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CHAPTER 3. SPECTRUM REPRESENTATION58

3-4 Fourier Series

The examples in Sec. 3-3 show that we can synthesizeperiodicwaveforms by using a sum ofharmonically relatedsinusoids. Now, we want to describe a general theory that shows howany periodic signal can be

synthesized with a sum of harmonically related sinusoids,although the sum may need an infinite number of

terms. This is the mathematical theory ofFourier serieswhich uses the following representation: x.t/D1X kD1a kej.2=T0/kt(3.19)

whereT0is the fundamental period of the periodic signalx.t/. Thekthcomplex exponential in (3.19) has a

frequency equal tofkDk=T0Hz, so all the frequencies are integer multiples of the fundamental frequency

f0D1=T0Hz.7

There are two aspects of the Fourier theory: analysis and synthesis. Starting fromx.t/and calculatingfakg

is calledFourier analysis. The reverse process of starting fromfakgand generatingx.t/is calledFourier

synthesis. In this section, we will concentrate on analysis. The formula in (3.19) is the general synthesis formula. When the complex amplitudes areconjugate- symmetric,i.e.,akDa k, the synthesis formula becomes a sum of sinusoids of the form x.t/DA0C1X kD1A kcos..2=T0/ktCk/(3.20) whereA0Da0, and the amplitude and phase of thekthterm come from the polar form,akD12

Akejk. In

other words, the conditionakDa kis sufficient for the synthesized waveform to be arealfunction of time. By appropriate choice of the complex amplitudesakin (3.19), we can represent a number of interesting

periodic waveforms, such as square waves, triangle waves, rectified sinusoids, and so on. The fact that a

discontinuous square wave can be represented with an infinite number of sinusoids was one of the amazing

claims in Fourier"s famous thesis of 1807. It took many years before mathematicians were able to develop a

rigorous convergence proof to support Fourier"s claim.

3-4.1 Fourier Series: Analysis

How do we derive the coefficients for the harmonic sum in (3.19), i.e., how do we go fromx.t/toak? The

answer is that we use theFourier series integralto perform Fourier analysis. The complex amplitudes for any

periodic signal can be calculated with the Fourier integral a kD1T 0 T0Z 0 x.t/e j.2=T0/ktdt(3.21) whereT0is the fundamental period ofx.t/. A special case of (3.21) is thekD0case for the DC component a

0which is obtained by

a 0D1T 0T 0Z 0 x.t/dt(3.22)7

There are three ways to refer to the fundamental frequency: radian frequency!0in rad/sec, cyclic frequencyf0in Hz, or with

the periodT0in sec. Each one has its merits in certain situations. The relationship among these is!0D2f0D2=T0.

c J. H. McClellan, R. W. Schafer, & M. A. Yoder DRAFT, for ECE-2026 Fall-2012, September 8, 2012

CHAPTER 3. SPECTRUM REPRESENTATION59

A common interpretation of (3.22) is thata0is simply the average value of the signal over one period.

The Fourier integral (3.21) is convenient if we have a formula that definesx.t/over one period. Two

examples will be presented later to illustrate this point. On the other hand, ifx.t/is known only as a recording,

then numerical methods such as those discussed in Chapters 66 and??will be needed.

3-4.2 Fourier Series Derivation

In this section, we present a derivation of the Fourier series integral formula (3.21). The derivation relies on a

simple property of the complex exponential signal-the integral of a complex exponential over any number of

complete periods is zero. In equation form, T 0Z 0 e j.2=T0/ktdtD0(3.23) whereT0is a period of the complex exponential whose frequency is!kD.2=T0/k, andkis a nonzero integer. Here is the integration: T 0Z 0 e j.2=T0/ktdtDej.2=T0/ktj.2=T 0/k T 0 0 D ej.2=T0/kT01j.2=T 0/kD0

The numerator is zero becauseej2kD1for any integerk(positive or negative). Equation (3.23) can also be

justified if we use Euler"s formula to separate the integral into its real and imaginary parts and then integrate

cosine and sine separately-each one overkcomplete periods: T 0Z 0 e j.2=T0/ktdtDT 0Z 0 cos..2=T0/kt/dt CjT 0Z 0 sin..2=T0/kt/dtD0Cj0

A key ingredient in the infinite series representation (3.19) is the form of the complex exponentials, which

all must repeat with the same period as the period of the signalx.t/, which isT0. If we definevk.t/to be the

complex exponential of frequency!kD.2=T0/k, then v k.t/Dej.2=T0/kt(3.24)

Even though the minimum duration period ofvk.t/might be smaller thanT0, the following shows thatvk.t/

still repeats with a period ofT0: v k.tCT0/Dej.2=T0/k.tCT0/

Dej.2=T0/ktej.2=T0/kT0

Dej.2=T0/ktej2k

Dej.2=T0/ktDvk.t/

c J. H. McClellan, R. W. Schafer, & M. A. Yoder DRAFT, for ECE-2026 Fall-2012, September 8, 2012

CHAPTER 3. SPECTRUM REPRESENTATION60

where again we have usedej2kD1for any integerk(positive or negative).

Next we can generalize the zero-integral property (3.23) of the complex exponential to involve two signals:

8Othogonality Property

T 0Z 0 v k.t/v `.t/dtD( T

0ifkD`(3.25)

where the * superscript inv `.t/denotes the complex conjugate. Proof:Proving the orthogonality property is straightforward. We begin with T 0Z 0 v k.t/v `.t/dtDT 0Z 0 e j.2=T0/ktej.2=T0/`tdt D T 0Z 0 e j.2=T0/.k`/tdt

There are two cases to consider for the last integral: whenkD`the exponent becomes zero, so the integral is

T 0Z 0 e j.2=T0/.k`/tdtDT 0Z 0 e j0tdt D T 0Z 0

1dtDT0

T 0Z 0 e j.2=T0/.k`/tdtDT 0Z 0 e j.2=T0/mtdtD0 tion. If we assume that (3.19) is valid, x.t/D1X kD1a kej.2=T0/kt8

The integral in (3.25) is called the "inner product" betweenvk.t/andv`.t/, sometimes denoted ashvk.t/;v`.t/i.

c J. H. McClellan, R. W. Schafer, & M. A. Yoder DRAFT, for ECE-2026 Fall-2012, September 8, 2012

CHAPTER 3. SPECTRUM REPRESENTATION61

then we can multiply both sides by the complex exponentialv `.t/and integrate over the periodT0. T 0Z 0 x.t/ e j.2=T0/`tdtD T 0Z 0 1X kD1a kej.2=T0/kt! e j.2=T0/`tdt D 1X kD1a k0 @T 0Z 0 e j.2=T0/.k`/tdt1 A nonzero only whenkD`Da`T0(3.26)

Notice that we are able to isolate one of the complex amplitudes.a`/in the final step by applying the orthogo-

nality property (3.25).

The crucial step in (3.26) occurs when the order of the infinite summation and the integration is swapped.

This is a delicate manipulation that depends on convergence properties of the infinite series expansion. It

was also a topic of research that occupied mathematicians for a good part of the early 19th century. For our

purposes, if we assume thatx.t/is a smooth function and has only a finite number of discontinuities within

one period, then the swap is permissible.

The final analysis formula is obtained by dividing both sides of (3.26) byT0and writinga`on one side of

the equation. Since`could be any index, we can replace`withkto obtainFourier Analysis Equation a kD1T 0T 0Z 0 x.t/e j.2=T0/ktdt(3.27) where!0D2=T0D2f0is the fundamental frequency of the periodic signalx.t/. This analysis formula

goes hand in hand with the synthesis formula for periodic signals, which isFourier Synthesis Equation

x.t/D1X kD1a kej.2=T0/kt(3.28)

3-5 Spectrum of the Fourier Series

When we discussed the spectrum in Section 3-1, we described a graphical procedure for drawing the spectrum

whenx.t/is composed of a sum of complex exponentials. By virtue of the synthesis formula (3.28), the Fourier

series coefficientsakare, in fact, the complex amplitudes that define the spectrum ofx.t/. In order to illustrate

this general connection between the Fourier series and the spectrum, we use the "sine-cubed" signal. First, we

derive theakcoefficients forx.t/Dsin3.3t/, and then we sketch its spectrum. c J. H. McClellan, R. W. Schafer, & M. A. Yoder DRAFT, for ECE-2026 Fall-2012, September 8, 2012 CHAPTER 3. SPECTRUM REPRESENTATION62CDROMExample 3-7: Fourier Series without

Integration

Determine the Fourier series coefficients of the signal: x.t/Dsin3.3t/

Solution:There are two ways to get theakcoefficients: plugx.t/into the Fourier integral (3.27), or use

the inverse Euler formula to expandx.t/into a sum of complex exponentials. It is far easier to use the latter

approach. Using the inverse Euler formula for sin./, we make the following expansion of the sine-cubed

function: x.t/D ej3tej3t2j 3 D 18j ej9t3ej6tej3tC3ej3tej6tej9t D j8 ej9tC3j8 ej3tC3j8 ej3tCj8 ej9t(3.29) Weseethat(3.29)isthesumoffourtermswithfrequencies:!D 3and!D 9rad/s. Sincegcd.3;9/D

3, the fundamental frequency is!0D3rad/sec. The Fourier series coefficients are indexed in terms of the

fundamental frequency, so a kD8

:0forkD0

j38 forkD 1

0forkD 2

j18 forkD 3

0forkD 4;5;6;:::(3.30)

This example shows that it is not always necessary to evaluate an integral to obtain thefakgcoefficients.Now we can draw the spectrum (Fig. 3-13) because we know that we have four nonzeroakcomponents

located at the four frequencies:!D f9;3; 3; 9grad/sec. We prefer to plot the spectrum versus frequency in hertz in this case, so the spectrum lines

9are atfD 4:5;1:5; 1:5;and 4.5 Hz. The second

harmonic is missing and the third harmonic is at 4.5 Hz.CDROMEXERCISE 3.5:Use the Fourier integral to determine all the Fourier series coefficients of the "sine-

cubed" signal. In other words, evaluate the integral a kD1T 0T 0Z 0 sin

3.3t/ej.2=T0/ktdt

for allk.

Hints: Find the period first, so that the integration interval is known. In addition, you might find it easier

to convert the sin

3./function to exponential form (via the inverse Euler formula for sin./) before doing the

Fourier integral on each of four different terms. If you then invoke the orthogonality property on each integral,

you should get exactly the same answer as (3.30).9

Repeating this footnote from Section 3.1 here for convenience:Spectra of signals comprised of individual sinusoids are often

called "line spectra". The term "line" seems appropriate for us here because we plot the components as vertical lines positioned at the

individual frequencies. However, the term originated in physics where lines are observed in emission or absorption spectra formed with

optical prisms (which serve as optical spectrum analyzers). These lines correspond to energy being emitted or absorbed at wavelengths

that are characteristic of atoms or ions. c J. H. McClellan, R. W. Schafer, & M. A. Yoder DRAFT, for ECE-2026 Fall-2012, September 8, 2012 CHAPTER 3. SPECTRUM REPRESENTATION634.5 1.5 1.5 4.50 fquotesdbs_dbs5.pdfusesText_9

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