RCS Estimation

European Journal of Scientific Research ISSN 1450-216X Vol.75 No.3 (2012), pp. 434-447 © EuroJournals Publishing, Inc. 2012 http://www.europeanjournalofscientificresearch.com

Radar Cross Section Estimation Procedure using

Signal Processing Approach

P. Suresh Kumar Center for Advanced Research

Department of Electronics and Communication Engineering Muthayammal Engineering College, Rasipuram-637408, Tamilnadu, India

E-mail: [email protected]

M. Madheswaran Center for Advanced Research

Department of Electronics and Communication Engineering Muthayammal Engineering College, Rasipuram-637408, Tamilnadu, India

E-mail: [email protected]

Abstract

The Radar Cross Section (RCS) estimation for various arbitrarily shaped Perfect Electric Conductor (PEC) objects has been developed using signal processing techniques and presented in this paper. The estimation has been done by processing the FMCW radar receiver outputs with the necessary parameters. The algorithm has been applied to sphere, cube and cylinder shaped objects for validation. It has been found that the estimation procedure developed in this paper will facilitate for further estimation of radar cross section. More importantly the radar cross section has been estimated by analyzing the time delay between the backscattered pulses with the help of ESPRIT and DFT based technique to estimate the range and phase difference through Doppler frequency. Numerical results indicate that the hybrid process presented in this paper is more accurate compared to the conventional method without signal processing techniques. Indexterms: Direct Digital Synthesis (DDS), Perfectly Electric Conductor (PEC), Radar

Cross Section (RCS), Radar, Frequency Modulated Continuous Wave (FMCW), Estimation of Signal Parameters.

I. Introduction In the recent years, the advanced digital signal processing techniques have motivated the designers to exploit the available concepts for various applications including radar systems. The frequency Modulated Continuous Wave (FMCW) radar is being widely used in civil and military applications and gained much importance as they utilize the emerging technologies of high power solid state microwave transmitters, fast digital circuits capable of generating and processing complex signals in real time. These radars have versatile utilization in detection of targets with high range resolution measurement and target shape identifications. The FMCW radar sensors have excellent measurement range with the superior signal propagation compared to systems operating in the visible or infrared light spectrum [1].

Modern fighter aircrafts, ships and missiles need very low Radar Cross Section (RCS) designs to avoid detection by hostile radars [2]. Operators of communication satellites often request a

Radar Cross Section Estimation Procedure using Signal Processing Approach 435

complicated differential radar cross section in order to assist the tracking of the satellite. Hence accurate prediction of RCS of complex objects is essential to meet this requirement. The estimation of RCS for various arbitrary shaped objects has become more important in the recent years the measurement of RCS has been focused by many researchers and various methods have been discussed. Knott, E.F et al., [3] have reported that the moment and Finite Element Methods can be used to predict the RCS using Electric Field Integral and Differential Equations. The Field Integral and Differential Equations are complex for open and close bodies which require advanced techniques. S. G. Wang [4] has proposed a method to compute the RCS of simple objects using Higher order Method of Moments (HO-MOM). The range of frequency used is from 0.8MHz to 3.9 GHz Numerical results of sphere and plane objects are presented in low and resonance frequency domain. Virga, K.L. and Rahmat-Samii, Y [5] have presented a methodology to obtain monostatic radar cross section of a finite-size perfectly conducting flat plate using triangular patch mesh profile. The RCS obtained shows better results in low frequency region as compared to physical optics.

In practice range detection using FMCW radar utilize the triangular and saw tooth modulation. Sinusoidal modulation is rarely used because it needs complex processing. Cantrell et al., [6] have proposed an optimal range algorithm for FMCW radars applicable to all types of modulation techniques. The concept of cross correlation has been used to measure the range. However it has been found that the approach was not suitable for the input signals with low Signal to Noise Ratio (SNR).

The range estimation algorithm using discrete Fourier transforms (DFT) and Interpolation has been reported by Kaichun R., [7]. The fine and course frequency estimation increased the range resolution. The DFT based range estimation provides high accuracy with low computations. Later, Stelzer et al., [8] have proposed the FFT based range estimation algorithm using the windowing technique. It was reported that the spectrum measured between transmitted and received signal gets widens when target distance increases and contains more noise components. The signal processing technique has been utilized to determine the height of liquid using the Sine plus Noise FMCW radar and reported by Xiong et al., [9]. The input signal applied is considered as the combination of a sinusoidal component and band pass filtered white noise. The performance analysis of FMCW radar under different environmental constrains have been explained by Schuster et al., [10].

Scheiblhofer S et al., [11] have implemented a fast Direct Digital Synthesis (DDS) based frequency synthesizer for FMCW radar technology. It operates at high frequencies and generates fast, but nonlinear frequency chirps. However due to nonlinear frequency sweeps, the standard Fourier transform methods cannot be directly applied for the baseband signal evaluation.

The available DDSs are fully integrated and single chip solutions, which allow full control of the signal frequency and phase. It requires stable source for optimum and broadband DDS performance which impacts the total system costs significantly. The phase locked loop (PLL) also been widely used in wireless communication systems due to the high frequency resolution and the short locking time. Longjun et al., [12] have developed the system to have DDS signal which is mixed with the voltage-control oscillator output in the PLL feedback path to achieve a very high-frequency resolution, fast settling and spectral purity.

Bradley et al., [13] have demonstrated the super resolution signal processing technique for RCS measurement. It was reported that the combination of Fourier analysis and window will improve the accuracy of estimation. Eugin et al., [14] have discussed the influence of bias voltage on various windowing techniques in DFT spectrum measurement. A Cramer–Rao lower bound has been derived for DFT spectrum Vijay [15].

Estimation of Direction of Arrival (DOA) of Frequency Modulated wave has been presented by S. Ejaz and M. A. Shaq [16].The comparison results shows that ESPRIT based state space estimation provides better resolution compared to MUSIC and other spectral estimation techniques. Baha A. Obeidat [17] has proposed a technique for estimating the range and direction of-arrival (DOA) of narrow-band near-field sources. Numerical results shows that the ESPRIT based technique provides better range estimation compared with other spectrum estimation methods.At high frequencies the DFT based range estimation cannot predict the range accurately. The state space approach decomposed

436 P. Suresh Kumar and M. Madheswaran

original signal into signal subspace using singular value decomposition and detection has been done by total least square algorithm. Cherian et al., [18] have proposed a 2D DFT based ESPRIT (Estimation of Signal Parameters via Rotational Invariance Technique) algorithm for stationary signals. The Doppler radar image was created by applying two-dimensional spectral analysis to a sequence of wideband radar returns and the amplitude and phase of the frequency components have been returned separately. The phase information provides the cross and slant range.

Jian et al., [19] have presented the rank reduced ESPRIT algorithm to estimate principle signal components with low computational complexity. It was developed especially for high resolution identification of closely spaced wireless multipath channels. This algorithm transforms the generalized Eigen problem from an original high dimensional space to a lower dimensional space depending on the number of desired principle signals. Vincent et al., [20] have proposed a Modified ESPIRIT algorithm for time delay algorithm for time delay measurement of Ground Penetrating Radar (GPR) .The measurement has been done for pulse radar in noise environment. The measurement results also compared with other signal processing techniques. II. RCS Measurement using Signal Processing Algorithm The flow graph of the RCS estimation algorithm is given in Fig.1.The estimation of Radar Cross Section requires an evaluation algorithm which extracts the time delay and range information.

Figure 1: Flow Diagram of the Proposed Algorithm for prediction of Radar Cross Section

Estimation of Radar Range and RCS

Discrete Fourier transforming (DFT)

Zero padding and windowing

Application of Spectral estimation Techniques to find the fundamental

frequency components

Generation of frequency sweep using DDS

Read out of the scattered and reflected data

In a radar system, detection and range calculation of a target are done by transmitting electromagnetic energy and observing the back scattered echo signal. In FMCW modulation the frequency is swept linearly between the up chirp and down chirp directions .The down chirp


component of transmitted and reflected signal is in the order of MHz .This difference frequency between transmitted and received is known as Doppler frequency. It is easier to filter the sum frequency out of the signal if the input frequency is higher for a specified bandwidth to improve resolution. The ultimate goal in the determination of targets with low RCS is high range resolution. The DFT provides poor range resolution for data with a small bandwidth and causes spectral leakages. In order to solve this spectral leakage problem, a two stage algorithms can be developed based on DFT and parametric spectral estimation approach. The parametric spectral estimation methods provide a high resolution range profiles for all the targets even when the data size is small. The range of a target can be determined either by parametric based ESPRIT algorithm which is applied to extract the fundamental period of the signal or the DFT to extract the peak position of the scattered signal. The well known windowing technique can be applied before extracting the DFT to reduce the low frequency harmonics.

According to FMCW radar principle, the received signal is time delayed by the round trip delay

time c

R2 with c denoting the speed of light and R is Range of a target. It is also [8] given by,

2

0( ) cos 2 22r t

kx t A f t t

(1)

The transmitted signal is of the form 0( ) cos 2

2t t

kx t A f t

(2)

where sweep

Bk

T and Tsweep is the steepness of the sweep signal

The received signal and the transmitted signals are mixed and digitized signal is produced at the output of the mixer. It is clear that the mixer output has two components consisting of additive and difference of both transmitting and receiving signals. The difference component is considered for range estimation where as the added frequencies not preferred. The digitized value of the mixer output with P target is given as

1

[ ] cos 2p

t i ii

x n A n

(3)

where ii f 02 , ii N

B , p is the sampling length of Radar and N is the number of samples.

Proposed Algorithm

The algorithm for RCS measurement has been developed based on the shift invariance of the discrete time series. It causes a rotational invariance between the corresponding signal subspaces. Initially the down chirp signal is sampled with N samples and denoted as ][nx , where n=0, 1… N-1.

The vector of ][nx is given as

1

[ ]M

i ii

x n A S

(4)

where iS is the signal matrix consisting of all complex exponentials and is white noise with zero

mean and variance 20 .

The shifted vector ][ˆ nx is derived which consists of the samples x[1] to x[N]. The vector of ][ˆ nx is given as

'

1

ˆ[ ]M

i ii

x n A S

(5)


where iA is the amplitude of ith sample, iS ' is the signal matrix and is the additive noise. M is order

chosen for parametric estimation. The value of iS ' is given as ' ji iS e S (6)

The procedure for estimating the fundamental frequency is described as follows Step 1: Estimation of auto and cross correlation from the data values. The value of auto correlation of

signal x is given as

* 20

1

MT T

x i i ii

R PS S I

(7)

where iP is the variance of iA .

The cross correlation of x and 'x is given by

'

* 2 * *0 1 0

1

Mj i T T

i i ixxi

R Pe S S D SP S

(8)

where is the diagonal matrix containing all of the information about fundamental frequencies and

where D−1 is the matrix with all ones immediately below the main diagonal and zeros elsewhere. 1

2

0 . . . 0

0 . . . 0

. . . .

. . . .

0 0 . . .

j

j

j M

e

e

e

(9)

Step 2: Estimation of sR and 'SS

R is done by subtracting the noise variance 20 I and 2

0 1D from xR and

'xxR .The values of sR and 'SS

R is given as *

0SR SP S (10) * *

' 0T

ssR SP S (11)

where 0P is a diagonal matrix is contains all magnitudes of complex sinusoids.

Step 3: The fundamental frequencies are estimated by subtracting SR and 'ssR ,which forms an angular

circle with k j ke where k =1,2…..M are the angular positions of these Eigen values. The low frequency inter harmonic signal are removed with help of windowing .The concept of

zero padding is done to improve the accuracy of the estimated frequencies and to prevent uncertainties. The windowing can be applied to ensure resolution of individual targets as well as keeping the side lobe level low.

Further, it can be seen that windowing process adds frequency components to the signal that must be isolated from the target frequency. This can be achieved by signal power compensation algorithm based on mean and variance of the signal. The range resolution of a target can be estimated from the backscattered delay. It depends on the width of the transmitted signal, type and sizes of targets.

The round trip delay time can be estimated from the frequencies of the cosines of equ.(2) by determining the DFT (Discrete Fourier Transform). The data is padded with zeros to make equal to the sample length of DFT before the transformation is applied.

The Discrete Fourier Transform kx is generally defined as


1

0

2[ ] exp

N

n

x k x n j knN

(12)

The summation limit can be shifted from 0 to 12

N to avoid the phase uncertainty. Then the

DFT of x(n) can be written as

1

2

0

2[ ] exp

N

n

x k x n j knN

(13)

The delay time between the signals is estimated from the phase values of the spectrum. The difference between the phases of the two primary peaks can be determined with the help of peak picking algorithm and it is related to the range as

2Rt

C (14)

where R is the Range of a target. Using the range (R) the RCS of a target can be estimated using 24 r

i

PR

P (15)

where Pi is the incident field and Pr is the scattered field. The value of various objects depends on objects dimensions apart from reflected power density. The RCS of a sphere[3]is given as

429 r Kr (16)

where r is radius of a sphere and 2

K , is the wavelength. The RCS of a circular cylinder[3] is

given as 2

2 sin[ cos( )sin( )

cos( )

KLKL

KL

(17)

where L, are length and angle of circular cylinder. III. Results and Discussion The radar cross sections for various shaped objects have been estimated using Matlab and the proposed algorithm (Fig.1). The parameters used for estimation is given in Table.1.The backscattered magnitude response of a target at three different locations has been considered for analysis and shown in Fig.2. Table 1: Parameters used for RCS Measurement

S.No Parameters Values 1. Center frequency 2.45 GHz 2. Sampling length 1024 3. No of samples 16,384 4. Coupling coefficient 10dB 5. Antenna gain 20dB 6. Sweep time 16ms


Figure 2: Magnitude response of a target with three different positions

0 5 10 15 20 25 30 35 40 45 50-20

-15

-10

-5

0

5

10

15

20

time

mag

nitu

de(d

B)

The fundamental frequency component of the reflected signal is initially estimated using ESPIRT algorithm with the model order M = 52. The range profile of radar at different positions has been obtained from the magnitude of the reflected signal using ESPIRIT algorithm and shown in Fig.3. However, the prime aim is to estimate the peak value of reflected signal magnitude from the spectral estimated signal.

Figure 3: Radar Range profile for three different locations

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-5

0

5

10

15

20

mag

nitu

de(d

B)

Profile of Reflectors time (µs)


The suitable windowing function has been selected to estimate the round trip delay time ∆t for further analysis. It has been observed that the window function selection plays vital role to improve the recognition. An optimum detection of a scattered signal from the magnitude response has been obtained with help of DFT and spectral based analysis. The DFT response of the sinusoidal input consists of a main lobe and side lobes. The width of the main lobe indicates the Fourier domain resolution. Hence the windowing function with high main lobe and very low side lobe is needed for phase angle detection. The main and side lobe characteristics of various windows are given in Table 2. Table 2: Main and Side lobe values of Different Windowing Function

Window Functions Main Lobe Width(dB) First Side Lobe(dB) Rectangular 1 -13.46 Hamming 2 -41 Hanning 2 -32 Blackman- Harries 4 -70

It can be seen that rectangular window consists of side lobes at -13.46dB and Blackman Harries

window has side lobes at -70dB. The Blackman Harries window is found most suitable to prevent spectral leakages during peak estimation due to higher main lobe and less side lobes. Fig.4 shows the simulated time response of Blackman Harries window and Hanning window. It can be observed that Blackman Harries window has low side lobe compared to hanning window.

Figure 4: Measured time responses of the reflectors with different window functions

0 0.5 1 1.5 2 2.5 3 3.5

x 104

-2

0

2

4x 10

5

Blackman-Harris window

0 0.5 1 1.5 2 2.5 3 3.5

x 104

-2

0

2

4x 10

5

sequence length

sequ

ence

Hanning window

The time response of Blackman Harries and Hanning window have been obtained and shown in Fig.5. It shows that the spectrum of Blackman Harries window has wide width. The resolution of the radar can be further enhanced by increasing the bandwidth by adding zeros to the measured data. Table 3 shows the variation of signal amplitude with signal to noise ratio for both 28 points and 210 points data. If the signal is added with more number of zeros, linearity is observed for various SNR values.


Figure 5: Spectrum of different windowing Functions

Table 3: Variation of RTDT with zero padding

Signal to Noise Ratio(dB) Value of (µs)

Zero padded with 28 points Zero padded with 210 points 0.9 1.67675 1.67663 4.7 1.67677 1.67676 5.4 1.67655 1.67669 7.4 1.67680 1.67675 8.2 1.67655 1.67553

In order to find the time delays and phase variations, Discrete Time Fourier (DFT) is obtained

with a sample length N=1024. The range information can be obtained from the variation of phase angle values.The length of DFT has been taken from the samples N= -512 to 511 instead of N=0 to 1023 to avoid the spectral leakages. Fig.6 shows the variation of magnitude and phase spectrum of scattered signal. The phase angle variations are clearly observed in the phase spectrum.

Figure 6: Measured time and phase response using DFT

0 2000 4000 6000 8000 10000 12000 14000 16000 18000-20

-10

0

10

20

30

mag

nitu

de(d

B)

0 2000 4000 6000 8000 10000 12000 14000 16000 18000-4

-2

0

2

4

phas

e(ra

d)

Time (ns)


The fundamental frequencies and the peak values have been estimated with the help of ESPIRT based DFT algorithm. Fig.7 shows the comparison of ESPRIT with other parametric methods. The ESPIRT method produces narrower peak compared with MUSIC and Min Variance methods. All the low frequency harmonics have been removed successfully and the time difference between the two signals are measured for range estimation.

Figure 7: Comparison of Various Parametric estimation methods

The RCS prediction are very complex at high frequencies even for simple shaped objects because they require solution of either differential or integral equations that describe the scattered waves from the object under the set of boundary conditions. In fact, when exact solutions are achievable, it is difficult to interpret and program using digital computers. Hence the approximate methods are more widely used in high frequency region for objects such as aircrafts, ships, and missiles.

Fig.8 (a) shows the RCS of a sphere in the high frequency region. The frequency range considered for the measurement is 2.45 to 9 GHz. It can be seen that the RCS is independent of frequency due to symmetry of waves scattered from a perfectly conducting sphere which are co-polarized with the incident waves. The power density of a sphere is initially estimated by considering the aperture efficiency as 0.7 the effective and physical aperture area are considered equal. From the reflected power density the RCS value is estimated.

Fig.8.(b) shows the mesh profile of sphere and the RCS of a sphere for various wavelengths. The mesh profile shows a peak position for the given frequency range and it is determined with the help of peak position algorithms. This information is expected to recognize the object shape more accurate.


Figure 8 (a): RCS measurement of sphere using ESPRIT -DFT

Figure 8 (b): Mesh profile of sphere

Fig.9. shows the scattered radar cross section of circular cylinder with height h=1m and radius r=0.25m for the frequencies 2.5 GHz and 9GHz. The simulation results are obtained by varying the aspect angle from 0 to 180 degrees. It is found that, the RCS for a symmetrical cylinder has a maximum value at an aspect angle of 90 degrees and it has minimum value at 0 and 180 degrees. It is also observed that the magnitude of cylindrical RCS is proportional to the frequency.


Figure 9: Scattered cross section of a circular cylinder

0 20 40 60 80 100 120 140 160 180-50

-40

-30

-20

-10

0

10

20

30

40

Aspect angle - degrees

RC

S -

dB

sm

RCS at 2.5GHz

RCS at 9 GHz

The RCS of different objects like sphere, cone and sphere cone are presented in Fig.10. The angle of incidence and radius of all the objects are considered as 0 degrees and 1cm for all the three objects.

Figure 10: Scattered cross section of a various shaped objects

The monostatic RCS frequency and angular response of a cube is shown Fig.11.The dimensions of a cube are chosen as 0.5X0.5X0.5 for the frequency range from 2.5 GHz and the incident angle is chosen as 0 degree. This is compared with finite element method with 892 triangular elements resulting into 1338 unknown current coefficients. The frequency increments are considered as 0.1GHz and the


phase angle increment is 1 degree. The developed method consumes 48 min CPU time to obtain the monostatic RCS where as FEM takes 120 minutes of CPU time.

Figure 11: Cross section of cube

In addition, the performance of RCS measurement using the developed procedures has been estimated and compared with the theoretical values. Table –4 shows the comparison of proposed method with theoretical value. The measurement is done at two different frequencies 2.5 and 9GHz at three different locations. Table 4: Comparison of proposed method with theoretical values

Types of Object Frequency Range(GHz) Radar Cross Section m2

Theoretical Measurement Proposed Method

Sphere 2.5 2.54 2.3 9 2.32 1.98

Cylinder 2.5 34.5 33.2 9 28.2 26.4

Cube 2.5 0.54 0.45 9 0.26 0.26

IV. Conclusion The Radar Cross Section estimation using signal processing technique has been found to be an improved technique. The algorithm developed for RCS estimation has been tested for three shaped objects and SNR values. Based on the analysis it is found that the estimation using combination of Discrete Fourier Transform with ESPRIT algorithm provides a very high range resolution and high SNR. The peak position algorithm has been utilized to estimate the number of peaks for further analysis. It can also be applied for targets with high RCS and multiple targets. References [1] A.G.Stove, “Linear FMCW Radar Techniques”, IEE Proceedings - F, Vol 139, pp 343-350.

1992.


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[5] Virga, K.L. Rahmat-Samii, Y , “RCS characterization of a finite ground plane with perforated apertures: simulations and measurements”IEEE transaction on Antennas and wave propagation,Nov 1994

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RCS Estimation

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