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612 Chapter 12 APPLICATIONS IN COMMUNICATIONS
where r ss (m) isthe autocorrelation function of the sampled signal sequence
s(n), defined as
r ss (m) =
N∑
s (i) s (i + m) (12.10)
i=1
Minimization of E p with respect to the predictor coefficients {a i (n)} results
in the set of linear equations, called the normal equations,
p∑
a (i) r ss (i − j) =r ss (j) , j =1, 2,...,p (12.11)
i=1
or in the matrix form,
Ra = r =⇒ a = R −1 r (12.12)
where R is the autocorrelation matrix, a is the coefficient vector, and r
is the autocorrelation vector. Thus the values of the predictor coefficients
are established.
Having described the method for determining the predictor coefficients,
let us now consider the block diagram of a practical DPCM system,
shown in Figure 12.3. In this configuration the predictor is implemented
with the feedback loop around the quantizer. The input to the predictor
is denoted as ˜s(n), which represents the signal sample s(n) modified by
the quantization process, and the output of the predictor is
The difference
̂˜s =
p∑
a (i)˜s (n − i) (12.13)
i=1
e(n) =s(n) − ̂˜s(n) (12.14)
is the input to the quantizer, and ẽ(n) denotes the output. Each value of
the quantized prediction error ẽ(n) isencoded into a sequence of binary
FIGURE 12.3
Block diagram of a DPCM transcoder: (a) encoder, (b) decoder
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