Lecture 10: Discrete-time Fourier Series
10Discrete-TimeFourier SeriesIn this and the next lecture we parallel for discrete time the discussion of thelast three lectures for continuous time. Specifically, we consider the representation of discrete-time signals through a decomposition as a linear combination of complex exponentials. For periodic signals this representation becomes 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 combination of complex exponentials is identical in both continuous time and discrete 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 decomposed 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 periodic 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 frequency variable spans a range of 27r. Beyond that, we simply see the samecomplex exponentials repeated over and over. Consequently, when we consider 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 periodic 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 involves only N Fourier series coefficients, and thus determining the coefficients from the sequence corresponds to solving N equations in N unknowns.The resulting analysis equation is a summation very similar in form to the synthesis equation and suggests a strong duality between the analysis and synthesis equations for the discrete-time Fourier transform. Because the basic10-1
Signals and Systems10-2complex exponentials repeat periodically in frequency, two alternative interpretations arise for the behavior of the Fourier series coefficients. One interpretation is that there are only N coefficients. The second is that the sequencerepresenting the Fourier series coefficients can run on indefinitely but repeats 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 series 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 period. 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 Fourier 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 combination of a continuum of complex exponentials. The synthesis equation is thenthe limiting form of the Fourier series sum, specifically an integral. The analysis 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 synthesis equations, the duality is lost in the discrete-time Fourier transform sincethe synthesis equation is now an integral and the analysis equation a summation. This represents one difference between the discrete-time Fourier transform and the continuous-time Fourier transform. Another important difference is that the discrete-time Fourier transform is always a periodic functionof frequency. Consequently, it is completely defined by its behavior over a frequency 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 Exponentials, pages 293-294Section 5.2, Representation of Periodic Signals: The Discrete-Time FourierSeries, pages 294-306Section 5.3, Representation of Aperiodic Signals: The Discrete-Time FourierTransform, pages 306-314Section 5.4, Periodic Signals and the Discrete-Time Fourier Transform, pages314-321
Discrete-Time Fourier Series10-3MARKERBOARD10.11Note that dt should be added at the end of the last equation in column 1.MARKERBOARD10.2
Signals and Systems10-4Example 5.2:x[n] 1 sin 0 n 3 cos 2 0 n cos (2E2 0 n Re akTRANSPARENCY10.1Example of theFourier seriescoefficients for adiscrete-time periodicsignal.)iak3"1-0/kN0NkN0N %akir/2*-IIiI:,a1- -TRANSPARENCY10.2Comparison of theFourier seriescoefficients for adiscrete-time periodicsquare wave and acontinuous-timeperiodic square wave.0-TI--I-kN
Discrete-Time Fourier Series10-5TRANSPARENCY10.3Illustration of thediscrete-time Fourierseries coefficients assamples of an envelope. Transparencies10.3-10.5 demonstratethat as the periodincreases, theenvelope remains thesame and the samplesrepresenting theFourier seriescoefficients becomemore closely spaced.Na.Here, N 10.TRANSPARENCY10.4N 20.NaoEnvelope:0 27r20sin[(2N 1 1) 92/2]sin (W/2)-F.1N- 20
Signals and Systems10-6TRANSPARENCY10.5N 40.NaoEnvelope:Afr rsin[(2N 1 1) 92/2]sin (92/2).N - 402i1. x(t) APERIODICTRANSPARENCY10.6A review of theapproach todeveloping a Fourierrepresentation foraperiodic signals.- construct periodic signal X(t) forwhich one period isx(t)- x(t) has a Fourier series-as period of x(t) increases,x(t) -- x(t) and Fourier series ofx(t)Fourier Transform of x(t)
Discrete-Time Fourier Series10-71. x[n] APERIODICTRANSPARENCY10.7A summary of theapproach to be used toobtain a Fourierrepresentation ofdiscrete-timeaperiodic signals.- construct periodic signal x[n]forwhich one period isx[n]- x[n]has a Fourier series- as period of x[n] increases,i I[n] -.-x[n] and Fourier series ofx[n]--Fourier Transform of x[n]FOURIER REPRESENTATION OF APERIODIC SIGNALSTRANSPARENCY10.8Representation of anaperiodic signal as thelimiting form of aperiodic signal withthe period N-N10x[n] x[n]As N - oo- let N- N1In I 2x[n]o x[n]oo to represent x[n]- use Fourier series to represent x[n]Nn
Signals and Systems10-8Example 5.3:TRANSPARENCY10.9Transparencies 10.910.11 illustrate howthe Fourier seriescoefficients for aperiodic signalapproach thecontinuous envelopefunction as the periodincreases. Here, N 10. [Example 5.3 fromthe text.]60 9U-NI?. - N0-1-N-N,0- WNN,nNao/Envelope:/sin[ (2N 1 1)N 10/2]sin (W/2)Nai02r10Example 5.3:TRANSPARENCY10.10N 20. [Example 5.3from the text.]I*-NA'.ft---N,1-.--nNNao// -'TNNai0 2w20Envelope:sin( [(2N 1 1) R/2]sin (fl/2)N - 20I
Discrete-Time Fourier Series10-9-NNa 1/-11 N1N11?.-N1N0Envelope:/in] N11XWkn)Ik N e jkE2Gn gon /n -N/2N - 40TRANSPARENCY10.12n N/2As N10.11N 40. [Example 5.3from the text.]sin [(2NI 1) 2/2)sin (U/2)X(kg2o) N ak TRANSPARENCYx[n] e-jk onooLimiting form of theFourier series as theperiod approachesinfinity. [The upperlimit in the summation in the secondequation should ben (N/2) - 1.]Fourier Transform:x[n]X(92) 2'r 2ir ooX(92) ein" d92x[n] e-in"n -oc
Signals and Systems10-10DISCRETE-TIME FOURIER TRANSFORMTRANSPARENCY10.13The analysis andsynthesis equationsfor the discrete-timeFourier transform. [Ascorrected here, x[n],not x(t), has Fouriertransform X(D).]-x [n]2X(E2) en"dsynthesisx[n]analysis7 oX (2) n -oox[n] X(2) XGe X(2) j Im I X(&)4 X(S2)1 ei"(2)oIIIx [n]TRANSPARENCY10.14The discrete-timeFourier transform fora rectangular pulse.-20n2X (W)-27r27r1-2
Discrete-Time Fourier Series10-11x[n] an u[n]O a 1TRANSPARENCY10.15The discrete-timeFourier transform foran exponentialsequence. 00X(W)an u[n]e-innn -00an e-j2nn 011 -ax[n]2anu[n]0 a 1X(2)1a e-iW1X() ITRANSPARENCY10.16-2r-727r0, X(W)tan (a/v'Y 2)92Illustration of themagnitude and phaseof the discrete-timeFourier transform foran exponentialsequence. [Note that ais real.]
Signals and Systems10-122. R(t) PERIODIC, x(t) REPRESENTS ONE PERIODTRANSPARENCY10.17A review of somerelationships for theFourier transformassociated withperiodic signals.Fourier series coefficients of 2(t)times samples of Fourier. (1/T)transform of x(t)3. 2*(t) PERIODIC-Fourier transform of x(t) defined asimpulse train: ooX(W)27rak 6 (o - ko) k I-oo02. '[n] PERIODIC, x[n] REPRESENTS ONE PERIOD- Fourier series coefficients of 2[n]TRANSPARENCY10.18A summary of somerelationships for theFourier transformassociated withperiodic sequences. (1/ N ) times samples of Fouriertransform of x [n]3. X[n] PERIODIC-Fourier transform of x[n]defined asimpulse train:X(n) 27rak 5 (2 - k20 )
Discrete-Time Fourier Series10-13Fourier series coefficients equal1- times samples of Fouriertransform of one periodTRANSPARENCY10.19The relationshipbetween the Fourierseries coefficients of aperiodic signal and theFourier transform ofone period.x nJ-N?TYATTTT?flT?,?TTTT-N,x [n]06 one period ofx [n]N1Nnx[n]o akx [n]0X (9)kN2irkNTRANSPARENCY10.20Illustration of therelationship inTransparency 10.19.
Signals and Systems10-142. *[n] PERIODIC , x[n] REPRESENTS ONE PERIOD- Fourier series coefficients of *[n]TRANSPARENCY10.21A summary of some (1/ N ) times samples of Fourierrelationships for theFourier transformassociated withperiodic sequences.[Transparency 10.18repeated]transform of x [n]3. #x[n] PERIODIC-Fourier transform of xIn]defined asimpulse train:X (9) .1111I27rak 5(2 - k20 )11111N1 0N1 T1111"TRANSPARENCY10.22Illustration of theFourier seriescoefficients and theFourier transform fora periodic squarewave.01020-11111n
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Discrete-Time Fourier Series In this and the next lecture we parallel for discrete time the discussion of the last 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 e
Introduction of Chemical Reaction Engineering Introduction about Chemical Engineering 0:31:15 0:31:09. Lecture 14 Lecture 15 Lecture 16 Lecture 17 Lecture 18 Lecture 19 Lecture 20 Lecture 21 Lecture 22 Lecture 23 Lecture 24 Lecture 25 Lecture 26 Lecture 27 Lecture 28 Lecture
2.1 Sampling and discrete time systems 10 Discrete time systems are systems whose inputs and outputs are discrete time signals. Due to this interplay of continuous and discrete components, we can observe two discrete time systems in Figure 2, i.e., systems whose input and output are both discrete time signals.
Gambar 5. Koefisien Deret Fourier untuk isyarat kotak diskret dengan (2N1 1) 5, dan (a) N 10, (b) N 20, dan (c) N 40. 1.2 Transformasi Fourier 1.2.1 Transformasi Fourier untuk isyarat kontinyu Sebagaimana pada uraian tentang Deret Fourier, fungsi periodis yang memenuhi persamaan (1) dapat dinyatakan dengan superposisi fungsi sinus dan kosinus.File Size: 568KB
Jan 03, 2011 · Section 1.2.2, we recall the discrete-time Fourier transform for discrete-time, aperi-odic signals. The duality between these two situations is then readily apparent. In Section 1.2.3, we motivate the continuous-time Fourier transform by examin-ing the limiting form of the Fourier-series representation o
2.1 Discrete-time Signals: Sequences Continuous-time signal - Defined along a continuum of times: x(t) Continuous-time system - Operates on and produces continuous-time signals. Discrete-time signal - Defined at discrete times: x[n] Discrete-time system - Operates on and produces discrete-time signals. x(t) y(t) H (s) D/A Digital filter .
(d) Fourier transform in the complex domain (for those who took "Complex Variables") is discussed in Appendix 5.2.5. (e) Fourier Series interpreted as Discrete Fourier transform are discussed in Appendix 5.2.5. 5.1.3 cos- and sin-Fourier transform and integral Applying the same arguments as in Section 4.5 we can rewrite formulae (5.1.8 .
6 POWER ELECTRONICS SEGMENTS INCLUDED IN THIS REPORT By device type SiC Silicon GaN-on-Si Diodes (discrete or rectifier bridge) MOSFET (discrete or module) IGBT (discrete or module) Thyristors (discrete) Bipolar (discrete or module) Power management Power HEMT (discrete, SiP, SoC) Diodes (discrete or hybrid module)
Section 501 SECTION 501 5-3 1 2 LIME-TREATED SOIL 3 501-1 DESCRIPTION 4 Perform the work covered by this section including, but not limited to, treating the subgrade, 5 embankment, natural ground or existing pavement structure by adding water and lime in the 6 form specified herein, mixing, shaping, compacting and finishing the mixture to the required 7 density. Prepare the soil layer to be .
