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Linear Predictive Coding (LPC) of Speech 625
Then
N−1
∑
G 2
n=0
x 2 (n) =
N−1
∑
n=0
e 2 (n) (12.36)
If the input excitation is normalized to unit energy by design, then
G 2 =
N−1
∑
n=0
e 2 (n) =r ss (0) +
p∑
a p (k) r ss (k) (12.37)
Thus G 2 is set equal to the residual energy resulting from the least-squares
optimization.
Once the LPC coefficients are computed, we can determine whether
the input speech frame is voiced, and if so, what the pitch is. This is
accomplished by computing the sequence
where r a (k) isdefined as
r e (n) =
r a (k) =
k=1
p∑
r a (k) r ss (n − k) (12.38)
k=1
p∑
a p (i) a p (i + k) (12.39)
i=1
which is the autocorrelation sequence of the prediction coefficients.
The pitch is detected by finding the peak of the normalized sequence
r e (n)/r e (0) in the time interval that corresponds to 3 to 15 ms in the
20-ms sampling frame. If the value of this peak is at least 0.25, the frame
of speech is considered voiced with a pitch period equal to the value of
n = N p , where r e (N p ) /r e (0) is a maximum. If the peak value is less than
0.25, the frame of speech is considered unvoiced and the pitch is zero.
The values of the LPC coefficients, the pitch period, and the type of
excitation are transmitted to the receiver, where the decoder synthesizes
the speech signal by passing the proper excitation through the all-pole
filter model of the vocal tract. Typically, the pitch period requires 6 bits,
and the gain parameter may be represented by 5 bits after its dynamic
range is compressed logarithmically. If the prediction coefficients were to
be coded, they would require between 8 to 10 bits per coefficient for accurate
representation. The reason for such high accuracy is that relatively
small changes in the prediction coefficients result in a large change in the
pole positions of the filter model. The accuracy requirements are lessened
by transmitting the reflection coefficients {K i }, which have a smaller dynamic
range—that is, |K i | < 1. These are adequately represented by 6
bits per coefficient. Thus for a 10th-order predictor the total number of
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