信号与系统奥本海姆Lab 4.pptxVIP

Lab 4: The Continuous-Time Fourier TransformWANG RuiAssociate ProfessorDepartment of Electrical Electronic Engineering2013-FallOverviewPage CT Fourier Transform (CTFT): How to calculate via Matlab?Frequency Response:How to calculate via Matlab? How to convert to impulse responseApplication of CTFTAnalysis of amplitude modulation (AM)CT Fourier Transform?Definition of CT Fourier transformFunction of continuous frequency Represent the “spectrum” of a signalPlot of CTFTChoose a sequence of frequency (sampling)Calculate integration for each (approximation)Plot each sample and do interpolationCan we reduce the complexity??XPage Another Approach?X CT Fourier Transform:Looks similarCan we use fft() to calculate the CTFT??DT Fourier Series:Page DerivationSuppose the dominant region of x(t) is in [0,T], then we can use the following approximation?X()=Slice into N intervals, each with length ?? It could be observed that?X()?fft() of [??(0), ??(??), ??(2??), …, ??((N-1)??)] Conclusion: we can use fft() to calculate CTFT approximately, which could reduce the computation complexityPage ExampleSingle rectangular wave with width=2T should be larger than 3tau=0.5; % 1/tau should be integerN=201; % N should be odd number, N*tau=T3x=[zeros(1,1/tau),ones(1,2/tau+1),zeros(1,N-3/tau-1)];y=fftshift(fft(x));X=tau*y;lb = (1-N)*pi/N/tau;ub = (N-1)*pi/N/tau;step = 2*pi/N/tau;plot(lb:step:ub, abs(X));What happen if we change tau and N? Why?Page Any problem in the approximation with fft()?Page What’s Frequency ResponseFrequency responseA function of frequencySystem gain in frequency domainMath definition Fourier transform of impulse responseCT LTI System: H(jw)???Page ?= ??Proof: Page Differential Equation and Frequency ResponseMatlab Function: freqs()Description: generate the frequency response of LTI systemSyntax: freqs(b,a,w)Example:High-pass or Low-pass??a=[1 3];b=3;w=linspace(0,4*pi);H=freqs(b,a,w);subplot(2,1,1), plot(w,abs(H));subplot(2,1,2), plot(w,angle(H));Page Frequency Response