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Digital Communication Systems by Simon Haykin

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CHAPTER

2

Fourier Analysis of Signals and Systems

2.1  Introduction

The study of communication systems involves:

  • the processing of a modulated message signal generated at the transmitter output so as to facilitate its transportation across a physical channel and
  • subsequent processing of the received signal in the receiver so as to deliver an estimate of the original message signal to a user at the receiver output.

In this study, the representation of signals and systems features prominently. More specifically, the Fourier transform plays a key role in this representation.

The Fourier transform provides the mathematical link between the time-domain representation (i.e., waveform) of a signal and its frequency-domain description (i.e., spectrum). Most importantly, we can go back and forth between these two descriptions of the signal with no loss of information. Indeed, we may invoke a similar transformation in the representation of linear systems. In this latter case, the time-domain and frequency-domain descriptions of a linear time-invariant system are defined in terms of its impulse response and frequency response, respectively.

In light of this background, it is in order that we begin a mathematical study of communication systems by presenting a review of Fourier analysis. This review, in turn, paves the way for the formulation of simplified representations of band-pass signals and systems to which we resort in subsequent chapters. We begin the study by developing the transition ...

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