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 = 2b non-overlapping intervals {Θ1 … ΘL} of length {Δ1 … ΔL},

– numbering the partitioned intervals {i1 … iL},

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

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 = 2b}. 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 image from the set of reproduction values image 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:

image

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

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