Tools for Signal Compression: Applications to Speech and Audio Coding

Chapter 1

Scalar Quantization

1.1. Introduction

Let us consider a discrete-time signal x(n) with values in the range [−A, +A]. Defining a scalar quantization with a resolution of b bits per sample requires three operations:

– partitioning the range [−A, +A] into L = 2^b non-overlapping intervals {Θ¹ … Θ^L} of length {Δ¹ … Δ^L},

– numbering the partitioned intervals {i¹ … i^L},

– selecting the reproduction value for each interval, the set of these reproduction values forms a dictionary (codebook)¹ .

Encoding (in the transmitter) consists of deciding which interval x(n) belongs to and then associating it with the corresponding number i(n) ∈ {1 … L = 2^b}. It is the number of the chosen interval, the symbol, which is transmitted or stored. The decoding procedure (at the receiver) involves associating the corresponding reproduction value from the set of reproduction values with the number i(n). More formally, we observe that quantization is a non-bijective mapping to [−A, +A] in a finite set C with an assignment rule:

The process is irreversible and involves loss of information, a quantization ...