Sampling and Quantization - Digital Speech: Coding for Low Bit Rate Communication Systems - page 48

quantizer and the resulting error signal is then used in the input to a B 2 bit L 2
level second vector quantizer. The sum of the two quantized vectors results
in the quantized value of the input vector x .
The computation and storage costs for a k -stage cascaded vector quantiza-
tion are respectively,

Com cc = N(L 1 + L 2 + ... + L k ) multiply − add per input vector (3.57)

M cc = N(L 1 + L 2 + ... + L k ) locations (3.58)

2 B 1 ,L 2 = 2 B 2 and L k = 2 B k and the total number of bits per Assuming L 1 =

input vector B B k , we can see that the number of candidate

vectors searched in a cascaded codebook for each input vector is less than in
a full search codebook,

= B 1 + B 2 ... +

k k

2 B n < 2 B if B = B n and k > 1 (3.59)

n = 1 n = 1

We can also see that the storage of a cascaded codebook is less than that
required by a binary codebook,

2 B n

< N B

2 i

k

N for k > 1 (3.60)

n = 1 =

i 1

Given the condition that the total number of bits used at various stages of a
cascaded codebook is B , both computation and storage requirements reduce
with an increase in the number of stages.

Split Codebooks
In all of the above codebook types an N dimensional input vector is directly
matched with N dimensional codebook entries. In a split vector quantization
scheme, an N dimensional input vector is first split into P parts where P > 1.
For each part of the split vector a separate codebook is used and each part may
be vector quantized independently of the other parts using B p bits. Assuming
a vector is split into P equal parts and vector quantized using B p bits for each
part, the computation and storage requirements can be calculated as follows:

N
P (L 1

Com ss = + L 2 + ... + L P ) multiply − add per input vector (3.61)

2 B p

= =

where L p for p 1 , 2 , ... ,P . Similarly, the storage is given by:

N
P (L 1

M ss = + L 2 + ... + L P ) locations (3.62)