EECE 301 Signals & Systems Prof. Mark Fowler1/27EECE 301 Signals & SystemsProf. Mark FowlerNote Set #14• C-T Signals: Fourier Transform (for Non-Periodic Signals)• Reading Assignment: Section 3.4 & 3.5 of Kamen and Heck2/27Ch. 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 Signal ModelsCh. 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).3/274.3 Fourier TransformRecall: Fourier Series represents a periodic signal as a sum of sinusoidsNote: Because the FS uses “harmonically related” frequencies kω0, it can only create periodicsignals∑∞−∞==ktjkkectxω)(or complex sinusoidstjke0ωWith arbitrary discrete frequencies…NOTharmonically related∑∞−∞==ktjkkectxω)(The problem with is that it cannot include all possible frequencies!Q: Can we modify the FS idea to handle non-periodic signals?A:Yes!! What about ?That will give some non-periodic signals but not some that are important!!4/27How about:∫∞∞−=ωωπωdeXtxtj)(21)(Called the “Fourier Integral” also, more commonly, called the “Inverse Fourier Transform”Plays the role of ckPlays the role oftjke0ωIntegral replaces sum because it can “add up over the continuum of frequencies”!Okay… given x(t) how do we get X(ω)?∫∞∞−−= dtetxXtjωω)()(Note: X(ω) is complex-valued function of ω∈ (-∞, ∞)|X(ω)|)(ωX∠Yes… this will work for any practicalnon-periodic signal!!Called the “Fourier Transform”of x(t)Need to use two plots to show it5/27Comparison of FT and FSFourier Series: Used for periodic signalsFourier Transform: Used for non-periodic signals (although we will see later that it can also be used for periodic signals)∑∞−∞==ntjkkectx0)(ω∫+−=TtttjkkdtetxTc000)(1ω∫∞∞−=ωωπωdeXtxtj)(21)(∫∞∞−−= dtetxXtjωω)()(Synthesis AnalysisFourierSeriesFourier Series Fourier CoefficientsFourierTransformInverse Fourier Transform Fourier TransformFS coefficients ckare a complex-valued function of integer kFT X(ω) is a complex-valued function of the variable ω∈ (-∞, ∞)6/27Synthesis Viewpoints:We need two plots to show these∑∞−∞==ntjkkectx0)(ω|X(ω)| shows how much there is in the signal at frequency ω∠X(ω) shows how much phase shift is needed at frequency ω∫∞∞−=ωωπωdeXtxtj)(21)(We need two plots to show theseFS:|ck| shows how much there is of the signal at frequency kω0∠ckshows how much phase shift is needed at frequency kω0FT:7/27Some FT Notation:)()(ωXtx ↔1.If X(ω) is the Fourier transform of x(t)…then we can write this in several ways:{})()( txX F=ω2.⇒ F{ } is an “operator” that operates on x(t) to give X(ω)⇒ F-1{ } is an “operator” that operates on X(ω) to give x(t){})()(1ωXtx−= F3.8/27Analogy: Looking at X(ω) is “like” looking at an x-ray of the signal- in the sense that an x-ray lets you see what is inside the object… shows what stuff it is made from.In this sense:X(ω) shows what is “inside” the signal – it shows how much of each complex sinusoid is “inside” the signalNote: x(t) completely determines X(ω)X(ω) completely determines x(t)There are some advanced mathematical issues that can be hurled at these comments… we’ll not worry about them9/27FT Example: Decaying ExponentialGiven a signal x(t) = e-btu(t) find X(ω) if b > 0 Now…apply the definition of the Fourier transform. Recall the general form:dtetxXtj∫∞∞−−=ωω)()(1)(txtb controls decay rateThe u(t) part forces this to zeroWhat does this look like if b < 0???Solution: First see what x(t) looks like:10/27dtetueXtjbt∫∞∞−−−=ωω)()(Easy integral![]0)()(0)(11ωωωωωjbjbtttjbeejbejb+−∞+−∞==+−−+−=⎥⎦⎤⎢⎣⎡+−=[]101−+−=ωjbNow plug in for our signal:integrand = 0 for t < 0 due to the u(t)dtedteetjbtjbt∫∫∞+−∞−−==0)(0ωωSet lower limit to 0 and then u(t) = 1 over integration rangeωjb +=1NN⎥⎥⎦⎤⎢⎢⎣⎡−+−===∞−=∞−10101eeejbmagjbωωOnly if b>0… what happens if b<011/27ωωjbX+=1)(221)(ωω+=bX⎟⎠⎞⎜⎝⎛−=∠−bXωω1tan)((Complex Valued)MagnitudePhase)()( tuetxbt−=For b > 01)()( tuetxbt−=tb > 0 controls decay rate)(ωXωSummary of FT Result for Decaying Exponential12/27MATLAB Commands to Compute FTw=-100:0.2:100;b=10;X=1./(b+j*w);Plotting Commandssubplot(2,1,1); plot(w,abs(X))xlabel('Frequency \omega (rad/sec)')ylabel('|X(\omega|) (volts)'); gridsubplot(2,1,2); plot(w,angle(X))xlabel('Frequency \omega (rad/sec)')ylabel('<X(\omega) (rad)'); gridFourier Transform of e-btu(t) for b = 10Note that magnitude plot has evensymmetryNote that phase plot has oddsymmetryTrue for everyreal-valued signalNote: Book’s Fig. 3.12 only shows one-sided spectrum plots(volts)Technically V/Hz13/27-10 0 10 20 30 4000.51t (sec)x(t)-100 -50 0 50 1000510ω (rad/sec)|X(ω)|-10 0 10 20 30 4000.51t (sec)x(t)-100 -50 0 50 10000.51ω (rad/sec)|X(ω)|-10 0 10 20 30 4000.51t (sec)x(t)-100 -50 0 50 10000.050.1ω (rad/sec)|X(ω)|b=0.1 b=0.1 b=1 b=1 b=10 b=10 Note: As b increases…1. Decay rate in time signal increases 2. High frequencies in Fourier transform are more prominent.Time Signal Fourier TransformExploring Effect of decay rate bon the Fourier Transform’sShapeShort Signals have FTs that spread more into High Frequencies!!!14/27Example: FT of a Rectangular pulseGiven: a rectangular pulse signal pτ(t)t2τ2τ−)(tpττ= pulse widthRecall: we use this symbol to