[PDF] CHAPTER 3. SPECTRUM REPRESENTATION 58 - 3-4 Fourier Series

View - Download CHAPTER 3. SPECTRUM REPRESENTATION 58 - 3-4 Fourier Series

Previous PDF

Next PDF

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 0quotesdbs_dbs7.pdfusesText_5

[PDF] exponential fourier series of half wave rectifier

[PDF] exponential fourier series of square wave

[PDF] exponential function to log calculator

[PDF] exponential to log conversion calculator

[PDF] exponential to natural log calculator

[PDF] exponentielle de 0 7

[PDF] exponentielle de 0 valeur

[PDF] exponentielle de x u003d0

[PDF] exponentielle traduction en arabe

[PDF] exponents and logarithms worksheet

[PDF] exponents surds and logarithms pdf

[PDF] exposition art contemporain paris janvier 2019

[PDF] exposition art paris avril 2019

[PDF] exposition art paris mai 2019

[PDF] exposition art paris mars 2019