CHAPTER 16
Fourier Series
CHAPTER CONTENTS

16.1 Fourier Series Analysis: An Overveiw

16.2 The Fourier Coefficients

16.3 The Effect of Symmetry on the Fourier
Coefficients

16.4 An Alternative Trigonometric Form of
the Fourier Series

16.5 An Application

16.6 Average-Power Calculations with
Periodic Functions

16.7 The rms Value of a Periodic Function

16.8 The Exponential Form of the Fourier
Series

16.9 Amplitude and Phase Spectra
16.1 Fourier Series Analysis: An
Overview
A periodic waveform

A periodic function is a function that repeats
itself every T seconds.

A period is the smallest time interval (T)
that a periodic function can be shifted to
produce a function identical to itself.

The Fourier series is an infinite series used
to represent a periodic function.

The series consists of a constant term and
infinitely many harmonically related cosine
and sine terms.

The fundamental frequency is the frequency
determined by the fundamental period .

f0 = 1 / T or ω0 = 2 f0

The harmonic frequency is an integer
multiple of the fundamental frequency.
16.2 The Fourier Coefficients

The Fourier coefficients are the constant
term and the coefficient of each cosine and
sine tem in the series.
16.3 The Effect of Symmetry on the
Fourier Coefficients





Four types of symmetry may be used to
simplify the task of evaluating the Fourier
coefficients:
Even-function symmetry
Odd-function symmetry
Half-wave symmetry
Quarter-wave symmetry
An even periodic function, f ( t ) = f ( -t )
An odd periodic function f ( t ) = f ( -t )
(a)A periodic triangular wave that is neither even nor odd.
(b)The triangular wave of (a) made even by shifting the
function along the t axis.
(c) The triangular wave of (a) made odd by shifting the
function along the t axis.
(a)A function that has quarter-wave symmetry.
(b) A function that does not have quarter-wave
symmetry.
16.4 An Alternative Trigonometric
Form of the Fourier Series

In the alternative form of the Fourier Series,
each harmonic represented by the sum of a
cosine and sine term is combined into a
single term of the form An cos ( nω0t – θn ).
16.5 An Application

For steady-state response, the Fourier series
of the response signal is determined by first
finding the response to each component of
the input signal.

The individual responses are added (superimposed) to form the Fourier series of the
response signal.

The response to the individual terms in
the input series is found by either
frequency domain or s-domain
analysis.

The waveform of the response signal is
difficult to obtain without the aid of a
computer.

Sometimes the frequency response (or
filtering) characteristics of the circuit can be
used to ascertain how closely the output
waveform matches the input waveform.
The effect of capacitor size on the
steady-state response
16.6 Average-Power Calculations with
Periodic Functions

Only harmonics of the same frequency
interact to produce average power.

The total average power is the sum of the
average powers associated with each
frequency.
16.7 The rms Value of a Periodic
Function

The rms value of a periodic function can be
estimate from the Fourier coefficients.
16.8 The Exponential Form of the
Fourier Series

The Fourier series may also be written in
exponential form by using Euler’s identity
to replace the cosine and sine terms with
their exponential equivalents.
16.9 Amplitude and Phase Spectra
The plot of Cn versus n where τ = T/5
The plot of θ’n versus n for
θ’n = - (θ’n + n / 5 )
THE END