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BU EECE 301 - Note Set #23

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EECE 301 Signals & Systems Prof. Mark Fowler1/17EECE 301 Signals & SystemsProf. Mark FowlerNote Set #23• D-T Signals: DTFT Details• Reading Assignment: Section 4.1 of Kamen and Heck2/17Ch. 1 IntroC-T Signal ModelFunctions on Real LineD-T Signal ModelFunctions on IntegersSystem PropertiesLTICausalEtcCh. 2 Diff EqsC-T System ModelDifferential EquationsD-T Signal ModelDifference EquationsZero-State ResponseZero-Input ResponseCharacteristic Eq.Ch. 2 ConvolutionC-T System ModelConvolution IntegralD-T System ModelConvolution SumCh. 3: CT Fourier SignalModelsFourier SeriesPeriodic SignalsFourier Transform (CTFT)Non-Periodic SignalsNew System ModelNew SignalModelsCh. 5: CT Fourier SystemModelsFrequency ResponseBased on Fourier TransformNew System ModelCh. 4: DT Fourier SignalModelsDTFT(for “Hand” Analysis)DFT & FFT(for Computer Analysis)New SignalModelPowerful Analysis ToolCh. 6 & 8: Laplace Models for CTSignals & SystemsTransfer FunctionNew System ModelCh. 7: Z Trans.Models for DTSignals & SystemsTransfer FunctionNew SystemModelCh. 5: DT Fourier System ModelsFreq. Response for DTBased on DTFTNew System ModelCourse Flow DiagramThe arrows here show conceptual flow between ideas. Note the parallel structure between the pink blocks (C-T Freq. Analysis) and the blue blocks (D-T Freq. Analysis).3/17Sect 4.1 continued: The DetailsnjnenxXΩ−∞−∞=∑=Ω ][)(In rad/sampleradians∫∞∞−−= dtetxXtjωω)()(In rad/secradiansCompare to CTFT:Define the DTFT:Very similar structure… so we should expect similar properties!!!4/17Example of Analytically Computing the DTFTn][nx-3 -2 -1 1 2 3 4 5 6 With your brain, not a computer⎪⎩⎪⎨⎧>≤≤<=qnqnannxn,00,0,0][q = 3oscillates ][,0explodes"" ][,1decays ][,1nxaIfnxaIfnxaIf<><By definition:∑∞−∞=Ω−=ΩnnjenxX ][)(∑∑=Ω−=Ω−==qnnjqnnjnaeea00)(Given this signal model, find the DTFT. ()Ω−+Ω−−−=ΩjqjaeaeX11)(1rrrrqqqqnn−−=+=∑112121General Form for Geometric Sum:5/17Characteristics of DTFT1.Periodicity of X(Ω)X(Ω) is a periodic function of Ω with period of 2π)()2(Ω=+Ω⇒ XXπRecall pictures in notes of “DTFT Intro”:Note: the CTFT does nothave this property2. X(Ω) is complex valued (in general)∑Ω−=ΩnnjenxX ][)(complexUsually think of X(Ω) in polar form:)()()(Ω∠Ω=ΩXjeXXmagnitudephaseSameas CTFT⇒|X(Ω)| is periodic with period 2π∠X(Ω) is periodic with period 2π6/173. SymmetryIf x[n] is real-valued, then:)()( Ω=Ω− XX)()(Ω−∠=Ω−∠ XX(even symmetry)(odd symmetry)Same as CTFTInverse DTFTQ: Given X(Ω) can we find the corresponding x[n]?A: Yes!!∫−ΩΩΩ=πππdeXnxjn)(21][We can integrate instead over anyinterval of length 2π…because the DTFT is periodic with period 2π7/17Generalized DTFTPeriodic D-T signals have DTFT’s that contain delta functions⎪⎩⎪⎨⎧<Ω<−Ω=Ω↔∀=elsewhereperiodicXnnx,),(2)(,1][πππδWith a period of 2πX(Ω)π2π2π2π2π2Ωππ−π2−π2π3π3−Main part∑∞−∞=−Ω=ΩkkX )2(2)(πδπAnother way of writing this is:Example:8/17How do we derive the result? Work backwards!∫−ΩΩΩ=πππdeXnxjn)(21][∫−ΩΩΩ=πππδπdejn)(2210⋅=jne1=Sifting property9/17Transform Pairs: Like for the CTFT, there is a table of common pairs (See Web)Be familiar with themCompare and contrast them with the tableOf common CTFT’sTable 3.2Table 4.1Careful here… the book’s table doesn’t have this subscript… see next slide.10/17DTFT of a Rectangular Pulse (Ex. 4.3)⎪⎩⎪⎨⎧−−==otherwiseqqnnpq,0,,1,0,1,,,1][ as pulse T- D:Define……Book doesn’t use this subscript!∑−=Ω−=ΩqqnjnqeP )(So, by DTFT definition:Use “Geometric Sum” Result…see Eq. (4.5){}{}2/sin)2/1(sin1)()1(ΩΩ+=−−=ΩΩ−Ω+−ΩqeeePjqjjqqSee book for detailsUse DTFT Tables on my Website11/17Properties of the DTFT (See table on my website)Like for the CTFT, there are many properties for the DTFT. Most are identical to those for the CTFT!!But Note: “Summation Property” replaces IntegrationThere is no “Differentiation Property”Compare and contrast these with the table of CTFT propertiesMost important ones:-Time shift-Multiplication by sinusoid… Three “flavors”-Convolution in the time domain-Parseval’s Theorem12/17Table 3.1Table 4.2This one has no equivalent on CTFT Properties Table…See next exampleIt provides a way to use a CTFT table to find DTFT pairs…here is an exampleUse the Tables on my Web Site!!!13/17Example 4.7: Finding a DTFT pair from a CTFT pair π 2π-π-2πΩX(Ω)Say we are given this DTFT and want to invert it…The four steps for using “Relationship to Inverse CTFT” property are:1. Truncate the DTFT X(Ω) to the -π to π range and set it to zero elsewhere2. Then treat the resulting function as a function of ω… call this Γ(ω)B-BBook’s picture is not quite correct… the “B”is in the wrong placeπ 2π-π-2πωΓ(ω) = X(ω)p2π(ω)B-BΓ(ω) = X(ω)p2π(ω)14/174. Get the x[n] by replacing t by n inγ(t) From CTFT table:⎟⎠⎞⎜⎝⎛= tBBtππγsinc)(⎟⎠⎞⎜⎝⎛===nBBtnxntππγsinc)(][3. Find the inverse CTFT of Γ(ω) from a CTFT table, call it γ(t)15/17Example of DTFT of sinusoid?)()cos(][0=Ω↔Ω=XnnxNote that:)cos(1][0nnxΩ×=⎪⎩⎪⎨⎧−<Ω<−Ω=Ωelsewhereperiodic2π),(2)(πππδYSo… use the “mult. by sinusoid”property1][ ==Δny∑∞−∞=−Ω=ΩkkY )2(2)(πδπFrom DTFT TableAnother way of writing this:Y(Ω)Ωk = 0termk = 1termk = 2termk = -2termk = -1termπ 2π 3π 4π-π-2π-3π-4π16/17[])()(21)(00Ω−Ω+Ω+Ω=Ω⇒ YYX“mult. by sinusoid”property says we shift up & down by Ω0[]⎪⎩⎪⎨⎧−<Ω<−Ω−Ω+Ω+Ω=ΩelsewhereperiodicXπππδδπ2,)()()(00Recall:)cos(1][0nnxΩ×= so we can use the “mult. by sinusoid” resultUsing the second form for Y(Ω) gives:Or…using the first form for Y(Ω) gives:[]∑∞−∞=−Ω−Ω+−Ω+Ω=ΩkkkY )2()2()(00πδπδπ17/17To see this graphically:⎪⎩⎪⎨⎧−<Ω<−Ω=Ωelsewhereperiodic2π),(2)(πππδY[]⎪⎩⎪⎨⎧−<Ω<−Ω−Ω+Ω+Ω=ΩelsewhereperiodicXπππδδπ2,)()()(00Ω0-Ω0Red comes from Up-shifted Y(Ω)Blue comes from Down-shifted Y(Ω)X(Ω)Ωπ 2π 3π 4π-π-2π-3π-4πY(Ω)Ωπ 2π 3π


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BU EECE 301 - Note Set #23

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