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to design a vector quantizer called the
pyramid vector quantizer
for the Laplacian source that
looks quite similar to the quantizer described in Example
10.6.1
. The vector quantizer consists
of points of the rectangular quantizer that fall on the hyperpyramid given by
L
1
|
x
i
|
=
C
i
=
where
C
is a constant depending on the variance of the input. Shannon's result is asymptotic,
and for realistic values of
L
, the input vector is generally not localized to a single hyperpyramid.
For this case, Fischer first finds the distance
L
r
=
1
|
x
i
|
i
=
This value is quantized and transmitted to the receiver. The input is normalized by this gain
term and quantized using a single hyperpyramid. The quantization process for the shape term
consists of two stages: finding the output point on the hyperpyramid closest to the scaled
input, and finding a binary codeword for this output point. (See [
144
] for details about the
quantization and coding process.) This approach is quite successful, and for a rate of 3 bits
per sample and a vector dimension of 16, we get an SNR value of 16.32 dB. If we increase the
vector dimension to 64, we get an SNR value of 17.03. Compared to the SNR obtained from
using a nonuniform scalar quantizer, this is an improvement of more than 4 dB.
Notice that in this approach we separated the input vector into a
gain
term and a pattern or
shape
term. Quantizers of this form are called
gain-shape vector quantizers
,or
product code
vector quantizers
[
145
].
10.6.2 Polar and Spherical Vector Quantizers
For the Gaussian distribution, the contours of constant probability are circles in two dimensions
and spheres and hyperspheres in three and higher dimensions. In two dimensions, we can
quantize the input vector by first transforming it into polar coordinates
r
and
θ
:
x
1
+
x
2
=
(8)
r
and
tan
−
1
x
2
x
1
θ
=
(9)
r
and
can then be either quantized independently [
146
], or we can use the quantized value
of
r
as an index to a quantizer for
θ
[
147
]. The former is known as a polar quantizer; the
latter, an unrestricted polar quantizer. The advantage to quantizing
r
and
θ
θ
independently is
one of simplicity. The quantizers for
r
and
are independent scalar quantizers. However, the
performance of the polar quantizers is not significantly higher than that of scalar quantization
of the components of the two-dimensional vector. The unrestricted polar quantizer has a more
complex implementation, as the quantization of
θ
depends on the quantization of
r
. However,
the performance is also somewhat better than the polar quantizer. The polar quantizer can be
extended to three or more dimensions [
148
].
θ
