Chapter 2

Signal and interference models

In this chapter models for target, clutter, jammers and noise are derived which formthe basis for the numerical evaluations carried out in the subsequent chapters. To someextent we follow the derivation by VAN TREES [508, pp. 238].

Based on such models ROMAN et at. [440] developed state variable models whichindicate that multichannel models of low order can represent airborne ground cluttereffectively. This will be confirmed by the results obtained in Chapters 5, 6, 7, and 9.These chapters deal with order-reducing subspace techniques for clutter rejection withreal-time capability.

2.1 Transmit and receive process

Let us assume that the radar transmits a signal of the form

where o;c = 2TT/C denotes the carrier frequency, A t is y/2Pt with Pt being thetransmitted power, and E(t) is the envelope of the transmitted waveform for whichwe assume

(2.1)

Assuming an ideal point-shaped reflector at distance Rt from the radar the receivedsignal becomes

where Ar = \ /2P r . Pr includes the transmit power and the two-way propagation andreflection processes according to the radar range equation (for details see S KOLNIK[467, pp. 2-1,...,2-73], [468, pp. 2-1,...,2-68]). r is the round-trip delay of thetransmitted wave between radar and target

(2.4)

(2.2)

(2.3)

Tm = T is the time delay until the signal reaches the matched filter. The matched filteroutput becomes a maximum if r m = r. In this way the target range is estimated bymeasuring the echo delay. By use of (2.2) one then obtains

sr(r) = A1. (2.8)

This is the response to a target at a range defined by r m = r. For a detailed theory ofmatched filtering see RiHACZEK [432].

2.2 The Doppler effect

We now consider a target moving at constant radial velocity vra(\. Ait 0 the targetrange is R0. The target motion causes the target range to vary with t:

where c is the velocity of light. The received signal then undergoes a complexdemodulation which can be described simply by

(2.5)

The high-frequency term will be suppressed by a low-pass filter and the factor of ^ canbe incorporated in the amplitude. Then the demodulated signal becomes

(2.6)

The low-pass filter is usually followed by a band-limiting filter matched to the envelopefunction

(2.7)

A signal received at time t was reflected at t r(t)/2. The associated target range is

Inserting (2.4) into (2.10) and solving for r(t) yields

In practice target velocities are small compared with the velocity of light, i.e. v raci/c

The target motion has two effects on the received signal

The envelope is stretched or compressed. That means that the matched filter (2.7)is mismatched to the received echo signal.1 In radar this effect is negligible forrealistic target velocities and signal time-bandwidth products (s. VAN TREES[508, p. 241]).

The carrier frequency is shifted by

(2.14)

with (JJB being the angular Doppler frequency. This Doppler shift causes afrequency mismatch with the matched filter. As long as the range of the Dopplerfrequencies is smaller than the signal bandwidth, i.e.,

(2.15)

where B8 is the signal bandwidth, this mismatch effect can be neglected.

After demodulation and matched filtering (2.8) becomes for a single rangeincrement

2.3 Space-time signals

2.3.1 The spatial dimension: array geometry

Figure 2.1 shows the geometry of an airborne (or spaceborne) array radar. Withoutloss of generality we assume throughout this book that the radar platform movesin the x-direction. We further define the x-direction as zero azimuth. Instead ofan elevation angle we use the depression angle 9 which is more appropriate for theairborne configuration. R8 is the slant range, Rg the ground range, ip denotes azimuth,and vp the platform velocity.

Two basic types of array antenna are depicted: the linear sideways and the linearforward looking array. Other configurations are possible and will be treated in Chapter9. Most of the fundamental studies presented here are based on the two arrayconfigurations shown in Figure 2.1.

1In a paper by NAKHMANSON [364] this effect is taken into account for optimum reception of abroadband signal by an array.

where T0 is the delay associated with Ro. By inserting (2.12) into (2.3) the receivedtarget signal becomes

(2.13)

(2.16)

sidelooking forward looking

Figure 2.1: Geometry of airborne antenna arrays

Let us forget the platform motion for the time being. Consider a sensor withcoordinates x^y^Zi relative to the array origin 4O'. The signal according to (2.3)received by this sensor from a stationary scatterer on the ground is phase shifted relativeto the origin by

(2.17)

where iV is the number of array elements. The received signal after demodulation andmatched filtering according to (2.8) becomes

(2.18)

(2.19)

(2.20)

(2.21)

where

2.3.1.1 Sensor directivity patterns

For the sensors the following directivity patterns will be used in the sequel

where the angle (po denotes the direction of the maximum of the sensor pattern. For aforward looking array we have (fo = 0, for the sidelooking configuration ipo 90.The vertical look angle is assumed to be ^o 0 throughout the book. It has some slightinfluence on the results only in the context of multiple-time-around clutter which willbe treated in Chapter 15.

2.3.2 The temporal dimension: pulse trains

The frequency resolution of a single pulse is determined by the pulse length S. For arectangular pulse envelope the frequency response is

see SKOLNIK [467, p. 3-19]. As long as the range of Doppler frequencies is smallerthan the signal bandwidth, i.e.,

the Doppler information cannot be retrieved from a single pulse.In pulse Doppler radar the Doppler frequency is measured by phase comparison

between echo signals due to a transmitted coherent pulse train. The transmit signal is

(2.22)

(2.23)

(2.24)

where T is the pulse repetition interval (PRI) and M is the number of transmittedpulses. The inverse of T

is the pulse repetition frequency (PRF). Accordingly the receive signal due to a movingtarget becomes (after demodulation and matched filtering)

(2.25)

(2.26)

(2.27)

which reduces to

if the pulse length given by E(t) is short compared with T.

2.3.2.1 Response to a single clutter patch

For the geometry shown in Figure 2.1, the signal received at the origin due to astationary scatterer on the ground becomes

(2.28)

and for a sensor atXi,yi,Zi

(2.29)

since we assumed that the radar platform moves in the x-direction. For an equispacedarray one obtains

(2.30)

where dx,dy,dz are the spacings in the x-, y-, z- directions.

2.3.2.2 Response to a moving target

For a moving target with radial velocity vrad (2.29) becomes

(2.31)

where vrad is the radial target velocity which includes the target-radar geometry aswell as the target's velocity and cruise direction, and

2.4 Interference

2.4,1 Ground clutter

A model for airborne clutter has been presented by RiNGEL [433]. In this model theclutter power contributions of individual range-Doppler cells of a single-channel radarare computed. Due to the non-linear boundaries of the range-Doppler cells this is acomplex procedure. On the other hand the RlNGEL model computes clutter power only.TOMLINSON [497] describes a STAP modelling computer code which can be used formonostatic and bistatic operation as well as for effects such as crabbing. Processing isdone in the frequency domain (post-Doppler architecture, see Chapter 9).

For the purpose of analysing space-time processing algorithms a simpler modelis sufficient. Since space-time clutter rejection takes place on a pulse-to-pulse basis,computation of the clutter echoes is carried out for one range increment only. Instedof absolute power values we determine a clutter-to-noise ratio at the individual sensorand echo pulse. Contrary to the RlNGEL model we have to generate complex clutterreturns due to a train of coherent transmit pulses. Moreover, we have to consider anarray antenna rather than a single channel radar.

For clutter echoes we make the following usual assumptions:

The contributions of different scatterers to the clutter echo are statisticallyindependent. Since the received clutter echoes are a sum over a large numberof scatterers they are asymptotically gaussian.

Temporal clutter fluctuations are slow compared with the observation time MT.

The total clutter echo is an integral over the various contributions from all groundscatterers in the visible range. Using (2.29) we obtain for a single range increment at

Notice that the space-time steering vector can be separated into a beamformer withlook direction ^ L . ^ L

(2.32)and a velocity (or Doppler) filter

(2.33)

the z-th sensor at the ra-th instant of time

(2.34)

where A is a circular complex gaussian distributed variable (gaussian amplitude anduniformly distributed phase). Like for the signal in (2.31), we omitted here thequadratic phase term due to the tangential component of the platform velocity. Itplays a role only if the clutter range is not too large and the number of echo pulsesis sufficiently large. It will play a role in the discussion of an ISAR problem in Chapter13.

The Doppler frequency of the individual clutter arrival is

This important relation was mentioned in the preface. It says that each direction isassociated with an individual clutter Doppler frequency. As an abbreviation we writethe temporal and spatial phase terms as follows

(2.35)

(2.36)

(2.37)

Including the sensor pattern2 D(ip) (2.34) becomes

The clutter model given by (2.37) can be slightly more refined by introducing twoquantities L((p) and G(

with (fL being the look direction and (p the angle of the individual clutter arrival, b isa beamformer vector with elements

(2.40)

where Nt is the number of transmit array elements.From the clutter echo (2.38) the elements of the clutter covariance matrix are

calculated as follows

where

and

are the indices of the NM x TVM covariance matrix, i, k denote sensors and m, pecho pulses. E{} means the mathematical expectation. We come back to the cluttercovariance matrix in Section 3.2.1.

2.4.2 Moving clutter

We now consider a clutter background that moves at the radial clutter velocity. Weassume that the clutter does not exceed a range increment during the observation timeMT. This clutter model may be used to simulate different kinds of moving clutter, forinstance sea, weather, chaff. In our model we introduce a radial clutter velocity v c.

By addition of radial platform and clutter velocities the temporal phase term in(2.36) becomes

(2.44)

The total clutter echo is given by replacing the temporal phase term $ Jn (, vc, vp) in(2.38) by (2.44).

Some calculations including moving clutter such as weather or chaff have beengiven by GHOUZ et al [150, 151].

2.4.3 Jamming

In addition to clutter we will consider CW (continuous wave) noise jamming. Thejammer signal received by the z-th sensor of the array due to J jammers is

(2.45)

where Ai are the jammer amplitudes and

Therefore, the jammer amplitudes Ai are modelled as complex gaussian variables. Weassume that

which means that there is no temporal correlation between subsequent pulse repetitionintervals. P\ is the power of the /-th jammer. We further assume that jammers aremutually uncorrelated

However, the sensor output signals due to the /-th jammer are mutually fully correlated3

according to

(2.48)

This expression includes the jammer power and the phase differences between differentsensors.

2.4.4 Noise

The main limiting factor in radar detection is the noise generated by the radar receiverwhich is dominated by the first amplifier in the receiver chain. We assume the noise tobe uncorrelated in time and space

(2.49)

(2.50)

2.5 Decorrelation effects

In this Section a few important effects are described which can degrade theperformance of anti-clutter and anti-jamming measures. Due to the two-dimensional(space-time) nature of the received signals there are two different effects, one applyingto the temporal, the other one to the spatial dimension.

2.5.1 Temporal decorrelation

Temporal decorrelation effects modify the space-time covariance matrix in thetemporal dimension, i.e., with respect to the index m in equation (2.34). There aretwo temporal effects: internal clutter motion and range walk.

3This is a narrowband assumption. For large radar bandwidth spatial decorrelation effects may occur.

(2.46)

(2.47)

2.5.1.1 Clutter bandwidth

The statistical behaviour of clutter returns depends very much on the nature ofthe individual scatterers. Hard reflectors such as buildings tend to produce stableecho signals which are reflected in a high temporal correlation between successiveecho samples. Equivalently, stable returns produce a narrow Doppler bandwidth.Soft scatterers such as vegetation produce less stable echoes with a larger Dopplerbandwidth. This is even more true when the scattering background exhibits internalmotion. Vegetation or weather under wind conditions are prominent examples for thiseffect.

It is common to model clutter fluctuations by means of a gaussian-shaped clutterspectrum.4 A gaussian correlation function is related to a gaussian power spectrum bythe equivalence (PAPOULIS, [397, pp. 25])

(2.51)

(2.52)

Defining a relative clutter bandwidth Bc = B / / P R BT (clutter bandwidthnormalised to the unambiguous Doppler band) one obtains with r = (m p)T thediscrete normalised autocorrelation function

(2.53)

This temporal decorrelation applies to clutter echoes only. Jammer and noise signalshave been supposed anyway to be uncorrelated in the time dimension. Target echoesare supposed to be deterministic and, hence, are fully correlated in time.

2.5.1.2 Range walk

Range walk can lead to temporal decorrelation of space-time clutter echoes. Theinfluence of range walk on space-time clutter covariance matrices and the associatedpower spectra has been analysed by KREYENKAMP [290]. The principle of rangewalk due to radar platform motion is illustrated in Figure 2.2. It shows in essence twocircular rings of width ,R0 - Ri displaced by

(2.54)

which is the distance the radar passes during the time mT. Both rings denote rangebins as seen by the radar at different instances of time.

4Under windy conditions clutter may exhibit an exponential rather than a gaussian shape, seeBlLLINGSLEYefa/. [40].

Let us define a clutter bandwidth B = 2a. Then (2.51) becomes

Figure 2.2: Geometry of range walk

With the usual assumption that the clutter background consists of a large number ofscatterers producing spatially uncorrelated echoes we can conclude that the correlationbetween the two instances of time is given by the area where both range rings overlap.At a particular azimuth, tp, the correlation will be:

(2.55)

with A R = R0 RI denoting the width of the range bin. Notice that, opposite todecorrelation by internal clutter motion, the decorrelation due to range walk is azimuthdependent. Obviously the correlation is a maximum in the broadsight direction of asidelooking array (SL) and minimum in the forward looking direction (FL).

The distance S((p) can be calculated as follows

where xp,yp and XQ^Q are the coordinates of the points P and Q in Figure 2.2:

(2.56)

For azimuth angles ^ > f the correlation measure S((p) can be obtained via symmetryconsiderations.

It should be noted that in practice the temporal decorrelation effect due to rangewalk will be even stronger than described above. The receive signal normally passesa bandwidth limiting matched filter so that the output signal is the autocorrelationfunction of the transmit signal. In the case of range walk, the individual shape of thematched filter response causes additional decorrelation which depends on the shape ofthe transmitted waveform. In the subsequent discussion we omit this aspect and assumethat the matched filter response is constant across the range bin.

2.5.2 Spatial decorrelation: effect of system bandwidth

The system bandwidth causes spatial decorrelation between signals appearing at theoutputs of the antenna array. The decorrelation is determined by the signal bandwidthand by the travel time of the incoming wave between the sensor positions.

The travel time between the i-th and the k-th sensor of a wave due to a source atposition ((/9,8) becomes for the configuration defined in Figure 2.1,

(2.58)

2.5.2.1 Rectangular impulse response

The system bandwidth is determined by the spectrum or, equivalently, by theautocorrelation function of the transmitted pulse. In narrowband radar systems themodulation of the carrier is often a rectangular function

(2.57)

(2.59)

(2.60)

The autocorrelation function of such a rectangular impulse is

otherwise

otherwise

Then the spatial decorrelation of a wave with complex amplitude A across the array isobtained by inserting (2.58) into (2.60)

(2.61)

Notice that the relative system bandwidth is related to the pulse length by

(2.62)

2.5.2.2 Rectangular frequency response

In broadband radar applications such as SAR frequently a linear chirp modulation of

the kind

is used. S is the pulse width as before, and a is the slope of the chirp. The frequencyresponse of a chirp is approximately rectangular. The autocorrelation function of sucha signal is (SKOLNIK [467, pp. 3-18])

(2.64)

with pr(Tik) being the autocorrelation function of a rectangular pulse as given by(2.60). The time-varying chirp frequency is /Ch = at so that the chirp bandwidthis B aS. Then we can relate the slope factor with the relative system bandwidth

(2.65)

with / c being the carrier frequency. As before r ^ is given by (2.58).

2.5.3 Doppler spread within range gate

For certain applications a large range gate equivalent to low range resolution isdesirable. Two examples are

Target search by a coherent sequential test (WlRTH [557]) is more efficient if therange gates are chosen large.

The omnidirectional low probability of intercept (OLPI) radar (WiRTH [558]) isbased on a CW pseudonoise transmit code in conjunction with extremely longtarget signal integration periods. The range bins are chosen sufficiently large inorder to prevent targets from range migration.

(2.63)

ParameterSystemcarrier frequencyPRFrel. system bandwidthpulse length

Operational dataplatform speedtarget velocity rangelook directionrangealtitudeflight path

Antennaarray orientationarray typeno. of elementstransmit arraysensor spacinghor. sensor patternvert, sensor pattern

Interferenceclutter/noise ratiorel. clutter bandwidthreflectivityclutter powerclutter phasejammer/noise ratioreceiver noisemultiple clutter

Signal Processingno. of temporal samplesspace-time channelsno. of antenna channelstemp. dim. of FIR filterDoppler filter lengthtapering

Symbol

/c

/PR

B8

6

VP

^radRH

NG(

If the clutter Doppler is range dependent5 a large range gate will produce a certainDoppler bandwidth instead of producing one Doppler frequency per direction. As thiseffect depends on the width of the range gate it depends in essence on the systembandwidth. Recall that decorrelation due to range walk increases as the range gate getssmaller. The effect of the Doppler spread behaves just the other way around because itincreases as the range gate gets larger.

Let us consider the clutter return from a single scatterer at the position ip, 9 as givenby (2.29). The space-time correlation between sensors and echo pulses is

Let us, for simplicity, consider linear arrays. Then zi = 0 , ^ 0- Extending thesingle clutter patch to a range interval (or, equivalently, to an interval in u cos 9) ofwidth Au gives

where U1V = cos 9 are directional cosines. Assuming that returns from differentscatterers are statistically independent gives E{A((p,u)A*(ip, v)} = 0, u ^ v, andthe clutter power is constant across the range gate so that E{A(ip,u)A*(ip, v)} =PC9 u v. Then (2.67) reduces to

(2.68)

5For details on clutter spectra, range dependence of the clutter Doppler and its effect on the detection ofmoving targets see Chapters 3 and 4.

(2.67)

(2.66)

where now ipr and

discussions are based on linear equispaced arrays in either the sidelooking or forwardlooking configuration. The look direction has been chosen as

7. For jamming we assume noise jammers whose bandwidth is large comparedwith the Doppler bandwidth (spatially correlated, uncorrelated in the timedimension).

8. Receiver noise is assumed to be white in space and time.

9. Clutter fluctuations are modelled by means of a gaussian correlation function.

10. Range walk is modelled by assuming uncorrelated scatterers and calculating theoverlap of range rings shifted against each other.

11. The impact of the system bandwidth is analysed for two different matchedfilters:

rectangular impulse response

rectangular frequency response

12. A model has been given for Doppler spread which may occur if the radar haslow range resolution.

13. The radar parameters used throughout this book, unless denoted otherwise, arelisted in Table 2.1.

Front MatterTable of Contents2. Signal and Interference Models2.1 Transmit and Receive Process2.2 The Doppler Effect2.3 Space-time Signals2.3.1 The Spatial Dimension: Array Geometry2.3.2 The Temporal Dimension: Pulse Trains

2.4 Interference2.4.1 Ground Clutter2.4.2 Moving Clutter2.4.3 Jamming2.4.4 Noise

2.5 Decorrelation Effects2.5.1 Temporal Decorrelation2.5.2 Spatial Decorrelation: Effect of System Bandwidth2.5.3 Doppler Spread within Range Gate2.5.4 Total Correlation Model

2.6 The Standard Parameter Set2.6.1 Multiple-time Around Clutter

2.7 Summary

Index