Digital Signal Processing Reference
In-Depth Information
TABLE 4.6
Fourier Coefficients for Example 4.7
k
H(jk
v
0
)
C
kx
C
ky
ƒ
C
kx
ƒ
ƒ
C
ky
ƒ
0
1
2
2
2
2
1
2
4
p
∠ -90°
4
1
2
∠ -45°
2
∠ -135°
1.273
0.900
p
2
1
2
4
3p
∠ -90°
4
3
10
∠ -71.6°
10
∠ -161.6°
0.424
0.134
3p
2
1
2
4
5p
∠ -90°
4
5
26
∠ -78.7°
26
∠ -168.7°
0.255
0.050
5p
2
Now,
1
1 + jk
=
1
1 + k
2
∠tan
-1
(-k).
H(jkv
0
)
|
v
0
= 1
=
2
For
k
odd, from (4.39),
1
c
4
pk
∠ -p/2 - tan
-1
(k)
C
ky
= H(jkv
0
)C
kx
=
d
1 + k
2
2
and
C
0y
= H(j0)C
0x
= (1)
(2) = 2.
Table 4.6 gives the first four nonzero Fourier coefficients of the output
y
(
t
). The Fourier co-
efficients for the output signal decrease in magnitude as for large
k
, since, for this case,
both and decrease as 1/
k
. Hence, the system attenuates the high harmonics
relative to the low harmonics. A system with this characteristic is called a
low-pass system
.
A MATLAB program that implements the complex calculations of Table 4.6 is
1/k
2
ƒC
kx
ƒ
ƒH(jkv
0
) ƒ
n = [0 1];
d = [1 1];
w = 1:2:5;
h = freqs (n, d, w);
ckx = 4 ./ (pi*w) .* exp(-j*pi/2);
cky = h .* ckx;
ckymag = abs(cky);
ckyphase = angle(cky)*180/pi;
results: ckymag = 0.9003 0.1342 0.0499
ckyphase = -135.0000 -161.5651 -168.6901
The period followed by a mathematical operator indicates the operation on the two arrays,
element by element. These symbols must be bracketed by spaces.
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