightingof spectrum estimate and circular estimate, by sinusoidalweighting.

34

m

m

variance reduction can be realized only if the received signal is coherent

over the additional lags. Lee (1978) proposed the "poly pulse pair"

algorithm for lidar signal processing. Velocity estimates can be found from

a weighted average of the estimate given at each lag, where the smaller lags

are given higher weighting since the correlations are higher. Poly pulse

pair velocity estimates (using a few lags) produce lower variance estimates

than the pulse pair estimates when the spectrum width of the signal is only

a few percent of the sampling frequency (Lee and Lee, 1980).

Strauch, et al. (1977) evaluated poly pulse pair for 3 cm radar processing.

They concluded that for typical velocity spectrum widths and PRF's (sample

rates) used with X-band Doppler radar, the coherence time was frequently too

short to give a significant improvement in the velocity estimates. However,

for infrared lidar the coherence times and sample rates permit a significant

improvement in reflectivity, velocity, and width estimates (Bilbro, et al.,

1984). Furthermore, Rastogi and Woodman (1974) and Srivastiva, et al.

(1979) use multiple lag estimates of the correlation function to estimate

moments of a Gaussian shaped spectrum. Several independent estimates of the

autocorrelation function can be found and a Gaussian shaped curve fitted to

these samples. Sato and Woodman (1982) have used this nonlinear curve

fitting technique to estimate signal, clutter, and noise parameters at the

Arecibo ST radar.

3.4 UNCERTAINTIES IN SPECThUM UMENT ESTIfM4AI

Any estimator has an associated uncertainty. In atmospheric radar signal

processing the velocity spectrum moments are being estimated with some

uncertainty that depends on the processing interval, the coherence time or

velocity width, and the SNR. Zrnic has published extensively on weather

radar spectrum estimator uncertainties and his results are succinctly

described in Doviak and Zrnic (1984). A summary is given here.

3.4.1 Reflectivity

Marshall and Hitschfeld (1953) describe the probability density function of

the distributed weather target. The received signal is a complex Gaussian

process which has a Rayleigh amplitude distribution and an exponential power

35

distribution. Thus, the mean received signal power is Ps with variance Ps2

and the coherence time is determined by the spectrum width of the signal.

The number of independent signal samples in a given integration time Td

seconds is approximately MI = 2/WTd (Doviak and Zric, 1984) where W is the

spectrum width (standard deviation) in Hertz. The number of independent

noise samples is just M = Td/Ts, the total number of samples in the dwell

time. Therefore, the variance of the mean power estimate is approximately

var(Pr) = Ps2/MI + Pn2/M . (3.23)

Doviak and Zrnic (1984) show that if the number of independent signal

samples is smaller than about 20 and a log receiver is used, the bias in the

estimated received power depends on MI and its variance is not exactly

proportional to 1/MI. A square law receiver does not encounter these

problems. Marshall and Hitschfeld (1953), as well as a recent review by

Ulaby, et al. (1982), show that the ratio of the mean power to the

fluctuating power associated with a single sample of a Rayleigh quantity is

5.6 dB. Therefore, for MI independent samples the signal power estimates

are known within 5.6/MjI dB. Averaging independent samples obtained in

range can further reduce the variance.

3.4.2 Velocity

Woodman and Hagfors (1969) used statistical analysis of Gaussian random

variables to estimate the uncertainty of pulse pair velocities. Berger and

Groginsky (1973) applied perturbation analysis to derive the variance of the

independent and contiguous pulse pair frequency estimators. Zrnic (1977b)

later extended their results to spaced but correlated pulse pairs. Two

conditions, both of which are usually satisfied for a large number of

samples (M), are necessary for the analysis to be accurate:

M >> A /47 W Ts (3.24a)

M >> (SNR-1 + 1)2 / p2(Ts) (3.24b)

where W is the velocity spectrum width and p(Ts) = R(Ts)/R(0) is the

autocorrelation function at lag Ts (the PRT) normalized to unity. At high

36

SNR and for large enough M that both conditions are satisfied, and for

contiguous pairs typical of radar Doppler processing, and for Gaussian

shaped spectra, the variance of the velocity estimate is

var(V) = f W/ 8/7 M Ts . (3.25)

Table 3 summarizes the velocity uncertainties at high SNR for three cases:

1) contiguous samples, 2) independent sample pairs, and 3) the minimum

variance bound. Expressions are given both in terms of the actual spectrum

width in ms -1 (W) and the width normalized to the Nyquist velocity interval

(Wn). Figure 3.5 shows the standard deviation of velocity estimates

normalized to the Nyquist velocity interval and to the square root of the

number of samples M as a function of the normalized spectrum width. The SNR

is a parameter for the two sets of curves -- those for the typical

contiguous pairs and for less typical spaced pairs of pulses (Campbell andStrauch, 1976; Doviak and Zrnic, 1984). Note that reasonably accurate

velocity estimates can be obtained for a given M doawn to SNR ~ 0 dB so long

as the Gaussian standard deviation velocity width is less than about 0.2 of

the Nyquist velocity interval 2Vmax.

Woodman (1985) discusses errors for multiple lag velocity estimators in

which the lags are statistically dependent. By weighting the correlation

estimates in an optimum fashion, he concludes that for high SNR only a few

(2 or 3) lags are necessary.

3.4.3 Velocity spectrum width

Benham, et al. (1972) and Berger and Groginsky (1973) applied a perturbation

analysis to the spectrum width estimator and Zrnic (1977b) later extended

their results to arbitrarily spaced pulse pairs. Their primary result for

high SNR, contiguous pairs, and narrow, Gaussian shaped spectra is that the

variance of the velocity width is

var(W) = 3 XW / 64/T M Ts . (3.26)

37

THBLE 3

Expressions for variance of velocity estimators at high SNR. AssumesGaussian spectra in white noise, low normalized velocity width (Wn=W/2Vmax)and large M. Expressions apply to both pulse pair and Fourier transformestimators.

Var(V) using W

Contiguoussamples(typical case)

x8]/7w MT

Independentpairs

2M

Minirumvariancebound

48 TS2

-MX2 W4M X2

Var(V) using Wn

W wn167r MTs 2

X2Wn2

8Mr 25

3 \ 2

Wn4

MTs2

38

C\M

Q<>

( '°V 1.0

z0

u 0.5Q0

Q0 0.1 0.2 0.3 0.4

NORMALIZED SPECTRUM WIDTH oavn

Fig 3.5 Velocity error as function of spectrum width and SNR. Spectrumwidth is normalized to Nyquist interval, vn=W/2Vn1 =2WTs/X. M isnumber of sample pairs and error is normalized to Nyquist velocityinterval, 2va = 2Vmax . Small circles represent simulation values(Doviak and Zrnic, 1984).

39

Table 4 summarizes the width uncertainties at high SNR for three cases: 1)

contiguous samples, 2) independent sample pairs, and 3) the minimum variance

bound. Expressions are given both in terms of the actual spectrum width in ms-1

(W) and normalized to the Nyquist velocity interval (Wn). Figure 3.6 shows the

normalized standard deviations of the width estimates as a function of normalized

spectrum width for a range of SNR's. The width estimator is relatively good ifthe normalized width is between 0.02 and 0.20 of the Nyquist interval and the SNR> 5 dB.

40

TABLE 4

Expressions for variance of width estimators at high SNR. Assumes Gaussianspectra in white noise, low normalized velocity width (Wn=W/2Vmax) and large M.Expressions apply to both pulse pair and Fourier transform estimators.

Var(W) using W Var(W) using wn

Contiguoussamples(typical case)

3w

64W,/642z MTs

3 X2

Wn128J7r MTS 2

Independentpairs

Minimumvariancebound

w2

2M

x 2 28MT2 Wn2

8MTs281]?

2880 T 44 W6

M X4

45 X2

MTS2

41

c

0

C

0.5az

cn n0 . 1 0.2 0.3 0.4

NORMALIZED SPECTRUM WIDTH, Ovn

Fig 3.6 Width error as a function of spectrum width and SNR. Spectrumwidth is normalized to Nyquist interval, vn=W/2Vmax=2WTs/X- M isnumber of sample pairs and error is normalized to Nyquistinterval, 2Vmax . Small circles represent simulation values(Doviak and Zrnic, 1984).

42

I

I

4. SIGNAL PROCESSING TO ETTMINATE BIAS AND ARIT'ACIS

The primary goal of an effective signal processing scheme is to provide

accurate, unbiased estimates of the characteristics of meteorological

echoes. This means that in addition to moment estimation, the signal

processing algorithms must also eliminate the degrading effects of ground

clutter targets, range aliasing and velocity aliasing. Indeed, this

challenging aspect of signal processing has received considerable attention

in the recent literature.

4.1 DOPPLER TECHNIQUES FM GROUND CLITER SUPPRESSION

Ground clutter poses a significant problem for both coherent and incoherent

radar applications. Clutter biases the reflectivity, mean velocity and

velocity spectrum width estimates. It significantly reduces the effective

area of coverage at close range where the azimuth resolution is best. Even

weak clutter can frequently mask clear air echoes. Fortunately, signal

processing can greatly reduce the effects of clutter. Zrnic and Hamidi

(1981), Zrnic, et al. (1982), and Evans (1983) address various aspects of

Doppler clutter cancellation.

Clutter cancellation is possible for both coherent (Doppler) and non-

coherent systems. Non-coherent techniques rely on the Rayleigh distribution

of the amplitude fluctuations of weather echo to differentiate between

clutter and weather (Geotis and Silver, 1976; Tatehira and Shimizu, 1978;

Aoyagi, 1983). The performance of this approach uses the correlation of

successive samples which depends on the Doppler spectrum width (Sirmans and

Dooley, 1980). Clutter cancellation on most modern systems is performed via

Doppler techniques. Coherent ground based systems rely on clutter being

nearly stationary and use high-pass digital filters to eliminate targets in

a narrow bandwidth near zero velocity. Groginski and Glover (1980) give

requirements and clutter filter specifications and design concepts

particular to weather radar systems.

43

4.1.1 Antenna and analog signal considerations

The first line of defense against clutter is an antenna with low sidelobes

and a good radar site. Main lobe clutter is very difficult to suppress

because clutter targets are usually much stronger than weather targets.

However, since sidelobes are usually down at least 20 dB (one way) from the

peak power, signal processing is effective in suppressing resulting clutter

power without problems caused by a saturated receiver.

Shorter wavelengths generally offer better signal-to-clutter ratios than

longer wavelengths given the same targets. This is because the power

returned from Rayleigh scatterers goes inversely as the 4th power of the

radar wavelength, while large clutter targets will behave more like specular

reflectors having a lesser wavelength dependence (Barton and Ward, 1984).

Superior clutter cancellation performance depends critically on the linear

dynamic range of the transmitter/receiver system. This dynamic range is

governed primarily by the system phase noise and the linear dynamic range of

the receiver itself. The phase stability of the oscillators used in the

radar will determine the degree of clutter cancellation that is possible.

The effect of phase noise is to spill power from a coherent target into

white noise. In the case of a strong clutter target and a weak weather

target, even a relatively small amount of phase noise can obscure a weather

target under the phase noise floor. For Gaussian distributed phase noise

and a coherent clutter target, the maximum clutter-to-phase noise power

ratio (CNR) that can be achieved for small phase errors is straight forward

to compute (Skolnik, 1980) as

CNR= exp(-/2 }/(l-exp{-p 2 } )

« P-2 for P<<1 (4.1)

where p is the pulse-to-pulse rms phase error in radians of the complex

(baseband) signal. The maximum CNR that can be tolerated is equal to the

clutter-to-signal ratio (CSR) that corresponds to a signal-to-phase noise

power (SNR) of about 0 dB. For example, a klystron transmitter can achieve

better than 0.1 degree rms phase error which corresponds to 55 dB CNR. A

44

signal at 55 dB CSR would have an SNR of 0 dB. If 55 dB of main lobe clutter

power could be cancelled, and only the phase noise power or clutter residual

remained, there would be an adequate SNR for Doppler processing. Some

coherent-on-receive magnetron systems may achieve only 5 degrees of phase

stability (21 dB CNR) depending on the quality of the phase lock loop that

synchronizes the receiver to the transmitted pulse. Therefore, it is

frequently not cost-effective to design a signal processor capable of more

than 20-25 dB of clutter cancellation for many magnetron systems. A well

designed magnetron system can achieve much better phase stability.

In many systems it is the dynamic range of the linear receiver that poses

the fundamental limit on the ability to separate weather signals from strongclutter signals. If the linear receiver has a dynamic range of 50 dB, then

this will be the order of the maximum clutter-to-signal ratio that can be

handled. High performance clutter cancellation that is commensurate with

the phase stability of a klystron typically requires a "fast AGC" gain

control and, essentially, floating point digital data conversion. Other AGC

techniques are less effective and may degrade the existing inherent quality

of a stable system. But because they introduce less noise than a typical

magnetron transmitter, they can be used in magnetron systems without

sacrificing overall system performance.

The simplest form of clutter cancellation by Doppler signal processing is to

simply ignore strongly reflecting narrow width targets that have velocities

near zero. On a color velocity display, for example, those bins can be

assigned the background color. More sophisticated processors use either time

domain digital filtering or frequency domain filtering. Which approach is

used depends on the general philosophy of signal processing that is employed

for spectrum moment estimation.

4.1.2 Frequency domain filtering.

Frequency domain processing was discussed earlier. Clutter is typically a

narrow spike (<1 ms-1) centered about zero frequency or DC (direct current).

Weather echoes are usually broader, so that it is possible to remove the

clutter and then interpolate the weather signal across the gap. The first

45

step in frequency domain filtering is to enter the frequency domain via some

spectrum estimation technique. This is usually done via an FFT. The choice

of the time-domain window is critical since the window sidelobes should be

matched to the dynamic range characteristics of the transmitter/receiver

system. For example, a 57 dB Blackman window (Harris, 1978) might be used

in a klystron system but it would not be justified for a magnetron system

that has a phase noise limited CNR of 25 dB.

Removal of clutter in the frequency domain is easily performed by the human

eye, and it is not difficult to develop algorithms that achieve ;30 dB

suppression. Passarelli, et al. (1981) discuss several algorithms for

frequency domain clutter cancellation and point out the adaptive nature of

the general technique, i.e., both the notch width and depth of the filter

can be adjusted to remove only the clutter that is present, with minimal

distortion of overlapped weather or noise. On the other hand, time domain

filters usually, but not necessarily, have a fixed notch width and stop band

attenuation.

4.1.3 Time domain filtering

Time domain digital filtering has been an active research area for over 20

years (Kaiser, 1966; Gold and Rader, 1969; Oppenhiem and Schafer, 1975;

Rabiner and Gold, 1975; Tretter, 1976; Roberts and Mullis, 1987). Precise

control of the digital transfer function allows filter characteristics not

obtainable with analog filters. Digital filters fall into two general

categories, finite impulse response (FIR) filters and infinite impulse

response (IIR) filters. Both of these are used in current weather radars

wherein the I and Q values are filtered separately. An example of a simple

IIR filter is an exponential average of the I and Q values to determine and

remove the DC offset. An example of a simple FIR filter is to calculate the

DC offset over a fixed number of pulses and then subtract this value from

the pulses. In practice, more general FIR and IIR filtering techniques are

used that attenuate not only the DC, but also the low frequency components

around DC to achieve clutter suppression of more than 40 dB. Figure 4.1

shows a typical high-pass filter. Filter design is fairly mechanical and

the parameters that are adjusted are the stopband attenuation, the stopband

46

0

-10

Z -20z0

-30

-50

0 I 2 3 4 5 6

VELOCITY (m s')

Fig 4.la Clutter filter frequency response for a 3 pole infinite impulseresponse (IIR) high pass elliptic filter. For ground clutterwidth of 0.6 ms-1 and scan rate of 5 rpm this filter gives about40 dB suppression. Vs = stop band. Vp = pass:band cutoff, Vmax= +16 ms-1 (Hamidi and Zrnic, 1981).

47

c·Vs-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~___I I I I

I ~~~~~~~~~~~~I I

I / V

TV__- II

I

I

I\ I ! ! I I, I I - I

Xk

Fig 4.lb Implementation of 3rd order IIR clutter suppression filter; z-1 is1 PRT delay. K1 - K4 are filter coefficients (Hamidi and Zrnic,1981).

48

width, the transition band width and the passband ripple which if too large,

can bias the mean velocity.

The IIR filter is computationally more efficient to implement than a

comparable FIR filter but, because of its transient response

characteristics, it is best run in a continuous mode with minimal

perturbation such as those caused by slow AGC changes or PRF changes.

Initialization of the filter can improve the transient response

characteristics. Hamidi and Zrnic (1981) and Groginsky and Glover (1980)

evaluate IIR filters for weather radar systems.

FIR filters offer linear phase performance and are well suited for batch

processing of pulses since they operate on a finite number of pulses. This

makes them well-suited to slow AGC or multiple PRF techniques (i.e., where

the PRF is held constant while a batch of pulses is collected and then

changed for the next batch).

There are other types of clutter suppression algorithms that should be

mentioned. Anderson's (1981) test of the mean block level subtraction

technique offers 20 to 30 dB of clutter cancellation. The parametric

clutter cancellation techniques described by Passarelli (1981, 1983) use

physical models of clutter and weather along with estimates of the

autocorrelation function at various lags to compute the clutter power and

then estimate various Doppler spectral moments. Clutter suppression of 30

dB or more has been achieved. Sato and Woodman (1982) use a nonlinear

processing scheme to fit the observed clutter spectrum and extract the

spectral moments when clutter is about 50 dB stronger than the signal.

When a separate calibrated log channel is used for reflectivity measurement,

an uncalibrated linear channel can be used to remove the clutter

contribution from the log channel power estimate. The ratio of the signal

power to the signal plus clutter power r = Pr[S]/Pr[ S + c3 is the same in

both the linear and the log channels. Therefore, after computing r from the

coherent (linear) channel data, the log channel signal power is

49

10 log Pr[S] = 10 log Pr[S + ] + 10 log r.

When multiple PRT measurements are made for the purpose of extending the

unambiguous velocity interval, nearly all clutter filters have problems.

Anderson (1987) describes an interpolation scheme for the dual PEF ASR-9

radar.

4.2 RANGE/VELOCTTY AMBIGUITY RESOLUDTCN

A fundamental tradeoff exists with constant PRF Doppler radar. A large

unambiguous range (IRma) requires a low PRF

PRF = ,/2RPx ; (4.3)

however, a large unambiguous velocity (Vmax) (and accurate spectral moment

calculations) requires a large PRF

PRF = 4 Vmax /A (4.4)

Another PRF tradeoff is that accurate measurement of the mean velocity

requires a high PRF since the Doppler spectrum width must be narrow relative

to the Nyquist interval (high coherency) whereas accurate intensity

measurements require a low PRF to acquire independent samples (low

coherency). Signal processing offers several techniques for expanding the

unambiguous range and unambiguous velocity. These tradeoffs illustrate that

the choice of PRF must be optimized for different applications.

A performance benchmark for comparison purposes is an S-band (10 cm) radar

operating at 1 KHz PRF with an unambiguous velocity range of ±25 ms'1 and an

unambiguous range of 150 km. This unambiguous range is too small for

assuring that second trip echoes will not be present. The unambiguous

velocity is also too small to ensure that aliasing will not occur, but large

enough that double aliasing (velocities greater than 75 ms-1) will be rare.

At C-band, the unambiguous velocity is halved so that double aliasing will

be fairly common and single aliasing will occur routinely. Reducing the PRF

50

(4.2)

to minimize second trip echoes, will make the velocity aliasing problem even

more serious at C-band.

Coherent lidar and profiler systems do not exhibit range/velocity

ambiguities. For Doppler lidar, the sampling rate during a single pulse can

be made sufficiently high with no impact on the unambiguous range. For the

case of a wind profiler operating at a high elevation angle, the long

wavelength and the steep angle of incidence provide such a large unambiguous

velocity that most profiler processing schemes utilize coherent averaging to

reduce the effective sample rate while simultaneously preserving processor

resources.

For microwave radar, range/velocity ambiguity is a serious problem in many

applications (Doviak, et al., 1978). Fortunately, there are several

techniques for mitigating these ambiguities and each technique has its

advantages and shortcomings. Selection of a technique is usually optimized

for specific applications.

4.2.1 Resolution of velocity ambiguities

There are several techniques for handling range/velocity aliasing that are

not truly signal processing techniques, but rather techniques that use

physical modeling to correct aliased data. Frequently, continuity can be

used to detect velocity folding. For example, one does not expect to see 25ms-1 discontinuities in velocity from bin to bin (in range or azimuth), so

they are assumed to be caused by aliasing. The disadvantage of this

approach is that one must have some region with a known velocity to

correctly invoke continuity. Also, this technique requires that the echo

coverage be fairly continuous and may need manual input to perform final

editing (Bargen and Brown, 1980). Hennington (1981) uses another physical

modeling approach by estimating the mean wind profile obtained from a

sounding or other source to correct aliased velocities. The technique works

well when the perturbation velocities are small compared to the Nyquist

interval. A similar technique described by Ray and Ziegler (1977) uses the

velocity distribution along a radial to dealias velocities. Merritt (1984)

employs both continuity and a wind field model to dealias isolated areas.

51

Boren, et al. (1986) describe an artificial intelligence approach. Bergen

and Albers (1988) have investigated 2 and 3 dimensional dealiasing for

NEXRAD algorithms.

There are several signal processing techniques for extending the unambiguous

range/velocity. The criteria useful in evaluating the techniques are:

1. The algorithm should not preclude the use of clutter cancellation

techniques.

2. The final moment estimates should have a conparable accuracy and be

made in a comparable time (number of pulses) to standard velocity

estimation techniques.

3. The cost of implementing the technique should be comparable to standard

velocity/range processing.

Batch PRT. One approach to velocity/range ambiguity resolution is to use

interlaced PRT sampling whereby a short PRT is used for velocity

measurements, and a long PRT is used for reflectivity estimates (Hennington,

1981). For example, several pulses at a short PTR are first transmitted,

followed by a clearing period (no transmission) and then one or two pulses

separated by a long PRr for the reflectivity estimate. The basic assumption

is that the PRT for the reflectivity estimate is sufficiently long so that

there are no second trip reflectivity echoes. The short PRr velocity

estimates will have two classes of range aliased echoes,- those that are

overlaid with the first trip echoes and those that are not overlaid with the

first trip echoes. When there is no overlap, the velocity estimates can

actually be assigned to the correct range. When first and higher trip

echoes are overlaid and one dominates the others in power by 10 dB or more,

then the velocity of the strong echo can be correctly estimated. The

disadvantages of this batch technique are:

1. Loss of velocity data where first and second trip echoes are overlaid

and powers are nearly equal.

2. The technique may preclude the use of effective clutter cancelling.

52

3. The data acquisition time is increased because the long PRT pulses are

unusable for making velocity estimates.

A similar approach is to have two radars share a common antenna which is

also known as a dual-frequency approach (Glover, et al., 1981). One radar

can sample at a long, constant PRT and the other can sample at a short,

constant PRT. Alternatively, two scans can be made at each elevation, a

long PRT scan for reflectivity and a short PRI scan for velocity. These

techniques are clearly more expensive but they allow excellent clutter

cancellation.

Multiple PRT and multiple PRF techniques can be used to dealias velocities.

Here, "multiple PRT" shall mean that the PRT is changed on a pulse-to-pulse

basis whereas "multiple PRF" shall mean that the PRF is fixed while a batch

of samples is collected and then changed for the next batch of samples. The

general technique is described by Sirmans, et al. (1976). Dazhang, et al.

(1984) and Zrnic and Mahapatra (1985) describe an actual implementation.

Dual PRT technique. In the dual PRT (or staggered PRT) method the two PRT's

usually are in ratios of either 3/2 or 4/3. First, one calculates the

first lag complex autocorrelation for each PRT, averaging over a number of

pulses. Then, the expanded velocity is calculated from

= (981 -2)/47r(T2-T 1) . (4.5)

The corresponding unambiguous velocity is

Vmax = + 4/4(T2-T1 ) . (4.6)

According to this expression, a 3/2 PRT ratio yields an unambiguous velocity

that is twice that corresponding to the short PRT, while for a PET ratio of

4/3, the expanded velocity range is 3 times. Why not expand further? Since

the variance of the expanded range velocity estimate is based on the

difference between the two fundamental estimates, its variance is roughly

proportional to twice that of each fundamental estimate. Fortunately, the

53

expanded velocity estimate need be used only to roughly dealias the two

fundamental estimates. The velocity estimate can be improved by averaging

the two velocity estimates to get the final estimator provided they have

been correctly dealiased. This averaging technique provides an estimator

that uses all available pairs of consecutive pulses, rather than half the

available pairs.

Since the dual PRT technique dealiases velocities by a large factor, one can

operate the radar at a lower PRF and thus have a larger unambiguous range.

Doviak and Zrnic (1984) point out that another advantage of the multiple PRT

technique is that second trip echoes will be incoherent or "whitened" and

thus not bias the first trip velocity estimates.

Dual PRF technique. A disadvantage of the dual PRT technique is that

standard clutter filters are very difficult to implement. This can be

overcome for some filtering schemes by using a dual PRF technique wherein a

sequence of pulses is collected at each of two PRF's and then each sequence

is processed separately. The data processing is identical to the standard

pulse pair processing except that the velocity from the previous sequence is

used along with the velocity from the current sequence to dealias the

current velocity. The sampling statistics are similar to the pulse pair,

except that for this technique to be viable the mean velocity change between

adjacent sequences must be small.

Because the PRF is fixed while each batch is collected, the dual PRF

technique can employ a batch processing clutter filter such as an FFT or an

FIR filter. An IIR filter can be used, but several pulses will be required

to clear the filter between PRF changes. Because the basic dual PRF

processing is essentially the same as standard pulse pair processing at a

constant PRF, it is easier to implement on an existing system.

Unfortunately, the dual PRT feature of "whitening" the second trip echoes is

lost when dual PRF sampling is used.

54

4.2.2 Resoluticn of range ambiguities

Low PRF radars minimize overlaid echo but require sophisticated velocity

dealiasing techniques. If we promote the occurrence of overlaid echoes by

using a higher PRF to provide a large unambiguous velocity, then the range

aliased echoes must be resolved.

Most range dealiasing techniques use phase codes to distinguish between

first and second trip echoes. The simplest is the "magnetron" technique for

which each transmitted pulse has a random phase. A typical magnetron is

coherent-on-receive only for the current pulse. This means that

contributions from multiple trip echoes are not coherent so that they appear

as increased white noise power. Consequently, the mean velocity and

spectrum width are unbiased by overlaid multiple trip echoes. A problem

with this technique is that the reflectivity cannot be deduced unless

various received noise sources can be evaluated quantitatively. Also, the

additional white noise that is caused by multiple-trip echoes reduces the

sensitivity to first trip echoes and degrades the accuracy of mean velocity

and width estimates.

A similar technique can be developed using a fully coherent system such as a

klystron in conjunction with a phase shifter to change the phase of the

transmitted pulse. This permits the transmission of pseudo-random phase

sequences that have "white" properties (Chakrabarti and Tomlinson, 1976;

Sawate and Dursley, 1980). The I and Q values can be "recohered" relative

to the first trip or the second trip, etc., by using the appropriate phase

shifts so that Doppler spectra can be evaluated for each trip (Laird, 1981).

This technique offers information for both the first and second trip

returns, but does not solve the problem of reduced sensitivity for overlaid

echoes.

Siggia (1983) addresses this issue by filtering the first trip echo from the

second trip echo and vice versa, to reduce noise contamination. The

technique works well as long as the two Doppler spectra (1t and 2nd trip)

are not so broad that they occupy a large fraction of the Nyquist interval.

Zrnic and Mahapatra (1985) have evaluated this technique.

55

Sachidananda and Zrnic (1986) describe a different technique where, instead

of inserting phase shifts to "whiten" the 2nd trip echo, the phase shifts

are inserted to cause the second trip Doppler spectrum to be a split bimodal

spectrum whose autocorrelation for lag 1 is zero. This means that the

second trip echo does not bias the first trip velocity estimates.

All of these "phase diversity" techniques are well suited for standard

clutter filtering techniques. However, there are substantial signal

processing computations to implement some of them.

4.3 POIARIZATION SWITCHING CO(SEQUENCES

Bringi and Henry (Chap. 19A) describe various polarization techniques which

provide valuable target information but make clutter suppression and

velocity dealiasing more difficult. Differential phase propagation,

scattering and instrumental effects preclude use of simple Doppler

processing techniques (Schnabl, et al., 1986). However, it is possible in

principle to extract both the Doppler information and differential phase

shift simultaneously (Sachidananda and Zrnic, 1989; Doviak and Zrnic, 1984).

Keeler and Carbone (1986) describe a dual PRT scheme which allows processing

two orthogonal polarization states separately prior to velocity dealiasing.

The alternating horizontal and vertical polarized pulse sequence mitigates

contamination caused by range aliasing since the overlaid second trip echo

is depolarized (Doviak and Sirmans, 1973).

Processing techniques to simultaneously provide clutter suppression,

velocity and range dealiasing, and polarization processing are just

beginning to receive serious attention.

56

5. EPDXRAfTORY SIGNAL PXCESSIN TECHXNIQUES

Implementations of modern signal processing algorithms on atmospheric radars

have evolved slowly in the last several years. Modern digital signal

processing algorithms have been difficult to implement for a variety of

reasons, but the algorithms are well known (Kailath, 1974). Programmable

processors with the speed to implement many of these algorithms and to

explore their application to distributed targets, rather than point targets,

is now possible.

5.1 PUISE OCfMPRESSICN

Pulse compression, or wideband waveform, schemes for improved radar range

resolution were first theoretically described by Woodward's (1953)

fundamental paper. Klauder, et al. (1960) and Cook (1960) later described

the linear FM (chirp) pulse which has been widely used in military radars.

Reid (1969) described a CW meteorological radar using pseudo-random coding.

Barton (1975) has edited a collection of pulse compression papers which

details the chirp technique. Lewis, et al. (1986) emphasize poly-phase

coded pulse compression waveforms.

Probably the first use of pulse compression for atmospheric distributed

targets was on the Arecibo ionospheric radar (Farley, 1969; Gray and Farley,

1973). The STORMY weather group at McGill University implemented a

compression scheme for reflectivity processing in the early 70's (Fetter,

1970; Austin, 1974). Their use was to provide many independent samples of

intensity within a given range cell to improve the reflectivity estimate.

They did not attempt any velocity measurements using their pseudo-random

phase coded pulse. In the late 70's Krehbiel and Brook (1979) reported

using a wideband noise waveform on the New Mexico Tech "Redball" radar to

provide reflectivity estimates during the short dwell time of their fast

scanning radar. Chadwick and Cooper (1972) and Keeler and Frush (1983a and

1983b) have described the principle of pulse compression Doppler

measurements on microwave weather radars using distributed targets.

Browning, et al. (1978) describe the 10 cm pulsed Doppler radar at Defford,

57

England which was modified to generate 4 Js, 5 MHz chirp pulses and measure

Doppler shifts from ice crystals at 8 km range. Chadwick and Strauch (1979)

demonstrated an FM-CW waveform on a 10 cm Doppler weather radar. Woodman

(1980b) shows how a continuous wave phase coded waveform was used in the

bistatic mode at Arecibo. Recently he has obtained full spectrum

information using this technique.

Pulse compression is a well established waveform design technique in the

military and aviation radar communities and has been used in the ST/MST

radar community (Crane, 1980; Gonzales and Woodman, 1984; Sulzer and

Woodman, 1985) and the lidar community (Oliver, 1979), but has not been

seriously investigated for microwave Doppler weather radar use. The reasons

for this are:

1. Range resolution and transmit power using standard high peak power

pulsed radars have been adequate to achieve the required scientific

goals.

2. Dwell times have been limited by mechanical scanning rates to tens of

milliseconds, thereby yielding the several independent samples of the

Rayleigh fluctuations necessary to obtain accurate reflectivity

estimates.

3. Presence of range time sidelobes on pulse compression waveforms causes

range smoothing and large bias errors in high reflectivity gradients.

5.1.1 Advantages of pulse omupression

The driving force for exploring pulse compression in weather radars is the

desire for ground based and airborne Doppler radars to rapidly sample the

volume at a spatial resolution adequate for mesoscale or cloud physics

analyses. These systems fall into the short dwell time category. Dwell

times of only a few milliseconds are insufficient for averaging independent

Rayleigh fluctuations to reduce the variance of parameter estimates.

Therefore, independence must be gained in some other way, in particular by

multiple frequency schemes or spatial averaging. Marshall and Hitschfeld

(1953) pointed out that frequency separations greater than the inverse pulse

width give independent Rayleigh returns. Pulse compression waveforms give

58

independent returns (to first order) at spatial resolution proportional to

the inverse bandwidth (Nathanson, 1969). Either technique gives independent

returns over short dwell times ( <5 ms) so that the antenna beam can be

scanned at least an order of magnitude faster than typical weather radars

(Keeler and Frush, 1983b). Strauch (1988) proposes a burst chirp waveform

relevant to short dwell time weather radars.

Another application of pulse compression waveforms is in solid state

transmitter systems which typically are peak power limited to low values

compared to klystron transmitters, but can sustain very long pulse widths

and generate average powers comparable to the tube systems with greater

reliability. Pulse compression techniques could be used with these high

duty cycle systems to achieve range resolution corresponding to a much

shorter pulse length. The NOAA network wind profilers will incorporate

pulse compression for this purpose.

5.1.2 Disadvantages of pulse ccmpression

There are tradeoffs associated with using pulse compression to achieve

faster scan time. The tradeoff involves reduced radar sensitivity with a

compressed pulse compared to a single frequency pulse of the same duration

and power. While the full benefit of the average transmitted power is

achieved, however the noise bandwidth must be increased to accommodate the

pulse bandwidth. Therefore, the SNR of the individual samples is degraded.

Keeler and Frush (1983a) describe how this tradeoff relates to the "time-

bandwidth product" (TB) of the compressed pulse. For the same average

transmitted power the increase in independence is TB and the decrease in SNR

is TB. For example, a chirp waveform 1 microsecond long sweeping 10 MHz of

bandwidth has a TB = 10. Range samples spaced by more than 15 m are

independent and have a SNR ten times lower than the uncompressed 1

microsecond pulsed waveform. Frequently, the independent range samples can

be averaged to provide estimates having a reduced variance while allowing

much faster scan rates.

The primary disadvantage is a contribution to the backscatter from range

time sidelobes. Because the receiver filter output is the cross-correlation

59

of the received waveform and the time reversed transmit waveform (a matched

filter), range time sidelobes will cause data "blurring" in range space

similar to that caused by antenna sidelobes in the transverse spatial

dimension. Range time sidelobes (and antenna sidelobes) are especially

troublesome in high reflectivity gradients. Because atmospheric targets are

distributed in space, it is the integrated sidelobes that contribute to the

distortion. They are analogous to the integrated antenna sidelobes which

contribute interference from distributed targets at the same range. The

contamination problem is particularly troublesome in downward looking radars

from air or space platforms when one desires to estimate precipitation

directly above the strongly reflecting earth surface. Careful waveform

design and tapering based on digital waveform generation rather than analog

devices may alleviate the range time sidelobe distortion (Farina, 1987).

For echoes with sufficiently long correlation times, as is the case of

ST/MST radars using long wavelengths, complementary codes (Golay, 1961;

Schmidt, et al., 1979; Woodman, 1980a; Gossard and Strauch, 1983; Wakasugi

and Fukao, 1985) completely cancel the range time sidelobes. However, more

robust schemes, like quasi-complementary codes (Sulzer and Woodman, 1984)

show good results in practice when non-linearities in the system distort the

desired pulse shape. The direct application of complementary codes is not

compatible with the shorter wavelength weather radar and lidar system.

The second disadvantage for pulse compression waveforms is the increase in

minimum range caused by transmitting a long pulse. Reception cannot begin

until the entire transmit waveform is finished. Pulses longer than several

microseconds are unacceptable for close ranges. The NWS wind profiler

solution is to extend the scan time using a short pulse mode for short

ranges and use a long pulse mode for long ranges. Other techniques also

exist.

A third disadvantage relates to the availability of bandwidth. Research

systems are not seriously constrained, but operational systems may require

bandwidths which do not fit into the channelized frequency assignments.

60

5.1.3 Ambiguity function

The tradeoff in sensitivity for a larger number of independent samples gives

considerable flexibility in waveform design - so much flexibility in fact

that the concept of the "ambiguity function" was developed by Woodward

(1953) to study the effects on range and velocity ambiguities for a specific

waveform. For our purposes this ambiguity function is indispensable for

understanding the receiver response to targets at other ranges and other

velocities from that to which the receiving filter is matched. Weather

targets are distributed in range and velocity by their very nature and are

especially sensitive to these undesirable responses.

The ambiguity function defines the ability of a waveform to resolve

different targets in range and velocity based on the power response of a

filter matched to some specific range time and Doppler shift (Nathanson,

1969; Skolnik, 1980; Brookner, 1977). Figure 5.1 shows the ambiguity

diagram for a single FM chirp waveform in range (r) and velocity (¢) space.

Note that targets having non-zero velocities at ranges different from the

desired range (r=0) contribute significantly to the filter output. The

function evaluated along the r axis (i.e., 0=0) is identically the

autocorrelation function of the waveform (Frank, 1963; Cook and Bernfeld,

1967; Barton, 1975).

Atmospheric radars involve estimation of the return power and velocity

rather than detection of such a target at some position in range-velocity

space. Our primary interest in the ambiguity diagram is to study the range

time sidelobes as a function of Doppler offset. It is easy to show that the

plot of the ambiguity function along the range axis is simply the

autocorrelation function. Real weather targets having Doppler shifts of

order only 103 Hz compared to pulse bandwidths of 107 Hz allow us to

concentrate our attention to this narrow strip of the ambiguity function

along the range axis. All the range time sidelobes in this strip must be

kept small to avoid contamination of targets at the desired range and

velocity. Known waveform design techniques may allow tailoring of the

waveform to our "small velocity" case to keep sidelobes in this narrow

61

Ijx (r, ) I

Io1O6 /

Fig 5.1 Ambiguity diagram for single FM chirped pulse waveform with TB=10.

r is range dimension. 0 is velocity dimension. Targets

distributed in (r,¢) space contribute to the filter output

proportional to the ambiguity function. For atmospheric targets,

Doppler shift frequencies are typically very small relative to

pulse bandwidth (Rihaczek, 1969).

62

I

ambiguity region acceptably small (Deley, 1970; Kretschmer and Lewis, 1983;

Costas, 1984; and Lewis, et al., 1986).

5.1.4 CCpariso with multiple frequency sdceme.

Krehbiel and Brook (1968) and Keeler and Frush (1983a) show that a pulse

compression waveform with time-bandwidth product TB has characteristics

similar to a multiple frequency radar using the same time and bandwidth

factors. Consecutive pulses may be generated at different frequencies and

processed in separate receivers tuned to the different frequencies. This

scheme yields the same number of independent samples for the same total

pulse duration and total bandwidth. The advantage of the multi-frequency

scheme, aside from the straightforward parallel receiver implementation, is

reduced range time sidelobes.

5.2 AiAPTIVE FJIIERING AIXIcTHMS

At Stanford University in the early 1960's, Widrow and his colleagues

(Widrow and Hoff, 1960) developed a class of filters that could "learn"

their received signal environment and, in time, adapt their characteristics

to optimally filter an incoming signal. Initial applications were in

pattern classification (Widrow, 1970), but use in adaptive antennas (Widrow,

et al., 1967) and the closely related field of spectrum line enhancement

(Zeidler, et al., 1978) and noise (interference) cancelling (Widrow, et al.,

1975a) quickly followed. Griffiths (1975) has described instantaneous

frequency estimation techniques applicable to Doppler radars. Atmospheric

radar applications (i.e., non-military) have been sparse mainly because the

computational load associated with constantly changing filter coefficients

could not be accommodated until recently. Keeler and Griffiths (1977) have

reported adaptive frequency estimation schemes applied to acoustic radars

sensing boundary layer winds.

With the advent of fast programmable signal processors, we can expect to see

a rash of new applications in radar for adaptive filtering techniques.

Adaptive filter systems are characterized by both a time variable transfer

function and the ability to self adjust, or be trained, to their environment

for optimizing some measurement criterion (Alexander, 1986b). A common

63

index for optimization is the minimum mean squared error (mmse) between the

processed output signal and a known desired output (or at least one which is

correlated with the desired signal). Figure 5.2 depicts a 2 dimensional (2

weight) error surface. Widrow's (1970) popular Ieast Mean Square (IMS)

algorithm estimates the gradient of the quadratic error surface and steps

the weights toward the minimum error value.

Nearly identical adaptive techniques have been developed for antenna beam

steering by Howells (1976), Gabriel (1976, 1980), Appelbaum (1976), Monzingo

and Miller (1980), and Compton (1988). Adaptive antenna systems have the

capabilities of tracking desired signals in space, maximizing the SNR, and

nulling out undesired interfering signals. The optimization criterion is

maximization of signal to interference plus noise ratio, which for many

cases is identical to the IMS criterion. For radar applications the beam

can be steered to the desired direction and the adaptation can

simultaneously maximize the SNR by spectral shaping and spatially nulling

any interfering sources. Van Veen and Buckley (1988) give a tutorial review

of spatial beam forming techniques.

5.2.1 Adaptive filtering applications

The structure for a performance feed back adaptive system is shown in Figure

5.3 where we note the input signal xk, the adaptive processor output yk, the

yet to be defined desired response dk, and the error signal, ek = dk - Yk.This error signal drives an adaptive algorithm which controls the transfer

function of the adaptive processor, and its output yk. Various closed loop

structures are possible as are a variety of adaptive algorithms. Widrow and

Stearns (1985), Honig and Messerschmitt (1984), Alexander (1986b), and

Haykin (1986) give excellent overviews of these structures and algorithms.

Widrow, et al. (1976) describe the learning characteristics of IMS adaptive

filters in both stationary environments when the filters converge to an

optimal setting and non-stationary environments where the filter continues

to adapt to the time variable input signal statistics. Adaptive filters

have found application in data prediction schemes, system identification or

modeling, parameter tracking, deconvolution and equalization, and

interference (clutter) cancelling (Alexander, 1986a). Usually the

64

LU()2

.0

WIWI W2

Fig 5.2 Prediction error surface for 2 weight adaptive filter. The LMSalgorithm estimates the negative gradient of the quadratic errorand steps toward the minimum mean square error (mse). The optimumweight vector is W* = (0.65,-2.10). If the input statisticschange so that the error surface varies with time, the adaptiveweights will track this change (Widrow and Stearns, 1985).

65

Inni it

Xk

Errork

Fig 5.3 Adaptive filter structure. The desired response (dk) isdetermined by the application. The adaptive filter coefficients(Wk) and/or the output signal (Yk) are the parameters used forspectrum moment estimation (Widrow and Stearns, 1985).

66

application determines the origin of the reference signal and the specific

adaptive algorithm to be used. Sibul (1987) has edited a collection of

application papers for adaptive filters. Further applications in neural

networks and fault tolerant computing are being explored (Lippmann, 1987;

Shriver, 1988).

As an example of an atmospheric radar application, an adaptive linear

prediction filter will improve the SNR of the received signal so that the

moment estimation will yield improved estimates. In the frequency domain

the prediction filter acts as a narrow band pass filter having time variable

center frequency which passes the received signal while suppressing the

spectral noise components. Tufts (1977) and Anderson, et al. (1983)

describe this enhancement procedure. The input signal xk is the desired

signal, dk. The previous input samples (xkl, xk_2, ..., xk-L} = XT are

filtered to predict, or estimate, the present sample xk. The error signal

is the difference between Xk=dk and its estimate Yk, i.e. ekx-yk. The

filter is adjusted using the IMS algorithm so that the mean squared error

signal <ek2> is minimized. Sequentially then, the filter adjusts itself to

predict the input signal more accurately. Some error will be present but

the predicted signal will have an improved SNR over the input itself. In

this sense, we have an adaptive matched filter which can track the input

signal as its characteristics (e.g., its Doppler shift and width) change

with time (Tufts and Rao, 1977).

Probing deeper into the mathematics, we find that the algorithm is

estimating the negative gradient of a quadratic error surface in the L

dimensional adaptive filter weight vector space and adjusting the filter

weight vector Wk to step towards the minimum mean squared prediction error

with every iteration. This operation plus some supporting mathematics

defines the highly efficient steepest gradient descent IMS adaptive

algorithm (Widrow, 1970; Widrow, et al., 1975b).

Wk+l = Wk + 2gek Xk, (5.1)

67

where g is a precisely defined constant which determines the convergence

rate and the excess noise generated by the adaptation process.

It is easy to show that the one step prediction structure leads to the Lth

order maximum entropy (ME) spectrum estimate (Lang and McClellan, 1980;

Griffiths, 1975). Keeler and Lee (1978) have shown how the complex, first

order, one step prediction filter yields the pulse pair frequency estimator,

which has been made adaptive. Keeler (1978) further reports a bias and

variance of an adaptive ME frequency estimator.

What makes adaptive prediction and SNR enhancement possible is the

difference in correlation time of the desired narrow band signal (or

sinusoid) and the unpredictable white noise. Similarly, the long coherence

time of clutter input components may allow these interfering signals to be

rejected using adaptive interference (noise) cancelling filters (Widrow, et

al., 1975a). For example, airborne Doppler clutter can be represented by a

strong, narrow spectral return having a variable Doppler shift and sea

clutter may be sufficiently offset from zero Doppler that an adaptive scheme

may provide adequate suppression in both cases.

5.2.2 Adaptive antenna applications

Adaptive beamforming was motivated by a desire to steer the main beam in a

desired direction while simultaneously nulling interfering sources and

maximizing the signal to interference plus noise ratio at the output of the

adaptive beamformer (Haykin, 1985a; Compton, 1988). Atmospheric radars are

troubled by interfering ground clutter returns and could benefit from using

an adaptive antenna. For example, an RHI scanning radar could dynamically

place a line of nulls along the dominant ground clutter return angles near

0° elevation. Or a ground reflected multipath ray could be suppressed. UHF

radio communications present slowly time varying interfering sources for

wind profilers which could be suppressed by adaptive array techniques.

Forming nulls in the array antenna patterns in real time as the interferers

become active or as the antenna elevation increases may be feasible in many

cases. Constraints on the adaptation speed and antenna scan rates may limit

68

performance of these proposed systems since stationarity over a finite time

period is usually required. Furthermore, narrowbeam systems require several

thousand array elements and a digital control system for a truly adaptive 3

dimensional beam. Cost is a limiting factor in this regard (Mailloux,

1982).

Array processing utilizes multi-channel processing algorithms to process the

individual signals from each element to effect both spatial beamforming and

temporal filtering. Vector and matrix based algorithms introduce special

difficulties. Haykin (1985a) describes array signal processing algorithms

which have been applied to a variety of fields, e.g. seismology, radio

astronomy, tomographic imaging, sonar, and radar. Recently, Sachidananda,

et al. (1985) have proposed sequentially changing (at the pulse repetition

rate) the pattern of a phased array antenna. Subsequent Doppler processing

allows contributions to velocity estimates entering through the antenna

sidelobes to be whitened and/or removed (Zrnic and Sachidananda, 1988).

5.3 UITI-CHANNEL PRDCESSING

As atmospheric remote sensors become more sophisticated and programmable

processors achieve greater computational power, multi-channel processing

algorithms will become more common. The signals from separate input

channels can be thought of as a vector time series and processed, or

filtered, collectively by using the correlated information in the channels

to produce more accurate parameter estimates than if they were processed

separately (Marple, 1987). The coefficients of these multi-channel filters

are found by solving a set of linear equations similar to the single channel

equations used in linear prediction filtering and associated applications.

Wiggins and Robinson (1965) give a recursive technique for solving these

"normal" equations. Strand (1977) and Morf, et al. (1978) describe multi-

channel maximum entropy spectrum estimation, which is a direct result of

solving the normal equations.

In addition to radar array antenna data, dual polarization data is another

example of a multi-channel complex input signal. Horizontal and vertical

channels of a dual linear polarization radar can be processed to yield cross

69

parameters. Each input data point can be thought of as a 2x2 matrix, the

polarization matrix, rather than a complex I and Q estimate. The set, or

vector of these matrix inputs is then processed using complex matrix

algorithms which are designed to optimally and jointly estimate target

parameters. Processing both channels simultaneously yields additional

information that could not be obtained if they were processed independently.

Integrated sensor systems can benefit by multi-channel processing schemes.

A multi-channel algorithm might make use of 10 minute wind profiler data and

1 minute radar or lidar data. Wind profiles on multiple scales would be

produced with lower error than either system operating alone. Application

of coherence functions to these multi-channel sensors provides an analytic

tool for correlated data which improves the analysis.

5.4 A IPRICI INFR4ATICN

Information that is known in advance, a priori information, can be used to

improve atmospheric parameter estimates. Most remote sensors treat each

spatial resolution volume independently from all others. However, there are

physical constraints in the atmosphere that limit the rates of change of

certain parameters. These constraints are known in advance and can be used

to constrain the processing algorithm to produce better estimates of

velocity, for example, than if they were ignored. To be most effective this

a priori knowledge should be used as early in the processing chain as

practicable. For example, if one knows (or is confident that the received

signal consists of a Gaussian shaped signal spectrum in white noise, then

one should be able to use this prior information to generate a lower error

velocity estimate than if the information were ignored.

Signal processing algorithms constrained by known a priori information

typically yield simpler and faster algorithms that give lower variance

estimates than unconstrained estimators. Frequently these estimators are

maximum likelihood, i.e., minimum variance, and can be readily computed

using modern processing hardware.

70

6. SIGNAL PFXFESSCR IMPTrFMTATION

Signal processing encompasses analog and digital processing of both the

transmitted and received radar signal. Because of timing requirements, most

pulse-to-pulse control functions are also handled by the signal processing

system. In this section we discuss the signal processing implementations

that are found on modern radars and the tasks typically allocated to the

signal processor.

6.1 SIGNAL PROCESSING CONTROL FUNCTICNS

Signal processors usually perform a variety of radar control functions and

serve as the interface between the radar system and the radar data

processing system (usually a host computer). These control tasks include:

1. Pulse waveform selection

2. Polarization switching

3. Phase sequencing

4. Pulse sequence generation

5. Range gate trigger generation

6. Linear channel gain control

7. Calibration pulse injection

Radar control starts at the transmitter. The signal processor usually

generates the PRF, although good practice dictates that the basic clock be

derived from a reference oscillator that is shared between the processor and

the radar. PRF control by the processor minimizes the possibility of range

bin jitter caused by timing uncertainties in the A/D sampling and is

particularly important if a multiple PRF processing scheme is employed since

the processing must be synchronized with the PRF.

Because of the need to preserve the duty cycle limit of the transmitter, it

is a safety feature and a convenience to have the signal processor also

control the pulse width and bandwidth filter selection.

71

Since the signal processor is in control of the PRF, it is typically

assigned the task of controlling all pulse-to-pulse functions such as phase

control for pseudo-random phase processing and polarization switch control.

This approach assures that the processing is properly synchronized with all

aspects of the transmit-receive sequence.

Built-in calibration test units that operate during normal data collection

are now found on some systems. The idea is to inject a pulse of known power

and phase characteristics in the last few range bins for each transmitted

pulse or during antenna repositioning intervals with the transmitter off.

These bins are then processed identically to all other bins. The output

values can be monitored in real time to verify that the system is

functioning properly, and for system power calibration. In addition, the

injected signal can be made coherent so that the Doppler processing can be

checked. The advantage of this approach is that the entire receiver and

processing system can be verified without interrupting normal operations.

The remainder of this section is devoted to linear channel gain control

techniques. Currently, the receiver systems for most applications use

analog signal -processing techniques for deriving the linear channel I and Q

(in-phase and quadrature) and log channel outputs. The log channel output

is typically used for quantitative power measurements because of its dynamic

range capabilities (90-100 dB). The linear channel measurements are used

for extracting information related to the phase of the signal, i.e., mean,

velocity, spectral width and clutter measurements, and can provide power

estimates as well. The linear channel measurements operate over a more

restricted dynamic range, typically z40-60 dB, that is usually shifted by

means of an automatic gain control (AGC) loop over a range of ~100 dB. It

is the linear channel gain control problem where digital signal processing

often makes its first appearance in the radar processing chain.

Linear receiver gain control is typically performed via one of the following

methods:

1. IF limiting

72

2. Sensitivity time control (STC)

3. Slow AGC

4. Fast AGC

5. Multiple receivers

In the first case, a "soft" limiter is inserted at IF before phase detection

(Nathanson, 1969; Zeoli, 1971; Frush, 1981 ). The advantage of this

technique is that it is extremely simple to implement and permits the linear

receiver to operate over a fairly wide dynamic range with good mean velocity

retrieval. However, if the Doppler spectrum is bimodal, such as for ground

clutter mixed with a weather spectrum, this technique tends to "capture" the

stronger signal and suppress the weaker one. This behavior makes it

unsuitable for systems that require clutter cancellation.

For the STC case, the linear channel gain is increased with range in an

attenpt to represent the average characteristics of weather and clutter.

Since there is no feedback based on actual power measurements, it is easy to

implement. However, it is a near certainty that strong clutter targets will

cause saturation of the linear receiver at close range unless an IF limiting

approach is used as well. Likewise, weak clear air echoes that would be

detectable at full gain at close range, will be attenuated beyond

detectability.

For the slow AGC, the log receiver measurements from the previous ray are

used to optimize the linear receiver gain for the targets that are actually

present at each range. The samples for an integration period are collected

while the gain is held constant. If the log receiver is used for

quantitative power measurements, the actual gain does not need to be known

with great precision (within 3 dB is usually satisfactory). Also, since the

gain is held constant, the phase shifts that are introduced by the gain

control are constant from pulse-to-pulse so that these do not have to be

corrected. The primary drawback is that the ability to distinguish between

the clutter and weather components of the signal may be limited by the

fundamental dynamic range of the linear receiver. Furthermore, strong

reflectivity gradients will cause erroneous gain settings.

73

The fast AGC, or instantaneous AGC (IAGC) approach, for which the gain of

the linear receiver is adjusted for each range and each pulse, is used where

there is a high degree of phase purity in the transmitted pulse (e.g.,

klystron systems). The power measurement for either the previous pulse, orthe current pulse (in which case a delay line is required) is used to set

the receiver gain. This is the most complicated form of AGC to implement

since it requires a very accurate calibration of both the amplitude and

phase response of the receiver as a function of gain and the input power.

Mueller and Silha (1978) employ a real-time calibration and correction

scheme so that the output phase of the linear receiver requires no

correction. Properly implemented this approach provides wide dynamic range

linear response for high-performance clutter cancellation and more accurate

estimates of the power than a log channel.

Another approach is to employ multiple receivers, each optimized for a fixed

range of input power with the advantage that all samples can be digitized

and the optimal receiver can be decided with a digital algorithm. Moreover,

switching transients and calibration procedures are minimized.

6.2 SIGNAL A/D CONVERSIC N AND CAIBRATICN

Figure 6.1 shows a block diagram of a typical digital, time domain Doppler

signal processor. The digital signal processor provides the interface to

the radar I, Q and log signals, and connects to a host computer that

provides the user interface, data processing, display and data

communications.

After analog phase detection, the I, Q and log values are digitized. In the

case of a fast AGC, a digital AGC value may also serve as an "exponent" for

a floating point representation. The precision that is required for

digitizing the I and Q values depends primarily on the underlying precision

of the linear receiver and the dynamic range limitations imposed by ground

clutter induced phase noise. In computing dynamic range, an additional bit

amounts to 6 dB more power measurement capability. However, because the

receiver noise level requires about two bits to coherently integrate weak

74

g

ication

Fig 6.1 Block diagram of a typical signal processor.

75

signals and one bit denotes the sign of bipolar data, the usable

instantaneous dynamic range is limited to ~54 dB for 12 bit samples. This

range provides a margin for an AGC that may not optimize its use of the

receiver dynamic range and offers reasonable clutter rejection. For the log

channel, the quantization of the digitized signal determines, to some

extent, the accuracy of the final power estimates. However, it is usually

the inherent large fluctuation of «30 dB for Rayleigh signals (Nathanson,

1969) that imposes the more fundamental limit.

The A/D converter values should not saturate. I and Q saturation causes

harmonic generation in the frequency domain. Furthermore, image spectrum

generation about DC in the spectrum is frequently caused by imbalance in the

amplitude and/or phase of the I and Q signals (Hansen, 1985).

Time domain averaging is an important step in processing ST/MST radar

signals to reduce the noise (Strauch, et al., 1984). The averaging not only

increases the SNR by N, but also increases the dynamic range by 10 log N.

The discussions above illustrate the need for time series and power spectrum

displays to optimize radar performance. Just as important, the host

processor must be equipped with software to provide the interactive displays

that are required for accurate system adjustment and verification.

6.3 REEIETlVlTY PROCESSING

The precise measurement of the received power is an important objective for

most weather radar systems, and for noncoherent systems, this is the primary

measurement. In the pre-Doppler era, there was interest in the so-called

"power-fluctuation spectrum" and spectrum width estimates (Rutkowski and

Fleisher, 1955; Atlas, 1964;). Most radar systems, whether Doppler or

noncoherent, employ a wide dynamic range log receiver that operates at IF.

These systems merely average the log values which results in an asymptotic

(with the number of independent samples) 2.51 dB bias in the estimation of

the average power for Rayleigh distributed targets (Doviak and Zrnic, 1984).

There are other types of receiver responses, such as the linear and square

law receivers, and the log receiver has the largest standard deviation for

power estimates (Zrnic, 1975a). However, in view of calibration errors and

76

the uncertainties in relating power measurements to rainfall rate, the log

receiver performance is adequate for many applications. When differential

reflectivity measurements are required, one attempts to measure small

differences in power so that the square law receiver is preferred (Bringi,

et al., 1983; Chandrasekar, et al., 1988).

Two common techniques that are used for power averaging are the exponential

average (Zrnic, 1977a) and the uniformly weighted average. Exponential

averaging is calculated using

Pk = Pk-1 *(1-C) + Pk *C (6.1)

where Pk is the current estimate of average power based on the new sample pk

and the previous estimate Pk-1 . C is a weighting constant between 0 and 1.

When C is close to 1, the current pulse is more strongly weighted. This

technique is extremely simple to implement in real time and provides a new

estimate for each pulse. Since real time digital processing capabilities

have improved, and analog CRT displays are rapidly being replaced by color

displays, this technique has been largely replaced by a simple uniformly

weighted average over a fixed number of pulses.

Averaging of independent samples is required to obtain accurate reflectivity

estimates. Since. independence is governed by the coherence time this

imposes a fundamental constraint on the scan rate for data collection. For

example, at 3 rpm and 500 Hz PRF, one can average only 27 pulses per degree

of antenna rotation. Depending on the wavelength and the spectrum width of

the scatterers, not all of these pulses will be independent. A technique

for increasing the number of independent pulses is to average in range using

a range bin spacing that is greater than the pulse width. This requires

somewhat more processing power, but results in more accurate reflectivity

estimates. Also, averaging can be adjusted as a function of range so that

the resulting average range interval is comparable to the beamwidth

dimension.

77

The conversion from dBm to dBZ is done via the radar equation which involves

the radar constant and range normalization. The term "STC" is sometimes

inappropriately used to refer to the digital range normalization that isperformed in the processor. This term is a reference to the analog

technique that was used in the past to represent the radar reflectivity on

CRT display. Digital range normalization merely adjusts the output values

appropriately without causing the loss of sensitivity at short range.

6.4 'THRESHOEIDfING R DATA QUALITY

The goal of thresholding is to have the signal processor flag data that

may be corrupted by bias and artifact. Clarity of presentation of the

spectral moments is important to a user trying to interpret a display. Forsubsequent data processing and product generation (e.g., CAPPI's, cross-

sections, rainfall accumulations), noise, bias and other artifacts increase

the computational demand on the data processor and degrade the final

product. Finally, thresholding followed by run length encoding for data

compression can greatly reduce the communications bandwidth requirements for

transmitting radar data and products and reduce the archive resources that

are required to store them.

There are numerous thresholding criteria and variables that are employed in

modern radars:

1. Incoherent signal-to-noise power

2. Coherent signal-to-noise power

3. Doppler spectrum width

4. Clutter-to-signal power

5. Zero velocity

6. Geometric criteria

7. Statistical criteria

The incoherent signal-to-noise power is calculated by comparing the received

power at a range bin with the system noise power (S+N/N). This criterion is

most commonly used to threshold the wide dynamic range power measurements

(e.g., from a log receiver). The coherent signal-to-noise power is the area

78

under the signal portion of the power spectrum divided by the total noise

power (S/N). It can be calculated directly from the spectrum, or using the

measured autocorrelations. Similarly, the spectrum width itself can be used

as an indicator of the accuracy of the Doppler mean velocity and spectrum

width.

Both a low coherent signal-to-noise ratio and a large spectrum width

contribute to a large variance in the velocity and width estimators.

Ideally, thresholding should be made at a constant variance level, e.g.,

velocity is accepted if it's expected error is less than 1 m/s.

Unfortunately the relationship that governs the effect of SNR and width on

the variance of the velocity estimator is not a simple one (Zrnic, 1977b),

hence it is usually not implemented as a real time thresholding criterion.

Instead, the typical approach is to use either the coherent SNR and/or the

width separately and adjust the threshold until the displays are reasonably

free of speckles.

A popular measure of the quality of velocity and width estimates, which

accounts for the effects of both the coherent SNR and the spectral width is

the normalized first lagged autocorrelation magnitude IR(1)|/R(O). It is

easily computed, conveniently bounded between 0 and 1 and thresholds

unreliable estimates reasonably well.

The measured clutter-to-signal ratio (CSR) is often calculated for the

purpose of correcting the log receiver power for the effects of clutter.

When the actual CSR exceeds the dynamic range capabilities of the receiver

or the ability of the clutter filter to accurately remove clutter, then the

data should be discarded. The calculated CSR can then be used as the

thresholding criterion.

Another method of thresholding range bins that are affected by clutter is to

simply not display bins that have a mean velocity within a narrow band about

zero velocity. This technique is effective for Doppler radars that have no

clutter filter, or Doppler radars of limited linear dynamic range available

for cancelling clutter. Both the velocity and reflectivity can be

79

thresholded using this criterion. Unfortunately, any weather that falls

into the threshold velocity band is also rejected.

Simple geometric considerations can be used for thresholding data that are

not physically reasonable. A very simple threshold is to eliminate all data

that are above a fixed height where weather echoes are assured not to occur,

e.g., 20 km. Another threshold that is easily implemented in a processor is

a "speckle remover" that eliminates all isolated range bins that have no

nearest neighbors in range or azimuth. Use of a speckle remover eliminates

aircraft and point clutter targets. It also allows other thresholds to be

set to lower values for greater sensitivity since only double speckles will

be passed.

Finally, statistical criteria involve considerations of local continuity and

rejection of data that are a few standard deviations away from local mean

values. Strauch, et al. (1984) utilize a very effective "consensus

averaging" technique (Fischler and Bolles, 1981) to delete wild points or

outliers for time domain integration of wind profiler processing. One or

two dimensional median filtering techniques also allows deletion of

individual or isolated groups of anomalous data.

The application of thresholding requires caution. One common problem

develops when a linear channel index is used to threshold both the velocity

and the reflectivity. If this is done it is not uncommon to observe "black

holes" of rejected reflectivity echo (so called if the display background is

black). These often occur in regions of large shear or turbulence such asthunderstorm cores (Hjelmfelt, et al., 1981) where there is ample

reflectivity present. This points out that different threshold

combinations, and perhaps threshold levels, should be used for the different

spectral moments. For example, an acceptable threshold for velocity will

generally not be appropriate for spectrum width since spectrum width

requires a stronger signal for proper estimation.

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7. ITREND IN SIGNAL PROCESSING

7.1 REALIZATIC[N FACT

Several key components comprise a realizable signal processing system--

chips, memory, and a large bandwidth output device. This digital technology

has found wide applications in modern radars (Rabinowitz, et al., 1985).

7.1.1 Digital signal processr chips

In the last 5 years integrated circuit chips specially optimized for digital

signal processor (DSP) operations such as multiply-accumulate, on-chip

memory, and the supporting logic have developed computational power

exceeding hardwired processors of several years ago. These DSP chips are

available from a variety of manufacturers and can be installed on

commercially available high speed busses, such as VME and Multibus II. As

integrated circuit developments in memory continue, on-chip memory will

expand to allow caching and make DSP algorithm's more efficient.

Interconnectability using multiple fast busses and fast communication ports

still allow full implementations of many DSP algorithms. The commercial

availability of families of DSP chips and busses provides documentation,

technical support, and probable upgrades for faster and compatible

processing speed.

Current 32 bit DSP chips are based on silicon technology (TTL and CMOS) and

can achieve clock rates of tens of MHz and execution rates of a few Million

Instructions Per Second (MIPS). The next generation of microprocessor and

DSP chips will be fabricated from gallium arsenide (GaAs) and will allow

several processors to be attached to a single chip component. Clock rates

for these advanced devices will be a few hundred MHz with instruction rates

exceeding 100 MIPS. This technology is growing rapidly. However, within

the next several years the number of components per chip will be limited by

fabrication processes and shortly thereafter by physical constraints within

the chip itself (Aliphas and Feldman, 1987).

81

An important factor that will allow rapid expansion of radar processing

power is the trend of D6P chip manufacturers to develop higher performance

chips that are compatible with previous versions. Thus, a relatively simple

redesign of the processor board using the same basic architecture, combined

with reprogrammed algorithms, offers greatly enhanced processing power at

low cost.

The ready availability of the processing power obviates a move towards more

real time processing. For example, as multi-parameter radars and faster

scanning radars evolve, more processing power will be necessary to compute

the quality-checked, auto-edited data that is so valuable to real time

observations. The real time processing can perform all the "signal

processing" plus an increasing amount of the "data processing" tasks.

7.1.2 Storage media

External devices for mass storage have long been dominated by magnetic tape.

The half inch tape is the standard, but various other tape-based media and

technologies are being explored. These include special high density tapes

such as NCAR's obsolete TBM (terra-bit memory), magnetic tape cartridges,

video cassettes, and the digital audio tape (DAT) devices using helical scan

technology. All of these tape storage media suffer from serial access

delays and are undesirable for on line, fast access storage. However, they

are extremely well suited for "write-once" archiving applications such as

radar data acquisition. Storage capacities of two or more gigabytes can be

achieved today. Higher capacity and faster transfer rates will continue to

evolve. Winchester disks using "vertical recording" techniques allow high

density and fast access and fit many applications which require fast, random

access storage.

The thrust in storage media development now seems to be in optical recording

techniques. Compact disk (CD) technology, being a consumer product, has

become relatively inexpensive. The data capacity of optical media is

approaching several Gbytes on a 5.25" CD and data transfer rates of several

Mbytes/sec are possible. Random access times are being reduced to the

millisecond range.

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7.1.3 Display technology

Real time color radar displays have become an important component of remote

sensor technology since their first implementation by Gray, et al. (1975).

Intensity modulated PPI and RHI scopes show high resolution reflectivity

displays, but digital color displays show all the directly measured

variables (e.g., velocity) as well as derived variables such as differential

reflectivity, phase, and depolarization quantities. Plotting data from

multiple sensors in real time, zooming into specific areas of interest,

generating time lapsed images, and defining special overlays provides a

measure of flexibility not available only a few years ago. Special purpose

programmable graphics processors allow these new, yet fairly simple, image

processing capabilities. The next generation of graphics processors will

accommodate 3 dimensional real time image generation, color images with

transparency, easily manipulated images to change the viewing angle, and

programmability in high level languages to allow a high degree of user

interaction. The display is the investigator's or the user's contact to the

environment being studied or watched. Particular emphasis should be placed

on this aspect of the remote sensor to extract its maximum utility.

7.1.4 omrmercial radar processors

Radar processors have historically been developed by the organization

responsible for the entire remote sensor system. Recently, however, digital

signal processors have become commercially available as special purpose

computers for Doppler lidars (Bilbro, et al., 1984), and weather radars

(Siggia, 1981; Chandra, et al., 1986, and Schroth, et al., 1988). The

specialized processing algorithms being developed and applied to atmospheric

remote sensors can be efficiently integrated into many types of remote

sensors and customized to the specific application by different software.

System engineering of signal processors is changing because of the

improvements in hardware technology and architectures (Allen, 1985).

However, the biggest change is occurring because of changes in the system

engineering methodology. Open software standards for operating systems

83

(e.g., POSIX), for computer language (e.g., ANSII standards, Ada, etc.), and

run-time environments (e.g., X-OPEN) are being developed and applied. Data

bus standards, (e.g., VME) are being clarified, updated and adhered to by

board and peripheral manufacturers. Open software standards and workable

data bus standards facilitate cost-effective development and manufacture of

special signal processing boards that integrate and can be upgraded to the

latest DSP chip sets.

7.2 IREN1&S IN I EGRA4TBILr[TY OF DSP

The new generation of digital signal processors for atmospheric remote

sensors is programmable. This is a marked contrast to early hardwired

processors in which the algorithms could be modified only with great

difficulty and most often resulting in the loss of the original capability.

Programmable processors allow algorithm modifications, processing

experiments, diagnostic testing, and system testing while still retaining

the capability of returning to a pre-existing mode of operation. Modern

digital filtering and waveform processing using advanced algorithms is now

possible without the constraints imposed by physical limitations of hardware

devices. Schmidt, et al. (1979) and Woodman, et al. (1980) describe

programmable signal processors for VHF Doppler wind profilers. These

present day DSP systems are directly programmable in modern languages, such

as "C".

Advanced processing algorithms using matrix methods, such as singular value

decomposition, orthogonalization, multichannel optimization techniques, and

non-linear processing algorithms using adaptive and data compression

techniques (Haykin, 1985a; Kay, 1987; Marple, 1987) can be coded and tested

on line in real time, without destroying the original algorithm

implementation. Standard algorithms can be as easily replaced as they can

be modified. Optimization may become an easier task.

As the DSP chips support higher level languages, algorithm portability

becomes easier to achieve. Reproducability of clone processors and

algorithms, for example in a radar network, is feasible. However,

programmable hardware leads to a new set of development and maintenance

84

problems. A higher level of training and maintenance equipment is required

for trouble-shooting a malfunctioning radar processor. Board level

maintenance may require a more expensive spare inventory. Programmability

brings new headaches as well as many new features.

Another area of rapid development important to distributed signal processor

architectures is the application of multiprocessor operating systems.

Distributed computing power on a common high speed bus requires an operating

system capable of controlling data transfers and bus arbitration and memory

management. Presently these operating systems are targeted towards more

general purpose processor chips (e.g., the Motorola 68030), but future

application will find them on distributed DSP processors as well. Software

development is a key issue in generating efficient realizations of the DSP

algorithms. UNIX is presently becoming accepted as the common operating

system of choice for many applications programs and for development of real

time software, which then typically run under a UNIX compatible real time

operating system (e.g., VxWorks, PDOS).

7.3 SHIOR TERM EXPECCTATICNS

During the next 5 years we may expect a revolution in atmospheric digital

signal processor technology. However, this technology will tend to leave

the atmospheric science community behind unless we prepare ourselves to take

advantage of the evolving hardware and software advances. We have lived by

the pulse pair processor for over a decade. Other techniques have been

explored that in same instances provide better parameter estimates but have

not been feasible to implement in the past. This constraint is rapidly

disappearing.

7.3.1 Rarge/velocity ambiguities

Within the next 2 or 3 years we may expect several research groups to

implement new pulsing and processing schemes for range and velocity

dealiasing. These schemes, driven by the FAA's Terminal Doppler Weather

Radar (TDWR) procurement, as well as the Nexrad implementations, will allow

ground clutter suppression simultaneously with velocity dealiasing and

overlaid echo suppression algorithms. There will be exploration of

85

polarization processing improvements combined with resolving range and

velocity ambiguities and clutter suppression.

7.3.2 Ground clutter filtering

Effective clutter filtering will be readily implemented on conventional

Doppler radars. However, efforts to integrate clutter suppression with

other processing improvements will likely encounter several technical

obstacles involving analog components (e.g., polarization switches, IF

amplifiers, and transmitter instabilities). Fundamental limitations related

to the narrow clutter spectra may well limit clutter suppression for radars

using dwell times shorter than the clutter correlation time. Yet to be

explored nonlinear filtering techniques may allow effective suppression even

under these conditions.

7.3.3 Waveforms for fast scanning radars

A major limitation of existing Doppler meteorological radars is their

inability to scan a solid angle in space fast enough to measure a rapidly

evolving atmospheric event with adequate temporal resolution. A dwell time

of a few milliseconds is desired. The proper long term solution requires an

electronically scanned phased array antenna - a very expensive item. The

mechanical solution of simply scanning faster and using short dwell times is

insufficient to preserve the parameter measurement accuracy. Scan rates

greater than about 100 degrees per second for a 1° beamwidth cause spectrum

spreading due to antenna motion that rapidly degrades the measurement

accuracy. A reasonable alternative is to rapidly scan mechanically at a

rate such that the spectrum spread is not dominated by the scan induced

component and to use a wideband waveform (pulse compression or multiple

frequency) that allows a reasonably large number of independent parameter

estimates to be made in the short dwell time imposed by the coherence time

of the return signal. Some research groups are testing short dwell time

waveforms (Keeler and Frush, 1983b; Strauch, 1988) on both airborne and

ground-based weather radars.

86

7.3.4 Data cmpression

Data compression algorithms are an important aspect of signal processing.

Data compression can be divided into two classes -- "truncation" for any

range gates at altitudes greater than the tropopause and "run length

encoding" or "compaction" for strings of data having the same value.

Typically parameter estimates not passing some threshold test are

arbitrarily set to zero and run length encoded. Data truncation will become

more common as programmable processors are installed.

7.3.5 Artificial intelligence ased feature extraction

Future computing will be directed at enhancing man's analytical and

inferential skills, rather than routine physical or mental activities.

Symbolic programming techniques combined with knowledge engineering and

artificial intelligence techniques show potential for rapid advance; the

same is true for meteorological image processing and automated recognition

and extraction of atmospheric features. Two dimensional signal and image

processing algorithms will be implemented using programming architectures,

reducing development time and extracting more meteorological information

from remote sensor data sets.

7.3.6 Real time 3D weather image processing

Relatively new computing hardware allows ready implementations of various

symbolic object processing systems that can be applied to problems in

atmospheric science. Coupled with fast graphics processors we can expect

real time 3D images produced with the latest image rendering techniques

which allow reconstructed radar data fields overlaid with in-situ

measurements from airborne and ground based meteorological stations.

Graphics computers with large video memories allow time lapsing of high

resolution 3D images and arbitrary cross sections to be displayed using a

variety of techniques currently being developed. Transparency of data

elements near the viewer allows observation of the storm interior.

7.4 IDNG TERM EXPECIATICNS

Several years from now we can expect revolutionary changes in the way signal

processing will increase our ability to understand atmospheric dynamics in

87

real time. Combining new hardware forms and more efficient softwaredevelopment techniques with evolving communications technology and thetumbling cost of computing power will allow remote sensor systems to present

readily assimilated graphical formats. These systems will provide an

interactive user interface taking forms that are only dreamed about today.

For example, tactile feedback technology will allow a meteorologist to

manually pick up a "thunderstorm" and manipulate it to better examine the

evolving towers and outflows.

7.4.1 Advance hardware

The present development of GaAs (gallium arsenide) computing elements may

replace silicon dominated chips if the promised five fold speed increases

and higher reliability in thermal and radiation extremes are realized.

7.4.2 Optical interconnects and processing

Fiber optical communication is capable of extremely high bandwidth. Data

rates and parallel processing using optical techniques can accommodate

processing algorithms having throughput many orders of magnitude higher than

serial and most existing parallel digital signal processing schemes. Fiber

optic back planes for computers are available now.

7.4.3 Ctumunicatians

Processing of atmospheric radar signals has many concepts in common with

communications processing and the same technologies can be incorporated. By

logically combining the processing functions with the communications link,

both locally and over long distance, new capabilities will be possible.

7.4.4 Electronically scanned array antennas

Military budgets have financed the development of highly efficient, very low

sidelobe, multiple beam, two dimensional electronically scanned array

antennas. The computing power necessary to control the beams is available

but the communications to each array element, the phase shifters capable of

handling high peak powers for radar systems, and the sheer number of

elements required (several thousand) are very costly. These step scan

antennas will allow more rapid volume coverage while retaining parameter

88

accuracy and will reduce the deleterious effects of antenna sidelobes. The

very high cost of this performance increase must be justified for

atmospheric radar applications.

7.4.5 Adaptive systems

Self learning, time variable processing systems will allow a degree of

optimization that is not possible today. Neural networking concepts utilize

interconnected arrays of processing elements which share the processing and

communications load so that the overall computational efficiency is

maximized. The algorithms used in these adaptive systems can be defined by

a training sequence or can be self learning during the processing time.

Research is concentrating on integrating distributed processing concepts

with expected hardware.

89

8. CoNCrI3SIONS

8.1 ASSESSMENT OF CUR PAST

Radar signal processing engineers, in the meteorological radar community at

least, have taken a somewhat narrow view of signal processing in the past.

A large effort has been dedicated to using the pulse pair algorithm for

estimating the first two or three spectral moments, largely because the

existing processing power has been rather limited to these simple algorithms

and because for an important class of signals the pulse pair algorithm is

optimum. Advances have been made in the ST/MST radar community in pulse

compression, coherent averaging, and non-linear least squares parameter

fitting techniques, and in the lidar community in multiple lag processing.

Other techniques have been ignored or rejected simply because the scientific

need for these advances did not exist, or if it did, the risk of undertaking

such a development was not warranted.

The operational radar community and many researchers have been unable to

explore weak echoes because of inadequate sensitivity. There are better ways

of improving radar sensitivity than brute force techniques of more power and

larger antennas. Advanced signal processing techniques must be explored

more thoroughly to achieve these sensitivity gains. Modern spectrum

analysis methods for modeling distributed target echoes in strong clutter

and multi-channel processing techniques to extract better information from

collections of remote sensors is an area ripe for extensive research.

The digital boundaries of the signal processor are being extended in both

directions. Digital IF quadrature mixers are presently available which will

accept IF and local oscillator analog signals and put out digitized I and Q

samples. Digital matched filters operating at IF rather than baseband (DC)

will became a reality. The radar engineering community is ready to

integrate these new components where warranted.

91

8.2 RECEMMENDATIONS R C FOR U RE

Aside from continuing to actively explore many of the modern signal

processing techniques, there are two general recommendations we would

encourage for utilizing modern signal processing algorithms.

First, many universities have active digital signal processing groups in the

Electrical Engineering departments and many industries have vast experience

in radar signal processing techniques. Our research community should strive

to interact more strongly with these two on an international scale. The

university cooperative education programs should be explored and encouraged.

University exchange programs involving signal processing experts as well as

meteorologists should be encouraged. Industrial contacts with radar

manufacturers and systems producers, such as NEXRAD and TDWR should be

maintained so as to exchange signal processing expertise as well as

meteorological expertise.

Second, the meteorological radar community should maintain the lead in

sponsoring signal processing sessions at AMS radar conferences and sponsor

participation in other signal processing related meetings. Members of the

ST/MST radar and coherent lidar communities should be encouraged to attend

these sessions (and vice versa) since our target models, our propagation

medium, our processing problems, and our techniques are nearly identical.

As noted before members of these communities have successfully explored

modern algorithms and predated weather radars use of the pulse pair and poly

pulse pair velocity estimators as well as use of pulse compression and

complementary coding schemes.

Finally, as R.W. Lee of the Signal Processing panel stated once, we can now

build processors with "megaflops to burn". We can use them very easily by

implementing new processing algorithms, for example, using a priori

knowledge to improve estimates. Computing special diagnostic outputs which

have no bearing on the data collected, but simply allow the operator to

adjust processing parameters, is an effective use of processing power.

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8.3 ACCEPiANCE OF NEW TECHNIQUES

New techniques are not usually accepted easily by any scientific community.

Twenty years ago, Doppler processing using the now standard pulse pair

estimator was not readily accepted. Why should any new signal processing

algorithms using only statistical concepts improve the accuracy of moment

estimates? Skepticism is healthy in science. Accepting a new technique

requires four critical conditions:

1. An important application, a problem which needs to be solved.

2. An intuitive, familiar basis for understanding the concepts involved in

the new technique, which includes a convenient interface for exploring

the innards of the new technique.

3. A field demonstration to convince the community that the new technique

is indeed an improvement over the former.

4. Real, live funding for development and demonstration.

8.4 ACKNoICMrEDGMENT

The authors wish to thank the panel members for their verbal and written

contributions to this report. D. Zrnic and R. Serafin have been especially

helpful with comments on various drafts. V. Chandrasekar, J. Evans, G.

Gray, J. Klostermeyer, F. Pratte, R. Strauch, R. Wiesenberg, and R. Woodman

and have provided helpful written comments that have been incorporated into

this signal processing review. J. Devine provided expert assistance with

integrating the text, the figures, and the references.

93

ACIHNYM isr

A/D - analog to digital

AGC - automatic gain control

CNR - clutter to noise ratio

CSR - clutter to signal ratio

DFT - discrete Fourier transform

DSP - digital signal processor

FFT - fast Fourier transform

FIR - finite impulse response

FM-CW - frequency modulated continuous wave

IIR - infinite impulse response

IF - intermediate frequency

I/Q - in-phase / quadrature

IMS - least mean square

ME - maximum entropy

ML - maximum likelihood

PRF - pulse repetition frequency

PRT - pulse repetition times

SNR - signal to noise ratio

95

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