CHAPTER 16CHAPTER 16

Fourier Fourier SeriesSeries

CHAPTER CONTENTSCHAPTER CONTENTS 16.1 Fourier Series Analysis: An Overv16.1 Fourier Series Analysis: An Overv

eiweiw

16.2 The Fourier Coefficients16.2 The Fourier Coefficients

16.3 The Effect of Symmetry on the Fo16.3 The Effect of Symmetry on the Fourier urier CoefficientsCoefficients

16.4 An Alternative Trigonometric For16.4 An Alternative Trigonometric Form of m of the Fourier Seriesthe Fourier Series

16.5 An Application16.5 An Application

16.6 Average-Power Calculations with 16.6 Average-Power Calculations with Periodic FunctionsPeriodic Functions

16.7 The rms Value of a Periodic Func16.7 The rms Value of a Periodic Functiontion

16.8 The Exponential Form of the Fou16.8 The Exponential Form of the Fourier rier SeriesSeries

16.9 Amplitude and Phase Spectra16.9 Amplitude and Phase Spectra

16.1 Fourier Series 16.1 Fourier Series Analysis: An OverviewAnalysis: An Overview

A periodic A periodic waveformwaveform

A A periodic functionperiodic function is a function is a function that repeats itself every T that repeats itself every T seconds.seconds.

A A period period is the smallest time is the smallest time interval (T) that a periodic interval (T) that a periodic function can be shifted to function can be shifted to produce a function identical to produce a function identical to itself.itself.

The Fourier series is an infinite The Fourier series is an infinite series used to represent a periodic series used to represent a periodic function.function.

The series consists of a constant The series consists of a constant term and infinitely many term and infinitely many harmonically related cosine and sine harmonically related cosine and sine terms.terms.

The The fundamental frequencyfundamental frequency is the fre is the frequency determined by the fundamenquency determined by the fundamental period .tal period .

ff00 = 1 / = 1 / T T or or ωω 00 = 2 = 2ff00

The The harmonic frequencyharmonic frequency is an intege is an integer multiple of the fundamental frequer multiple of the fundamental frequency.ncy.

16.2 The Fourier 16.2 The Fourier CoefficientsCoefficients

The Fourier coefficients are the The Fourier coefficients are the constant term and the coefficient of constant term and the coefficient of each cosine and sine tem in the each cosine and sine tem in the series.series.

16.3 The Effect of 16.3 The Effect of Symmetry on the Fourier Symmetry on the Fourier

CoefficientsCoefficients FourFour types of symmetry may be types of symmetry may be

used to simplify the task of used to simplify the task of evaluating the Fourier evaluating the Fourier coefficients:coefficients:

Even-function symmetryEven-function symmetry Odd-function symmetryOdd-function symmetry Half-wave symmetryHalf-wave symmetry Quarter-wave symmetryQuarter-wave symmetry

An even periodic function, An even periodic function, f ( t ) = f ( t ) = f ( -t f ( -t ))

An odd periodic function An odd periodic function f ( t ) = f f ( t ) = f ( -t ( -t ))

(a)(a)A periodic triangular wave that is neither A periodic triangular wave that is neither even nor odd.even nor odd.

(b)(b)The triangular wave of (a) made even by The triangular wave of (a) made even by shifting the function along the shifting the function along the tt axis. axis.

(c)(c) The triangular wave of (a) made odd by The triangular wave of (a) made odd by shifting the function along the shifting the function along the tt axis. axis.

(a)(a)A function that has quarter-A function that has quarter-wave symmetry.wave symmetry.

(b)(b) A function that does not have A function that does not have quarter-wave symmetry.quarter-wave symmetry.

16.4 An Alternative 16.4 An Alternative Trigonometric Form of the Trigonometric Form of the

Fourier SeriesFourier Series In the alternative form of the Fourier In the alternative form of the Fourier

Series, each harmonic represented by Series, each harmonic represented by the sum of a cosine and sine term is cthe sum of a cosine and sine term is combined into a single term of the forombined into a single term of the form m AAnn cos ( cos ( nnωω 00t – t – θθ n n ).).

16.5 An Application16.5 An Application

For steady-state response, the For steady-state response, the Fourier series of the response signal Fourier series of the response signal is determined by first finding the is determined by first finding the response to each component of the response to each component of the input signal.input signal.

The individual responses are added The individual responses are added (super-imposed) to form the Fourier (super-imposed) to form the Fourier series of the response signal.series of the response signal.

The response to the individual The response to the individual terms in the input series is terms in the input series is found by either frequency found by either frequency domain or domain or ss-domain analysis.-domain analysis.

The waveform of the response The waveform of the response signal is difficult to obtain signal is difficult to obtain without the aid of a computer.without the aid of a computer.

Sometimes the frequency Sometimes the frequency response (or filtering) response (or filtering) characteristics of the circuit can characteristics of the circuit can be used to ascertain how closely be used to ascertain how closely the output waveform matches the the output waveform matches the input waveform.input waveform.

The effect of capacitor size on The effect of capacitor size on the steady-state responsethe steady-state response

16.6 Average-Power 16.6 Average-Power Calculations with Periodic Calculations with Periodic

FunctionsFunctions Only harmonics of the same Only harmonics of the same

frequency interact to produce frequency interact to produce average power.average power.

The total average power is the The total average power is the sum of the average powers sum of the average powers associated with each frequency.associated with each frequency.

16.7 The rms Value of a Periodi16.7 The rms Value of a Periodic Functionc Function

The rms value of a periodic function The rms value of a periodic function can be estimate from the Fourier coecan be estimate from the Fourier coefficients.fficients.

16.8 The Exponential Form 16.8 The Exponential Form of the Fourier Seriesof the Fourier Series

The Fourier series may also be written The Fourier series may also be written in exponential form by using in exponential form by using Euler’s Euler’s identityidentity to replace the cosine and sine to replace the cosine and sine terms with their exponential terms with their exponential equivalents.equivalents.

16.9 Amplitude and 16.9 Amplitude and Phase SpectraPhase Spectra

The plot of The plot of CCnn versus versus nn where where ττ = = TT/5 /5

The plot of The plot of θθ’’nn versus n f versus n f

or or θθ’’nn = - ( = - (θθ ’n’n + + n n / 5 )/ 5 )

THE ENDTHE END