The amplitude, A(f
k
, τ
n
), is related to the radar cross section of the target while the phase is dep endent on the
frequency of each sample and on the differential range, ∆R(τ
n
), given by
∆R(τ
n
) = d
a
0
(τ
n
) − d
a
(τ
n
). (6)
Equation (5) assumes that each pulse is motion compensated such that a scatterer at the scene origin will have
zero phase for all f
k
and τ
n
. The actual receiver output is thus the sum of the contributions of all scatterers in
the scene.
The frequency samples, {f
k
}, have a minimum value denoted by f
1
, a median value denoted by f
c
, a maximum
value denoted by
f
K
, and a step size denoted by ∆
f
. The frequency step size is inversely related to the maximum
alias-free range extent of the image, W
r
, by
W
r
=
c
2∆f
. (7)
Thus, the frequency step size is chosen to match the size of the scene to be imaged. The total bandwidth, B, of
the received pulse is B = (K − 1)∆f. The range resolution is thus
δ
r
=
c
2B
=
c
2(K − 1)∆f
. (8)
In a similar manner as above, the azimuth angle traversed during the synthetic aperture determines the
cross-range resolution, and the azimuth angle from pulse to pulse determines the maximum alias-free cross-range
extent of the image, W
x
. Given an azimuth step size of ∆θ,
W
x
=
λ
min
2∆θ
(9)
where λ
min
is the minimum wavelength such that λ
min
= c/f
K
. The total azimuth angle, θ
a
, traversed during
the synthetic aperture is θ
a
= (N
p
− 1)∆θ. Thus, the cross-range resolution, δ
x
, is given by
δ
x
=
λ
c
2θ
a
=
λ
c
2(N
p
− 1)∆θ
(10)
where λ
c
is the center wavelength such that λ
c
= c/f
c
.
To utilize the imaging algorithms described in this paper, one can select any arbitrary pixel locations. How-
ever, one should be careful when selecting these locations to avoid aliasing of the image or the frequency support.
The pixel spacing should be finer than the resolution defined in Equations (8) and (10) and the overall scene size
should be less than the maximum scene size defined in Equations (7) and (9).
3. MATCHED FILTER ALGORITHM
The most straightforward method for forming a SAR image is to perform a matched filter. One can build the
matched filter to any kind of scatterer, but here we will assume an isotropic point scatterer. The received signal
from a point scatterer at location r is given in Equation (5). An isotropic scatterer will have a constant amplitude
and thus A(f
k
, τ
n
) = A
0
. Therefore, the matched filter response, denoted by I(r), at location r is given by
I(r) =
1
N
p
K
N
p
∑
n=1
K
∑
k=1
S(f
k
, τ
n
) exp
(
+j4πf
k
∆R(τ
n
)
c
)
= A
0
, (11)
assuming a single scatterer in the scene.
To form an image using this method, Equation (11) is applied for each pixel in the image. This requires
calculation of the differential range, ∆R(τ
n
), for every pixel for every pulse. The algorithm has a computational
complexity of O(N
4
) for 2D images, which makes it impractical for most applications. However, Equation (11)
forms the basis for the derivation of the backprojection algorithm in Section 4. MATLAB code for the matched
filter algorithm is provided in Appendix A.1.