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ELG 3120 Signals and Systems Chapter 3

1/3 Yao Chapter 3 Fourier Series Representation of Period Signals

3.0 Introduction

· Signals can be represented using complex exponentials - continuous-time and discrete-time

Fourier series and transform.

· If the input to an LTI system is expressed as a linear combination of periodic complex exponentials or sinusoids, the output can also be expressed in this form.

3.1 A Historical Perspective

By 1807, Fourier had completed a work that series of harmonically related sinusoids were useful in representing temperature distribution of a body. He claimed that any periodic signal could be represented by such series - Fourier Series. He also obtained a representation for aperidic signals as weighted integrals of sinusoids - Fourier Transform.

Jean Baptiste Joseph Fourier

3.2 The Response of LTI Systems to Complex Exponentials

It is advantageous in the study of LTI systems to represent signals as linear combinations of basic signals that possess the following two properties: · The set of basic signals can be used to construct a broad and useful class of signals.

ELG 3120 Signals and Systems Chapter 3

2/3 Yao · The response of an LTI system to each signal should be simple enough in structure to provide

us with a convenient representation for the response of the system to any signal constructed as a linear combination of the basic signal. Both of these properties are provided by Fourier analysis. The importance of complex exponentials in the study of LTI system is that the response of an LTI system to a complex exponential input is the same complex exponential with only a change in amplitude; that is

Continuous time: ststesHe)(®, (3.1)

Discrete-time: nnzzHz)(®, (3.2)

where the complex amplitude factor )(sH or )(zH will be in general be a function of the complex variable s or z. A signal for which the system output is a (possible complex) constant times the input is referred to as an eigenfunction of the system, and the amplitude factor is referred to as the system's eigenvalue. Complex exponentials are eigenfunctions. For an input )(tx applied to an LTI system with impulse response of )(th, the output is ttttttttt ttt dehedehdehdtxhty ssttsts , (3.3) where we assume that the integral tttdehs

¥--)( converges and is expressed as

tttdehsHs

¥--=)()(, (3.4)

the response to ste is of the form stesHty)()(=, (3.5)

It is shown the

complex exponentials are eigenfunctions of LTI systems and )(sH for a specific value of s is then the eigenvalues associated with the eigenfunctions. Complex exponential sequences are eigenfunctions of discrete-time LTI systems. That is, suppose that an LTI system with impulse response ][nh has as its input sequence

ELG 3120 Signals and Systems Chapter 3

3/3 Yao n

znx=][, (3.6) where z is a complex number. Then the output of the system can be determined from the convolution sum as -¥===-=kkn kkn kzkhzzkhknxkhny][][][][][. (3.7) Assuming that the summation on the right-hand side of Eq. (3.7) converges, the output is the same complex exponential multiplied by a constant that depends on the value of z. That is, nzzHny)(][=, (3.8) where =kk zkhzH][)(. (3.9)

It is shown the

complex exponentials are eigenfunctions of LTI systems and )(zH for a specific value of z is then the eigenvalues associated with the eigenfunctions nz. The example here shows the usefulness of decomposing general signals in terms of eigenfunctions for LTI system analysis:

Let tststseaeaeatx321

321)(++=, (3.10)

from the eigenfunction property, the response to each separately is tstsesHaea11)(1111® tstsesHaea22)(2222® tstsesHaea33)(3333® and from the superposition property the response to the sum is the sum of the responses, tststsesHaesHaesHaty321)()()()(333222111++=, (3.11) Generally, if the input is a linear combination of complex exponentials, =kts kkeatx)(, (3.12) the output will be

ELG 3120 Signals and Systems Chapter 3

4/3 Yao å

=kts kkkesHaty)()(, (3.13) Similarly for discrete-time LTI systems, if the input is =kn kkzanx][, (3.14) the output is =kn kkkzzHany)(][, (3.15)

3.3 Fourier Series representation of Continuous-Time Periodic Signals

3.31 Linear Combinations of harmonically Related Complex Exponentials

A periodic signal with period of T,

)()(Ttxtx+= for all t, (3.16) We introduced two basic periodic signals in Chapter 1, the sinusoidal signal ttx0cos)(w=, (3.17) and the periodic complex exponential tjetx0)(w=, (3.18) Both these signals are periodic with fundamental frequency

0w and fundamental period

0 /2wp=T. Associated with the signal in Eq. (3.18) is the set of harmonically related complex exponentials tTjktjk keet)/2(0)(pwf==, ......,2,1,0±±=k (3.19) Each of these signals is periodic with period of T(although for 2³k, the fundamental period of )(tkf is a fraction of T). Thus, a linear combination of harmonically related complex exponentials of the form

ELG 3120 Signals and Systems Chapter 3

5/3 Yao åå+¥

-¥===ktTjk k ktjk keaeatx)/2(0)(pw, (3.20) is also periodic with period of T.

· 0=k, )(tx is a constant.

· 1+=k and 1-=k, both have fundamental frequency equal to 0w and are collectively referred to as the fundamental components or the first harmonic components. · 2+=k and 2-=k, the components are referred to as the second harmonic components. · Nk+=and Nk-=, the components are referred to as the Nth harmonic components.

Eq. (3.20) can also be expressed as

==ktjk keatxtx0*)(*)(w, (3.21) where we assume that )(tx is real, that is, )(*)(txtx=.

Replacing k by k- in the summation, we have

-¥=-=ktjk keatx0*)(w, (3.22) which , by comparison with Eq. (3.20), requires that kkaa-=*, or equivalently kkaa-=*. (3.23) To derive the alternative forms of the Fourier series, we rewrite the summation in Eq. (2.20) as -++=1)/2(

00)(ktTjk

ktjk keaeaatxpw. (3.24)

Substituting

ka* for ka-, we have ++=1)/2( 0*)(0 ktTjk ktjk keaeaatxpw. (3.25) Since the two terms inside the summation are complex conjugate of each other, this can be expressed as +=100Re2)(ktjk keaatxw. (3.26)

ELG 3120 Signals and Systems Chapter 3

6/3 Yao If

ka is expressed in polar from as k j kkeAaq=, then Eq. (3.26) becomes +=1)(

00Re2)(ktkj

kkeAatxqw.

That is

++=100)cos(2)(kkktkAatxqw. (3.27) It is one commonly encountered form for the Fourier series of real periodic signals in continuous time.

Another form is obtained by writing

ka in rectangular form as kkkjCBa+= then Eq. (3.26) becomes -+=1000sincos2)(kkktkCtkBatxww. (3.28) For real periodic functions, the Fourier series in terms of complex exponential has the following three equivalent forms: -¥===ktTjk k ktjk keaeatx)/2(0)(pw ++=100)cos(2)(kkktkAatxqw -+=1000sincos2)(kkktkCtkBatxww

ELG 3120 Signals and Systems Chapter 3

7/3 Yao 3.3.2 Determination of the Fourier Series Representation of a Continuous-Time Periodic

Signal

Multiply both side of å+¥

-¥==ktjk keatx0)(w by tjne0w-, we obtain -¥=--=ktjntjk ktjneeaetx000)(www, (3.29)

Integrating both sides from 0 to

0/2wp=T, we have

tnkj k kT tjntjk kT tjndteadteeadtetx0)(

000000)(wwww, (3.30)

Note that

nknkTdteTtnkj ,0, 0)(0w

So Eq. (3.30) becomes

dtetxTa Ttjn n

ò-=00

)(1w, (3.31) The Fourier series of a periodic continuous-time signal -¥===ktTjk k ktjk keaeatx)/2(0)(pw (3.32) dtetxTdtetxTa TtTjk Ttjk k -==)/2()(1)(10pw (3.33) Eq. (3.32) is referred to as the Synthesis equation, and Eq. (3.33) is referred to as analysis equation. The set of coefficient {}k a are often called the Fourier series coefficients of the spectral coefficients of )(tx.

The coefficient

0a is the dc or constant component and is given with 0=k, that is

ELG 3120 Signals and Systems Chapter 3

8/3 Yao ò

=TdttxTa)(10 , (3.34)

Example: consider the signal ttx0sin)(w=.

tjtjejejt00 21

21sin0www--=.

Comparing the right-hand sides of this equation and Eq. (3.32), we have j a211 =, j a211

0=ka, 11-+¹ork

Example: The periodic square wave, sketched in the figure below and define over one period is <<<=2/,0,1)(11

TtTTttx, (3.35)

The signal has a fundamental period T and fundamental frequency T/20pw=. 1 T1T-2 TT 2

T-T-T2T2-)(tx

To determine the Fourier series coefficients for

)(tx, we use Eq. (3.33). Because of the symmetry of )(tx about 0=t, we choose 2/2/TtT££- as the interval over which the integration is performed, although any other interval of length T is valid the thus lead to the same result.

For 0=k,

TT dtTdttxTa T TT T1

021)(11

11 1 ===òò--, (3.36)

For 0¹k, we obtain

ELG 3120 Signals and Systems Chapter 3

9/3 Yao p

w wwww wwww k Tk TkTkj ee Tke

TjkdteTa

TjkTjkT

T tjkT Ttjk k )sin()sin(22 211
10 01000
10101
1 0 1 10 (3.37) The above figure is a bar graph of the Fourier series coefficients for a fixed

1T and several

values of T. For this example, the coefficients are real, so they can be depicted with a single graph. For complex coefficients, two graphs corresponding to the real and imaginary parts or amplitude and phase of each coefficient, would be required.

3.4 Convergence of the Fourier Series

If a periodic signal

)(tx is approximated by a linear combination of finite number of harmonically related complex exponentials

å-==N

Nktjk kNeatx0)(w. (3.38)quotesdbs_dbs17.pdfusesText_23

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