Fourier Transforms in Radar and Signal Processing 2 doc
Fourier Transforms in Radar And Signal Processing_2 pdf
... y)* dy (2. 24)
28 Fourier Transforms in Radar and Signal Processing
4. ͐
∞
−∞
sinc (x − m) sinc (x − n) dx =
␦
mn
Using the result in item 3 above, if m = n the integral is
͵
∞
−∞
sinc
2
(x − n) ... from the rules. Writing
out Rule 7 using the definitions of Fourier transform and convolution [ (2. 1)
and (2. 15)] gives
26 Fourier Transforms in Radar and Si...
Fourier Transforms in Radar And Signal Processing_3 pot
... of the
signum function used in obtaining P2a, and the comb and rep transforms.
Figure 2B.1 Contour for integral.
46 Fourier Transforms in Radar and Signal Processing
Figure 3.8 Asymmetric trapezoidal ... h
ͩ
t −
T
2
ͪ
(3.9)
Figure 3.6 Rising edge of width
.
52 Fourier Transforms in Radar and Signal Processing
Figure 3.13 Power spectra for rect and e...
Fourier Transforms in Radar And Signal Processing_4 ppt
... the information in the waveform is retained
by sampling it at a rate 2rW, where r is given in (4.13) above.
56 Fourier Transforms in Radar and Signal Processing
Using T
0
and ⌬T as given in Figure ... for t > 0
(t ∈ޒ) (3 .21 )
72 Fourier Transforms in Radar and Signal Processing
Figure 4.7 Maximum sampling rate.
2f
u
/k = 2( k +
␣
)W /k ≤ 2W ′ 2( k −...
Fourier Transforms in Radar And Signal Processing_5 doc
... sampling with a finite window width on a high IF may require
82 Fourier Transforms in Radar and Signal Processing
above, in particular from the point of view of finding the minimum sampling
rate needed ... discussed
88 Fourier Transforms in Radar and Signal Processing
v(t) = cos 2
f
0
t + i sin 2
f
0
t = exp 2
if
0
t
and so V ( f ) =
␦
( f − f
0
)...
Fourier Transforms in Radar And Signal Processing_6 ppt
... by
98 Fourier Transforms in Radar and Signal Processing
Figure 5.8 FIR interpolation weights with oversampling.
Figure 5.9 Trapezoidal spectral gate.
1 02 Fourier Transforms in Radar and Signal Processing
Figure ... exp
ͫ
−
2
√
t
2
ͬ
= exp (−4
2
2
t
2
) (5.48)
110 Fourier Transforms in Radar and Signal Processing
p =
͵
∞
−∞
|...
Fourier Transforms in Radar And Signal Processing_7 docx
... =
∑
∞
n
=
0
(−1)
n
(
y)
2n
(2n + 1)!
(6 .23 )
snc
1
( y) =
∑
∞
n
=
0
(−1)
n
2n(
y)
2n
−
1
(2n + 1)!
=
∑
∞
n
=
1
(−1)
n
2n(
y)
2n
−
1
(2n + 1)!
(6 .24 )
snc
2
( y) =
∑
∞
n
=
0
(−1)
n
2n(2n − 1)(
y)
2n
−
2
(2n + ... 1)!
=
∑
∞
n
=
1
(−1)
n
2n(2n − 1)(
y)
2n
−
2
(2n + 1)!
(6 .25 )
In general, we can put
snc
r
( y) =
∑
∞
n
=
[(r
+
1) /2]
(−1)
n
2n!(
y)
2n
−
r
(2n − r)!(2n + 1)...
Fourier Transforms in Radar And Signal Processing_8 pdf
... ramp
2
( f /F ) with
rect[2f /(1 + a)] is [(1 + a) /2]
2
ramp
2
[2f /(1 + a)], and, again using (6.18),
the transform is given by
c
(t) =−
ͩ
1 + a
2
ͪ
2
snc
2
ͫ
(1 + a)Ft
2
ͬ
sinc
ͫ
(1 − a)Ft
2
ͬ
(6.47)
Using ... taps, 20 % oversampling; and (b) four
taps, 50% oversampling.
156 Fourier Transforms in Radar and Signal Processing
Figure 6.15 Difference beam ga...
Fourier Transforms in Radar And Signal Processing_9 doc
... raised cosine weighting (solid line).
180 Fourier Transforms in Radar and Signal Processing
and the weight function, given by the Fourier transform of (7 .28 ), is
a(x) = (u
0
/U ) comb
1/U
[sinc (u
0
x) ... ]
ͮ
(7.9)
168 Fourier Transforms in Radar and Signal Processing
Figure 7.3 Beam patterns for uniform linear array with raised cosine shading: (a) u -space;...
Fourier Transforms in Radar And Signal Processing_10 pot
... 127
coefficients, 119, 121 gain response against frequency offset,
156 Gaussian, 120
for interpolation, 91, 109with narrowband weights, 154
196 Fourier Transforms in Radar and Signal Processing
Least ... 125
resampling and, 120 21
rep, 17–18
spectrum independent, 90–107
repeated, overlapping, 169
summary, 122 23
sinc, 3, 13–15, 125
worst case for, 93
sketches, 4
Inverse...
Fourier Transforms in Radar And Signal Processing_1 ppt
... exp ( 2
ixy) dx =
͵
1 /2
−
1 /2
exp ( 2
ixy) dx
=
exp ( 2
ixy)
2
iy
|
1 /2
−
1 /2
=
exp (−
iy) − exp (
iy)
2
iy
=
−2i sin (
y)
2
iy
= sinc ( y)
P3b:
From P3a and R4, using rect ... transforms.
Figure 2B.1 Contour for integral.
32 Fourier Transforms in Radar and Signal Processing
This is in the form of a Fourier series, with period 1/X = Y...