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Australian Guvmuiiciit 


Peparl mein nr Helmet 1 

Defense Science and 

Technology Organisation 


Approximate Invariance of the Inverse of the Covariance 
Matrix and the Resultant Pre-built STAP Processor 

Yunhan Dong 

Electronic Warfare and Radar Division 
Systems Sciences Laboratory 

DSTO-RR-0291 


ABSTRACT 

Space-time adaptive processing (STAP) has been proven to be optimum in scenarios where an 
airborne phased-array radar is used to search for moving targets. The STAP requires the 
inverse of the covariance matrix (ICM) of undesired signals. The computation of the real-time 
ICM is impractical at current computer speeds. Proposing two Theorems, this report indicates 
that the ICM is approximately invariant if radar and platform parameters remain unchanged. 
A pre-built STAP (PSTAP) processor is then proposed. Both the simulated data from a generic 
airborne phased array radar model and real data collected by the multi-channel airborne 
radar measurement (MCARM) system are processed to verify the processor. Results indicate 
that the performance of the proposed PSTAP processor is the same as that of the real-time 
STAP processor. 


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Approved for public release 



Published by 

DSTO Systems Sciences Laboratory 
PO Box 1500 

Edinburgh South Australia 5111 Australia 

Telephone: (08) 8259 5555 
Fax: (08)8259 6567 

© Commonwealth of Australia 2005 

AR-013-364 

March 2005 


APPROVED FOR PUBLIC RELEASE 



Approximate Invariance of the Inverse of the 
Covariance Matrix and the Resultant Pre-built STAP 

Processor (U) 


Executive Summary 

The central task for an airborne early warning & control (AEW&C) radar is to detect 
moving targets in the presence of undesired signals such as surface clutter and 
directional broadband noise jamming. Within certain conditions space-time adaptive 
processing (STAP) has been proven to be optimum in scenarios where an airborne 
phased-array radar is used to search for moving targets. If the covariance matrix (CM) 
of undesired signals including clutter, jamming and thermal noise can be determined 
(measured or estimated), the upper limit of the coherent processing gain for desired 
target signals can be achieved using STAP while the undesired signals are sufficiently 
suppressed. However, computation of the inverse of the covariance matrix (ICM) 
becomes a bottleneck which prevents the fully adaptive STAP from real-time 
implementation at current computer speeds. Hence many operational systems must 
use alternative less computationally intensive processes, such as the one proposed in 
this report. 

Proposing two Theorems, this report indicates that the ICM is approximately invariant 
when radar and platform parameters remain unchanged. Based on this, the following 
conclusions are deduced: 

• For clutter, STAP needs to adapt only to system parameters (both radar and 
platform). Variations in clutter intensity, which may incur significant variations 
in the elements of the CM, cause little variations in the elements of the ICM; 

• For jamming, STAP needs to adapt only to the bearing of jamming. Variations 
in jamming intensity, which may incur significant variations the in elements of 
the CM, causes little variations in the elements of the ICM. 

Because the construction of optimum weighting vectors only requires knowledge of 
the ICM, the approximate invariance of the ICM naturally results in a proposal for 
constructing a pre-built space-time non-adaptive processing (PSTAP) processor. 
PSTAP is not adaptive processing in the broad meaning, as no adaptive processor can 
be built a priori. However, because it does not require knowledge of the clutter 
environment, the PSTAP processor can be pre-built, and still perform the same as the 
STAP processor as long as the system (radar and platform) parameters are known. 

Prior to the mission, libraries of the jamming filters as well as optimum weighting 
vectors may be constructed either purely from modelling or based on test flight 
measurements. Each weighting vector is optimal to a particular operational situation. 
During the mission, first the collected data are analysed to determine the presence of 
jamming by other means. Appropriate jamming filters are called from the library or 



beamforming techniques are used to suppress the jamming. The jamming-free data are 
then simply multiplied with the appropriate optimum weighting vectors from the 
library to generate the result. 

If a real radar system is difficult to model (for instance, the array may not be rigorously 
linear and the effect of the platform on the antenna system may be difficult to model 
etc), the library may be pre-built based on data collected from previous missions. 
Because of the nature of its approximate invariance, the ICM from different clutter 
environments should be approximately the same for the same set of radar and platform 
parameters. Therefore, the optimum weighting vectors obtained from previous 
missions can be directly used for future missions, irrespective of clutter environments, 
provided that the radar and platform parameters remain unchanged. 

There are usually a large number of possible combinations of radar and platform 
parameters, so a large number of pre-built optimum weighting vectors will be 
required. If the radar pulse repetition frequency (PRF) is linked to the platform velocity 
to maintain the number of clutter foldovers to be constant, the number of combinations 
can be significantly reduced. Also because the ICM is not sensitive to changes in 
steering angle which varies from range bin to range bin, the same set of optimum 
weighting vectors can be applied to many range bins with little signal to interference 
and noise ratio (SINR) loss. 

A robust analysis for PSTAP has been carried out. Uncertainties incurred in steering 
angles possibly caused by an undulating terrain surface have been studied. It has been 
shown that the PSTAP SINR loss compared to STAP SINR is normally only about 
0.2-0.5dB even for a highly undulating terrain. 

Numerical examples are presented. The first numerical example is for a generic 
airborne radar model. Cases of clutter coefficients randomly fluctuating up to ±15 dB 
(simulating extreme inhomogeneous environments) are compared to the usual case of 
the constant clutter coefficient (simulating an ideally homogenous clutter environment) 
showing that the resultant ICMs are almost identical. The correlation coefficient r 2 for 
all pairs of ICMs compared are higher than 0.99. The performance of the PSTAP 
processor is compared to the real-time STAP processor, and results are the same for all 
scenarios. 

The report then examines the PSTAP processor applied to real data collected by the 
multi-channel airborne radar measurements (MCARM) system. Temporal and spatial 
correlation is considered in order to build a realistic and appropriate PSTAP processor. 
It is shown that decorrelation in the MCARM data is primarily caused by range 
ambiguity and clutter intrinsic motion. Finally the PSTAP processor is compared to the 
conventional STAP processor for detecting embedded small target signals. It is shown 
that the performance of PSTAP is the same as that of STAP for the cases studied. A 
moving target has been first detected in the MCARM data set by the application of the 
PSTAP process. 

The PSTAP is also applied to process airborne data generated by the high fidelity 
airborne radar simulation software, Rome Laboratory Space-Time Adaptive Processing 
(RLSTAP), details will be reported in the future. 



Author 


Yunhan Dong 

Electronic Warfare and Radar Division 

Dr Yunhan Dong received his Bachelor and Master degrees in 
1980s in China and his PhD in 1995 at UNSW, Australia, all in 
electrical engineering. He then worked at UNSW from 1995 to 
2000, and Optus Telecommunications Inc from 2000 to 2002. He 
joined DSTO as a Senior Research Scientist in 2002. His research 
interests are primarily in radar signal and image processing, and 
radar hackscatter modelling. Dr Dong was a recipient of both 
Postdoctal Research Fellowships and Research Fellowships from 
the Australian Research Council. 



Contents 


1. INTRODUCTION.1 

2. FORMULATION OF THE COVARIANCE MATRIX OF UNDESIRED SIGNALS2 

2.1 Clutter.3 

2.2 Jamming.7 

2.3 Thermal Noise.8 

2.4 Covariance Matrix of Undesired signals.8 

3. SPACE-TIME ADAPTIVE PROCESSING.9 

3.1 Fully Adaptive STAP.9 

3.2 Partially Adaptive STAP.11 

4. APPROXIMATE INVARIANCE OF THE INVERSE OF THE COVARIANCE 

MATRIX .16 

4.1 Theorem 1.16 

4.1.1 Example 1.18 

4.1.2 Example 2.20 

4.1.3 Examples 3, 4 and 5.22 

4.2 Theorem 2.25 

4.2.1 Examples.27 

4.3 Robust Analysis.28 

4.4 Significance.34 

4.5 Interpretation.34 

4.6 Implementation.35 

4.6.1 Prior to the mission.35 

4.6.2 During the mission.36 

5. NUMERICAL RESULTS.36 

5.1 Results from a Generic Model.36 

5.2 Results from MCARM Data.39 

5.2.1 Forming the Covariance Matrix Using the SMI Method.40 

5.2.2 Crab Angle Correction.42 

5.2.3 Temporal and Spatial Decorrelation Effects.43 

5.2.4 Results.48 

6. CONCLUSIONS.56 

7. ACKNOWLEDGEMENT.58 

8. REFERENCES.59 


APPENDIX A: MCARM SYSTEM 


61 



































DSTO-RR-0291 


1. Introduction 


The central task for an airborne early warning & control (AEW&C) radar is to detect 
moving targets in the presence of undesired signals such as clutter (echoes from the Earth 
surface) and directional broadband noise jamming. Subject to certain conditions, space- 
time adaptive processing (STAP) has been proven to be optimal in such scenarios where 
an airborne pulsed Doppler phased array radar is used in searching for moving targets. If 
the covariance matrix (CM) of the undesired signals including clutter, jamming and 
thermal noise can be determined (measured or estimated), the upper limit of the coherent 
processing gain for the desired target signal can be achieved using STAP while the 
undesired signals are sufficiently suppressed (Ward, 1994, Klemm, 2002, Wirth, 2001). 
However, in real-time implementation of STAP, the computation of the inverse of the 
covariance matrix (ICM) becomes a bottleneck which makes the fully adaptive STAP 
infeasible at current computer speeds. Numerous algorithms, aimed at reducing the 
dimensionality of the CM in both the spatial and temporal domains, have then been 
proposed to minimise the computation time to satisfy the real-time requirement while 
maintaining the coherent processing at or close to the optimum level. 

This report presents a significant finding, the approximate invariance of the ICM. Based 
on this, use of a pre-built STAP (PSTAP) processor for real-time processing is proposed. 
The PSTAP is not an adaptive processor in the broad meaning of adaptive processing 
where the processor automatically adapts to any changes in both the system (radar and 
platform) parameters and the environment (clutter). However, it can cope with changes in 
clutter provided that the radar and platform parameters remain unchanged. For 
convenience, the name of PSTAP is used. The prefixal 'pre-built' indicates that the 
processor is not adaptive in a broad sense as no adaptive processor can be pre-built, while 
the suffixal 'STAP' implies that the processor is constructed in a STAP fashion. The idea of 
PSTAP is to construct a set of weighing vectors a priori, either by theoretical modelling or 
using flight data. Each weighting vector is equivalent to a fully STAP weighting vector for 
a specific situation. Using such a processor the computational bottleneck is eventually 
removed thoroughly. Theoretically, if the system (radar and platform) parameters are 
known, the proposed PSTAP can achieve the same coherent processing gain as real-time 
STAP (the knowledge of clutter is not required). In reality, due to the fact that the CM for 
the real-time STAP is difficult to obtain accurately, the PSTAP may even achieve better 
results if all system parameters required for the construction of the processor are known 
precisely. 

Various tapering (window) functions can be applied to both the spatial and temporal 
domains in the STAP. The pros and cons of window functions are well known. This report 
does not include the application of window functions, as their effects can be readily 
anticipated. 

Section 2 starts formulating the CM of undesired signals for a generic airborne phased- 
array antenna in the side-looking situation. Section 3 briefly summarises the fully and 
partially adaptive STAP algorithms. A unique characteristic, the invariance of the ICM, is 
presented in Section 4. Two Theorems are given which in turn form a base for the 


1 



DSTO-RR-0291 


construction of the PSTAP processor. A robust analysis for the PSTAP processor is also 
presented in Section 4 to deal with uncertainties. Numerical examples of applying the 
PSTAP processor to a generic model (the same model used by Ward, 1994) and to the data 
collected by a real airborne system, the multi-channel airborne radar measurement 
(MCARM) system, are presented in Section 5 in supporting verification of the PSTAP 
processor. Section 6 concludes the report. 


2. Formulation of the Covariance Matrix of Undesired 

Signals 

The undesired signals in the radar environment under consideration include clutter 
(echoes from the Earth surface), directional broadband noise jamming, and thermal noise. 
Their modelling has been well documented (Ward, 1994, Klemm, 2002) and is briefly 
summarised in this Section. 

Mathematical notations follow the convention (Ward, 1994). Italic typeface symbols stand 
for scalars; lowercase and uppercase boldface symbols for vectors and matrices, 
respectively. Subscripts T , * and H denote the operations of transpose, complex conjugate 
and Hermitian transpose, respectively. The symbol <E> refers to the Kronecker matrix 
product. Finally the symbol e{ ) denotes the expected value of a random quantity. 

An airborne linear phased array antenna looking at the broadside is the default 
configuration. In particular, let a platform be moving in the x -direction at a speed of v a , a 
linear antenna array with N elements be parallel with the direction of motion and look at 
the broadside y -direction, as shown in Figure 1. Symbols H , R , 6 and <]) denote platform 
height, range, elevation angle and azimuth angle, respectively. 



Figure 1: Geometry of a linear airborne antenna array. 


2 



DSTO-RR-0291 


Because the receive antenna consists of N elements, the received signal, after being down 
processed to the baseband, for a given range bin, also consists of N complex components 
(I and Q values) for every transmitted pulse. Assuming that there are M pulses in a 
coherent processing interval (CPI), the received signal, which is comprised of MN complex 
components in total, is usually expressed by a MN x 1 vector (often referred to as a space- 
time snapshot) as. 


Z = , 


~[Xo,o Xo,N-\ 


Xm- 1,0 


Xm-\,n 


1 

- 1 . 


(1) 


where x mn is the received signal at the n th antennal element ( n = 0,• • •, N - 1 ) for the m th 
pulse (m = 0,---,M -1). Without loss of generality, a snapshot, /, may contain the desired 
signal component, i,, as well as the undesired signal components of clutter, ■/,, jamming, 
Z/, and noise, 


2.1 Clutter 

We assume at this stage of the analysis that the pulse repetition frequency (PRF) is 
sufficiently low that range is sampled unambiguously. We further neglect the effects of the 
intrinsic motion of clutter. These assumptions lead to the first-order general clutter model 
(Ward, 1994). Spatial and temporal correlations due to range ambiguity (range fold-over), 
the intrinsic motion of clutter and the motion of platform will be dealt with later in Section 
5 where live airborne radar data are processed. The space-time snapshot of clutter for the 
first-order general clutter model can be written as a summation (integral) of clutter echoes 
from a constant range ring as (Ward, 1994), 

N c 

Z c = (2) 

k=\ 


where a k is a random clutter echo from the k th clutter patch at the steering angle 9 k , and 
v(t9^, m k ) the corresponding space-time steering vector. 

v(^,^) = b(cr i .)®a(6i i .) (3) 

where n(9 k ) and b (tu k ) are spatial and temporal steering vectors, respectively, as 


*&)=[! 

exp(- jl7z9 k ) ••• 

exp(- jlxiN -\)9 k )Y 

(4) 

b(®> ) = [l 

ex p(- j2xP9 k ) 

■■■ ex p(- ~k)9 k )] r 

(5) 


3 



DSTO-RR-0291 




— COS <9 COS (4 
A 


( 6 ) 


where d is the antenna element space, A the radar frequency, 9 and <j> k are the elevation 
angle and the azimuth angle, respectively, for the kth clutter patch. p = 2 v a T r Id is the ratio 
of the normalised Doppler frequency to the normalised spatial frequency. p = 1 means that 
the platform moves one element space d , in one pulse repetition interval (PRI). Sometimes 
f) is also referred to as the number of clutter fold-overs. 


The clutter amplitude satisfies 2s{| a k | 2 ]= a 2 ^ k , where <r 2 is the system thermal noise, and 
4 k is the single-pulse clutter-to-noise ratio (CNR) for a single antenna element on receive, 
whose value may be determined from the radar equation as. 


P,T p G{9, <f> k )g(G, 4> k )A 2 <j 0 (9, <j> k )A t 

(4 


(7) 


where P, is the transmit power, T the pulse width. G and g are the full array transmit 
power gain and the element receive power gain, respectively. A k is the clutter patch area 
and <t 0 the backscattering coefficient for the area. L s denotes the total loss in the radar 
system. 

With a further assumption that clutter echoes are uncorrelated from patch to patch, the 
clutter CM is therefore. 


R 


C 



= VL C V H 


( 8 ) 


where V = (v, ••• j and E c =cr 2 diag( 4) ••• 4 Nc ). 

Since antenna gain patterns in (7) are system dependent, we need to further define system 
parameters in order to compute the clutter CM. Parameters for the generic radar system 
model used in this report are the same as those used by Ward (1994), and are given below. 

• Equal-spaced linear planner antenna array with half wavelength element spacing 
(w = d / A = 0.5); 

• Number of elements in azimuth: 18; 

• Number of elements in elevation: 4; 

• Element pattern: cosine with a -60dB backlobe level; 

• Transmitter tapering: uniform; 

• Platform height 9000 m; 

• Range 130 km; 

• PRF: 300 Hz; 

• Number of clutter fold-over p = 1; 


4 




DSTO-RR-0291 


• Number of pulses in a CPI M = 18 . 

Based on these parameters, it is straightforward to obtain the transmit antenna pattern 
G{G,^> k ), and the receive element pattern g(#,^.) (Wirth, 2001). 

With the assumption of the clutter coefficient a 0 to be constant, G(0,</> k ) and g(0.(j> k ) for a 
patch positioned at (G, <j> k ) can be determined from the geometry relationship between the 
illuminated patch and the antenna orientation (Note that the antenna's elevation angle and 
azimuth angle are generally not coincident with the elevation angle and azimuth angle of 
the illuminated patch, but can be numerically or analytically determined for the given 
geometry). Figure 2 shows the normalised clutter pattern H c for a ring of R = 130 km, 
assuming the clutter coefficient a a {6,<j> k ) to be constant and the antenna looking 
horizontally. It can be seen that the backlobe is 120dB down (two-way) compared to the 
frontlobe, because we assume the backlobe of the antenna element pattern is 60dB down 
(one-way) from the frontlobe. In reality, the measured backlobe level will be limited by the 
receiver's thermal noise level, and will never be as low as those shown in Figure 2. 
However, because the thermal noise is included in the modelling, there is no need to 
further modify these theoretical backlobe values. Levels of backlobe, if below the thermal 
noise level, will eventually be masked. 



Figure 2: Clutter pattern received from a ring of R = 130km, assuming the clutter coefficient 
c 0 (o, (f k ) , k = \,---,N c , to be constant. 


The structure of R ( , given by (8) has a so-called Toeplitz-block-Toeplitz pattern (Ward 
1994). It is also a Hermitian and usually low rank matrix. The rank of R c is given by 
Brennan's Rule as (Brennan and Staudaher, 1992, Ward, 1994), 

N r = int {/V + p(M - l)} (9) 


5 





































DSTO-RR-0291 


Figure 3 shows the Toeplitz-block-Toeplitz pattern of the clutter CM. Only the magnitude 
(normalised to 1) is shown in the figure. Shown in Figure 4 is the rank of R r which varies 
as a function of /?. 



Figure 3: Clutter covariance matrix has a Teoplitz-block-Toeplitz pattern. 



Figure 4: The number of eigenvalues of the clutter covariance matrix varies with the number of 
clutter fold-overs (side-looking array, N = M = 18). Only the first 100 of 324 eigenvalues are 
shown. 


6 
































DSTO-RR-0291 


2.2 Jamming 

Only directional broadband noise jamming is considered. The jamming signal is assumed 
to be a point target spatially, correlated from element to element spatially but uncorrelated 
from pulse to pulse temporally. The space-time snapshot of the jamming signal can be 
written as, 

Xj =(i j ®a ; (10) 

where a. is the spatial steering vector corresponding to the jammer's location, and 
a.j = [a 0 a, ■■■ a M | ] T is a M x 1 vector containing random jamming signals measured by 
the pulse train in a CPI, and satisfies 

E {a m \ a m2 1 = V 2 Zj$m\-m2 (H) 

where c 7 is the jammer-to-noise ratio (JNR). The jamming CM is therefore, 

( 12 ) 


where I M is an M xM identity matrix. 

Multiple jamming signals are normally uncorrelated, so the CM of multiple jamming 
signals is the sum of the CM of each jamming signal as, 

R j = £ tc;Xf }=o- 2 Zly I M ®( a y a f) ( 13 ) 

j 


Figure 5 shows the normalised jamming CM for the case of two jamming signals of equal 
intensity, located at (o°,65°) and (o°, 130°), respectively. 


7 



DSTO-RR-0291 



M IDO IS) J» 30 BO 


Figure 5: The covariance matrix of two independent jamming signals, each of which is assumed to 
be spatially correlated and temporally uncorrelated. 

2.3 Thermal Noise 

The dominant thermal noise is the receiver noise (usually limited by the thermal noise of 
the first amplifier). If each element has its own receiver, then the noise process in each 
element is mutually uncorrelated. The snapshot of noise can be written as. 

In = i a 0,0 a 0.N-l a M- 1,0 1 (U) 

Its temporally and spatially uncorrelated nature satisfies, 

E{ a m\,n\ a m2,n2 1 = °’ 2 ^ml-m2 < ^nl-n2 (15) 

The CM is therefore. 



(16) 


2.4 Covariance Matrix of Undesired signals 

Clutter, jamming and thermal noise are mutually uncorrelated, so the CM contributed 
together by clutter, jamming and thermal noise is. 


R„ - R f +R y +R„ 


(17) 


8 







DSTO-RR-0291 


From now on, unless specified, the covariance matrix given by Equation (17) is simply 
referred to as the CM, and its inversion, the ICM. 


3. Space-Time Adaptive Processing 

3.1 Fully Adaptive STAP 

Suppose that the data snapshot at range of interest contains a target signal as, 

l = lt+lc+lj+ln ( 18 ) 


where 

1, =a t y t (19) 


a, and v f are the amplitude and steering vector, respectively, of the target signal. 

It is well known that the optimum space-time weighting vector for detecting the desired 
target signal with a steering vector of \ t is (Compton, Jr., 1988), 

^apt=/ R u 1 ^t ( 20 ) 


where y is an arbitrary scalar. The output of the STAP processor is the product of the 
Hermitian transpose of the weighting vector times the data snapshot, 

y = ^o P ti ( 21 ) 


The coherent processing gain of the optimum processor approaches the upper limit, 
10 l()g| 0 (MV) dB, for deterministic signals whose bearings differ from the jamming bearings, 
and whose Doppler frequencies differ from the clutter Doppler frequency at the looking 
direction. 


The signal to interference and noise ratio (SINR) is defined as the output target power to 
the output interference and noise power. 


SINR = 



" opt R u " opr 


( 22 ) 


9 



DSTO-RR-0291 


where E{\a t \ 2 } = a 2 £ t . 

The performance of the processor sometimes is evaluated by the so-called improvement 
factor (IF), which is defined as the ratio of the SINR of output to the SNR of input. If we 
assume the SNR of input to be 1 (OdB), then IF will be identical to SINR of output as given 
by Equation (22). Both the IF and SINR are used interchangeably in this report. Figure 6 
shows the SINR of the optimum processor given by Equation (22) when the antenna looks 
at the broadside. It can be seen that the coherent processing gain achieves the upper limit 
of the system (in our case, it is 101og 10 (18xl8) = 25.1 dB) except around the Doppler 
frequency of 0 Flz, where the filter notches the clutter. 




20 







50 







S' 

in 



y 

j 



z 
c n 

0 







-10 







-i 

50 -1 

oo -e 

0 0 5 

Doppler frequency (Hz) 

0 1C 

jo i; 


Figure 6: SINR of the fully adaptive STAP when the antenna looks at the broadside. 

The response of the weight vector to the angle and Doppler is referred to as the adapted 
pattern and defined by (Ward, 1994), 

P w = |w"v(5,cj)| (23) 

Figure 7 shows an adapted pattern for the optimum STAP. The assumed target is at the 
broadside (90° azimuth) with a Doppler frequency of 100 Hz and has a SNR of OdB at the 
input. The clutter and jamming signals are above the noise 47dB (CNR) and 38dB (JNR), 
respectively. It can be seen that clutter (the diagonal ridge) and jamming (two constant 
azimuth ridges) have been well suppressed. Two principal cuts at the target azimuth and 
Doppler are shown in Figure 8. 


10 





















DSTO-RR-0291 



Figure 7: Adaptive pattern of the optimum processor. The horizontal abscissa is cosine of azimuth 
angle and the vertical abscissa the Doppler frequency. 




Figure 8: Principal cuts of the adapted pattern at the target azimuth and Doppler. 


3.2 Partially Adaptive STAP 

Fully adaptive STAP given in Section 3.1 is optimum for detecting desired target signals 
embedded in interference and noise signals. To implement the fully adaptive STAP in real¬ 
time, however, the CM with dimensions of MN x MN is often too large to be inverted in the 


11 














































































DSTO-RR-0291 


available time. In addition a large dimensionality also requires a large number of 
identically independent distribution (iid) samples to compute the ensemble average of the 
CM. The sample matrix inversion (SMI) method requires at least 2 MN samples to obtain a 
reliable ensemble average of the CM (Reed, et al, 1974), which is often impractical. 

As discussed, the CM has a Toeplitz-block-Toeplitz pattern and it is also Hermitian. Figure 
9 depicts the pattern of R„ assuming M = 4. Each small block denotes a N x N matrix. The 
Toeplitz-block-Toeplitz pattern means that each N*N matrix itself is Toeplitz and all 
diagonal matrix blocks are identical. Since the matrix is Hermitian, the total number of 
different elements 1 in the whole matrix includes: 

• N elements in main diagonal blocks, and 

• (2N - l)(M -1) elements in remaining diagonal blocks. 

Therefore the information presented in the CM is highly repetitive, which mathematically 
provides a base for possible reductions in the dimensionality of the matrix with little loss 
of IF. 



Figure 9: Covariance matrix has a Toeplitz-block-Toeplitz pattern, and it is also Hermitian. 

In practice, the reduction of the dimensionality can be realised in the temporal domain by 
processing a sub-CPI (a few pulses) at a time and/or in the spatial domain by 
beamforming 2 . Compared to the fully adaptive STAP (no dimensionality reduction), the 
process with the reduced dimensionality is commonly referred to as partially adaptive 
STAP or sub-space STAP, which can normally greatly reduce the total computational cost 
with little loss of IF. 

Numerous algorithms have been published in the area of partially adaptive STAP aiming 
at reducing computation cost. Based on the domains in which the reduced dimensionality 
applies, these algorithms may be grouped into four categories (Ward 1994, Ward and 
Kogon, 2004) 


1 The element value might differ from the others in this context. 

2 Beamforming in this context is a spatial-only operation. 


12 














































DSTO-RR-0291 


• Element-space pre-Doppler; 

• Element-space post-Doppler; 

• Beam-space pre-Doppler; and 

• Beam-space post-Doppler. 

The element-space pre-Doppler algorithm uses K, pulses (usually 2 or 3) at a time, and 
applies the familiar 2 or 3 pulse clutter canceller principle to suppress clutter. The adaptive 
process is then applied to a reduced dimension of K,N instead of a full dimension of MN. 
Depending on whether the Doppler filtering is applied prior or posterior to space 
processing, the algorithm is referred to as element-space pre- or post-Doppler. In the later 
category, in order to make the Doppler processing adaptive, usually two or more Doppler 
filters have to be adaptively combined for each element. 

The dimensionality can be further reduced by beamforming prior to adaptation. The 
reduction hence can undergo two ways. First the element data is processed with 
beamforming to produce a small number K s of beam outputs, and second only K, pulses 
are adaptively processed at a time, so the adaptive problem dimensionality becomes K S K,. 
Similarly the Doppler processing can be combined with this beam-space architecture in a 
prior or posterior manner. A universal beamformer matrix is difficult to form in the 
presence of jamming, if the direction of the jamming is unknown a priori. A typical way is 
to filter the jamming signals before implementing the beam-space architecture. 

Typical algorithms of these four categories have been discussed in detail (Ward, 1994). We 
only briefly present simulation results here for comparison purposes. Figure 10 to Figure 
13 show the performance of four categories of the partially adaptive STAP algorithms in a 
comparison with the fully adaptive STAP algorithm. It can be seen that for the given 
conditions, all four partially adaptive STAP algorithms perform similarly, and their SINRs 
approach the optimum SINR. The ripples in the SINRs are due to the straddling loss of the 
Doppler filters. In the simulation, the number of Doppler filters used was equal to the 
number of pulses in a CPI, which is 18. 

The necessary condition for partially adaptive STAP to have little IF loss is that the CM 
must be of strict Toeplitz-block-Toeplitz form. This is the basic assumption in deriving the 
various partially adaptive STAP algorithms. Therefore, if the measured CM is not strict 
Toeplitz-block-Toplitz, the partially adaptive STAP may have much lower gains than 
expected depending upon the degree of the departure from a Toeplitz-block-Toeplitz 
structure. 


13 



DSTO-RR-0291 



Doppler frequency (Hz) 


Figure 10: Performance of element-space pre-Doppler algorithm in comparison with that of the fully 
adaptive STAP. 



- Optimum, fully 
adaptive 
STAP 

- PRI-staggered 
post-doppler, 
Kt = 2 

- PRI-staggered 
post-doppler, 
Kt = 3 


Doppler frequency (Hz) 


Figure 11: Performance of element-space post-Doppler algorithm in comparison with that of the 
fully adaptive STAP. 


14 










































DSTO-RR-0291 



-Optimum, fully adaptive 

STAP 


- Beams pace tw o-step 

pre-Doppler, Kt = 2, Ks 
= 3 

-Beams pace, two-step 

pre-doppler, Kt = 2, Ks 
= 5 


Doppler Frequency (Hz) 


Figure 12: Performance of two-stage beam-space pre-Doppler algorithm in comparison with that of 
the fully adaptive STAP. 



- Optimum, fully 
adaptive STAP 


- beams pace post- 
doppler, Ks = 2; 

Kt = 2 

Beamspace, post- 
doppler, Ks = 3, 

Kt = 3 


Doppler frequency (Hz) 


Figure 13: Performance of two-stage beam-space post-Doppler algorithm in comparison with that of 
the fully adaptive STAP. 


15 











































DSTO-RR-0291 


4. Approximate Invariance of the Inverse of the 

Covariance Matrix 


Because the optimum processing given in Equation (20) only requires knowledge of the 
ICM, we are more interested in the characteristics of the ICM rather than the CM itself. In 
this Section we give two Theorems in support of a claim that the ICM is approximately 
invariant, based on which, a concept of PSTAP is proposed. 

Mathematically we know that any changes in a matrix normally introduce changes in its 
inverse. However because of its structure it is possible that R„ can vary in such a way that 
the resultant variations in R are ignorable. The following two Theorems specify the 
changes in R„ which introduce ignorable changes in R „ 1 . 

It is useful to introduce the matrix inversion lemma before introducing two Theorems. 

Matrix Inversion Lemma : Let A and C be square and invertible matrices but need not to be of 
the same dimension, and B matrix have appropriate numbers of rows and columns, then we have 
(Mardia et al, 1979, Wirth, 2001 and Ward, 1994), 

(a + BCB") -1 = a -1 -A _1 b(b"A _1 B + C _1 ) _1 B"A _1 (24) 

It is also useful to note a matrix proposition. 

Proposition: The trace of a matrix, ie, the sum of the elements on the main diagonal, is equal to the 
sum of all eigenvalues of the matrix (Bodewig, 1959), 

trace( A)= (25) 

/Ij-eA 


With the assistance Equation (24) and Equation (25), we give the following two theorems. 

4.1 Theorem 1 

Let the covariance matrix of the undesired signals be the sum of the covariance matrices of clutter, 
jamming and thermal noise, 

R„=R C +R,+R„ (17) 

where R„ = cr 2 I and R e is expressed as, 

R c = VL c V h (8) 


16 



DSTO-RR-0291 


V contains N c space-time steering vectors, V = [v! v 2 ••• v N \, and each \ k (k = 1,2, ■■■,N C ) is 
a MN xl vector. T, c =a 2 diag([| 1 | 2 "■ vj) (4k > 0, k = 1,2, •■■,N C ) contains the mean clutter 
power measured from N c patches by the receiver. The diagonal elements of can be divided into 
two groups, one for those illuminated and received by frontlobes of the transmit and receive 
antennas, which are assumed to be much greater than the thermal noise, and the other for those 
illuminated and received by backlobes, which are assumed to be close to zero. 

If R R„ and V remain unchanged, then R“ 1 remains approximately invariant irrespective of 
changes in £ c . 

Proof 

The proof has two steps. First the proof is given for the case without the presence of 
jamming. The proof for the case with jamming then follows. 

Because \ k ( k = 1,2,-• -,iV c ) is a steering vector, so \ k \ k = MN (MN is the dimension of R f ). 
We normalise V and express R as, 

Rf = V 0 E 0 Vo" (26) 


where V 0 = V / f NM is the normalised steering vector matrix, and E 0 = MNL C . 

Applying the matrix inversion lemma to the sum of the clutter and noise covariance 
matrices, A = R H + R e = a 2 1 + VoLA',^, we have 


A- 1 = 


(o- 2 I + R c ) * =-t ! —T 


4 V 0 


r | \ _1 

-V^Vo+IV 1 

Vu 2 j 


(27) 


The clutter values are only relevant to the diagonal elements of V 0 + E 0 1 j, but with 

the condition of the theorem, for the clutter patches measured by frontlobes, the diagonal 
elements simplify to. 


1 1 ~ 1 
<J 2 MNo 2 $ k ~ cr 2 


(as 4 »1) 


(28) 


17 



DSTO-RR-0291 


whereas for the clutter patches measured by backlobes, the diagonal elements simplify to, 

—r- +-—— = M q oo (as % k « 0, see note 3 ) (29) 

<j- MNa~^ k 

From Equations (28) and (29) we can see that since expressions for the diagonal elements 
of j^—+ E 0 _1 J are approximately independent to specific clutter values c, 

(i = l,2,-,N c ), all elements of A 1 should also be approximately independent to specific 
clutter values 4,- in £ c . 

With the presence of jamming, 

R„=A + R,. (30) 


Applying the matrix inversion lemma to Equation (30), because A -1 is approximately 
invariant to changes in £ c , we immediately deduce that R,, 1 is also approximately 
invariant to changes in E c . The proof is complete. 

Rigorously speaking, the above proof is only approximately and qualitatively. It is difficult 
to quantitatively indicate how R „ is invariant to the changes in L c , and give the error 
analysis for the ICM. Therefore we present several numerical examples below to verify 
Theorem 1, and demonstrate that R „ is invariant to variations that are likely to be 
observed in practice. 

4.1.1 Example 1 

The clutter pattern received from a ring at R = 130 km, assuming the clutter coefficient 
(T () {0, </>,. ), k = \,---,N c , to be constant, is shown in Figure 2. We denote this as clutter pattern 
case 1 for convenience. The corresponding clutter power component in Y. c is denoted as 
41(i) (k = l,---,N c ). Now assume the mean clutter power 4»(i) 1° undergo a random 
variation as, 

4u 2) = IOO|m«(/«0|4 t(l) k = N c (31) 


3 Because R,, 1 exists, so all its elements are finite. To express the relationship between an element of 
R J and M 0 , in general, it can be considered as a ratio of two polynomial functions of M 0 as 


ci„M 0 +a n _ x M 0 


n -1 


b,„M'n +b„ 


-l M o' 1 + ■ ■ • + b i 


-, with m > n . There are three possible results of the ratio: (1) = a 0 /b 0 if 


m = n = 0; (2) « a„ / b n , if m = n ; and (3) » 0 if m > n . 


18 



DSTO-RR-0291 


where randn() is the (0, l) Gaussian random function. The resultant clutter pattern will be a 
random pattern, with most measures % k[T) higher than by 0 to 20dB (corresponding to 
the \randnQ\ function varying in the range of (0.01,l)). We denote this clutter pattern as case 
2. The clutter pattern comparison between case 1 and case 2 is shown in Figure 14. Clutter 
pattern case 1 may simulate returns from a homogeneous surface, such as bare soil, as 
(j o [0,</> k ), k = 1,■ ■ ■ ,N C , is assumed to be constant. In contrast, clutter pattern case 2 may 
simulate returns from inhomogeneous forests, as the majority returns are 0 to 20dB 
randomly higher than that of case 1. 



Azimuth angle (degree) 


Figure 14: Clutter pattern comparison of case 1 and case 2. 

With the usual assumption of R ; , R„ and V remaining unchanged, all 324-by-324 

elements of R„ and R„ for clutter pattern case 1, and case 2, respectively, are numerically 
compared. The one-one plot of the element-to-element comparison (real and imaginary 
parts separately) for the two CMs is shown in Figure 15, and little correlation between the 
two CMs is observed. Because the CM is Hermitian and has a Toeplitz-block-Toeplitz 
pattern, it only has N + (2N -1 )(M - I) = 613 elements whose values might differ from others 
at the most. This is why there are not many circles shown in Figure 15 (many are 
overlapped). On the other hand, the element-to-element comparison for the two ICMs is 
shown in Figure 16. The comparison indicates that the two ICMs, from the engineering 
point of view, are almost identical, and the differences are so small and insignificant that 
they can be ignored. 


19 





























































DSTO-RR-0291 



(a) Real part 



(b) Imaginary part 


Figure 15: Element-to-element comparison of the two covariance matrices for clutter case 1 and case 
2. Little correlation between the two covariance matrices is observed (note the scale differences in 
abscissas). 



(a) Real part (b) Imaginary part 

Figure 16: Element-to-element comparison of the two inverses of the covariance matrices for clutter 
case 1 and case 2. 


r.1.1 

w* I] £■ 
Wia = ID 


.100 


iU ffl NO ED 


4.1.2 Example 2 

In this example clutter pattern case 3 is defined as, 

4(3)=10 15W " ()/10 4(1) k = l,-,N c (32) 


The resultant clutter pattern will be random, with most measures 44(3) varying around 
44 (1) by ±15 dB (corresponding to the randnQ function varying in the range of (—1, l)). The 
clutter pattern comparison between case 1 and case 3 is shown in Figure 17. Clutter case 3 


20 









DSTO-RR-0291 


may be considered as an extreme example of variation in clutter as measured clutter 
patterns are most unlikely to be worse than this. 



Figure 17: Clutter pattern comparison of case 1 and case 3. 

With the usual assumption of R ; , R„ and V remaining unchanged, the element-to- 
element comparison (real and imaginary parts separately) for all 324-by-324 elements of 
R„ for clutter case 1, and case 3 is plotted in Figure 18. Again, no significant differences 
are found. 




Figure 18: Element-to-element comparison of the two inverses of the covariance matrices for clutter 
case 1 and case 3. 


21 

















































































DSTO-RR-0291 


4.1.3 Examples 3, 4 and 5 

Each of the following comparisons is with the usual assumption of R /7 R„ and V 
remaining unchanged. 

Example 3 considers a scenario in which the antenna is steered 40° from the broadside, and 
the clutter pattern follows case 3 but with a random variation of only ±10 dB. The clutter 
pattern comparison is shown in Figure 19. The element-to-element comparison for the two 
ICMs is shown in Figure 20. 

Examples 4 and 5 consider scenarios in which the antenna element interval is not a half¬ 
wavelength, and the clutter fold-over (i is not equal to 1. In particular. Example 4 
compares clutter pattern case 1 to case 3 (random ±10 dB) with d! X = 0.4 and /? = 0.8 , while 
Example 5 compares clutter pattern case 1 and case 3 (random ±10 dB) with dtX = 0.5 and 
P = 1.2. The element-to-element comparisons for the two ICMs for Examples 4 and 5 are 
shown in Figure 21 and Figure 22, respectively. It can be seen that element differences 
increases for both examples, but overall, the linear correlation is still very high. 



Figure 19: Clutter pattern comparison of case 1 and case 3. In case 3, not only is the clutter 
random, but the antenna is also steered 40° from the broadside (mainlobe points to 50°azimuth). 


22 






























DSTO-RR-0291 




Figure 20: Element-to-element comparison of two inverses of the covariance matrices for clutter 
case 1 and case 3 with antenna steered 40° from broadside. 




Figure 21: Element-to-element comparison of two inverses of the covariance matrices for clutter 
case 1 and case 3 with d IA = 0.4 and p = 0.8 . 


23 








DSTO-RR-0291 




Figure 22: Element-to-element comparison of two inverses of the covariance matrices for clutter 
case 1 and case 3 with d 12 = 0.5 and p = 1.2. 


The familiar statistical parameter, the correlation coefficient, r 2 , usually measures the 
linear correlation between two vectors, x and y, with the same length. The correlation 

coefficient, r 2 , is defined as 


r 


2 


n 


n T x iyi 

i =1 



\i=l 




( 

n n 

r 

Yl 1 

/ 

n 

r 

n \ 

«2>, - 

2>, 

"ity? - 


/=1 

V 

Ui ) j 

i =1 

V 

) ) 


(33) 


The linear correlation between two ICMs (each ICM is stacked as a vector) may be 
measured using the same parameter, r 2 . Table 1 lists the values of r 2 for the examples we 
have used. It can be seen that even for the worst case of Example 5, the values of r 2 are 
still as high as 0.9982 and 0.9983 for the real part and the imaginary part, respectively. 

Table 1: Statistical measure r 2 for the five examples. 


Example 

Correlation coefficient r 2 


Real part 

Imaginary part 

Example 1 

0.9996 

0.9999 

Example 2 

0.9996 

0.9999 

Example 3 

0.9996 

0.9999 

Example 4 

0.9990 

0.9980 

Example 5 

0.9982 

0.9983 


24 


























DSTO-RR-0291 


4.2 Theorem 2 

Let the covariance matrix of the undesired signals be the sum of the covariance matrices of clutter, 
jamming and thermal noise, 

R« = R c + R/ +R« (17) 

where R„ = a 2 1. R c and R, are positive semi-definite but low rank matrices. Let rank(R c ) = r c , 
rank{Rj)=rj, A c and A f be the smallest of the non-zero eigenvalues of R c and R ; , respectively. 

Suppose that R„ undergoes a change as, 

R„ =k c R c +kjRj +k n R n (34) 


If A c » a 2 , Aj » a 2 , and k c , kj and k n are real and positive scalars whose values satisfy the 
conditions k c A c »k n a 2 and k^Aj »k n a 2 , then 


R 


-l 

u' 


. 1 R-! 

X " 


(35) 


The condition of A c » a 2 and Aj » a 2 is generally true as clutter and jamming are much 
stronger than the thermal noise. Now let us prove Theorem 2. 

Proof 

For simplicity, first consider a covariance matrix comprising two components, 

R„ =R„+R,=u 2 I + R c (36) 

According to the matrix inversion lemma, we have, 

E^E + A^j E" (37) 

where R e is expressed by its eigenvector decomposition as R = EAE " (note A c is a r c x r ( 
diagonal matrix). 

Now letting k c = 1 and k n = k , we have 

R„. = ^ R (I + R c = ka 2 l + R c (38) 


r; 


= (a 2 I + R f ) _1 =(u 2 I + EAE // ) _ 1 =X I _ _L 


E! 


Vcr- 


25 



DSTO-RR-0291 


A:R m .' = A'(^cj 2 I + R c ,) ' =k 


7 2 7 2 4 

K(J k G 


-E 




-E^E + A -1 I E" 


(39) 


1 , 1 


I—— E 


— E^E + ArA -1 
.2 


V <7~ 


E 


Comparing Equations (37) and (39), the difference exists only between the corresponding 


diagonal elements of 


-E // E +A -1 


X f 

and 


—lyE^E + A: A 1 . Also the diagonal elements of 

Vcr~ ) 

E W E are all equal to 1 as E is the eigenvector matrix of R . The z th diagonal elements of 


1 -E // E +A -1 I and 


V <T~ 


1 E"E + £A 1 I, respectively, are. 


VCr' 


1 1 A, i + g 1 
t 2 T,- a 2 Aj 


(40) 


and 


1 k _A j +ka~ 
t 2 A: a 2 A, 


(41) 


The assumption of At »a 2 , i = l,2,--,r c , results in that both expressions of Equations (40) 
and (41) approach 1 / a 2 . Therefore we have. 


r: 1 ^-r: 1 =— r: 1 


(42) 


If k c ^ 1 

R»" = k n a 2 \ + KK = K (ka 2 1 + R c )= k c R„. 


(43) 


where k = k n /k c . Using the result of Equation (46), we have, 


P 1 _ ^ p 1 ^ ^ ^c p 1 1 p -1 

K h" K «' ~~ — K » K » 

k c k c k„ k„ 


(44) 


26 



DSTO-RR-0291 


In the presence of jamming, we may need to further assume that the smallest non-zero 
eigenvalue of (r c + R y ) is still much greater than <r 2 . This assumption is generally true, by 
noting, 

rank[ R f + Ry) < rank( R c ) + rank ^R j ) (45) 

and 

trace [R c + R ; ) = trace (R c ) + tmce{R j ) (46) 


Therefore the number of non-zero eigenvalues of (R r +R-) is no more than the total 
number of the non-zero eigenvalues of R and R . On the other hand, the sum of the 
eigenvalues of (r c + R ( ) are equal to the sum of the eigenvalues of R c and R ; . 


Reorganising (34) as. 


R„, = k 


^a 2 1 + 
k r 


k: ^ 

R c + y R j 


(47) 


let R . = R + —R ,, 

c c b i 


the remaining proof is the same as the proof for Equation (38). The 


proof is complete. 


Similarly, the proof of Theorem 2 is only approximately and qualitatively. Numerical 
examples are therefore presented below to verify Theorem 2. 


4.2.1 Examples 

The CMs of clutter case 1, jamming and noise presented in Section 4 are referred to as the 
original case. Two other cases are considered, as. 

Case la: k c = 50.0 , kj = 1.5 , k n = 0.5 ; and 
Case lb: k c = 0.5 , k ; = 30 , k n = 2.0. 


The element-to-element ICM comparisons between the original case and case la, as well as 
between the original case and case lb are shown in Figure 23. It can be seen from the 
figure that the two ICMs for each case are highly linearly correlated. The slope of the 
correlation is equal to 1 / k n irrespective of k c and kj. 


27 



DSTO-RR-0291 



(a) Real part 



50 fi 5(i n5i) 

Qnitf&iMTicaM i 

(b) Imaginary part 




I ■ I li e ' LlY 

i da- I 0 


i 

r 

u ■ 


m- J u 


Figure 23: The linear correlation of the inverses of the covariance matrices. Two cases of k c =50.0, 
kj =1.5 and k n = 0.5, as well as k c = 0.5 , k } = 30 and k n = 2.0 are compared to the original case of 
k c = kj = k n =1. The linear correlation slope is 1 / k n irrespective of k c and kj. 


The statistical measure r 2 , measuring the linear correlation between the two ICMs (each 
ICM is stacked as a vector) for the above two cases are given in Table 2. 

Table 2: Statistical measure r 2 for the two cases. 


Case 

Correlation coefficient r 2 


Real part 

Imaginary part 

Case 1 

0.9996 

0.9999 

Case 2 

0.9992 

0.9998 


4.3 Robust Analysis 

The preceding subsection has discussed the invariance of the ICM if the radar and 
platform parameters remain unchanged. In reality, there always exist uncertainties. For 
instance, air turbulence could cause variations in drift angle, roll angle and velocity etc for 
the platform. We assume that all system parameters can be measured at a relatively 
accurate level, and the effects due to system uncertainties are therefore insignificant and 
not discussed further. However there still exist other uncertainties beyond our control. A 
typical uncertainty is that an undulating terrain induces a random variation in the steering 
angle for the clutter patch. As depicted in Figure 24, for a given range R , surface scatterers 
within the range resolution A p make their clutter contributions to the same range bin. A 
flat terrain surface corresponds to a constant elevation angle 6 for a given range R , 
whereas an undulating terrain surface introduces a random variation Ad which in turn 
introduces a variation to the steering vector defined by (3)-(6). Because the height variation 
Ah is random from patch to patch for an undulating terrain surface, so is the 
corresponding A 6 . According to the geometry shown in Figure 24, the expression for A 6 
is. 


28 










DSTO-RR-0291 


A0 


Ah 

Rcosd 


(48) 


In general, for the same Ah , AO has bigger values for shorter ranges. For instance, a 
variation of Ah = ±\0km 4 corresponds to a variation of A<9 = ±4.4° for R = 130km and 
H = 9km. On the other hand, the same variation of Ah = ±10 km incurs a variation of 
A 6 = ±11.5° for R = 50 km and H = 5 km . 



Figure 24: Uncertainty of the steering vector due to an undulating terrain. 


Without loss of generality, we define the elevation angle for the k th clutter patch as, 

0 k = 0 + Ad\l randQ -1] (49) 

where randQ is a random function with a uniformly distributed interval of (o,l). The 
corresponding steering angle specified by Equation (6) becomes, 

9 k = — cos 0 k cos (pk (50) 


A random variation in elevation angle ( A0 = 10 °) simulating an undulating terrain is 
shown in Figure 25, in comparison with a constant elevation angle simulating a flat terrain 
surface with R = 130 km and H = 9 km . 


4 The highest peak of the Himalayas is only 8796m. 


29 




DSTO-RR-0291 


15 



. tO-1 - i -- i--- ... 

n gn tan i$g jgra ;nn inn ing 
AzJmuBi angto (degs) 

Figure 25: Constant elevation angle for a flat terrain surface versus random elevation angle (a 
variation of ±10°) for an undulating terrain surface. 

To examine the effect of an undulating terrain surface on the ICM as well as the IF 
performance of the PSTAP, the clutter pattern case 3 is again used in comparison with the 
clutter pattern case 1. However this time for case 3, not only does its intensity randomly 
vary ±15 dB as shown in Figure 17, but also its elevation angle randomly varies ±10° as 
shown in Figure 25. This scenario is referred to as clutter case 3a in order to differentiate it 
from the previously defined case 3. Due to the random variation in elevation angle, 
elements of the steering matrix V also alter accordingly, which is shown in Figure 26 
where the elements of the steering matrix corresponding to a flat terrain surface are used 
for the comparison. It can be seen that there is little correlation between the two steering 
matrices. The element-to-element ICM comparison between case 1 (constant clutter 
coefficient and constant elevation angle) and case 3a (random clutter coefficient with a 
variation of ±15 dB and random elevation angle with a variation of ±10°) in Figure 27 
shows that the two ICMs are still strongly correlated. Finally the IF of PSTAP (using the 
optimum weighting vector obtained from clutter case 1 to compute the IF values for clutter 
case 3a, see (52)) is shown in Figure 28 together with the IF of STAP (using the optimum 
weighting vector obtained from clutter case 1 to computer the IF values for clutter case 1) 
for comparison. It clearly shows that even for such a scenario with severely uncertain 
conditions, PSTAP still possesses a nearly identical IF performance as STAP for the whole 
Doppler frequency spectrum. The PSTAP SINR loss, compared to the STAP SINR, due to 
the uncertainties in clutter intensity and steering angle as specified as case 3a, ie, the 
difference of the two SINR curves in Figure 28, is shown in Figure 29. It can be seen that 
the SINR loss is insignificant (usually less than 0.2dB) except at the centre frequency of the 
notch where a significant loss happens. However, the loss at the centre frequency of the 
notch means that the notch is deeper and has even a greater capability of clutter 
suppression. 


30 




































































DSTO-RR-0291 



CUKr rttlsM rwi i CUr n»i>yn i 


(a) real part (b) imaginary part 

Figure 26: Elements of the steering matrix V for a randomly undulating terrain surface compared 
with the corresponding elements for a flat terrain surface. 


m 

m 

rt 

I" 

i" 
s ■■ 

M 


Fy-i py; 

• IMj ---I- |i|(J,(T, ‘1 

*:Q5 a 

trails 




■ 200 


(a) real part 


0 4X K» BOD 

LIA* | -11 ir-H I 



ClMSf paBfl CJM 1 
(b) imaginary part 


Figure 27: Element-to-element ICM comparison between clutter case 1 (constant clutter coefficient 
and constant elevation angle) and clutter case 3a (random clutter coefficient with a variation of 
±15 dB and random elevation angle with a variation of ± 10 °). 


31 







DSTO-RR-0291 



Figure 28 IF values of PSTAP (for clutter case 3a) are nearly identical to that of STAP for the 
whole Doppler spectrum. The maximum IF loss of PSTAP is about 0.2dB. 



Doppler frequency (Hz) 


Figure 29: PSTAP SINR loss, compared to the STAP SINR, due to uncertainties in clutter 
intensity and steering angle as specified in case 3a. The loss is usually less than 0.2dB. A 
significant loss only happens at the centre frequency indicating that the notch of the PSTAP SINR 
is even deeper. 


The above analysis assumes that the height H of the platform is the mean height of the 
constant range ring on the terrain surface. If the height H is assumed to be referenced to 
sea level, then the elevation angle for an undulating terrain surface normally can only vary 
on one side of the nominal elevation angle. The elevation angle for the k th clutter patch can 
be written as, 

6 k = 9 + AO rand () (51) 


32 

































DSTO-RR-0291 


Let Ad = 10°, the random variation of the elevation angle for an undulating terrain surface 
is shown in Figure 30 in comparison with the constant elevation angle for a flat terrain 
surface. We define clutter case 3b as having an intensity variation the same as case 3 as 
shown in Figure 17 and an elevation angle variation as shown in Figure 30. The scenario is 
compared to the scenario of clutter pattern case 1 (constant clutter coefficient and constant 
elevation angle). Again the two ICMs are nearly identical. The IF of PSTAP (using the 
optimum weighting vector obtained from clutter case 1 to computer the IF values for 
clutter case 3b, see (52)) is shown in Figure 31 together with the comparison to the IF of the 
STAP (using the optimum weighting vector obtained from clutter case 1 to computer the 
IF values for clutter case 1). Again it clearly shows that even for such a scenario with 
severely uncertain conditions PSTAP still possesses the nearly identical IF performance as 
STAP for the whole Doppler frequency spectrum. 



Figure 30: Constant elevation angle for a flat terrain surface versus random elevation angle (a 
variation of + 10°) for an undulating terrain surface. 



■ Fully STAP, i 

= j = 1 


o Fully FSTAP, 
i = 1;j= 3 


Doppler frequency (Hz) 


Figure 31: IF values of PSTAP (for clutter case 3b) are nearly identical to that of STAP for the 
whole Doppler spectrum. The maximum IF loss of PSTAP is about 0.4dB. 


33 
































































































DSTO-RR-0291 


Based on the above simulations, it can be concluded that the invariance of the ICM is 
robust and not sensitive to uncertainties such as random variations in clutter intensities 
and in elevation angles due to height variations of an undulating terrain surface. This 
implies that the performance of the PSTAP is stable and suffers little, even used in severely 
uncertain conditions. 

4.4 Significance 

The implications of Theorems 1 and 2 are significant. The main points are: 

• Theorem 1 indicates that for clutter, adaptive processing needs to adapt only to 
system parameters (both radar and platform) but not the clutter itself. The fact that 
the ICM is approximately invariant implies that the ICM is approximately 
independent of the clutter environment, so that the optimum weighting vectors can 
be pre-built, as long as system parameters are known. 

• Theorem 1 indicates that for jamming, adaptive processing needs to adapt only to 
the jamming bearing, but not the magnitude of the jamming. The ICM is 
approximately invariant as long as the steering matrix of jamming is unchanged. 
Therefore, jamming filters can also be pre-built. 

• Theorem 2 indicates that the effect of changes on the power of the radar (which 
normally incurs a variation in k c ), and/or the jammer power (which normally 
incurs a variation in k,), and/or the thermal noise levels, on the optimum 

processor as given in Equation (20), is insignificant and can be ignored. Such 
variations only introduce an arbitrary scalar to the processor. As we know, this 
arbitrary scalar has no effect on the processor. This also suggests that in 
constructing PSTAP processor, there is no need to know the exact and absolute 
thermal noise level. 

• The invariance of the ICM is robust and able to sustain elevation angle 
uncertainties induced by undulation of the terrain surface. 

4.5 Interpretation 

Figure 32 depicts clutter and jamming ridges in the Doppler-azimuth plane. To reject the 
interference of clutter and jamming, the optimum processor places deep notches in the 
same position of the clutter and jamming ridges as shown in Figure 7 (note that Figure 7 
shows two notches for two jamming signals, whereas only one jamming signal is depicted 
in Figure 32). The position of the clutter ridge is solely determined by the parameters of 
radar system and platform, but not the clutter itself (limited to the first-order general 
clutter model). Although the clutter intensity profile may vary significantly corresponding 
to different clutter environments, the position of the clutter ridge remains unchanged if the 
parameters of radar and platform are kept unchanged. Because the clutter notch is 
normally deep enough, the same notch can be used to reject different clutter intensity 


34 



DSTO-RR-0291 


profiles. That is, the adaptive processing needs to adapt only to the radar and platform 
parameters. 

Similarly, the position of jamming ridge is solely determined by the jammer bearing and 
has little to do with the intensity of the jamming signal. The same notch can be used to 
reject jamming signals with different intensities, as long as their bearing is the same. 

4.6 Implementation 

With the assurance of Theorems 1 and 2, the implementation of PSTAP can be separated 
into two stages: building jamming filters and optimum weighting vectors prior to the 
mission, and simply calling the appropriate jamming filters and optimum weighting 
vectors to process data snapshots during the mission. 

4.6.1 Prior to the mission 

• For jamming: pre-build a library of jamming filters corresponding to different 
bearings in conjunction with different radar and platform parameters. 

• For clutter: pre-build a library of optimum weighting vectors corresponding to 
different combinations of radar and platform parameters, search angles and 
Doppler frequencies. If the PRF is linked to platform velocity in such a way that 
fl = 2v a T r /d remains constant, the total number of possible combinations can be 
significantly reduced. 


CO 

X. 

z 

CO 


Figure 32: Clutter and jamming ridges in the D op-pler-azimuth plane. 



35 





DSTO-RR-0291 


4.6.2 During the mission 


• Determine the presence of jamming and its bearing, and call the appropriate 
jamming filters to filter the jamming interference. The rejection of jamming can also 
be achieved using beamforming techniques. 

• Determine the radar and platform parameters and call the appropriate optimum 
weighting vectors to process the jamming-filtered snapshots to determine the 
presence of moving targets. 

• If necessary, update the jamming filter library and optimum weighting vector 
library to compensate for changes in the radar and platform parameters. 


5. Numerical Results 


This Section presents some numerical results using both a generic airborne side-looking 
phased-array model and real airborne radar data collected by the MCARM system. 
Various issues including range ambiguity, clutter Doppler spectrum and platform motion 
which incur temporal and spatial decorrelation are discussed, and appropriate formulas 
are used to process the MCARM data for detecting desired signals which are many tens of 
dB below the clutter. 


5.1 Results from a Generic Model 


The generic model presented in Section 3, as well as clutter pattern cases 1, 2 and 3 
presented in Section 4 were used for the simulation. The CMs R„ (1) , R„ (2) and R h(3) 

correspond to clutter pattern cases 1, 2, and 3, respectively. Jamming and thermal noise 
described in Section 3 remain unchanged for all calculations. To examine the performance 
of the PSTAP processor, its SINR is compared to that of the real-time STAP processor. In 
general, the SINR given previously in Equation (22) may be re-expressed as. 


SINR = 


O’ 2 !/ w" f(0 v, 

^ opt (;') ® - u(j) ^ opt(i) 


(52) 


If i = j in Equation (52), the processor becomes the familiar real-time STAP, as the 
optimum weighting vector is calculated from the corresponding CM. If i * j in Equation 
(52), the processor can be considered as PSTAP, because the optimum weighting vector 
vto pt (i) resulting from clutter case i is used to process data obtained from clutter case j. 

Because the ICM correlation is the least when p = 1.2, as shown in Figure 22, we use this 
scenario to compare SINRs computed from real-time STAP and PSTAP. The results are 


36 



DSTO-RR-0291 


shown in Figure 33. It can be seen that all three fully adaptive curves are nearly identical, 
so are all three partially adaptive curves, irrespective of whether it is STAP (i = j) or 
PSTAP (i * j). For the partially adaptive, the algorithm of element-space, pre-Doppler 
with K t = 2 (Ward, 1994) was used in the simulation. 



Fully STAP, i = j = 1 

-Fully PSTAP, i = 1; j 

= 2 

-Fully PSTAP, i = 1, j 

= 3 

-Partially STAP, i = j 

= 1 

- Partially PSTAP, i = 

i; j = 2 

Partially STAP, i = 
1. j = 3 


Doppler frequecy 


Figure 33: Comparison of SINK for STAP (i = j ) and PSTAP (i* j ). It seems that there are only 
two curves instead of six shown, because the three fully adaptive SINRs are nearly identical, so are 
the three partially adaptive SINRs. 

Snapshots of clutter, jamming, noise and target are generated, in which the real part and 
imaginary part of clutter, jamming and noise are random and obey the (o,l) Gaussian 
distribution, and the target signal has a constant Doppler and a constant magnitude. These 
snapshots are shown in Figure 34. The SNR of target and noise is OdB, clutter is 47dB 
above the noise, and two equal jamming signals are 38dB above the noise. Various 
processors were applied to these snapshots of individual signals as well as the sum of the 
signals. The results are given in Table 3 and Table 4 corresponding to the fully adaptive 
and partially adaptive processors, respectively. It can be seen from the tables, the SINR of 
the sum of the signals may vary and be about 2 dB lower than the optimum value, which 
might be due to the fact that simulated interference signals are not perfectly Gaussian (this 
can be confirmed that the SINR of / c is not very low). But an important point we want to 
make is that the performance of STAP and PSTAP processor are about the same. 


37 
























DSTO-RR-0291 



Figure 34: Snapshots of clutter, jamming, thermal noise and target signals. 


Table 3: SINR values for various signals calculated using the fully adaptive STAP processors (R/T: 
real time STAP; P/B: pre-built PSTAP). 


Clutter 

pattern 

Processor 

SINR (dB) 

Xt+Xc+Xj+Xn 

It 

X c 


In 

Case 1 

R/T (w opr(1) ,R„ (1) ) 

22.5 

25.0 

14.9 

-67.8 

4.0 

Case 2 

R/T (w oM 2 ) ,R„ (2) ) 

25.3 

25.0 

-2.8 

-63.3 

3.8 

P/B (w 0 / ,, ( 1 ) ,R„ (2) ) 

22.4 

24.9 

14.8 

-67.9 

4.0 

Case 3 

R/T (w oM 3 ) ,R„ (3) ) 

23.5 

25.0 

16.5 

-66.8 

3.9 

P/B (w 0 / , /( 1 ) ,R„ (3) ) 

22.4 

25.0 

14.9 

-67.8 

3.9 

R/B ( w o/«(2pR»(3)) 

25.3 

25.0 

-2.8 

-63.3 

3.8 


38 






















































DSTO-RR-0291 


Table 4: SINR values for various signals calculated using partially adaptive processors (element- 
space, pre-Doppler K = 2, R/T: real time STAR; P/B: pre-built PSTAP). 


Clutter 

pattern 

Processor 

SINR (dB) 

Xr + Xc + Xj + Xn 

X, 

Xc 

X j 

X n 

Case 1 

R/T (w oM 1 ) ,R 1((1) ) 

24.5 

22.8 

10.9 

-79.6 

1.2 

Case 2 

R/T (W 0/ ,,(2),R i( (2)) 

22.6 

22.7 

9.5 

-44.8 

0.9 

P/B (w oM i),R„ (2 )) 

23.9 

22.1 

10.3 

-80.2 

0.6 

Case 3 

R/T («V(3),R u( 3)) 

24.4 

22.7 

10.8 

-79.7 

1.1 

P/B (w oM i),R„(3)) 

24.4 

22.7 

10.8 

-79.7 

1.1 


P/B (W 0/ ,((2pR w (3)) 

22.5 

22.7 

9.5 

-44.9 

0.9 


5.2 Results from MCARM Data 

Detailed descriptions of the multi-channel airborne radar measurement (MCARM) system 
can be found elsewhere (Sloper etc, 1996, Fenner, 1996). Some of the MCARM data 
analyses are also available (MITRE, 1999, RAFDCI, 1999, Sarker et al, 2001). A brief 
summary of the MCARM system is given in Appendix A. 

This subsection focuses on the performance comparison between the real-time STAP and 
PSTAP processors. In doing so, some issues not considered in the previous sections, 
including range ambiguity, clutter Doppler spread and platform motion which incur 
temporal and spatial decorrelation, are discussed, and their effects are taken into account 
in order to construct a proper PSTAP processor. 

The data cube #5-575 of Flight 5 was used for the analysis. The radar and platform 
parameters of #5-575 are given in Table 5 and Table 6, respectively. 


Table 5: MCARM radar parameters 


Frequency/ 

polarisation 

PRF 

CPI 

Pulse 

width 

Duty 

Range 

resolution 

PRI 

(us/gates) 

1240 MHz/VV 

1984 

128 

50.4 us 

10% 

0.8us 

504/630 


Table 6: MCARM platform parameters 


Height 

Velocity 

Illumination 

Crab angle 

Antenna tilt angle 
from horizontal 5 

3488 m 

100.1 m/s 

Side-looking 

7.28° 

5.11° 


5 This angle is the sum of the antenna tilt angle relative to the platform plus the recorded platform 
roll angle. 


39 






DSTO-RR-0291 


5.2.1 Forming the Covariance Matrix Using the SMI Method 

The SMI method uses sampled snapshots to compute the CM by (Ward and Kogon, 2004), 
R U=^iuik (53) 

& k =1 

The commonly accepted number of snapshot samples for forming the CM using the SMI 
method for the STAP analysis is 2 MN (Reed et al, 1974, Ward 1994 Ward and Kogon, 
2004). As described in Appendix A, for the Flight 5 medium PRF MCARM data, the 
number of receiver elements is N = 22 , the number of pulses in a CPI is M = 128. Therefore, 
approximately, a total of 2 MN = 5632 snapshots would be required to form the CM for 
STAP analysis. For the given data cube of #5-575, the total range bins are only 630. Besides, 
the first 200 bins or so have to be excluded, due to either the range less than the height (so 
the data do not reflect true clutter) or a leakage in range bin 68 (refer to the signal range 
profile shown in Figure A6). The usable range bins therefore, are far less that the 
theoretically required number. 

To reduce the dimensionality, we only used one row of the receive modules (N = 11), and 
50 pulses (M = 50 ) in this report. 

The CM of the so-called diagonally loaded SMI method is given by (Carlson, 1988, Ward 
and Kogon, 2004), 

R u =^Ixaf+M (54) 

& k =1 


where 8 is a small value of the order of the system thermal noise level. 

The diagonally loaded SMI method consistently outperforms the classical SMI method 
(Carlson 1988). It generally has a better adaptive si delobe pattern, is more robust to the 
mismatch, and needs fewer snapshots (Ward and Kogon, 2004). Mathematically, by 
adding a small positive value to all the diagonal elements, the diagonally loaded SMI 
method guarantees the CM to be positive definite and invertible. Our simulation has 
confirmed the above reports, and therefore only the results of the diagonally loaded SMI 
will be reported. 

According to MITRE (1999), a moving target with a large radar cross-section (RCS), 
referred to as a Sabreline, having a Doppler of approximate 522 Hz should appear in range 
bin 320. Unfortunately, for unknown reasons, the target signal has not been convincingly 
detected from the recorded data (MITRE, 1999, Sarkar, et al, 2001). For range bin 320 as the 
range bin of interest, initially we selected the sample range bins from 200 to 600 with the 
exclusion of range bins of interest itself and its 5 near and 5 far range bins to compute the 
CM. That is. 


40 



DSTO-RR-0291 


R 


1 


u(k) 


( 600-200 + 1 - 11 ) 


6 »° H k + 5 H 

Lhli ~ Llil, 

i =200 i=k—5 


+ si 


(55) 


where k is the range bin of interest, and 8 used in the simulation was 70dB below the 
mean clutter. 

It can be seen from Equation (55) that a total about 400 snapshots were used for computing 
the CM with a dimension of 11 x 50 = 550 . The number of samples seems to be still less than 
the required 2MN = 1100 . In fact the sample requirement of twofold the dimension of the 
CM can be relaxed. Because the CM of the MCARM data is assumed to have a structure of 
R„ = R c +cr 2 I, the diagonally loaded SMI method usually only requires a minimum of 2 r c 
iid samples to form a reliable CM, where r c is the rank of R e , and normally 2 r c «2MN 
(Steiner and Gerlach, 1998 and 2000, Gerlach and Picciolo, 2003, Bresler, 1988). The number 
of sample snapshots in Equation (55) is 390, much greater than 2 r c . Indeed, we found that 
a further increase/decrease in the number of sample snapshots makes little difference in 
the CM and the ICM. In fact more samples are available. Because only the data of the first 
50 pulses are used, the data of the next 50 pulses could provide another set of iid samples. 
Furthermore, the data collected by the second row of the antenna can also be considered as 
another set of iid samples, since the second row of the antenna is identical to the first row 
of the antenna. Because the complex conjugate operation is involved in the computation, 
the phase difference between the first and the second rows due to their positions relative 
to clutter is not of concern in calculating the CM. 

The SINR of STAP calculated from the CM obtained by Equation (55) is shown in Figure 
35. Initially we thought that the small notch with the Doppler frequency about 300 Elz next 
to the clutter notch (note that the clutter Doppler frequency is not zero due to the crab 
angle) might be due to unknown interferences such as the interference of the platform to 
the radar. Careful examination using PSTAP (because PSTAP does not require any sample 
data) later indicated that it is due to a target signal in the range bin 299. With the exclusion 
of target range bin 299 and its 10 near and 10 far neighbouring bins, the CM is re¬ 
calculated and the resultant SINR of STAP is also shown in Figure 35. The disappearance 
of the small notch can be viewed. This also indicates one of the potential problems of STAP 
using the SMI method. If target range bins are included in the sample data without 
awareness, STAP will treat the target signal as an undesired signal, the performance of the 
processor then degrades. 

The choice of 8 in (55) for the diagonally loaded SMI method may be justified by 
examining the eigenvalues of the CM. The value of 8 should be smaller than those 
significant eigenvalues of the CM. The eigenvalues of the CM without diagonally loading 
are shown in Figure 36 together with the comparison of the eigenvalues of the CM with 
diagonally loading 8 1, where 8 was 70dB below the mean value of the diagonal elements 
of the CM. It can be seen that adding 8 1 only affects the eigenvalues ranked beyond 300 or 
so. Theoretically the rank of R c should be much less than 300, according to Brenna's Rule 
given in (9). We also tried different values of 8 , the results are not sensitive to 8 in the 
range of 60-80dB below the mean value of the diagonal elements of the CM. Further 


41 



DSTO-RR-0291 


decease in 8 did not seem to improve the results as in that case the numerical error would 
become dominant. 



Figure 35: STAP SINR comparison using the diagonally loaded SMI method. If target bins are 
included in the sample data without awareness, the performance of the processor might degrade. 



Figure 36: Eigenvalues of the sample covariance matrix compare to eigenvalues of the diagonally 
loaded covariance matrix. 


5.2.2 Crab Angle Correction 

A comparison between the preliminary SINR computed from the simulated CM presented 
in Section 2 and the SINR computed from the measured CM is shown in Figure 37. 
Because various gains and losses in process stages are unknown, the value of the 
measured SINR is adjusted and matched to the theoretical value of the simulated SINR. 


42 





























































DSTO-RR-0291 


Due to the crab angle, the Doppler frequency for the centre of the notch drifts from 0 Hz 
even though the antenna was looking at broadside. It can be noted that the centre of the 
notch for the simulated SINR does not point to the same Doppler frequency as the 
measured SINR does. This may be explained as the measurement error in the crab angle, 
as it is difficult to measure accurately. In the simulation a further 1° was added to the 
recorded crab angle of 7.28°, so the centre of the notch points to the same Doppler 
frequency as the measured SINR, as shown in Figure 37. It should be pointed out, 
however, that the effect of such a small adjustment in the crab angle on the performance of 
the PSTAP would be little. The point here is to demonstrate that the accurate system 
parameters but no single real-time sample data are required in forming the PSTAP 
weighting vectors. On the other hand, ample real-time sample data but not system 
parameters are required in forming the STAP weighting vectors. 

The notch of the preliminarily simulated SINR is much narrower than that of the 
measured SINR. It can be expected that if such a preliminarily simulated SINR is used to 
process the measured data, the clutter would not be sufficiently suppressed. Therefore the 
effects of temporal and spatial decorrelation due to range ambiguity, clutter Doppler 
spread and platform motion has to be taken into account to modify the simulated SINR. 
The following subsection addresses this issue. 



Figure 37: Comparison of the preliminarily constructed PSTAP SINR to the STAP SINR. 


5.2.3 Temporal and Spatial Decorrelation Effects 

Klemm (2002) lists four contributions to the decorrelation. They are, 

1. Range walk (platform motion, temporal correlation); 

2. System bandwidth (ambiguity function /(r,0) for antenna elements, spatial 
correlation); 


43 



































DSTO-RR-0291 


3. Clutter intrinsic motion (clutter temporal correlation); 

4. Doppler spread in range (temporal and spatial correlation, only if the Doppler is 
range dependent). 

Klemm (2002) has reported detailed simulations of the above four decorrelation effects. 
According to Klemm (2002), 

• The effect of range walk is insignificant for radars with coarse range resolution 
(e.g., >100 m, the range resolution of MCARM data is 150m). 

• The effect of system bandwidth is insignificant if the length of the antenna is small. 
The effect may become significant for synthetic aperture radar (SAR) data. 

• There is no Doppler spread in range for the side-looking scenario, so the last term 
is also insignificant. 

Therefore, among the four correlation models, we only need to consider the effect due to 
clutter intrinsic motion. 

However, for medium PRF radar, range ambiguity is unavoidable. The steering angle of 
ambiguous range is different from the steering angle of the true range. The effect of clutter 
contribution due to range foldover can be considered as an effect of decorrelation, since 
the effect of range foldover also widens the clutter notch. 

Effect of range ambiguity 

For a medium PRF, the range ambiguity is unavoidable. Clutter returns from different 
ranges have different steering matrices. Assuming that the clutter returns from different 
patches are uncorrelated, the CM for the multiple range case is, 

R c = tv A , 2 : fc v" (56) 

k =1 


where r is the number of multiple ranges. For instance, for range bin 320 of MCARM data 
#5-575, the true range is 38.4 km, and the 1 st and 2 nd ambiguous ranges are 114 km and 
189.6 km, respectively. Assuming clutter against grazing angle obeys the sind function, 
where 6 is the grazing angle, and the clutter power decays as 1 / R 3 as specified in radar 
Equation (7) (the cutter patch area A is proportional to R ), the clutter profiles for the 
above three different ranges are obtained as shown in Figure 38. 


44 



DSTO-RR-0291 



— True range 

1st 

ambiguous 

range 

— 2nd 
ambiguous 
range 


Figure 38: Clutter profiles corresponding to the true range, 1 st and 2 nd ambiguous ranges (only the 
frontlobe part is shown). 


The SINR with the range ambiguity being taken into account is shown in Figure 39. It can 
be seen that the notch has been widened, but is still not as wide as obtained from the 
STAP. 



Doppler freq (Hz) 


Figure 39: Constructed PSTAP SINR with the range ambiguity being taken into account in 
comparison with the STAP SINR. 

Due to range foldover, theoretically, data collected from the first pulse, which do not 
contain components of multiple range clutter, should be statistically different from the 
data collected from the following pulses. This difference is not of concern to STAP if the 
SMI method is used, as the difference is also sampled. For PSTAP, this might be an issue 


45 



























































DSTO-RR-0291 


unless the model takes the difference into account. Because the model assumes that data 
collected by all pulses are statistically the same, in the process the data collected by the 
first pulse were excluded, and the actual data used were the data collected by the 2 nd to 51 st 
pulses for PSTAP. For the comparison purpose, the same data were used for STAP, 
although there is no need to exclude the data of the first pulse for STAP. 

Next we also include the effect of clutter intrinsic motion. 

Clutter intrinsic motion 

Under windy conditions surface clutter echoes fluctuate due to the motion of moving parts 
of scatterers. The Doppler spectrum of land clutter is exponentially distributed and there is 
a large dc component in the spectrum (Billingsley, 2002), from which the correlation can be 
derived. Such a correlation is then generally non-Gaussian. Other researchers argue that a 
radar system performs linear averaging at many stages including antenna, IF bandpass 
filters, baseband anti-aliasing filters, pulse compression filters etc, so that the central limit 
theorem applies in general (Goldstein and Guerci, 2004). The resultant correlation is 
Gaussian. Figure 40 shows the measured temporal correlation at L-band due to clutter 
intrinsic motion under different wind conditions (Billingsley, 2002). It can be seen that 
these correlations are close to Gaussian distributions. 



Figure 40: Measured temporal correlation at L-band due to clutter intrinsic motion (Billingsley, 
2002 ). 

The Gaussian distributed correlation model due to clutter intrinsic motion is given by 
(Klemm, 2001), 


46 




DSTO-RR-0291 


Pmp = ex P 


B ci m ~Pf 


(62) 


where B c = B / f PRF is the normalised clutter bandwidth, and B is the clutter bandwidth. 

According to Long (2004), the Doppler velocity of land clutter with a 60dB spectrum 
(signal decays 60dB from the centre frequency of 0 Hz) is often <1 mis , giving, 

B < 2 — = 2-^—!- = 16.7 Hz (63) 

2 0.24 v ’ 


Another way of calculating the clutter bandwidth is (Ward, 1994), 
8 ncr s , 


B = - 


2 


(64) 


where a v is the standard deviation of clutter velocity, whose value typically varies from 0 
to 0.5m/s for land clutter (Nathanson, 1999). If we select <t„ = 0.5 m/s , we have. 


8/rx 0.5 
0.24 


= 52 Hz 


(65) 


In the construction of the PSTAP processor, the bandwidth given in Equation (65) was 
used, because it seems that the calculated SINR of PSTAP is closer to the SINR of STAP for 
the MCARM data. The SINR comparison among the SINR of STAP and the SINR with 
range foldover only, and the SINR with multiple range as well as clutter Doppler spread of 
PSTAP is shown in Figure 41. It can be seen that the SINR of PSTAP, after the effects of 
range foldover and clutter Doppler spread are taken into account, is very close to the SINR 
of STAP. The result of PSTAP therefore can be expected to be approximately the same as 
the result of STAP. The PSTAP SINR has a deeper notch than STAP SINR indicating the 
PSTAP has a greater capability to suppress clutter than required, which is however not our 
concern. STAP (PSTAP) is not designed for detecting targets whose Doppler is in the 
region of clutter Doppler. 


47 



DSTO-RR-0291 



Doppler freq (Hz) 


Figure 41: SINR comparison among the STAP S1NR and the PSTAP SINR with range foldover 
only and the PSTAP SINR with range foldover as well as clutter intrinsic motion. 


5.2.4 Results 


Nobody was able to find the supposed target. Sabreline in range bin 320 or elsewhere with 
an approximate Doppler frequency of 522 Hz (MITRE, 1999, Sarkar, et al, 2001). Our 
process shows that although there is no target in range bin 320, a target with a Doppler 
frequency about 300 Hz, indeed has been found in range bin 299. If this target is the 
Sabreline, what causes the range bin and frequency shift is unknown, possibly due to 
experimental measurement errors. This data set has been studied in many cases, but as far 
as we are aware, this is the first time that a target has been declared found in the data set. 
The spectrum of clutter plus target signals in range bin 299 is shown in Figure 42. It can be 
seen that the target cannot be identified without clutter suppression. The results of clutter 
being suppressed using PSTAP and STAP are shown in Figure 43. Comparing the results 
of PSTAP and STAP, we can seen that the sidelobe level of PSTAP much more regular and 
lower than that of STAP. This might be attributed to the fact that the CM of PSTAP 
approaches ideal while the CM of STAP is only obtained from limited and inhomogeneous 
sample data. 


48 



































DSTO-RR-0291 



Doppler frequency (Hz) 


Figure 42: Spectrum of clutter plus target signals in range bin 299. 



Doppler frequency (Hz) 


Figure 43: A target with Doppler frequency of 300 Hz in range bin 299 has been detected using 
both STAP and PSTAP processors. 

Because each range bin corresponds to a specific elevation angle, so each range bin should 
have a corresponding set of weighting vectors of PSTAP. However, we have shown that 
the ICM is insensitive to a certain variation of the elevation angle (the penalty is only 
about 0.5dB). Therefore the same set of weighting vectors of PSTAP can be used to process 
different range bins. Figure 44 shows the result of PSTAP, where only one single set of 


49 


































































DSTO-RR-0291 


weighting vectors, corresponding to the middle range bin, was used to process all range 
bins 200-600. The target in the range bin 299 is obvious. 



di -19 
T7 

-20 

ifs 

1-30 


3 CO 


600 


1COO 


500 


-500 

10CO DopFJer^req(Hz) 


range- bin 


Figure 44: A target is detected using PSTAP. The same set of weighting vectors was used for all 
range bins. 


In order to further assess the performance of PSTAP, range bins 320 and 350 were selected 
as range bins of interest. Because no target presents in these range bins, two artificial target 
signals, each with constant amplitude and constant Doppler frequency (Swirling 0 model), 
were injected into the original signal. In particular, the amplitudes of the two injected 
signals are the same with levels at 40 and 30dB below the mean clutter, respectively, for 
each test case. Their Doppler frequencies are -300 Hz and +600 Hz, respectively. The mean 
clutter amplitude of a snapshot is defined as, 

„ 1 MN 

£<llzll>= w§w (68) 

The amplitude of the clutter snapshot in range bin 320 is shown Figure 45. Shown in the 
figure is also the injected target signal with a constant Doppler frequency and a constant 
amplitude but 40dB below the mean clutter. The Doppler spectrum of clutter in range bin 
320 is shown in Figure 46. 


50 




DSTO-RR-0291 


-20 


s - 40 

<u 

■O 

D 

ro -60 


-80 

0 100 200 300 400 500 600 

Space-time No 





y \jmMi 





Clutter 

Signal 
















Figure 45: Snapshots of clutter in range bin 320 and the injected target signal which is 40dB below 
the clutter. 



Figure 46: Doppler spectrum of clutter in range bin 320. The Doppler spread of the mainlobe clutter 
is about 200 Hz. 

The performance of two processors, ie, diagonally loaded SMI STAP and PSTAP, are 
compared. The results are shown in Figure 47 to Figure 48. Observing these plots we can 
conclude that the performance of the PSTAP is about the same as that of the diagonally 
loaded SMI STAP. The sidelobes of PSTAP near the notch are a few decibels higher than 
that of STAP especially when the target signal is weak, which can be easily overcome if 
necessary. In fact if we re-examine the STAP SINR and PSTAP SINR in Figure 41, we can 
see that the PSTAP SINR is a few decibels higher than that of the STAP SINR in that 


51 



















































DSTO-RR-0291 


Doppler region. This may be due to some decorrelation effect which has not been 
considered and included. If the PSTAP SINR is further modified, these high sidelobes 
would be suppressed. This issue will be further investigated in the future. 

In order to further examine the performance of the PSTAP for the whole Doppler 
frequency spectrum, targets with low and high frequencies were injected into the original 
signal, and the results compared. Figure 49 shows the processor comparison for detecting 
two injected low frequency (-50 Hz and +300 Hz) and low magnitude (30dB below clutter) 
target signals, while the processor comparison for detecting two injected low frequency 
(-100 Hz and +350 Hz) and low magnitude (30dB below clutter) target signals is shown in 
Figure 50. Finally compares the processors for detecting two injected high frequency 
(-900 Hz and +900 Hz) and low magnitude (40dB and 30dB below clutter) target signals. 
These comparisons again indicate that the performance of the PSTAP is about the same as 
that of the diagonally loaded SMI STAP. 


52 



DSTO-RR-0291 



Doppler frequency (Hz) 



Doppler frequency (Hz) 


Figure 47: Processor comparison for detecting targets 40dB (top) and 30dB (bottom) below the 
clutter at -300 Hz and +600 Hz in range bin 320. 


53 











































































































DSTO-RR-0291 



Doppler frequency (Hz) 



Doppler frequency (Hz) 


Figure 48: Processor comparison for detecting targets 40dB (top) and 30dB (bottom) below the 
clutter at -300 Hz and +600 Hz in range bin 350. 


54 


































































































DSTO-RR-0291 



Doppler frequency (Hz) 


Figure 49: Processor comparison for detecting targets 30dB below the clutter at -50 Hz and 
+300 Hz in range bin 320. 



Doppler frequency (Hz) 


Figure 50: Processor comparison for detecting targets 30dB below the clutter at -100 Hz and 
+350 Hz in range bin 320. 


55 












































































































DSTO-RR-0291 



Doppler frequency (Hz) 



Doppler frequency (Hz) 


Figure 51: Processor comparison for detecting targets 40dB (top) and 30dB (bottom) below the 
clutter at -900 Hz and +900 Hz in range bin 320. 


6. Conclusions 


With phased-array radar technologies, STAP is optimal in terms of detecting moving 
target signals embedded in much stronger interference signals such as surface clutter. 
Implementation of real-time STAP algorithms is however not practical due to the time 
taken to compute the covariance matrix and its inverse. Partially adaptive (reduced 
dimensionality) STAP algorithms may significantly reduce the computation requirement, 
but the algorithms are based on a necessary condition that the structure of the covariance 
matrix is Toeplitz-block-Toeplitz. If the covariance matrix is not strictly Toeplitz-block- 
Toeplitz, which is almost always the case for the measured data, the expected processing 
gain of the partially adaptive STAP processors greatly degrades. 


56 





























































































DSTO-RR-0291 


A significant contribution of this report is the indication of the inverse of the covariance 
matrix (ICM) to be approximately invariant to variations of clutter. Two Theorems have 
been proposed and their mathematical proofs provided. Various numerical examples have 
been given for confirmation of the Theorems. The following conclusions have been 
deduced: 

• For clutter, STAP needs to adapt only to system parameters (both radar and 
platform). Variations in clutter intensity, which will result in significant variations 
in the elements of the covariance matrix, cause little variations in the elements of 
the inverse of the covariance matrix; 

• For jamming, STAP needs to adapt only to the bearing of jamming. Variations in 
jamming intensity, which will result in significant variations in the elements of the 
covariance matrix, cause little variations in the elements of the inverse of the 
covariance matrix. 

In consequence, a pre-built space-time non-adaptive processor (PSTAP) has been 
proposed. The PSTAP is not an adaptive processor in the broad meaning of adaptive 
processing as no adaptive processor can be pre-built. Flowever, it does have a capability to 
cope with changes in clutter, due to the invariance of the inverse of the covariance matrix 
(the knowledge of the clutter environment is not really required). Therefore the 
performance of the PSTAP may approach that of the STAP as long as the system (radar 
and platform) parameters are known. 

The idea of PSTAP is to construct a library of weighing vectors a priori, either by 
modelling or using flight data. Each weighting vector corresponds to a specific set of the 
radar and platform parameters. Using such a processor the computational bottleneck is 
completely removed. Theoretically, if the system parameters are known, the proposed 
PSTAP can achieve the same coherent processing gain as real-time STAP. In reality, due to 
the fact that the CM for real-time STAP is difficult to obtain accurately, PSTAP may even 
achieve better results if all the system parameters required for the construction of the 
processor are known precisely. 

Before the mission, a library of optimum weighting vectors corresponding to various 
combinations of radar and platform parameters can be pre-built without knowledge of the 
actual clutter environment. A library of jamming filters can also be pre-built 
corresponding to various jamming bearings. During the mission, the collected data are 
first analysed to determine the existence of the jamming. Either appropriate jamming 
filters or beamforming techniques are applied if jamming is detected. The jamming-free 
data are then multiplied with appropriate weighting vectors from the library to generate 
the output. The pre-built libraries may need updating once the radar and platform 
parameters undergo changes. 

If a real radar system is difficult to model (for instance, the array may not be rigorously 
linear and the effect of the platform on the antenna system may be difficult to model etc), 
the library may be pre-built based on data collected from previous missions. Because of 


57 



DSTO-RR-0291 


the nature of its invariance, the ICM from different clutter environments should be 
approximately the same. Therefore, the optimum weighting vectors obtained from 
previous missions can be directly used for future missions, irrespective of clutter 
environments, as long as the radar and platform parameters remain unchanged. 

There are usually a large number of possible combinations of radar and platform 
parameters, and so a large number of weighting vectors have to be pre-built. If the radar 
PRF is linked to the platform velocity in such a way that p = 2 v a T r Id remains constant, the 
number of combinations can be significantly reduced. Also because the ICM is not very 
sensitive to the changes in steering angle varying from range bin to range bin, the same set 
of weighting vectors can be used to process different range bins, which has been 
demonstrated in the process of the MCARM data as shown in Figure 44. 

A robust analysis for PSTAP has been carried out. Uncertainties incurred in the steering 
angles caused by possible undulating terrain surfaces as well as variations in clutter 
intensities have been studied. It has been shown that the SINR loss of PSTAP is normally 
only about 0.2-0.5dB even for the most extreme variations in the terrain surface. 

Numerical examples have been presented. The first numerical example was for a generic 
airborne radar model. Cases of clutter coefficients randomly fluctuating up to ±15 dB 
(simulating an environment much worse than any inhomogeneous environment) were 
compared to the usual case of the constant clutter coefficient (simulating a homogenous 
clutter environment) showing that the resultant ICMs are almost identical. The correlation 
coefficient r 2 for all pairs of ICMs compared is higher than 0.99 showing that the ICM is 
insensitive to variations in the clutter environment. The PSTAP processor performs the 
same as the STAP processor for all scenarios simulated. 

The second numerical example was to apply the PSTAP processor to MCARM data. In 
order to process the real data, some assumptions in the generic model have been modified. 
Temporal and spatial correlations due to both the radar system and clutter have been 
taken into account in order to built a realistic and appropriate PSTAP processor. It has 
been shown that the dominant decorrelation mechanisms include multiple range and 
clutter intrinsic motion. A target in the #5-575 data set has been convincingly detected 
using PSTAP. Finally the PSTAP processor has also been compared to the real-time STAP 
processor (the diagonally loaded SMI method) for detecting injected small target signals. It 
has been shown that the performances are about the same. 

In conclusion, the PSTAP processor can be constructed purely from modelling, or 
alternatively based on date collected from previous missions. 


7. Acknowledgement 

The discussions with Drs D Madurasinghe, T Winchester, A Shaw and J Whitrow help to 
improve the readability of the report. Thanks are also due to Dr Madurasinghe for the 
indication of an error in the draft of the report. Finally Dr Whitrow helped correcting 


58 



DSTO-RR-0291 


grammar errors. Creative comments by the vetting officer. Dr J Fabrizio, are also 
acknowledged. The MCARM data was supplied by the Rome Laboratory, Air Force 
Material Command. 


8. References 

Billingsley, J B, Low-Angle Land Clutter Measurements and Empirical Models, William Andrew 
Publishing, New York, 2002. 

Brennan, L E, and Staudaher, F M, "Subclutter visibility demonstration". Technical Report, 
RL-TR-92-21, Adaptive Sensors Incorporated, March 1992. 

Bresler, Y, "Maximum likelihood estimation of a linearly structured covariance with 
application to antenna array processing". Proceedings of the Fourth Annual ASSP 
Workshop on Spectrum Estimation and Modelling, pp. 172-175, 3-5 Aug 1988. 

Bodewig, E , Matrix calculus, 2 nd edition, North-Holland Publishing, Amsterdam, 1959. 

Carlson, B D, "Covariance matrix estimation errors and diagonal loading in adaptive 
arrays", IEEE Trans on Aerospace and Electronic Systems, vol. 24, no. 4, pp. 397-401,1988. 

Compton, Jr, R T, Adaptive Antennas, Concepts and Performance, Prentice Hall, New Jersey, 
1988. 

Dong, Y, "Models of land clutter vs grazing angle, spatial distribution and temporal 
distribution - L-band VV polarisation perspective". Research Report, DSTO-RR-0273, 
DSTO, 2004. 

Fenner, D, and Hoover, W F, "Test results of a space-time adaptive processing system for 
airborne early warning radar". Proceedings of IEEE 1996 National Radar Conference, Ann 
Arbor, Michigan, 13-16 May 1996. 

Gerlach, K, and Picciolo, M L, "Airborne/ spacebased radar STAP using a structured 
covariance matrix", IEEE Trans on Aerospace and Electronic Systems, vol. 39, no. 1, pp. 
269-281, January 2003 

Goldstein, I C, Guerci, J R, "STAP II: Advanced techniques", tutorial slides, CD of the 
Proceedings of 2004 IEEE Radar Conference, Philadelphia Pennsylvania, 26-29 April 2004. 

Klemm, R K, Principles of Space-Time Adaptive Processing, 2 nd Edition, IEE, London, 2002. 

Kreyenkamp, O, "Clutter covariance modelling for STAP in forward looking radar", 
DGON International Radar Symposium 1998, Munich, 15-17 Sept 1998. 

Long, M W, "Radar clutter". Tutorial slides, CD of the Proceedings of 2004 IEEE Radar 
Conference, Philadelphia Pennsylvania, 26-29 April 2004. 


59 



DSTO-RR-0291 


Mardia, K V, Kent, J T, and Biddy, J M, Multivariate Analysis, Academic Press, London, 
1979. 

MITRE, "STAP processing monostatic and bistatic MCARM data". Final report prepared 
by MITRE, Centre for Air Force C2 Systems, Bedford, MA, 1999. 

Nathanson, F E, Reilly, J P, and Cohen, M N, Radar Design Principles, 2 nd Edition, Scitech 
Publishing Inc., 1999. 

RAFDCI, "MCARM/STAP data analysis". Final report. Part I and II, prepared by Research 
Associates for Defence Conversion Inc., New York, 1999. 

Reed, I S, Mallett, J D, and Brennan, L E, "Rapid convergence rate in adaptive arrays", 
IEEE Trans on Aerospace and Electronic Systems, vol. AES-10, no. 6,1974. 

Sarkar, T K, Wang, H, Park, S, Adve, R, Koh, J, Kim, K, Zhang, Y, Wicks, M C, and Brown, 
R D, "A deterministic least-squares approach to space-time adaptive processing", 
IEEE Trans on Antennas and Propagation, vol. 49, no. 1, January 2001. 

Skolnik, M I, Radar Handbook, McGraw-Hill, New York, 1970. 

Sloper, D, Fenner, D, Arntz, J, and Fogle, E, "Multi-channel airborne radar measurement 
(MCARM), MCARM flight test", Westinghouse Electronic Systems, final technical 
report, RL-TR-96-49, vol. 1, April, 1996. 

Steiner, M, and Gerlach, K, "Fast converging maximum-likelihood interference 
cancellation," Proceedings of IEEE 1998 National Radar Conference, pp. 117-122, Dallas, 
TX, 12-13 May 1998. 

Steiner, M, and Gerlach, K, "Fast converging adaptive processor for a structured 
covariance matrix", IEEE Trans on Aerospace and Electronic Systems, vol. 36, no. 4, pp. 
1115-1126, Oct 2000. 

Thomas, V, private communications, 2003. 

Ward, J, "Space-time adaptive processing for airborne radar". Technical report TR-1015, 
Lincoln Laboratory, MIT, 1994. 

Ward, J, and Kogon, S M, "Space-time adaptive processing (STAP) for AMTI and GMTI 
radar", tutorial slides, CD of the Proceedings of 2004 IEEE Radar Conference, 
Philadelphia Pennsylvania, 26-29 April 2004. 

Wirth, W D, Radar Techniques Using Array Antennas, IEE, London, 2001. 


60 



DSTO-RR-0291 


Appendix A: MCARM System 

Detailed descriptions of the Multi-channel airborne radar measurement (MCARM) system 
may be found in the relevant documents (Sloper, et al, 1996, Fenner, 1996). Some MCARM 
STAP data analyses are also available (MITRE, 1999, RAFDCI, 1999, Sarkar, et al, 2001,). 
This Appendix only provides a brief summary of the system. 

The system is L-band, vertically polarised. The transmitter consists of 32 modules in two 
rows. Each module is formed by 4 antenna elements as depicted in Figure Al. Therefore, 
the transmitter pattern can be simulated by a phased array comprising 32-by-8 antenna 
elements. The azimuth spacing and elevation spacing are 4.3 and 5.54 inches, respectively. 
The azimuth tapering coefficients are close to uniform and their values from the centre are: 
OdB, -0.0626dB, -0.1260dB, -0.1904dB, -0.2553dB, -0.3217dB, -0.3891dB and -0.4576dB 
(Thomas, 2003). A 20dB Taylor taper is used across the elevation elements. The actual 
tapering coefficients from the centre are: OdB, -1.488dB, -4.662dB and -8.449dB (Sloper, et 
al, 1996). The simulated and the measured transmitter patterns (array as a whole) are 
shown in Figures A2-A3. It can be seen that the agreement between the simulated and 
measured patterns is reasonably good. 


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Figure Al: Structure of the antenna system: the transmitter consists of 32 modules while only the 
central 22 modules are used as the receiver elements. 

The receiver (element) is modelled by a cos 04 6/;) function (frontlobe) which gives a 6dB 
beamwidth of 160° for both the azimuth and elevation directions (Sloper, et al, 1996). A 
comparison between the simulated and measured module (4-by-l elements) patterns is 
shown in Figures A4-A5. Similarly the agreement between the simulation and 
measurements is reasonably good. An exception is the azimuth pattern for the angle 
beyond the ±60° region. However, the steering angle is normally within ±60°, so there is 
no need to modify the simulated receiver pattern to fit the measured pattern beyond the 
±60° region. 


61 



































DSTO-RR-0291 



Measured 

Figure A2: Simulated and measured transmitter azimuth patterns. Note that the measurement and 
the simulation define the broadside direction as 0° and 90°, respectively. 


62 
























































DSTO-RR-0291 



Measured 

Figure A3: Simulated and measured transmitter elevation patterns. 


63 


















































DSTO-RR-0291 



Simulated 

, 0 ,—-■---■-—— 


5- 



Measured 

Figure A4: Simulated and measured receiver azimuth patterns. The steering angle is normally 
within ± 60°, so there is no need to modify the simulated pattern to fit the measured pattern beyond 
the ±60° region. 


64 





























DSTO-RR-0291 



Simulated 



Measured 

Figure A5: Simulated and measured receiver elevation patterns. 


65 



































DSTO-RR-0291 


During Flight 5 only the signals of the central 22 receiving modules were digitally 
recorded. The flight path of Flight 5 is in the Baltimore-Washington area. The mainlobe 
area for the data cube #5-575 is mainly flat farmland (range bins 200-400, 500 - 600 for 
example) and bay water (range bins 400-500, for example). The clutter profile against 
range in the analogue sum channel is shown in Figure A6. There is a signal leakage in 
range bin 68, and the useful clutter data are from around range bin 200 to the end. 


-20 









00 -Ad 




F }’ u | * 



, J 


<D 

3 -fin 







\M Lw 1 ftf 


o bU 

-80 








-100 









0 100 200 300 400 500 600 

Range gate 


Figure A6: Clutter profile against range. 

From the radar equation, clutter power is proportional to 1 / 2? 3 . If the SMI method is used 
to compute the covariance matrix, the 1/7? 3 range effect on the clutter data is better to be 
removed. Otherwise, statistics of the close range data play a more important role in the 
covariance matrix than that of the far range data. The clutter profile against range, after the 
1/7? 3 range effect is moved, is shown in Figure A7. 



Figure A7: Clutter profile against range after the 1/7? 3 range effect is moved. 


66 





































DSTO-RR-0291 


The resulting clutter profile is almost levelled after the range effect is removed, though the 
clutter intensity is still slightly lower at the far range. This is contributed by two factors: 
the shape of the mainlobe of the transmitter antenna elevation pattern and the clutter 
profile against grazing angle (Dong, 2004). However, these two factors are not further 
taken into account when the SMI method is applied to the data to compute the covariance 
matrix. 


67 



Invariance of the Inverse of the Covariance Matrix and the Resultant Pre-built STAP Processor 


Yunhan Dong 

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Approximate Invariance of the Inverse of the Covariance Matrix and 
the Resultant Pre-built STAP Processor 


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4. AUTHOR(S) 
Yunhan Dong 


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Systems Sciences Laboratory 
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6a. DSTO NUMBER 
DSTO-RR-0291 


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March 2005 


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18. DEFTEST DESCRIPTORS 

Adaptive signal processing 
Airborne radar 
Phased arrays 
Covariance 


19. ABSTRACT 

Space-time adaptive processing (STAP) has been proven to be optimum in scenarios where an airborne phased-array radar is used to 
search for moving targets. The STAP requires the inverse of the covariance matrix (ICM) of undesired signals. The computation of the 
real-time ICM is impractical at current computer speeds. Proposing two Theorems, this report indicates that the ICM is approximately 
invariant if radar and platform parameters remain unchanged. A pre-built STAP (PSTAP) processor is then proposed. Both the 
simulated data from a generic airborne phased array radar model and real data collected by the multi-channel airborne radar 
measurement (MCARM) system are processed to verify the processor. Results indicate that the performance of the proposed PSTAP 
processor is the same as that of the real-time STAP processor. 


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