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Differential PCM (DPCM) 613
digits and transmitted over the channel to the receiver. The quantized
error ẽ(n) isalso added to the predicted value ̂˜s(n) toyield ˜s(n).
At the receiver the same predictor that was used at the transmitting
end is synthesized, and its output ̂˜s(n) isadded to ẽ(n) toyield ˜s(n). The
signal ˜s(n) isthe desired excitation for the predictor and also the desired
output sequence from which the reconstructed signal ˜s (t) isobtained by
filtering, as shown in Figure 12.3b.
The use of feedback around the quantizer, as described, ensures that
the error in ˜s(n) issimply the quantization error q(n) = ẽ(n) − e(n)
and that there is no accumulation of previous quantization errors in the
implementation of the decoder. That is,
q(n) =ẽ(n) − e(n) =ẽ(n) − s(n)+̂˜s(n) =˜s(n) − s(n) (12.15)
Hence ˜s(n) =s(n) +q(n). This means that the quantized sample ˜s(n)
differs from the input s(n) by the quantization error q(n) independent
of the predictor used. Therefore the quantization errors do not
accumulate.
In the DPCM system illustrated in Figure 12.3, the estimate or predicted
value ˜s(n) ofthe signal sample s(n) isobtained by taking a linear
combination of past values ˜s (n − k) , k =1, 2,...,p,asindicated by
(12.13). An improvement in the quality of the estimate is obtained by
including linearly filtered past values of the quantized error. Specifically,
the estimate of s(n) may be expressed as
p∑
m∑
̂˜s(n) = a (i)˜s (n − i)+ b (i)ẽ (n − i) (12.16)
i=1
where b (i) are the coefficients of the filter for the quantized error sequence
ẽ(n). The block diagram of the encoder at the transmitter and the decoder
at the receiver are shown in Figure 12.4. The two sets of coefficients
a (i) and b (i) are selected to minimize some function of the error e(n) =
˜s(n) − s(n), such as the sum of squared errors.
By using a logarithmic compressor and a 4-bit quantizer for the error
sequence e(n), DPCM results in high-quality speech at a rate of 32,000
bps, which is a factor of two lower than logarithmic PCM.
i=1
12.2.1 PROJECT 12.2: DPCM
The objective of this project is to gain understanding of the DPCM encoding
and decoding operations. For simulation purposes, generate correlated
random sequences using a pole-zero signal model of the form
s(n) =a (1) s (n − 1) + b 0 x(n)+b 1 x (n − 1) (12.17)
where x(n) isazero-mean unit variance Gaussian sequence. This can be
done using the filter function. The sequences developed in Project 12.1
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