Wireless Communications

2Mathematical Background

Abstract

This chapter briefly outlines some mathematical skills, techniques, and methodology to be used in the subsequent study of the book, with emphasis on their applications. Readers with a good mathematical background only need to go through the section on congruence mapping and signal spaces, which supplies a powerful means for understanding many communication systems and relevant signal processing schemes involved therein. Other material in this chapter can be skipped for the first time of reading and revisited whenever necessary.

2.1 Introduction

It is impossible, and not intended, to systematically describe the mathematical background for communications in a limited space. But rather, the purpose of this chapter is to present mathematical results and skills of relevance.

2.2 Congruence mapping and signal spaces

Congruence mapping is a transformation that maps elements from one metric space to another such that distance between the elements is preserved in the new space. This implies that the shape and size of a geometric figure or object remain unchanged before and after mapping.

As a powerful tool, congruence mapping is widely used in wireless communications. In wireless systems, message data to be handled is often in discrete form. They must be converted into appropriate waveforms to acquire sufficient energy and to match a given physical channel for efficient transmission. At the receiver end, continuous waveforms are converted back to ...