When the resolution is low, it is natural to group several samples x(n) in a vector and to find a way to quantize them together. This is known as vector quantization. The resolution b, the vector dimension N, and the size L of the codebook are related by:
In this case, b does not have to be an integer. The product bN must be an integer or even, more simply, L must be an integer. Vector quantization therefore enables the definition of non-integer resolutions. However, this is not the key property of vector quantization: vector quantization allows us to directly take account of the correlation contained in the signal rather than first decorrelating the signal, and then quantizing the decorrelated signal as performed in predictive scalar quantization. Vector quantization would be perfect were it is not for a major flaw: the complexity of processing in terms of the number of multiplications/additions to handle is an exponential function of N.
Vector quantization is an immediate generalization of scalar quantization. Vector quantization of N dimensions with size L can be seen as an application of RN in a finite set C which contains L N-dimensional vectors:
The space RN is partitioned into L regions or cells defined by: