The samples in the prediction residual are now assumed to be uncorrelated and therefore may be coded independently. The problem of residual coding is therefore to find an appropriate form for the probability density function (p.d.f.) of the distribution of residual values so that they can be efficiently modelled. Figures 2 and 3 show the p.d.f. for the segmentally normalized residual of the polynomial predictor (the full linear predictor shows a similar p.d.f). The observed values are shown as open circles, the Gaussian p.d.f. is shown as dot-dash line and the Laplacian, or double sided exponential distribution is shown as a dashed line.
These figures demonstrate that the Laplacian p.d.f. fits the observed distribution very well. This is convenient as there is a simple Huffman code for this distribution . To form this code, a number is divided into a sign bit, the th low order bits and the the remaining high order bits. The high order bits are treated as an integer and this number of 0's are transmitted followed by a terminating 1. The low order bits then follow, as in the example in table 1.
As with all Huffman codes, a whole number of bits are used per sample, resulting in instantaneous decoding at the expense of introducing quantization error in the p.d.f. This is illustrated with the points marked '+' in figure 3. In the example, giving a minimum code length of 4. The error introduced by coding according to the Laplacian p.d.f. instead of the true p.d.f. is only 0.004 bits per sample, and the error introduced by using Huffman codes is only 0.12 bits per sample. These are small compared to a typical code length of 7 for 16 kHz speech corpora.
This Huffman code is also simple in that it may be encoded and decoded with a few logical operations. Thus the implementation need not employ a tree search for decoding, so reducing the computational and storage overheads associated with transmitting a more general p.d.f.
The optimal number of low order bits to be transmitted directly is linearly related to the variance of the signal. The Laplacian is defined as:
where is the absolute value of and is the variance of the distribution. Taking the expectation of gives:
For optimal Huffman coding we need to find the number of low order bits, , such that such that half the samples lie in the range . This ensures that the Huffman code is bits long with probability 0.5 and long with probability , which is optimal.
Solving for gives:
When polynomial filters are used is obtained from using equation 21. In the LPC implementation is derived from which is obtained directly from the calculations for predictor coefficients the using the autocorrelation method.