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Differential PCM (DPCM) 611
the signal-to-quantization noise ratio (SQNR) in dB as
( ∑ )
N
n=1
SQNR = 10 log s2 (n)
10 ∑ N
n=1 (s(n) − s q(n)) 2 .
For different b-bit quantizers, systematically determine the value of µ
that maximizes the SQNR. Also plot the input and output waveforms
and comment on the results.
12.2 DIFFERENTIAL PCM (DPCM)
In PCM each sample of the waveform is encoded independently of all the
other samples. However, most signals, including speech, sampled at the
Nyquist rate or faster exhibit significant correlation between successive
samples. In other words, the average change in amplitude between successive
samples is relatively small. Consequently, an encoding scheme that
exploits the redundancy in the samples will result in a lower bit rate for
the speech signal.
A relatively simple solution is to encode the differences between successive
samples rather than the samples themselves. Since differences between
samples are expected to be smaller than the actual sampled amplitudes,
fewer bits are required to represent the differences. A refinement
of this general approach is to predict the current sample based on the
previous p samples. To be specific, let s(n) denote the current sample of
speech and let ŝ(n) denote the predicted value of s(n), defined as
ŝ(n) =
p∑
a (i) s (n − i) (12.8)
i=1
Thus ŝ(n) isaweighted linear combination of the past p samples, and
the a (i) are the predictor (filter) coefficients. The a (i) are selected to
minimize some function of the error between s(n) and ŝ(n).
A mathematically and practically convenient error function is the sum
of squared errors. With this as the performance index for the predictor,
we select the a (i) tominimize
[
△ N 2
∑ N∑
p∑
E p = e 2 (n)= s(n) − a (i) s (n − i)]
(12.9)
n=1
n=1
=r ss (0) − 2
i=1
p∑
a (i) r ss (i)+
i=1
p∑
i=1 j=1
p∑
a (i) a (j) r ss (i − j)
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