Databases Reference
In-Depth Information
3
2
1
0
−
1
−2
−3
500
1000
1500
2000
2500
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3500
4000
F I GU R E 11 . 8
The residual sequence using a third-order predictor.
Using these autocorrelation values, we obtain the following coefficients for the three differ-
ent predictors. For
N
=
1, the predictor coefficient is
a
1
=
0
.
66; for
N
=
2, the coefficients
are
a
1
=
0
.
596 and
a
2
=
0
.
096; and for
N
=
3, the coefficients are
a
1
=
0
.
577
,
a
2
=−
0
.
025,
and
a
3
=
204. We used these coefficients to generate the residual sequence. In order to see
the reduction in variance, we computed the ratio of the source output variance to the variance
of the residual sequence. For comparison, we also computed this ratio for the case where the
residual sequence is obtained by taking the difference of neighboring samples. The sample-to-
sample differences resulted in a ratio of 1.63. Compared to this, the ratio of the input variance
to the variance of the residuals from the first-order predictor was 2.04. With a second-order
predictor, this ratio rose to 3.37, and with a third-order predictor, the ratio was 6.28.
The residual sequence for the third-order predictor is shown in Figure
11.8
. Notice that
although there has been a reduction in the dynamic range, there is still substantial structure
in the residual sequence, especially in the range of samples from about the 700th sample to
the 2000th sample. We will look at ways of removing this structure when we discuss speech
coding.
Let us now introduce a quantizer into the loop and look at the performance of the DPCM
system. For simplicity, we will use a uniform quantizer. If we look at the histogram of the
residual sequence, we find that it is highly peaked. Therefore, we will assume that the input
to the quantizer will be Laplacian. We will also adjust the step size of the quantizer based on
the variance of the residual. The step sizes provided in Chapter 9 are based on the assumption
that the quantizer input has a unit variance. It is easy to show that when the variance differs
from unity, the optimal step size can be obtained by multiplying the step size for a variance of
one with the standard deviation of the input. Using this approach for a four-level Laplacian
quantizer, we obtain step sizes of 0.75, 0.59, and 0.43 for the first-, second-, and third-order
predictors, and step sizes of 0.3, 0.4, and 0.5 for an eight-level Laplacian quantizer. We
measure the performance using two different measures, the signal-to-noise ratio (SNR) and
0
.
