Department of Electrical and Computer Engineering
Digital Speech Processing
Homework No. 8
Problem 1
– The uniform probability density function is defined as
p
(
x
) =
1
Δ

x

<
Δ
/
2
0
otherwise
Find the mean and variance of the uniform distribution.
Problem 2
– Consider the Laplacian probability density function
p
(
x
) =
1
√
2
σ
x
e

√
2

x

/σ
x
Find the probability that

x

>
4
σ
x
.
Problem 3
– A speech signal is bandlimited by an ideal lowpass filter, sampled
at the Nyquist rate, quantized by a uniform
B
bit quantizer, and converted back
to an analog signal by an ideal
D/A
converter, as shown in the figure above.
Define
y
[
n
] =
x
[
n
] +
e
1
[
n
] where
e
1
[
n
] is the quantization error. Assume that
the quantization step is Δ = 8
σ
x
/
2
B
and that
B
is large enough so that we can
assume:
1.
e
1
[
n
] is stationary
2.
e
1
[
n
] is uncorrelated with
x
[
n
]
3.
e
1
[
n
] is a uniformly distributed white noise sequence
We have seen that, under these conditions, the signaltoquantizing noise ratio
is:
1
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SNR
1
=
σ
2
x
σ
2
e
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 Fall '08
 Rabiner,L
 Digital Signal Processing, Signal Processing, 2bit, Department of Electrical and Computer Engineering, quantization error

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