```강원대학교 지구물리학과 대학원 ‘영상레이더’. 2005년 1학기. 담당교수 이훈열
Focusing by
Rectangular Algorithm
a
H
p
x
y
a
References:
1. Curlander and McDonough, 1991. Synthetic Aperture Radar Systems and Signal
Processing, John Wiley &amp; Sons, Chapter 4.
2. ASF Scientific User’s Manual
1. Wave

T
Spatial measurement of wave at a time (photo)
Time measurement of wave at a point
Description of wave:
 

s(r , t )  Ae j  A exp[ j (k  r  t )]

r : location vector
t : time
A : amplitude of the wave
j  1
 : phase of the wave
Frequency: f 
Period: T 

2
1
f
Angular frequency:   2f

k : wave number vector

k  (k x , k y , k z ) in Cartesian coordinate.
Wave number: k  2 / 
k x  2 / x , k y  2 /  y , k z  2 / z
k  k x2  k y2  k z2
2. Range Compression
As the vehicle moves along its path, the radar transmits a linear-FM, or “chirp”,
pulse
s(t )  exp[ j 2 ( f c t  Kt 2 / 2)],
t  p 2
(1)
where, frequency (time derivative of phase) f  f c  Kt , which is a linear function of
time over the pulse duration  p with chirp rate K .
The chirp bandwidth (in range direction) is BR  K p since the frequency starts at
f c  K p / 2 , sweeps through all intermediate frequencies, and ends at f c  K p / 2 .
The multi-pulse real transmitted signal is then
p(t )   s(t  nT p ) ,
(5)
n
where T p is the pulse repetition period and the sum includes all pulses for which the
target is in the radar beam.
At an arbitrary time t, the radar is at some slant range R (t ) from a target point with
image coordinates ( x c , Rc ) at the radar beam centre. The received pulse train is then
v(t )   a n s[t  nT p  2 R(t ) / c] , t  nTp  2 R(t ) / c 
n
p
2
.
(6)
Range compression is to correlate the received pulse with s * (t ) (equivalently
matched filtering with s * (t ) ) as:

g (t )   s* (t   t )v(t )dt 

  an  s* (t   t ) s t   nTp  2 R(t ) / c dt 

n

(7)
As shown in the figure below, the variation of R (t ) within a chirp pulse is very small,
so that we can assume that R(t )  Rn , the distance between the sensor and the target at
chirp centre time. This means that the slant range R (t ) can be assumed constant during
the time of one pulse width. This assumption enables the general two-dimensional
compression problem to be de-coupled into a sequence of two one-dimensional
compression operations, one in fast time (range) and the other in slow time (azimuth).
Since slow time measures a coordinates orthogonal to fast time, this process sequence is
Vs , Satellite Velocity
t  s , slow time
Point target
( xc , Rc )
Rn
R (t )
nth chirp pulse
R(t )  Rn
Emitted chirp signal centre at nT p
Received signal centre at t n  nTp  2 Rn / c , fast time
called the rectangular algorithm. The centre of the returned chirp signal from a point
target is at t n , where
tn  nTp  2 Rn / c .
(8)
The matched filtering becomes:

g (t )   an  s* (t   t ) s(t   tn )dt 
n

(9)
The actual integration is not  ,  as there is a time limit in both s (t ) and v (t ) .
The range of effective integration also depends on t . The range of integration can be
obtained from the diagram below.
t  tn
2
t  tn  p  t  tn

2
2
t  tn  p  t  tn

2
2
v(t )  s(t 'tn )
s (t 't )
tn
tn 
p
t
2
t
t
tn 
p
p
2
t
p
2
2
Considering carefully the limit of the integration gives:
1) If t  tn   p , then
t tn  p  t tn

2
 p  t tn
n

2
2
g (t )   an t 2t
n
 


exp  j 2 f c (t   t )  K (t   t ) 2 / 2


 exp j 2 f c t   tn   K t   tn  / 2 dt 
2
2) else g (t )  0 .
(10)
We only evaluate the first case.



 
g (t )   an exp  j 2  f ct  Kt 2 / 2  exp j 2  f ctn  Ktn2 / 2

n
t tn  p  t tn

2
2
t tn  p  t tn

2
2



  

exp  j 2 f ct   K (t 2  2tt ) / 2 exp j 2 f ct   K (t 2  2tnt ) / 2 dt 
 
  an exp j 2 f c (t  tn )  K (t  t ) / 2
2
2
n
n
Let t  
t tn  p  t tn

2
2
t tn  p  t tn

2
2

exp  j 2K (t  tn )t dt 
(11)
t  tn
 t  , then
2
 
g (t )   an exp j 2 f c (t  tn )  K (t 2  tn2 ) / 2
n

 p  t tn
2
 p  t tn

2
t  tn 

exp  j 2K (t  tn )(t  
) dt 
2 

 p  t tn
  an exp  j 2f c (t  tn )  p 2t tn exp  j 2K (t  tn )t dt 

n
(12)
2
The integration gives the sinc function:
g (t )   an exp  j 2f c (t  tn )
sin  p  t  tn K (t  tn )
n
(13)
K (t  tn )
Remember it holds if t  tn   p otherwise g (t )  0 . The signal near to the maximum
power of the sinc function ( t  tn   p ) can be rewritten as:
g (t )   an exp  j 2f c (t  tn )
n
sin  pK (t  tn )
(14)
K (t  tn )
Dropping the summation, the range compressed result of nth pulse, g n (t ) can be
written as,
g n (t )  an exp  j 2f c (t  tn )
sin  pK (t  tn )
(15)
K (t  tn )
In other form using tn  nTp  2 Rn / c ,   2f and c  f ,
g n (t )  an exp( j ct ) exp(  j 2nTp ) exp(  j 4Rn /  ) 
sin K p (t  tn )
K (t  tn )
(16)
We assume synchronization of the detailed pulse waveform, that is the radar is coherent.
In effect, the term exp(  j 2nTp ) can be neglected as it is the same for all pulses. We
obtain:
g n (t )  an exp( jct ) exp(  j 4Rn /  ) 
sin K p (t  tn )
K (t  tn )
(17)
The envelope of this has a 3 dB width (half the maximum power)
t 
1
1
.

K p BR
(18)
This is the slant range time resolution of the SAR system, which corresponds to slant
range spatial resolution (or, range resolution in short) as:
R  c / 2BR ,
where
(19)
BR  K p is the range chirp bandwidth. Therefore, a large chirp bandwidth
gives enhanced range resolution for a SAR system. Note that the range resolution holds
with positive K,
‘up-chirp’. In case of ‘down-chirp’, K should be replaced with K .
3. Azimuth Compression
After range compression, the data is transformed into the azimuth frequency domain
via FFT. There are two important parameters for azimuth compression: Doppler
centroid frequency f Dc and Doppler rate f R . They can be derived from the satellite
state vectors including satellite’s position and velocity vectors. Sometimes, however,
those values need to be recalculated more precisely, which can be done using the data
itself. The Doppler centroid frequency f Dc is determined from the data via the
clutterlock procedure. The Doppler rate f R can be determined by an autofocus
procedure using the azimuth subaperture correlation method. Multiple pulse repetition
frequency (PRF) ambiguity of Doppler centroid, which is important for range migration,
can be solved by the range subaperture correlation method. Azimuth spectral filtering
can be applied on two SAR signals forming an interferometric pair to compensate
decorrelation from different Doppler centroids of the two images. The signal is then
azimuth compressed using the same matched filtering as the case of range compression.
A Hamming filter can then be applied to reduce the sidelobe effect similar to the one
during the range compression. The following is the detailed procedures for azimuth
compression.
Azimuth Matched Filtering
The angular carrier frequency  c from the range-compressed signal can be
eliminated by demodulation. Selecting the value at t  t n that gives maximum g n (t )
by precise sampling and representing t as slow time s gives:
gˆ (s xc , Rc )  exp[  j 4R(s) /  ] .
(20)
The range function R(s) can be expanded as a Taylor series around sc  xc / Vs , the
slow time at which the centre of the radar beam crosses the target, where Vs is the
speed of the radar platform relative to the target point.
 ( s  s ) 2 / 2   .
R( s )  Rc  R c ( s  sc )  R
c
c
(21)
The Doppler frequency, by definition, is the time rate of phase  (s )
 ( s)   4 R( s)  ,
(22)
f D ( s)   2   2 R ( s)  ,
(23)
( s )  .
fD ( s )   2   2 R
(24)
We can define the Doppler centroid and Doppler rate at s  sc as
f Dc   2 R c  ,
(25)
  ,
f R   2R
c
(26)
R( s)  Rc  (f Dc / 2)( s  s c )  (f R / 4)( s  s c ) 2 .
(27)
so that
The range-compressed signal can be expressed in terms of f Dc and f R as


gˆ (s sc , Rc )  exp(  j 4Rc /  ) exp j 2 [ f Dc (s  sc )  f R (s  sc ) 2 / 2] ,
s  sc  S 2 ,
(28)
where S is the azimuth integration time. This is a linear FM wave with centre frequency
f Dc and frequency rate f R . The azimuth compression is to compute the correlation

 (s)   h1 (s  s) gˆ (s)ds .

(29)
Similar to range compression, the azimuth compression can be realised approximately
using a correlator function
h 1 ( s )  exp[  j 2 ( f Dc s  f R s 2 2)] .
(30)
The output of azimuth compression is then
 ( s | sc , Rc )  exp( j 2f D s) exp(  j 4Rc /  ) 
c
sin f R S ( s  sc )
f R ( s  sc )
(31)
Derivation of this equation is similar to that of range compression by the following
replacement of variables:
Range Compression
Azimuth Compression
t : fast time
s : slow time
t
s
t 
s
t n : two-way travel time
sc : beam centre time
 p : pulse duration
S : coherent integration time
f c : carrier frequency
f Dc : Doppler centroid
K : chirp rate
f R : Doppler rate
The peak of this pulse occurs at s  sc , the target azimuth location. The 3 dB width of
this pulse, that determines the azimuth time resolution, is
s  1/ f R S  1 BD
where
(32)
BD  f R S is the Doppler bandwidth. The azimuth spatial resolution (or,
azimuth resolution) is then
x  Vss  Vs BD  Vs f R S .
(33)
From the simple geometry of a radar antenna, with physical length La along track,
the nominal beam width is  H   La so that any particular target point at range Rc
is illuminated for a nominal time S  Rc Vs La .
For the geometry of the figure below, given a squint angle  s , the Doppler
parameters can be geometrically determined as
R 2 ( s)  Rc2  Vs2 ( s  sc ) 2  2 RcVs ( s  sc ) sin  s
(34)
R( s)  Rc  Vs2 ( s  sc ) 2 / 2 Rc  Vs ( s  sc ) sin  s
(35)
f Dc  (2Vs  ) sin  s ,
(36)
f R   2Vs2 Rc .
(37)
In case of a real aperture radar, azimuth resolution changes inversely with the physical
length of its antenna as x  Rc La . For SAR, the Doppler bandwidth is therefore
BD  2Vs La and the system azimuth resolution is
x  La 2 .
x changes in proportion to the physical length of SAR antenna.
(38)
(39)
With intensive signal compression processing, high azimuth resolution can be achieved
with a SAR in a significantly smaller physical dimension than that in a real aperture
Vs
sc
s0
s
Rc
R(s)
s
( sc , Rc )
Appendix A.
Dirac Delta Function
 ( x)  0, x  0






 ( x)dx  1
f ( x) ( x)  f (0)
No such function exists.
Sinc function – one approximate of Dirac Delta Function
sin nx
1
 n ( x) 

x
2



n
e
ixt
dt
n
 n ( x)dx  1

lim  
n   
n
sinc( x) 
( x) f ( x)dx  f (0)
sin x
x
Appendix B.
SAR Focusing – Point Target
azimth
range
original
After migration
After range compression
After azimuth compression
```