Sampling and Quantization - Digital Speech: Coding for Low Bit Rate Communication Systems - page 42

Furthermore, they correlatewell with subjective judgements of speechquality.
However, as the bit rate decreases and distortion increases, simple distortion
measures may not be related to the subjective quality of speech. Since the
main application of vector quantization is expected to be at low bit-rates,
it is very important to develop and use distortion measures that are better
correlated with human auditory behaviour. A number of perceptually based
distortion measures have been developed [10, 11, 12]. Since the main aim is to
produce the highest speech quality possible at a given bit rate, it is essential
to use a distortion measure that correlates well with human perception.

3.4.2 CodebookDesign
When designing an L level codebook, N dimensional space is partitioned into
L cells C i , 1

≤ ≤

L , and each cell C i is assigned a vector y i . The quantizer
chooses the codebook vector y i if x is in C i . To optimize a quantizer, the
distortion in equation (3.41) is minimized over all L levels. There are two
necessary conditions for optimality. The first condition is that the optimum
quantizer finds a matching vector for every input vector by minimizing the
distortion criterion. That is, the quantizer chooses the codebook vector that
results in the minimum distortion with respect to x [13].

i

q( x ) = if d [ x , y i ] ≤ d [ x , y j ] , j = i, 1 ≤ j ≤ L. (3.48)

y i

The second necessary condition for optimality is that each codebook vector
y i

is optimized to give the minimum average distortion in cell C i .

D i = E { [ d( x , y i ) |

x C i ]

}= d [ x , y i ] p(x)dx (3.49)

x C i

where p(x) is the probability density function of vectors that result in the
quantized vector y i in cell (cluster) C i .
Vector y i is called the centroid of the cell C i . Optimization of the centroid
of a particular cell depends on the definition of the distortion measure. For
either the mean squared error or the weighted mean squared error, distortion
in each cell is minimized by,

M i

1
M i

=

y in x kn

x C i

(3.50)

k = 1

is the n th element of the centroid y i of the cluster C i .
That is, y i is simply the sample mean of all the training vectors M i contained
in cell C i . One of the most popular methods for codebook design is an iterative
clustering algorithm known as the K-means algorithm [13] (also known as

where y in { n = 1 , 2 , ... ,N }