Tools for Signal Compression

Applications to Speech and Audio Coding

Nicolas Moreau

• Publisher: Wiley-ISTE
• Year: 2013
• Print ISBN: 9781848212558
• eBook ISBN: 9781118616628
• Edition: 1
• Topic: Technology & Engineering
• Subtopic: Signals & Signal Processing
• Index: Technology & Engineering

This book presents tools and algorithms required to compress/uncompress signals such as speech and music. These algorithms are largely used in mobile phones, DVD players, HDTV sets, etc.

In a first rather theoretical part, this book presents the standard tools used in compression systems: scalar and vector quantization, predictive quantization, transform quantization, entropy coding. In particular we show the consistency between these different tools. The second part explains how these tools are used in the latest speech and audio coders. The third part gives Matlab programs simulating these coders.

PART 1

Tools for Signal Compression

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 error which is defined as

. The definition of a distortion measure

is required. We use the simplest distortion measure, quadratic error:

This measures the error in each sample. For a more global distortion measure, we use the mean squared error (MSE):

This error is simply denoted as the quantization error power. We use the notation

for the MSE.

Figure 1.1(a) shows, on the left, the signal before quantization and the partition of the range [−A, +A] where 6 = 3, and Figure 1.1(b) shows the reproduction values, the reconstructed signal and the quantization error. The bitstream between the transmitter and the receiver is not shown.

Figure 1.1. (a) The signal before quantization and the partition of the range [−A, +A] and (b) the set of reproduction values, reconstructed signal, and quantization error

The problem now consists of defining the optimal quantization, that is, in defining the intervals {Θ¹ … Θ^L} and the set of reproduction values

to minimize

.

1.2. Optimum scalar quantization

Assume that x(n) is the realization of a real-valued stationary random process X(n). In scalar quantization, what matters is the distribution of values that the random process X(n) takes at time n. No other direct use of the correlation that exists between the values of the process at different times is possible. It is enough to know the marginal probability density function of X(n), which is written as px(.).

1.2.1. Necessary conditions for optimization

To characterize the optimum scalar quantization, the range partition and reproduction values must be found whic...

Table of contents

• Cover
• Title Page
• Copyright
• Introduction
• Part 1: Tools for Signal Compression
• Part 2: Audio Signal Applications
• Part 3: MATLAB® Programs
• Bibliography
• Index