Digital Signal Processing Reference
In-Depth Information
24” (610 mm) diameter primary parabolic reflector and a 1
75” (44.5 mm) diameter
secondary hyperbolic reflector is mounted to the front of the radiometer receiver.
The position of the hyperbolic secondary is variable but fixed in our system such
that the effective focal length is 6” (152.4 mm) (i.e., the system is
f
/4) and the
image distance is 5.81” (147.6 mm).
One can model the 94-GHz imaging system as a linear, spatially incoherent,
quasi-monochromatic system [85]. The intensity of the detected image can be
represented as a convolution between the intensity of the image predicted by the
geometrical optics with the system point spread function [66]
.
2
ii
(
x
,
y
)
|
i
(
x
,
y
)
|
=
o
g
(
x
,
y
)
∗∗
h
(
x
,
y
)
,
(3.14)
where
represents
the inverted, magnified image of the object that a ray-optics analysis of the system
predicts.
The second term in Eq. (
3.14
),
h
∗∗
represents two-dimensional convolution. The function
o
g
(
x
,
y
)
, is the incoherent point spread function
(PSF) that accounts for wave propagation through the aperture
(
x
,
y
)
p
−
2
1
x
f
,
−
y
(
,
)=
,
h
x
y
(3.15)
(
λ
f
)
4
λ
λ
f
where
p
(
x
/
λ
f
,
y
/
λ
f
)
is the coherent point spread function. The function
p
(
x
,
y
)
is
the inverse Fourier transform of the system pupil function
P
(
u
,
v
)
,
FT
−
1
p
(
x
,
y
)=
[
P
(
u
,
v
)]
.
Assuming that object and image are
N
×
N
arrays, one can then rewrite Eq. (
3.14
)
in matrix notation as
i
=
Ho
g
,
(3.16)
where
i
and
o
g
are
N
2
×
1 lexicographically ordered column vectors representing
, respectively, and
H
is the
N
2
N
2
matrix that
the
N
×
N
arrays
ii
(
x
,
y
)
and
o
g
(
x
,
y
)
×
models the incoherent point spread function
h
.
Displacement of an object from the nominal object plane of the imaging system
introduces a phase error in the pupil function that increases the width of a point
response and produces an out of focus image. The system's depth-of-field is defined
as the distance in object space over which an object can be placed and still produce
an in-focus image.
For a 94 GHz imager with an aperture diameter
D
(
x
,
y
)
=
24” and object distance
d
o
=
180” (4572 mm),
DoF
≈
17
.
4” (442 mm) which ranges from 175
.
2” (4450.1
mm) to 192
6” (4892 mm).[85]
In [85], it was demonstrated how to extend the DoF using a cubic phase element
in conjunction with post-detection processing. The cubic phase element
P
c
(
.
u
,
v
)
is
))
rect
u
v
ξ
v
P
c
(
u
,
v
)=
exp
(
j
θ
c
(
u
,
v
ξ
u
,
,
(3.17)
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