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Department of Electrical and Computer Engineering
Digital Speech Processing
Homework No. 6
Problem 1
The complex cepstrum, ˆ
x
[
n
], of a sequence
x
[
n
] is the inverse Fourier transform
of the complex log spectrum
ˆ
X
(
e
jω
) = log

X
(
e
jω
)

+
j
arg[
X
(
e
jω
)]
Show that the cepstrum
c
[
n
], deﬁned as the inverse Fourier transform of the log
magnitude, is the even part of ˆ
x
[
n
]; i.e., show that
c
[
n
] =
ˆ
x
[
n
] + ˆ
x
[

n
]
2
Problem 2
A linear timeinvariant system has the transfer function:
H
(
z
) = 8
1

4
z

1
1

1
6
z

1
(a) Find the complex cepstral coeﬃcients,
ˆ
h
[
n
], for all
n
.
(b) Sketch
ˆ
h
[
n
] versus
n
for the range

10
≤
n
≤
10.
(c) Solve for the (real) cepstrum coeﬃcients,
c
[
n
], for all
n
.
Problem 3
Consider a
ﬁnite length
minimum phase sequence
x
[
n
] with complex cepstrum
ˆ
x
[
n
], and a sequence
y
[
n
] =
α
n
x
[
n
]
with complex cepstrum ˆ
y
[
n
].
(a) If 0
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 Fall '08
 Rabiner,L

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