Digital Communication Techniques

Appendix 3Fourier Analysis

A3.1 Introduction

Baron Jean Baptiste Joseph Fourier (1768–1830) introduced the idea that any periodic function can be represented by a series of sines and cosines which are harmonically related.

We are therefore interested in series: cosθ, cos2θ, cos3θ… and sinθ, sin2θ, sin3θ… .

Figure A3.1. Some sines and cosines representations. For a color version of this figure, see www.iste.co.uk/gontrand/digital.zip

for all values of m.

a₀ is the mean value (DC) of the f (θ)

a_n Multiply (1) by cos mθ.

Then integrate on both sides;

Integration on the right,

First term,

Second term,

Second term,

Third term,

Integration on the right,

First term,

Second term,