Digital Signal Processing Reference
In-Depth Information
x(n)
index i
x
Input Vector
Buffer
Vector
Matching
y
i
Codebook
Y
Figure 3.9
Block diagram of a simple vector quantizer
superscript
T
denotes transpose in vector quantization), this vector ismatched
with another real-valued, discrete-amplitude,
N
dimensional vector
y
.Hence,
x
is quantized as
y
,and
y
is used to represent
x
. Usually,
y
is chosen from
afinitesetofvalues
Y
[
y
i
1
,y
i
2
, ... .,y
iN
]
T
.Theset
Y
is called the codebook or reference templates where
L
is the size of the
codebook, and
y
i
are the codebook vectors. The size of the codebook may be
considered to be equivalent to the number of levels in a scalar quantizer. In
order to design such a codebook,
N
dimensional space is partitioned into
L
regions or cells
C
i
,
1
=
y
i
,
1
≤
i
≤
L
,where
y
i
=
L
and a vector
y
i
is associated with each cell
C
i
.
The quantizer then assigns the codebook vector
y
i
≤
i
≤
if
x
is in
C
i
,
q(
x
)
=
if
x
C
i
(3.40)
y
i
The codebook design process is also known as training or populating
the codebook. Figure 3.10 shows an example of the partitioning of a two-
dimensional space (
N
2 ) for the purpose of vector quantization. The filled
region enclosed by the bold lines is the cell
C
i
. During vector quantization,
any input vector
x
that lies in the cell
C
i
is quantized as
y
i
. The other codebook
vectors corresponding to the other cells are shown by dots.
If the vector dimension,
N
, equals one vector quantization reduces to scalar
quantization. Scalar quantization has the special property that whilst cells
may have different sizes (step sizes) they all have the same shape. In vector
quantization, however, cells may have different shapes which gives vector
quantization an advantage over scalar quantization.
When
x
is quantized as
y
, a quantization error results and, to measure
the performance of a specific codebook, an overall distortion measure
D
is
defined as,
=
M
1
M
D
=
d
i
[
x
,
y
]
(3.41)
i
=
1
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