Digital Signal Processing Reference
In-Depth Information
n
=
1
x(n)x(n
−
1
)
Substituting
ρ
=
(first order normalized autocorrelation coeffi-
n
=
1
x
2
(n)
cient) in (3.34) gives,
σ
r
σ
x
+
a
2
σ
x
−
2
aσ
x
ρ
=
(3.35)
The prediction gain
G
p
is then found as,
σ
x
1
G
p
=
σ
r
=
(3.36)
a
2
1
+
−
2
aρ
To maximize the prediction gain, the denominator of equation (3.36) should
be minimized with respect to
a
,hence,
a
2
∂(
1
+
−
2
aρ)
=
=
+
−
0
(
0
2
a
2
ρ)
(3.37)
∂a
which gives,
a
=
ρ
(3.38)
Substituting
a
=
ρ
in (3.36)
1
1
G
p
=
2
ρρ
=
(3.39)
ρ
2
ρ
2
1
+
−
1
−
The above result shows that if the correlation between the adjacent samples
is high, then a differential quantizer will perform significantly better than a
nondifferential quantizer. In fact, if the signal to be quantized is a nonvarying
DC signal, where
ρ
=
1, the gain of the prediction process will be infinite, i.e.
no residual error will be left and, hence, no residual information will need to
be transmitted. A typical
ρ
for speech is between 0.8 and 0.9 which may result
in 4-7 dB signal reduction before quantization, hence achieving significant
increase in quantization performance.
3.4 Vector Quantization
When a set of discrete-time amplitude values is quantized jointly as a single
vector, the process is known as vector quantization (VQ), also known as block
quantization or pattern-matching quantization. A block diagram of a simple
vector quantizer is shown in Figure 3.9.
If we assume
x
[
x
1
,x
2
, ... .,x
N
]
T
is an
N
dimensional vector with real-
valued, continuous-amplitude (short or float representation is assumed to
be continuous amplitude) randomly varying components
x
k
,
1
=
≤
k
≤
N
(the
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