Digital Signal Processing Reference
In-Depth Information
1. Zero-pad h to get h
z
= [h
o
h
1
h
2
000000]andx to get x
z
= [1291013869
0 0].
2. Compute: H
z
= fft(h
z
), X
z
= fft(x
z
), and Y
z
= X
z
H
z
.
3. Find
y
z
= ifft(Y
z
)
to
get
y
z
¼
y
¼
6
:
88
:
51011
:
59
:
77
:
583
:
41
:
2
0 1 2 3
4 567 8
Consider only the central 5 values of {y(n)}, i.e., items no. 2, 3, …,6.
Note: the coefficients {h(n)} can be adaptive based on change of market variables.
Tutorial 41
Q:
Find x(n)if
(A) X
ð
z
Þ¼
3
z
2
;
(B) X
ð
z
Þ¼
1
z
2
3z
þ
2
:
Solution:
(A) X
ð
z
Þ¼
3
z
2
¼
3z
1
z
2
¼
3z
1
R
ð
z
Þ:
z
r(n) = 2
n
From
Tables
we
find:
u(n)
and
z
1
R
ð
z
Þ
ZT
r
ð
n
1
Þ¼
2
n
1
u
ð
n
1
Þ
.
Hence we have x(n) = 3r(n - 1) = 3
2
n-1
u(n - 1).
(B) z
2
3z
þ
2
¼
0
)
p
1
;
2
¼
ð
3
Þ
p
ð
3
Þ
2
4
ð
1
Þð
2
Þ
¼
1
;
2
:
2
ð
1
Þ
1
ð
z
1
Þð
z
2
Þ
¼
a
1
z
1
þ
b
X
ð
z
Þ¼
ð
z
p
1
Þð
z
p
2
Þ
¼
(where we used partial fraction
z
2
expansion).
To find a:
ð
z
1
Þð
z
2
Þ
¼
a
þ
b
z
1
z
1
1. Multiply by z
1 :
z
2
2. Put z
¼
1 :
ð
1
2
Þ
¼
a
þ
0
!
a
¼
1.
Similarly we find b = 1.
1
X
ð
z
Þ¼
1
z
1
þ
1
z
z
1
þ
z
1
z
z
2
:
z
2
¼ð
1
Þ
z
1
)
From Tables and (A) above we find:
x
ð
n
Þ¼
u
ð
n
1
Þþ
2
n
1
u
ð
n
1
Þ¼½
2
n
1
1
u
ð
n
1
Þ:
Note: Normally in such questions, it is better to expand X(z)/z rather than X(z).
Applying this to Q2 above, we get x
ð
n
Þ¼
2
d
ð
n
Þþ½
2
n
1
1
u
ð
n
Þ:
Show that the
two answers are equivalent.
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