(contd.)

To prove :

We find an expression for the distortion and then find the bit allocation that minimizes the distortion. We perform the minimization using the method of Lagrange multipliers. Let average number of bits/simple to be used by the vector source be R=B/N; and average number of bits/sample used by the k th scalar be , then,

	(3.3.1)

where N is number of scalars in vector source. The MSQE i.e. reconstruction error variance is given by

	(3.3.2)

where is a factor that depends on the input distribution and the quantizer; and is variance of scalar . The total reconstruction error variance,

	(3.3.3)

The objective of bit allocation procedure is to find that minimizes eqn(3.3.3), subject to constraint given by eqn (3.3.1). Let us assume that is a constant for all k.

We can set up the minimization problem in terms of Lagrange's multipliers as

	(3.3.4)

Taking the derivative of J wrt and setting it equal to zero, we obtain,

	(3.3.5)

Substituting equ (5) in (1) we get a value for as:

	(3.3.6)

Substituting (6) in (5) we get

	(3.3.7)

It is to be noted that the values are not guranteed to be integers or even positive. The standard approach is to set the negative to Zero.

This will increase the average bit rate above the constrain. Therefore, the non zero are uniformly reduced until the average rate is equal to R.

Substituting for and R in equ (7), in terms of and B, we have