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624 Chapter 12 APPLICATIONS IN COMMUNICATIONS
and the corresponding error between the observed sample s(n) and the
predicted value ŝ(n) is
p∑
e(n) =s(n)+ a p (k) s (n − k) (12.29)
k=1
By minimizing the sum of squared errors, that is,
[
2 N∑
N∑
p∑
E = e 2 (n) = s(n)+ a p (k) s (n − k)]
(12.30)
n=0
n=0
we can determine the pole parameters {a p (k)} of the model. The result
of differentiating E with respect to each of the parameters and equating
the result to zero, is a set of p linear equations
p∑
a p (k) r ss (m − k) =−r ss (m) , m =1, 2,...,p (12.31)
k=1
where r ss (m) isthe autocorrelation of the sequence s(n) defined as
r ss (m) =
k=1
N∑
s(n)s (n + m) (12.32)
n=0
The linear equation (12.31) can be expressed in matrix form as
R ss a = −r ss (12.33)
where R ss is a p × p autocorrelation matrix, r ss is a p × 1 autocorrelation
vector, and a is a p × 1vector of model parameters. Hence
a = −R −1
ss r ss (12.34)
These equations can also be solved recursively and most efficiently, without
resorting to matrix inversion, by using the Levinson-Durbin algorithm
[19]. However, in MATLAB it is convenient to use the matrix inversion.
The all-pole filter parameters {a p (k)} can be converted to the all-pole
lattice parameters {K i } (called the reflection coefficients) using the
MATLAB function dir2latc developed in Chapter 6.
The gain parameter of the filter can be obtained by noting that its
input-output equation is
p∑
s(n) =− a p (k) s (n − k)+Gx(n) (12.35)
k=1
where x(n) isthe input sequence. Clearly,
p∑
Gx(n) =s(n)+ a p (k) s (n − k) =e(n)
k=1
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