2.9 DOWN-RANGE AND CROSS-RANGE IMAGING – SIGNAL DEFINITION
We now want to extend our previous work to both down-range and crossrange imaging. We will also extend the problem to include a more general case
of squinted SAR. Squinted SAR is normally associated with strip-map SAR but
the development here also applies to spot-light SAR. As before, we will start by
defining the signal that the SAR processor must work with since this will give
insight into how to process the signal.
2.9.1 Signal Definition
The geometry of interest is a modification of the geometry of Figure 13
and is contained in Figure 24. The main difference between Figure 13 and
Figure 24 is that in Figure 13 the center of the imaged area lies on the x-axis of
the coordinate system while in Figure 24 it does not. This offset of the imaged
area center will result in additional Doppler considerations plus a phenomena
termed range cell migration (RCM), both of which complicate SAR processing.
Another, minor, difference is that the coordinates of the scatterer are relative to
the center of the imaged area. We did this as a convenience.
Figure 24 – Geometry for Down and Cross Range Imaging
Since we are considering both down-range and cross-range imaging the
transmit waveform will be pulsed instead of CW. In practical SAR, the pulses
are phase coded, usually with LFM, to achieve the dual requirements of large
bandwidth to achieve fine range resolution and long duration to achieve
sufficient energy. In this development we will use narrow, uncoded
(unmodulated) pulses to avoid complicating the development with pulse coding
30
and the associated matched filter or stretch processing. The extension to coded
pulses is relatively straightforward.
Given the above we write the transmit signal as
 t  kT 
vT  t   e j 2 fct  rect 

k
  p 
(68)
where
1
rect  x   
0
x 1 2
x 1 2
(69)
and  p is the pulse width. The sum notation means a sum over all k and is
used to indicate that the waveform is, in theory, infinite duration. We will later
make it finite duration.
The signal from a single scatterer at  xn , yn  (see Figure 24) is
vnRF  t  
 t  kT  2rn  t  c 
PSn j 2 fc t  2 rn t  c 
e
rect



rn2  t 
p
k


(70)
where
rn  t  
 x0  xn    y0  yn  Vt 
2
2
.
(71)
2.9.1.1 Removal of the Carrier and Gross Doppler
As before, the first operation we will perform is removal of the carrier.
However, in addition, we will also remove what we term gross Doppler. Removal
of gross Doppler is necessary in some applications in that this Doppler is large
relative to the PRF and has the potential of causing problems with aliasing and
Doppler ambiguities.
To determine the gross Doppler we examine the phase of the returned
(RF) signal. From Equation (70) this phase is
 RF  t   2 fc t  2rn t  c   2 fct  4 rn t   .
(72)
We can find the frequency from
f RF 
2r  t 
1 dRF  t 
 fc  n
.
2
dt

(73)
The first term is the carrier frequency and the second is the Doppler frequency.
We define the gross Doppler as the Doppler frequency at xn  0 , yn  0 and
t  0 . This yields
31
f dg 
2V y0 2V

sin  .
 ro

(74)
In Equation (74),  is the squint angle. For the unsquinted SAR we considered
in the CW development,  was zero because y0 was zero.
Given the above, we remove f c and f dg from the received signal by
multiplying vnRF  t  by the heterodyne signal,
vh  t   e


 j 2 f c  f dg t
.
(75)
We also “normalize away” rn2  t  as before to leave the baseband signal
vn  t   PSn e
 j 2 f dg t  j 4 rn  t  
e
 t  kT  2rn  t  c 
.
p


 rect 
k
(76)
2.9.1.2 Single-pulse Matched Filter
The next step in processing is to send vn  t  through a matched filter
matched to the transmit pulse. The (normalized) output of the matched filter is
vn  t   PSn e
 j 2 f dg t  j 4 rn  t  
e
 t  kT  2rn  t  c 

p


 tri 
k
(77)
where
1  x
tri  x   
 0
x 1
.
x 1
(78)
2.9.1.3 Generation of the Sampled Signal
Recall that for the CW case we sampled vn  t  at intervals of T . We will
do the same for the pulsed case. However, for each pulse (each T ) we will also
subsample vn  t  at intervals of  p , the pulse width. We will start sampling,
relative to each transmit pulse, at some
 min 
2  x0  l 2 
c
(79)
and continue sampling to some
 max 
2rmax
c
(80)
where
32
rmax 
 x0  l 2    y0
2
 w 2  L 2 .
2
(81)
Between  min and  max we obtain approximately
M
 max   min
p
(82)
range samples. We will do this 2 K L  1 times to form  2K L  1 M samples,
which we will collect into a 2 K L  1 by M element array for further processing.
Mathematically, we sample vn  t  at
t  kT   min  m p
(83)
to yield
vn  k , m   PSn e

 j 2 f dg kT  min  m p

e

 j 4 rn kT  min  m p


 min  m p  2rn  kT   min  m p  c  .
tri 

p


(84)
2.9.2 Preliminary Processing Considerations
If we were to directly extend our CW processing methodology we would,
for each m , remove a quadratic phase term and then perform a DFT across k .
Unfortunately, the situation is complicated by the range sampling so that this
straightforward approach is not directly applicable. We will need to perform an
interim step first.
2.9.2.1 Range Cell Migration Correction
Figure 25 contains a plot of vn  k , m  for the parameters of EXAMPLE 1
(Section 2.8, Table 1) with the added condition that we are using a pulse with a
pulse width of 0.5 m or  p  3.33 ns . We are also considering a single target at
xn  0 and yn  0 . With this resolution we will have 50/0.5 = 100 down-range
cells. We force this to an odd number of 101 and index the range cells from 50 to 50. Thus m goes from -50 to 50 and k goes from -120 to 120. (We
already determined that we had 2 K L  1  241 cross range samples.)
Since we are considering a single scatterer in the center of the imaged
area we expect, at first blush, the return to be at range sample 0. However, this
is not the case because the location of the target return depends upon range to
the scatterer, not the x location of the target. Since the range is given is given
by
33
rn  kT  
 x0  xn    y0  yn  VkT 
2
2
(85)
it will vary as a function of k . This is why we see a curved line in Figure 25
instead of the straight line we would like to see.
The problem with the curved line comes when we try to apply the
quadratic phase correction and take the Fourier transform to form the image.
Specifically, we want to perform both of these operations across k for each
range cell, i.e., each m . In fact, we should apply the quadratic phase correction
and Fourier transform along the curved line. We get around this problem by
“warping” the plot of Figure 25 so that the curved line becomes a straight line.
The method we use is interpolation. A straight forward method of performing
this interpolation is via the Fourier transform method. Specifically, compute
the Fourier transform (using the FFT) for each k , apply the appropriate linear
phase shift and take the inverse Fourier transform. The amount of linear phase
shift depends upon the distance, in time, we want to move the samples. With
some thought, it will be obvious that all range samples at the particular k will
be moved by the same amount.6
The algorithm we use is as follows: From Equation (85), we note that the
minimum value of rn  kT  , when xn  yn  0 , occurs when y0  kVT  0 and is
equal to x0 . We decide that we want this range to correspond to a down-range
value of   0 . For each k we compute
  k   2

x02   y0  VkT   x0
2
c
.
(86)
This   k  then becomes the range correction based on the assumption that
  0 when y0  kVT  0 . We use this with the Fourier transform method to
move the samples in range. As an implementation note, for the 101 range cells
of this example, I used a 128 point FFT. After the inverse FFT I simply
discarded the last 27 elements of the array.
For small squint angles, this works well. If the squint angle gets above a few degrees this is not
a good approach because different range cells should be moved different amounts. For further
discussion of this the reader is referred to the text by Cumming and Wong entitled “Digital
Processing of Synthetic Aperture Radar Data” by Artech House, or other advanced SAR
references.
6
34


Figure 25 – Plot of vn  k , m  for a single scatterer at xn , yn   0, 0 
The result of applying the above methodology to the plot of Figure 25 is
shown in Figure 26. As can be seen, the curved line of Figure 25 is now a
straight line, albeit somewhat distorted. The distortion is due to the fact that
we sample in range at multiples of the pulse width,  p . It turns out that this
distortion causes problems in the image. We will address this later.
35


Figure 26 – Plot of vn  k , m  for a single scatterer at xn , yn   0, 0  , with RCM
Correction (RCMC)
The methodology discussed above will adjust the returns from all
scatterers by the same amount for each value of k . Furthermore, the amount
that the returns are adjusted is determined by the properties of a hypothetical
scatterer located at the center of the imaged area. As an example, we consider
three scatterers that are located at the same yn  0 but at xn values of -23, 0
and 23 meters (range sample numbers of -46, 0 and 46). The resulting
uncorrected plot of vn  k , m  is shown in Figure 27. It will be noted that this
figure uncovers a problem: The curved trace of the scatterer at xn  23 is cutoff.
To correct this problem we find that we need to sample over a larger number of
range cells. From Figure 25 we note that the trace spans about 12 or 13 range
samples. Thus, to be sure we get the complete trace for a scatterer at the farrange end of the imaged area we need to extend the range samples to an upper
limit of 50+12. We will round this to 65. The result of this is shown in Figure
28. The resulting RCM corrected image is shown in Figure 29. It will be noted
that there are three straight lines located at m  46, 0, and 46 , as they should.
36
Figure 27 – Plot of vn  k , m  for a three scatterers at
x
n,
yn    23, 0  ,  0, 0  ,  23, 0 
It will be noted that the curved lines of Figure 27 are the same, as are the
three straight lines of Figure 28. As another example we place the three
scatterers at xn , yn   23, 25  ,  0, 0  ,  23, 25  . That is, at diagonal corners and


the center of the imaged area. The resulting uncorrected and corrected plots of
vn  k , m  are shown in Figures 30 and 31. Careful examination of Figure 30
shows that the three curved lines are not exactly the same. Also, the top and
bottom straight lines of Figure 31 are not exactly horizontal. It turns out that,
in some applications, this can cause problems and an interim processing step
must be used to eliminate the problem. This interim step is discussed in
Section 2.12.
37
Figure 28 – Plot of vn  k , m  for a three scatterers at
x
n,
yn    23, 0  ,  0, 0  ,  23, 0  , with range extension
2.9.3 Quadratic Phase Removal and Image Formation
Now that we have an algorithm that performs RCMC we need to develop
an algorithm for removing the quadratic phase. We will want to remove the
quadratic phase from the RCMCed signal. The information we need is in the
phase of vn  k , m  (Equation (84)) at the peak of the tri( ) function (i.e. along the
curved ridge before RCMC).
If we refer to vn  t  of Equation (77) we find we want to examine the
information in the phase of vn  t  at
t  kT  2rn  t  c .
(87)
The problem with this equation is that t appears on both sides of the equation
and is embedded in a square root on the right side. As a result, solving for t
will involve the solution of a rather complex quadratic equation. To avoid this
38
we seek a simpler approach. Specifically, we ask the question: Does the phase
of vn  t  vary slowly enough to allow the use of an approximate value of t ?
Figure 29 – Plot of vn  k , m  for a three scatterers at
x
n,
yn    23, 0  ,  0, 0  ,  23, 0  , with range extension and RCMC
We can write the phase of vn  t  , from Equation (77), as
  t   2 f dg t  4 rn t   .
(88)
From calculus we know that we can relate variations in   t  to variations of t
by
  t0  
  t 
t .
t t
(89)
0
If we perform the partial derivative we get

2V  y0  Vt0  
  t0   2  f dg 
 t .
 rn  t0  

(90)
39
Figure 30 – Plot of vn  k , m  for a three scatterers at
x
n,
yn    23, 25  ,  0, 0  ,  23, 25  , with range extension
We are interested in the variation of   t  over the times we are taking
measurements. Specifically, from t  kT   min to t  kT   max . Thus, we use
t0  kT   min . Further, we let t   max   min   . With this we have

2V  y0  V  kT   min  
  kT   min   2  f dg 

2
2


x

y

V
kT






0
0
min


 t .



(91)
Figure 32 contains a plot of   kT   min  vs. k as the top plot. For reference,
the bottom curve is a plot of pulse-to-pulse phase change vs. k . As can be seen
the pulse-to-pulse phase change ranges between about -1000 and +1000
degrees while the phase variation, or phase error, over  is between -6×10-3
and +6×10-3 degrees. This says that   t  varies slowly over  , and thus, that
it will be reasonable to compute   t  at kT   min , or even kT rather than via the
more accurate form of Equation (88).
40
Figure 31 – Plot of vn  k , m  for a three scatterers at
x
n,
yn    23, 25  ,  0, 0  ,  23, 25  , with range extension and RCMC
Given this we can now examine   kT  to formulate a quadratic phase
correction scheme. We can write
  kT   2 f dg kT  4 rn  kT  
2

2 x02   y0  VkT  


 2 f dg kT 





2

2 y0VkT VkT   
2
 2  f dg kT   ro 



 
ro
ro  



4 ro


2y V
  f dg  0
 ro


VkT 
 kT  2
 ro

(92)
2
41
Figure 32 – Phase Change and Phase Error vs. Pulse Number
We recognize the first term of the last equality of Equation (92) as a
constant phase that we do nothing about. The second term is zero since, by
Equation (74), f dg  2Vy0   ro  . Finally, the third term is the quadratic phase
that we want to eliminate. It will be noted that this quadratic phase term is
exactly the same as the quadratic phase term in the CW problem. Thus, to
perform the quadratic phase correction we multiply each row of the RCMCed
signal space array by
vh  k   e
V2 
2
j 2 
 kT 
  ro 
.
(93)
We are now in a position to formulate an algorithm for creating a cross/downrange image.
42
2.10 ALGORITHM FOR CREATING A CROSS- & DOWN-RANGE IMAGE

We assume we have a sampled base-band signal of the form given by
Equation (84). Note: this signal has had the gross Doppler, f dg ,
removed.

Perform RCMC using the FFT methodology with the corrections given in
Equation (86). The RCMC is applied to all range cells for each pulse
(each k ).

Perform the quadratic phase correction by multiplying the returns for
each range cell by the vh  k  of Equation (93).

Take the FFT across pulses, for each range cell.

Transform the frequency and range delay axes of the output of the FFTs
to cross-range and down-range, and plot the image.
2.11 EXAMPLE 2
As an example, we extend EXAMPLE 1, Section 2.8. Table 2 is a repeat
of Table 1 with additions and modifications consistent with the cross- & downrange methodology.
Table 2 – Parameters Used in SAR EXAMPLE 2
Parameter
Value
Width of image area, w
50 m
Depth of image area, l
50 m
SAR wavelength, 
0.03 m
Aircraft velocity, V
50 m/s
Synthetic array length, L
600 m
Number of scatterers, N s
3
Waveform PRI, T
50 ms
Down-range resolution,  x
0.5 m
Center of Imaged Area  x0 , y0  (m)
(20000,200)
Scatterer locations,  xn , yn  (m)
(-23,0), (0,0), (23,0)
Scatterer powers, PSn (w)
1, 1, 1
43
As with EXAMPLE 1, we have K L  120 so that k goes from -120 to 120
and we transmit 241 pulses over a time period of -6 to 6 seconds. Given the
down-range resolution of 0.5 m we compute a pulse width of
p 
2 x
 3.33 ns .
c
(94)
From EXAMPLE 1, we recall that the cross-range resolution is also 0.5 m.
From the previous examples, we recall that we need to extend  max by 6 to 8 m
to account for the RCM of scatterers near the far down-range of the imaged
area. These extra range cells need to be trimmed before we make the image.
We also recall that, since our T is smaller than the minimum dictated by the
width of the imaged area our SAR image will need to be trimmed in cross-range
before we make the image.
When we formed the image for EXAMPLE 1, we used an FFT length that
was longer than the number of samples because we wanted a smooth linear
plot. Since we are only forming an image for this example, we can limit the FFT
length to the nearest power of two greater than 2 K L  1 . Since 2 K L  1 is 241, a
256 point FFT will suffice.
When I implemented the aforementioned algorithm, with the additional
steps indicated in the previous paragraphs, the image of Figure 33 was the
result.
As can be seen the three dots are about where they should be. The
center dot is at (0,0) and is fairly sharp. This is expected since our RCMC and
quadratic phase correction is based on a scatterer at the center of the imaged
area. The other two dots are somewhat smeared and are offset slightly in the
cross-range direction. The Offset is due to a residual Doppler and the smearing
is due to a residual quadratic phase.
Figure 34 contains an image that resulted when the three scatterers were
placed at (-23,23), (0,0) and (23,-23) m. Again the center dot is reasonably
sharp but the other two dots are offset in the cross-range dimension and
smeared in both the cross- and down-range dimensions. The cross-range offset
is due to the aforementioned residual Doppler and the smearing is due to the
residual quadratic phase. The down-range smearing is due to the imperfect
RCMC discussed in association with Figure 30 and Figure 31.
44
Figure 33 – Image for EXAMPLE 2
45
Figure 34 – Image for scatterers at (-23,23), (0,0) and (23,-23) m
2.12 AN IMAGE-SHARPENING REFINEMENT
We noted in the generation of Figure 33 that there was a slight skewing of
the upper and lower dots. Given that the skewing was in opposite directions at
the top and bottom we surmise that it is due to a frequency shift, and possibly
FM slope variation, that is dependent upon the down-range location of the
scatterer, xn . We want to examine this further.
For a scatterer at  x0  xn , y0  we have
rn  t  
 x0  xn    y0  Vt 
2
2
(95)
where we are temporarily using t  kT for convenience. We can manipulate this
as
rn  t  
 x0  xn 
 rn 
2
 y02  2 y0Vt  V 2t 2  rn2  2 y0Vt  V 2t 2
y0V
V2 2
t
t
rn
2rn
.
(96)
With this we can write the phase of vn  t  as (see Equation (88))
46
 2rn

2y V
  f dg  0
 rn
  
  t   2  f dg t  2rn  t     2 
 V2 2
t  .
t 
  rn 
(97)
During the quadratic phase removal step we essentially add
q  t   2
V2 2
t
 ro
(98)
to the above phase to get a corrected phase of
 2rn 
2 y0V   V 2 V 2  2 
c  t   2 
  f dg 

t  
t  .
 rn    rn  ro  
  
(99)
We want examine the linear phase, or frequency, term first. We can
write it as

 f  t   2  f dg 

2 y0V
 rn

t .

(100)
Recalling that f dg  2 y0V  ro we have
 f  t   2
2 y0V  1 1 
  t .
  ro rn 
(101)
Now,
rn 
 x0  xn 
2
 y02  x02  2 xn x0  xn2  y02
 ro2  2 xn x0  xn2  ro 
where we have made use of ro2
With this we can write
 f  t   2
(102)
xn x0
 ro  xn
ro
2 xn x0  xn2 , 2 xn x0
xn2 and ro  x0 since x02
2 y0V  xn 
 t
  ro2 
y02 .
(103)
where we have used ro rn  ro2 .
From Equation (103) we see that we have a frequency of
f  
2 y0V  xn 
 .
 ro  ro 
(104)
When the scatterer is at scene center, xn  0 and thus f  0 . That is, there is
no frequency offset. When xn  0 there will be a frequency offset, which will
lead to a cross-range offset.
47
To see if the frequency offset could be the cause of the skewing in Figure
33 we recall that cross-range position is related to frequency by (see Equation
(45))
y
 ro
f.
2V
(105)
With this we can write
y 
 ro
2V
f  
xn y0
.
ro
(106)
In our case ro  20, 000 m , y0  200 m and xn  25 m , and
y  
25  200
 0.25 m
20, 000
(107)
or ½ of a cross-range resolution cell, which is about the shift noted in Figure
33. This leads us to conclude that it might be a good idea to include a rangecell-dependent frequency correction to the quadratic phase correction. When
such a correction was included the image of Figure 35 was obtained. As can be
seen, the skewing is no longer present.
Figure 35 – Case of Figure 33 with Additional Doppler Correction
48
Very careful examination of Figure 35 reveals a slight cross-range
smearing of the upper and lower dots, relative to the center dot. From our
experience with stretch processing we postulate that this could be due to the
residual quadratic phase term of Equation (99).
We can write the residual quadratic phase term as
 V2 V2  2
2

 t      t .
  rn  ro 
q  t   2 
(108)
With approximations similar to the previous development we can write
  
x
2V 2  1 1  2V 2 xn
 o n .
  
  rn ro   ro ro
ro
(109)
The result of applying a correction to remove this residual quadratic phase is
contained in Figure 36. Very careful examination of this figure reveals that the
all three dots are equally sharp in cross range.
Figure 36 – Case of Figure 35 with Added Residual Quadratic Phase
Correction
49
Figure 37 contains an image equivalent to Figure 34 with the
aforementioned residual frequency and quadratic phase corrections included.
As can be seen, the dots of Figure 37 seem to be slightly more focused than
those of Figure 34. However, the upper and lower dots are still smeared in the
down-range direction. As discussed earlier, this down-range smearing is
caused by the fact that the RCM is due to cross-range position of the scatterer,
whereas the RCMC is based on a scatterer at zero cross-range. Wong, see
footnote 1, presents an alternate RCMC algorithm that corrects this problem.
We will not discuss it here. The reader is referred to Wong’s book.
Figure 37 – Case of Figure 34 with Additional Doppler and Quadratic Phase
Correction
2.13 CLOSING REMARKS
The discussions presented in this chapter are very preliminary when
compared to the body of literature on SAR. The technique presented is a bare,
basic image formation method, with the exception of the image refinement
technique of Section 2.12. There are several texts that discuss other image
formation and sharpening techniques. Many of these provide sharper images
but are also more difficult to implement, and run slowly when compared to the
technique discussed herein.
The technique discussed herein is applicable to both stripmap and
spotlight SAR for the case where the SAR platform is moving in a straight line.
50
There is another class of spotlight SAR termed circular SAR. In this type of
SAR, the SAR platform follows a circular path relative to some point in the
imaged area. The techniques developed in this chapter are not applicable to
this type of SAR because the RCMC technique developed herein can’t, to my
knowledge, be extended to the circular SAR case. The most common
techniques applicable to circular SAR appear to be a matched filter technique
and a technique termed back projection, both of which require a large amount
of computations and computer time. These techniques are also applicable to
the type of SAR considered in this chapter.
In the derivations of this chapter, it was (somewhat unrealistically)
assumed that the SAR platform was flying in the x-y plane; i.e., at an altitude of
zero. The extension to a non-zero, but constant, altitude is straightforward. In
essence, when the non-zero altitude case is considered, the image that results
is in slanted plane. The points in this slanted plane can be mapped to the
ground by a coordinate transformation. This simplified approach makes the
assumption that SAR antenna length (distance the SAR platform travels) and
the dimensions of the imaged area are small compared to the slant range to the
imaged area. If this is not the case, a somewhat more complicated method of
accounting for SAR platform altitude must be used.
51