4.0 SPACE-TIME ADAPTIVE PROCESSING (STAP)
4.1 Introduction
We now want to briefly discuss space-time adaptive processing, or STAP.
When we discuss radars we normally consider the processes of beam forming,
matched filtering and Doppler processing separately. By doing this we are
forcing the radar to operate on only one domain at a time: space for beam
forming, fast time for matched filtering and slow time for Doppler processing.
At some time in the 1970’s or 1980’s radar analysts noted that this separation
of functions may be sacrificing capabilities because the processor was not
making use of all available information, or degrees of freedom. Suppose we
have a linear phased array that has N elements. In terms of beam forming to
maximize the target return and minimize returns from interference (e.g., clutter
and jammers) we say that we have N degrees of freedom. Suppose that in a
Doppler processor we process M pulses. In this case we say that we have M
degrees of freedom with which to maximize the signal return and minimize
interference (e.g., clutter, jamming and noise). In our normal methods whereby
we separate beam forming and Doppler processing we have a total of M+N
degrees of freedom. Now, if we were to consider that we could perform Doppler
processing at each antenna element and then form the beam we would have MN
degrees of freedom.
Figure 1 might help in visualizing this further. It contains a depiction of
angle-Doppler space. Each of the squares corresponds to a particular angle
and Doppler. There are N beam positions and M Doppler cells. The dark
square indicates an angle and Doppler that contains clutter. With our standard
processing techniques we could null the clutter by suppressing a beam position
and by suppressing a Doppler cell. We would do these independently. The
suppressed beam position is denoted by the crosshatched cells and the
suppressed Doppler is denoted by the dotted cells. It will be noted that in the
process of suppressing the clutter cell we suppress other cells. This happens
because we separately process in angle and Doppler space. With STAP we
would, ideally, simultaneously process in both angle and Doppler space and
suppress only the cell containing the clutter.
1
Figure 1 – Clutter Nulling Using Conventional Techniques
4.2 Spatial Processing
To begin talking about STAP techniques we need to establish the math
that seems to be used when discussing STAP. With STAP, the processor design
technique involves maximizing the ratio of signal power to interference plus
noise power (SINR) at the output of the processor. The maximization technique
is based on the Cauchy-Schwarz inequality, the same method we used in
matched filter theory.
4.2.1 Signal Only
To start the problem we consider the antenna problem first. In
particular, we consider a linear array of N elements as shown in Figure 2. In
that figure, it is assumed that the target is located at an angle of  s relative to
boresight. From linear array theory we can write the output of the linear array
as
N 1
V  s    wk Ps e
j  2 kd sin  s  
.
(1)
k 0
If we define
 w0 
 w 
w 1 




 wN 1 
(2)
2
Figure 2 – Linear Phased Array
and
1

 s0  


j 2 d sin  s  
 s 
e


1

s  s   








j 2 N 1 d sin  
 sN 1  e    s  
(3)
we can write V  s  as
V  s   Ps w H s  s 
(4)
where the superscript H denotes Hermetian, or conjugate-transpose, of the
vector.
In STAP, we say that we would like to choose w so as to maximize
V  s  subject to the constraint that the norm of w remain finite. In equation
2
form we want to solve the problem
3
max V  s   max
2
w
w
Ps w H s  s   Ps max w H s  s 
2
w
2
.
(5)
subject to w  constant  
2
To solve this problem we invoke one of the Cauchy-Schwarz inequalities.
Specifically, if w and s  s  are column vectors then
w H s  s   w
2
2
s  s 
2
(6)
with equality when
w   s s 
(7)
where  is an arbitrary constant. If we apply this to our problem with   1 we
get the familiar antenna solution of
1


 j  2 d sin s   
 e

w



e j  2  N 1d sin s   
(8)
or
1


  j  2 d sins   
 e

w  
.


e j  2  N 1d sin s   
(9)
We note that if we replace s  s  with s   , allow  to vary from  2 to  2
and compute
V    w H s  
2
2
(10)
we get the radiation pattern of the linear array. This pattern will have a peak at
   s and a sin Nx sin x shape.
4.2.2 Noise Only
To extend this idea to adaptive processing we next want to consider the
case where the input to each antenna element is only noise. This is illustrated
in Figure 3. Following the pattern from above, we can write the noise voltage at
the output of the summer as
N 1
VN   wk nk  w H n
(11)
k 0
4
where
 n0 
 n 
n  1 .




 nN 1 
(12)
Figure 3 – Array with only noise
Since the noise is a random variable, we can write the noise power at the
output of the summer as1
   E  w n   w E nn  w  w
Pno  E VN
2
H
2
H
H
H
Rn w .
(12)
In the above Rn is termed the interference (receiver noise only in this case)
covariance matrix.
4.2.3 Signal and Noise – SNR
With this we can write the SNR at the output of the summer as
H
Pso Ps w s  s 
SNR 

.
Pno
w H Rn w
2
(13)
If we assume that n is zero-mean and uncorrelated across the elements of the
array we can write
1
Implied in the above is that there is a receiver attached to each element of the array and that
the summing of the outputs of the elements is performed in the signal processor. Thus, the array
is an active array that contains T/R modules. This configuration is taken for granted in radars
that might employ STAP.
5
Rn   N2 I  PN I
(14)
and
SNR 
Ps w H s  s 
PN w
2
.
2
If we recognize w
2
(15)
as a scalar constant that we force to be non-zero and finite,
we can again invoke the Cauchy-Schwarz inequality to maximize SNR by
maximizing w H s  s  . Doing this leads to the same solution as given above in
2
Equation (7).
We point out that the above development is analogous to the derivation
of the matched filter in waveform theory. It also leads to the same result. That
is, the “filter” that maximizes SNR is matched to the signal.
4.2.4 Correlated Interference – SINR
We now consider the case where the interference is correlated across the
array. This interference could be clutter or jammers. The appropriate model we
need to work with is given in Figure 4. In this figure, nIn represents the
interference “voltage” and is a zero-mean, random variable. The subscript n is
used to represent the nth interference source (which we will need shortly when
we consider multiple interference sources). The fact that the same random
variable is applied to each of the antenna elements makes the outputs of the
elements random variables that are correlated. Thus, we can write VIn n  as
N 1
VIn n    wk nIn e
j  2 kd sin n  
k 0
 w H nIn
(16)
where
nIn  nInd n 
(17)
and
1


 j  2 d sin n   
 e

d n   
.


e j  2  N 1d sin n   
(18)
d n  is termed the steering vector for the nth interference source.
6
Figure 4 – Array with Interference
We allow for multiple interference sources by simply summing the
voltages for the multiple sources. Specifically, we write.
K
nI   nIn d n  .
(19)
n 1
We further assume that the interference sources are independent so that
P
E nIn nIk    In
0
n  k
.
n  k
(20)
We write the interference power (from the K interference sources) as

PI  E w H nI
2
  w E n n w  w R w .
H
I
H
I
(21)
H
I
In the above we write RI as
RI  E nI nIH    E nIn nIl  d n  d H l    PIn D n 
K
K
n 1 l 1
K
(22)
n 1
where we have made use of Equation (20) and
7
D n   d n  d H n 
1


 j  2 d sin n   
 e

 j  2 d sin n  

 1 e


e j  2  N 1d sin n   
.
 j 2  N 1 d sin n  

e 

(23)
If we combine Equation (12) with Equation (21), we get the total interference
power as
Pn I  Pno  PI  w H  Rn  RI  w  w H Rw .
(24)
With the above, the signal-to-interference-plus-noise ratio (SINR) at the
output of the summer is given by
Ps w H s  s 
Pso
.
SINR 

Pn  I
w H Rw
2
(25)
The objective of the STAP algorithm is to maximize the SINR. To do this via the
Cauchy-Schwarz inequality we need to manipulate Equation (25). We start by
noting that, because of the receiver noise, R will be positive definite. This
means that R1 2 is real and that R 1 2 exists, and is also real. We can use this
to write
SINR 
Ps w H R1 2 R 1 2 s  s 
w H Rw
2

Ps w RH sR  s 
wR
2
2
(26)
This is of the same form as Equation (15). With this we can state that the SINR
is maximized when
wR   sR s  .
(27)
If we let   1 and substitute for wR and sR  s  we get the solution
w  R1s s  .
(28)
The net effect of the above equation is that the antenna weights, w , are
selected to place the main beam on the target and simultaneously attempt to
place nulls at the angular locations of the interference sources. We will
demonstrate this through an example. Before doing so, we note that a critical
part of this development is the fact that the interference consists of both
receiver noise and other interference sources such as clutter or jammers. The
fact that we have included receiver noise is what makes the R matrix positive
definite, and more importantly, non-singular. Without this, R 1 would not exist
and we would need to use another approach for finding the weights, w . One of
the common approaches is to use a mean-squared criterion such as leastmean-squared estimation or pseudo inverse. Both of these approaches are
beyond the scope of this course.
8
4.2.5 Example 1
As an example we consider a 16 element linear array with ½ wavelength
element spacing ( d   1 2 ). We assume that we have a per-element SNR of 0
dB. We assume that we have two jammers with per-element JNRs of 40 dB.
The target is located at an angle of zero and the jammers are located at angles
of +18º and -34º. The selected jammer angles place the jammers on the second
and fourth sidelobes of the antenna pattern that results from using uniform
illumination (see Figure 5). The above lead to the following parameters
Ps  1 , PN  1, PI 1  10 4 , PI 2  104 , 1  18 and 2  34 .
If we assume that we choose w so as to only point the main beam to 0º, and
not attempt to reject the jammers, we obtain a SINR of about -24 dB. If we
choose w according to Equation (28), the SINR increases to the noise limited
case of about 12 dB (10log16). To accomplish this, the algorithm chose the
weights so as to place nulls in the antenna pattern at the locations of the
jammers. This is illustrated in Figure 5.
Figure 5 – Normalized Radiation Pattern with and without optimization – 16
element linear array
4.3 Temporal Processing
We can consider the above as the space part of STAP. We now want to
consider the time part. In this part we consider Doppler processing.
We let the transmit waveform be a string of M pulses. We can write the
transmit signal as
9
M 1
vT  t   e j 2 fct  p  t  lT 
(29)
l 0
where we have used p  t  as the general form of a pulse. If the pulse were a
simple unmodulated pulse we would have
t 
p  t   rect   .
 p 
(30)
For an LFM pulse we would have
t 
2
p  t   e j t rect   .
 p 
(31)
The exponential term in Equation (29) represents the carrier part of the
transmit signal.
The return signal, from a point target, is a delayed version of vT  t  . That is
vR  t   vT  t  2 R  t  c   e
j 2 f c  t  2 R  t  c 
M 1
 p t  2R t 
l 0
c  lT  .
(32)
If the target is moving at a constant range-rate we can write R  t  as
R  t   R0  Rt .
(33)
From earlier radar courses we recall that we can write a good approximation of
vR  t  as
M 1
vR  t   e  j 2 R0  e j 2 fct e j 2 f d t  p  t   R  lT  .
(34)
l 0
In Equation (34), f d  2 R  and  R  2 R0 c . From our experience with
waveforms, we recognize that we can further approximate vR  t  as2
M 1
vR  t   e  j 2 R0  e j 2 fct  e j 2 f d lT p  t   R  lT 
(35)
l 0
In the receiver we heterodyne to remove the carrier and normalize away
the first complex constant. We also match filter the signal. The result is
M 1
vM  t    e j 2 f d lT m  t   R  lT 
(36)
l 0
where m  t  is the response of the matched filter to p  t  .
The approximation is based on the assumption that the phase shift caused by the Doppler
doesn’t change across a compressed pulse width.
2
10
For the next step we sample vM  t  at times t   RC  lT . That is, we
sample the return from each transmitted pulse in some range cell,  RC . The
result is a sequence of samples that we denote as
Vl  f d   e j 2 fd lT mRC l 0, M  1 .
(37)
where mRC is the (generally complex) value of m  t   R  lT  evaluated at
t   RC  lT . If we sample m  t  R  lT  at it’s peak we will use mRC  PS for a
target (desired signal) and mRC  nIn for an interference signal. nIn is a zero
mean, random variable with a variance of PIn . In subsequent work we will
assume that we sample at the peak of m  t   R  lT  .
We assume that the Doppler processor is an M length finite impulse
response (FIR) filter. As such, we can write its output as
M 1
M 1
l 0
l 0
V  f d    lVl  f d   mRC  le j 2 f d lT .
(38)
It will be noted that Equation (38) is of the same for we used in the
spatial processing problem. Given this we can write the signal voltage as
V  f s   PS
M 1
 e
l 0

l
j 2 f d lT
 PS  H s  f s  .
(39)
Similarly, we can write the interference voltage3 as
M 1
VIn  f n   nIn  le j 2 fnlT  nIn H d  f n    H nIn .
(40)
l 0
In Equations (39) and (40) the vectors s  f s  and d  f n  are Doppler “steering”
vectors for the signal and interference, respectively. Finally, we write the
receiver noise at the FIR output as
M 1
VN   l nl   H n
(41)
l 0
where the nl are zero-mean, independent, random variables with a variance of
PN .
Since Equations (39), (40) and (41) are the same form as those used in
the spatial processing problem we can use the same techniques for finding the
 that maximizes SNR or SINR.
3
Note: to be meaningful in Doppler processing we restrict the interference to coherent types of signals, i.e.
clutter. If we have a broadband noise jammer we would treat it as receiver noise for the temporal
processing part of STAP.
11
4.4 Adaptivity Issues
With the above we have discussed both the space and time parts of
STAP. However, we haven’t really addressed the adaptive part. In concert with
the above methodologies for characterizing the target angle and Doppler, and
the angles and Dopplers of the interference sources, the adaptive part would
involve re-specifying the target steering vector and the R matrices on each dwell
(sequence of M pulses). Since we are changing these on each dwell we compute
new sets of weights on each dwell. Thus, we are changing the antenna (space)
and Doppler (time) processing so that it adapts to our interpretation of the
target and interference environment.
4.5 Space-Time Processing
We now want to address the issue of combined space and time
processing. In a classical sense, this would be considered true space-time
processing. In space-time processing, rather than form a function of angle or a
function of Doppler, we combine spatial and temporal equations for the signal
(Equation (1) and Equation (39)) to form a combined function of angle and
Doppler. In equation form we write
M 1
 N 1
j 2 kd sin  s   
 j 2 lf sT 
V  s , f s   Ps   wke 
  l e

 k 0
 l 0

N 1 M 1
 Ps   w  e
k 0 l 0

k

l
j  2 kd sin  s  
e
.
(42)
j 2 lf sT
We recognize the above as a sum of MN terms. If we generalize the product of
the weights to MN distinct weights we get
N 1 M 1
V  s , f s   Ps   wk,l e
j  2 kd sin  s  
e j 2 lf sT .
(43)
k 0 l 0
If we organize the weights into a general weight vector, w , and the
j  2 kd sin   
e
e j 2 lfT terms into a generalized steering vector, s , we can write
V  s , f s  in the same form we used above. Specifically,
V s , f s   w H s s , f s  .
(44)
Further, if we extend the interference representation from earlier we can write
the interference as
VN  I  w H nn  I
(45)
where
K
nn  I  nn  nI  nn   nIq d n , f n  .
(46)
n 1
12
In Equation (46), nn is the receiver noise and d n , f n  is the steering vector to
the interference in angle-Doppler space.4
Given the above, we can use the techniques discussed in Section 4.2 to
place the “main beam” in angle-Doppler space on the target and to place nulls
at the angle-Doppler locations of clutter.
It is reasonably straight forward to show that
Pn I  Pno  PI  w H  Rn  RI  w  w H Rw
(47)
where
Rn  PN I
(48)
and
K
RI   PIn D n , f n 
(49)
n 1
with
D n , f n   d n , f n  d H n , f n 
(50)
With some thought, it should be clear that the dimensionality of the
STAP problem has increased substantially. If we perform adaptation separately
in angle and Doppler, we would need to compute M+N weights. If we
simultaneously perform adaptation in angle and Doppler space, we must
compute MN weights. To complicate the problem further, we need to remember
that we need to compute a separate set of weights for each range cell that is
processed. This represents a considerable computational burden. To minimize
the burden, much of today’s research in STAP is concerned with avoiding the
computation of MN weights, while still trying to maintain acceptable
performance.
4.6 Example 2
As an illustration of how one might apply STAP we will extend the
problem discussed earlier. We again assume a 16 element array. We also
consider Doppler processing that uses 16 pulses. We will use the classical
STAP approach and process all 16×16=256 signal samples in one processor
with 256 weights. (Recall that we do this for each range cell of interest.) We
assume that the target is located at and angle of zero and a Doppler of zero. We
assume ½ wavelength element spacing and a normalized PRI of unity. The
single-pulse, per element SNR is 0 dB. We assume that we have two
interference sources (clutter). The interference sources are located at angles of
+18º and -34º as in the previous example. The Doppler locations of the
interference sources, corresponding to the above angles, are 0.217 Hz and
As with the temporal processing case we are limiting ourselves to clutter types of interference.
We will extend the representation to the noise jamming case shortly.
4
13
0.28 Hz respectively. The SCRs of the two clutter sources is 50 dB. With this
we get the following parameters
Ps  1 , PN  1, PI 1  105 , PI 2  105 , 1  18 , 2  34 , f1  0.217 Hz and
f 2  0.28 Hz .
We write
D n , f n   d n  d T  f n 
1


 j  2 d sin n   
 e

j 2  f nT

 1 e


e j  2  N 1d sin n   
e j 2  M 1 f nT 
.
(51)
and form the column vector d n , f n  by concatenating the columns of D n , f n 
which is an N by M matrix. The s  s , f s  vector is a column of 256 ones, which
puts the target at an angle of zero and a Doppler frequency of zero. We next use
the equations of Section 4.2 to find the 256 element weight vector, w . The
weight vector is formed into a two-dimensional weight matrix, W by reversing
the algorithm used to form d n , f n  from D n , f n  . Finally, we can find the
angle-Doppler image analogous to Figure 5 by using a two-dimensional, inverse
FFT with the weight matrix as the signal. The results of the above process are
shown in Figures 6 and 7. Figure 6 is the angle-Doppler plot for the case where
there is no interference. Figure 7 is the angle-Doppler plot for the case
described above. The presence of the two nulls is clearly visible on Figure 7 as
is the main beam at (0,0). The numbers beside the color bar to the right of the
figures denote relative amplitudes in dB. Note the difference in the scales
between Figure 6 and Figure 7. This difference indicates that the null depth in
Figure 7 is quite large.
14
Figure 6 – Angle-Doppler Map Without Optimization
Figure 7 – Angle-Doppler Map With Optimization
15
In the above examples, the SINR before optimization was -13.4 dB. After
optimization the SINR was 24.1 dB, the noise limited value (i.e. 10log(256)).
As an interesting experiment, the optimization was extended to include
two desired targets: one at (0,0) and another at (ang,Doppler)=(34º,-0.217 Hz).
Figure 8 contains the angle-Doppler map for the case of the two targets and
only receiver noise. It will be noted that the optimization placed peaks at the
locations of the two targets. Figure 9 corresponds to the case where the two
interference sources were included. It will be noted that the peaks at the
locations of the targets is still present. Also, upon careful examination, and
comparison to Figure 7, it will be noted that the two nulls are also present. In
this case, the SINR on each target before optimization was -13.4 dB, as with the
previous case. The SINR after optimization was 21.1 dB. This is not as high as
for the single-target case but is still very good.
In the above work, we assumed that we knew the locations and strengths
of the interference sources. In practice this may not be the case. If it isn’t one
must form the interference covariance matrix, R , by sampling the environment.
Discussions of this can be found in texts on STAP.
Figure 8 – Angle-Doppler Map Without Optimizatin – Two Targets
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Figure 9 – Angle-Doppler Map With Optimization – Two Targets
4.7 STAP With a Noise Jammer
To handle a noise type of jammer with space-time processing we need to
modify the definition of D n , f n  in Equation (51). This type of interference will
be correlated in angle but not across the pulses. Therefore, we will have a
spatial component to the steering matrix, D n , f n  , but the temporal
component will consist of independent random variables. To model this we
write the steering matrix as
D J , nJ   d J  nTJ
1


 j  2 d sin J   
 e


  nJ 1


e j  2  N 1d sin J   
nJ 2
nJ  M 1 
.
(52)
where the nJk are be zero-mean, independent, random variables with a variance
of PJ .
The resulting d J , nJ  would look like
17
 nJ 1d J  


nJ 2 d   J  

d  n , n J  




 nJ 2 d nJ  
(53)
which is vector of length MN. The resulting covariance matrix would be
0
 D  J 

0
D  J 
RJ  PJ 


0
 0





D J  
0
(54)
where D J  is defined in Equation (23). We note that to properly form
d J , nJ  we concatenate columns of D n , f n  and D J , nJ  . The weights that
are computed for a jammer will place a null at  J that spans across all
frequencies.
4.8 Adaptivity Again
Thus far we have discussed the space and time parts of STAP. We now
want to revisit issues related to adaptivity.
In our work so far we assumed that we knew the various parameters
needed to compute the optimum weights. In particular, we had enough
information to compute the R matrix. In some applications this is not the case
and we must estimate R through measurements. A potential procedure for
doing this follows.
For each antenna element (T/R module) and pulse we sample the
combined noise and interference in range cells we believe contains the
interference but not the target.5. We then use the samples to estimate R .
Specifically, if we write the combined noise and interference voltage on a
particular sample as Vnl I we can form an estimate of R as
H
1 L
Rˆ   Vnl I Vnl I 
L l 1
(55)
where L is the number of samples taken. As a point of clarification it should
be noted that Vnl I is an MN element vector.
A question that arises is: how large does L need to be. If L  1 we will be
multiplying an MN element vector by its Hermetian to produce an MN by MN
matrix. This matrix will have a rank of 1 since it has only one independent
In practice we can allow the range cells to contain the target return is the overall SNR and SIR is
very small for each antenna element/pulse.
5
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column. This means that R̂ has only one non-zero eigenvalue. Thus R̂ is
singular and R̂ 1 doesn’t exist. This means that solving for w by the previous
method will not work.
Given that Vnl I consists of a bunch of random numbers there is a chance
that R̂ will have a rank equal to the number of samples that are taken
(assuming L  MN ). Thus, to have any chance of obtaining a R̂ that is nonsingular, at least MN samples of Vnl I must be taken. As L becomes larger it
provides a better and better approximation of R 6.
Guerci has an equation that relates L to the SINR improvement.
Specifically, he denotes  as the SINR improvement actually obtained versus
the theoretical SINR improvement. He writes

L  MN  2
L 1
L  MN .
(56)
Thus, for L  MN we get

MN  MN  2
2

.
MN  1
MN  1
(57)
For the case of Example 2 we have MN  256 . Thus the expected SINR based
on 256 samples of Vnl I will be 2/257 or about 21 dB below the optimum SINR
improvement. If we increase L to 2MN or 512 samples we would get

2MN  MN  2 1
 .
2MN  1
2
(58)
Thus, the expected SINR improvement based on R̂ would be about 3 dB below
the optimum SINR improvement. However, we note that this represents a large
number of samples that will require extensive time and radar resources.
4.9 Practical Considerations
If practice we may be able to get by with using fewer samples of if we
have a reasonable estimate of the receiver noise power, and the interference
power is large relative to the receiver noise power. We would use the
aforementioned approximation to form and estimate of RI , the interference
covariance matrix. If we term this estimate Rˆ I we would form R̂ from
Rˆ  Rˆ I  PN I
(59)
where PN is the receiver noise power estimate (per antenna element and pulse).
Gureci, J. R., Space-Time Adaptive Processing for Radar, Artech House, 2003, ISBN 1-58053377-9.
6
19
This approach is termed diagonal loading. Adding the term PN I assures that R̂
will be positive definite and that R̂ 1 exists.
With this method the number of samples, L , needed is chosen to be at
least as large as the number of anticipated interference sources. As should be
obvious, this will generally be much smaller than MN .
This method can have problems in that sometimes the eigenvalues of R̂
become ill behaved. This, in turn, causes R̂ to become ill conditioned, and the
optimization to put nulls in the wrong locations. To get around this problem
one needs to take more samples and/or increase PN . The problem with taking
more samples is obvious in that it requires an extra expenditure of time and
radar resources. Increasing PN increases will cause the SIR, and SINR,
improvement to degrade, potentially to unacceptable levels.
More information about these and other practical aspects of STAP can be
found in Guerci’s book and other sources.
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