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608 Chapter 12 APPLICATIONS IN COMMUNICATIONS
Speech signals transmitted over telephone channels are usually limited
in bandwidth to the frequency range below 4 kHz. Hence the Nyquist rate
for sampling such a signal is less than 8 kHz. In PCM the analog speech
signal is sampled at the nominal rate of 8 kHz (samples per second), and
each sample is quantized to one of 2 b levels, and represented digitally by
a sequence of b bits. Thus the bit rate required to transmit the digitized
speech signal is 8000 b bits per second.
The quantization process may be modeled mathematically as
˜s(n) =s(n)+q(n) (12.1)
where ˜s(n) represents the quantized value of s(n), and q(n) represents the
quantization error, which we treat as an additive noise. Assuming that a
uniform quantizer is used and the number of levels is sufficiently large,
the quantization noise is well characterized statistically by the uniform
probability density function,
p(q) = 1 ∆ , −∆ 2 ≤ q ≤ ∆ 2
(12.2)
where the step size of the quantizer is ∆ = 2 −b . The mean square value
of the quantization error is
E(q 2 )= ∆2
12 = 2−2b
12
(12.3)
Measured in decibels, the mean square value of the noise is
( ) ( )
∆
2
2
−2b
10 log =10log = −6b − 10.8 dB (12.4)
12
12
We observe that the quantization noise decreases by 6 dB/bit used
in the quantizer. High-quality speech requires a minimum of 12 bits per
sample and hence a bit rate of 96,000 bits per second (bps).
Speech signals have the characteristic that small signal amplitudes
occur more frequently than large signal amplitudes. However, a uniform
quantizer provides the same spacing between successive levels throughout
the entire dynamic range of the signal. A better approach is to use
a nonuniform quantizer, which provides more closely spaced levels at the
low signal amplitudes and more widely spaced levels at the large signal
amplitudes. For a nonuniform quantizer with b bits, the resulting quantization
error has a mean square value that is smaller than that given
by (12.4). A nonuniform quantizer characteristic is usually obtained by
passing the signal through a nonlinear device that compresses the signal
amplitude, followed by a uniform quantizer. For example, a logarithmic
compressor employed in U.S. and Canadian telecommunications systems,
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