10Discrete-TimeFourier Series

In this and the next lecture we parallel for discrete time the discussion of thelast three lectures for continuous time. Specifically, we consider the represen-tation of discrete-time signals through a decomposition as a linear combina-tion of complex exponentials. For periodic signals this representation be-comes the discrete-time Fourier series, and for aperiodic signals it becomesthe discrete-time Fourier transform.

The motivation for representing discrete-time signals as a linear combi-nation of complex exponentials is identical in both continuous time and dis-crete time. Complex exponentials are eigenfunctions of linear, time-invariantsystems, and consequently the effect of an LTI system on each of these basicsignals is simply a (complex) amplitude change. Thus with signals decom-posed in this way, an LTI system is completely characterized by a spectrum ofscale factors which it applies at each frequency.

In representing discrete-time periodic signals through the Fourier series,we again use harmonically related complex exponentials with fundamentalfrequencies that are integer multiples of the fundamental frequency of the pe-riodic sequence to be represented. However, as we discussed in Lecture 2, animportant distinction between continuous-time and discrete-time complexexponentials is that in the discrete-time case, they are unique only as the fre-quency variable spans a range of 27r. Beyond that, we simply see the samecomplex exponentials repeated over and over. Consequently, when we con-sider representing a periodic sequence with period N as a linear combinationof complex exponentials of the form efkOn with o = 27r/N, there are only Ndistinct complex exponentials of this type available to use, i.e., efoOn is period-ic in k with period N. (Of course, it is also periodic in n with period N.) In manyways, this simplifies the analysis since for discrete time the representation in-volves only N Fourier series coefficients, and thus determining the coeffi-cients from the sequence corresponds to solving N equations in N unknowns.The resulting analysis equation is a summation very similar in form to the syn-thesis equation and suggests a strong duality between the analysis and syn-thesis equations for the discrete-time Fourier transform. Because the basic

10-1

Signals and Systems10-2

complex exponentials repeat periodically in frequency, two alternative inter-pretations arise for the behavior of the Fourier series coefficients. One inter-pretation is that there are only N coefficients. The second is that the sequencerepresenting the Fourier series coefficients can run on indefinitely but re-peats periodically. Both interpretations, of course, are equivalent because ineither case there are only N unique Fourier series coefficients. Partly to retaina duality between a periodic sequence and the sequence representing itsFourier series coefficients, it is typically preferable to think of the Fourier se-ries coefficients as a periodic sequence with period N, that is, the same periodas the time sequence x(n). This periodicity is illustrated in this lecture throughseveral examples.

Partly in anticipation of the fact that we will want to follow an approachsimilar to that used in the continuous-time case for a Fourier decompositionof aperiodic signals, it is useful to represent the Fourier series coefficients assamples of an envelope. This envelope is determined by the behavior of thesequence over one period but is not dependent on the specific value of the pe-riod. As the period of the sequence increases, with the nonzero content in theperiod remaining the same, the Fourier series coefficients are samples of thesame envelope function with increasingly finer spacing along the frequencyaxis (specifically, a spacing of 2ir/N where N is the period). Consequently, asthe period approaches infinity, this envelope function corresponds to a Four-ier representation of the aperiodic signal corresponding to one period. This is,then, the Fourier transform of the aperiodic signal.

The discrete-time Fourier transform developed as we have just describedcorresponds to a decomposition of an aperiodic signal as a linear combina-tion of a continuum of complex exponentials. The synthesis equation is thenthe limiting form of the Fourier series sum, specifically an integral. The analy-sis equation is the same one we used previously in obtaining the envelope ofthe Fourier series coefficients. Here we see that while there was a duality inthe expressions between the discrete-time Fourier series analysis and synthe-sis equations, the duality is lost in the discrete-time Fourier transform sincethe synthesis equation is now an integral and the analysis equation a summa-tion. This represents one difference between the discrete-time Fourier trans-form and the continuous-time Fourier transform. Another important differ-ence is that the discrete-time Fourier transform is always a periodic functionof frequency. Consequently, it is completely defined by its behavior over a fre-quency range of 27r in contrast to the continuous-time Fourier transform,which extends over an infinite frequency range.

Suggested ReadingSection 5.0, Introduction, pages 291-293Section 5.1, The Response of Discrete-Time LTI Systems to Complex Expo-

nentials, pages 293-294Section 5.2, Representation of Periodic Signals: The Discrete-Time Fourier

Series, pages 294-306

Section 5.3, Representation of Aperiodic Signals: The Discrete-Time FourierTransform, pages 306-314

Section 5.4, Periodic Signals and the Discrete-Time Fourier Transform, pages314-321

Discrete-Time Fourier Series10-3

MARKERBOARD10.11

Note that dt should be added at the end of the last equation in column 1. MARKERBOARD10.2

Signals and Systems

10-4

TRANSPARENCY10.1Example of theFourier seriescoefficients for adiscrete-time periodicsignal.

Example 5.2:

x[n] = 1 + sin £ 0n + 3 cos 2 0 n

+ cos (2E2 0n + )

Re ak iak

3

kN 0 N

*-IIi I:, a1

"1--

0/

N 0 N

<%ak

ir/2

0-TI - - --I - N

TRANSPARENCY10.2Comparison of theFourier seriescoefficients for adiscrete-time periodicsquare wave and acontinuous-timeperiodic square wave.

k

k

Envelope: sin[(2N1 + 1) 92/2]sin (W/2)

-F.1N - 20

Na.

Nao

0 27r20

Discrete-Time Fourier Series10-5

TRANSPARENCY10.3Illustration of thediscrete-time Fourierseries coefficients assamples of an enve-lope. Transparencies10.3-10.5 demonstratethat as the periodincreases, theenvelope remains thesame and the samplesrepresenting theFourier seriescoefficients becomemore closely spaced.Here, N = 10.

TRANSPARENCY10.4N = 20.

Signals and Systems

10-6

TRANSPARENCY10.5N = 40.

TRANSPARENCY10.6A review of theapproach todeveloping a Fourierrepresentation foraperiodic signals.

Nao

Envelope: sin[(2N1 + 1) 92/2]sin (92/2)

Afr~r . 2i

1. x(t) APERIODIC

- construct periodic signal X(t) for

which one period is x(t)

- x(t) has a Fourier series

- as period of x(t) increases,

x(t) -- x(t) and Fourier series of

x(t) Fourier Transform of x(t)

N - 40

Discrete-Time Fourier Series10-7

1. x[n] APERIODIC

- construct periodic signal x[n]for

which one period is x[n]

- x[n]has a Fourier series

- as period of x[n] increases,

i I[n] -.-x[n] and Fourier series of

x[n] -- Fourier Transform of x[n]

FOURIER REPRESENTATION OF APERIODIC SIGNALS

inJ

... *jaaTTT?? nTTITTT.?TTTATTTTTTL??T????TTrlTT?.-N -N1 0 N1 N n

x[n] = x[n] In I < 2

As N -+oo x[n] o x[n]

- let N -+ oo to represent x[n]

- use Fourier series to represent x[n]

TRANSPARENCY10.7A summary of theapproach to be used toobtain a Fourierrepresentation ofdiscrete-timeaperiodic signals.

TRANSPARENCY10.8Representation of anaperiodic signal as thelimiting form of aperiodic signal withthe period increasing.

Signals and Systems

10-8

TRANSPARENCY10.9Transparencies 10.9-10.11 illustrate howthe Fourier seriescoefficients for aperiodic signalapproach thecontinuous envelopefunction as the periodincreases. Here, N =10. [Example 5.3 fromthe text.]

Example 5.3:

-NI?.. - N 0-N -N, 0 N,

//

UN

Nao

Envelope: sin[ (2N 1 + 1) /2]sin (W/2)

Nai

0 2r10

TRANSPARENCY10.10N = 20. [Example 5.3from the text.]

Example 5.3:

-N

Nao

-'T NNai//

0 2w20

-N,

Envelope: sin( [(2N 1 + 1) R/2]

sin (fl/2)

n

N = 10

1N

-.-- n

N - 20

I

I* A'.ft - -

- W

6 0 9 -1

Discrete-Time Fourier Series

10-9

-N

Na1

-1 1 N1 N11? ...-N1 0 N1 N

Envelope: sin [(2NI + 1) 2/2)sin (U/2)

N - 40

TRANSPARENCY10.11N = 40. [Example 5.3from the text.]

in] = Ik=<N>

X(kg2o) = N ak =

As N oo

Fourier Transform:

x[n] = 2'r 2ir

1XWkn) e jkE2Gn go

n= N/2

n / x[n] e-jk on

n=-N/2

X(92) ein" d92

+oo

X(92) = x[n] e-in"n =- oc

TRANSPARENCY10.12Limiting form of theFourier series as theperiod approachesinfinity. [The upperlimit in the summa-tion in the secondequation should ben = (N/2) - 1.]

//

Signals and Systems

10-10

TRANSPARENCY10.13The analysis andsynthesis equationsfor the discrete-timeFourier transform. [Ascorrected here, x[n],not x(t), has Fouriertransform X(D).]

DISCRETE-TIME FOURIER TRANSFORM

x [n] - X(E2) en" d2 7

+ o

X (2) =n=-oo

x[n]

synthesis

analysis

x[n] < XG

X(2) = e X(2) + j Im I X(&)4

= |X(S2)1 ei"(2)

TRANSPARENCY10.14The discrete-timeFourier transform fora rectangular pulse.

X (W)

27r 1-2

x [n]

I IoI-2 0 2 n

-27r

Discrete-Time Fourier Series

10-11

x[n]= an u[n]

O < a < 1

an u[n]e-inn

an e-j2n

TRANSPARENCY10.15The discrete-timeFourier transform foran exponentialsequence.

anu[n] 0<a<1

1a e-iW

1 X()| I

27r 92

, X(W)

tan~ (a/v'Y 2)

TRANSPARENCY10.16Illustration of themagnitude and phaseof the discrete-timeFourier transform foran exponentialsequence. [Note that ais real.]

X( W)+00

n=-00n= 0

11 -a 2

x[n]

X(2)

-2r -7 0

Signals and Systems10-12

TRANSPARENCY10.17A review of somerelationships for theFourier transformassociated withperiodic signals.

2. R(t) PERIODIC, x(t) REPRESENTS ONE PERIOD

- Fourier series coefficients of 2(t)

= (1/T) times samples of Fourier.

transform of x(t)

3. 2*(t) PERIODIC

-Fourier transform of x(t) defined as

impulse train:

+oo

X(W) = 27rak 6 (o - ko)

TRANSPARENCY10.18A summary of somerelationships for theFourier transformassociated withperiodic sequences.

2. '[n] PERIODIC, x[n] REPRESENTS ONE PERIOD

- Fourier series coefficients of 2[n]

= (1/ N ) times samples of Fourier

transform of x [n]

3. X[n] PERIODIC

-Fourier transform of x[n]defined as

impulse train:

27rak 5 (2 - k2 0 )

k=I-oo0

X(n) =

Discrete-Time Fourier Series

10-13

Fourier series coefficients equal1- times samples of Fourier

transform of one period

x~nJ

??TYATTTT?flT??,??TTTT??-N, 0 N1 N n

x [n] 6 one period of x[n]

x [n]

x [n]o ak

0X (9)

k N 2irkN

TRANSPARENCY10.19The relationshipbetween the Fourierseries coefficients of aperiodic signal and theFourier transform ofone period.

TRANSPARENCY10.20Illustration of therelationship inTransparency 10.19.

-N

Signals and Systems10-14

TRANSPARENCY10.21A summary of somerelationships for theFourier transformassociated withperiodic sequences.[Transparency 10.18repeated]

TRANSPARENCY10.22Illustration of theFourier seriescoefficients and theFourier transform fora periodic squarewave.

2. *[n] PERIODIC , x[n] REPRESENTS ONE PERIOD

- Fourier series coefficients of *[n]

= (1/ N ) times samples of Fourier

transform of x [n]

3. #x[n] PERIODIC

-Fourier transform of xIn]defined as

impulse train:

27rak 5(2 - k2 0 )

..1111IN1 0 N1

0 1

~T1111"

0 20

11111

__ -11111

X (9) =

n

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