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MATLAB Session 5

MATLAB Session 5. ES 156 Signals and Systems 2007 HSEAS. Prepared by Frank Tompkins. Outline . Fourier transforms in MATLAB DTFT computation Two methods An example Digital filtering in MATLAB FIR and IIR. Fourier Transform. Fourier Transforms Continuous Time Fourier Transform (CTFT)

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MATLAB Session 5

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  1. MATLAB Session 5 ES 156 Signals and Systems 2007 HSEAS Prepared by Frank Tompkins

  2. Outline • Fourier transforms in MATLAB • DTFT computation • Two methods • An example • Digital filtering in MATLAB • FIR and IIR

  3. Fourier Transform • Fourier Transforms • Continuous Time Fourier Transform (CTFT) • Discrete Time Fourier Transform (DTFT)

  4. Fourier Transform Computation • How to compute transforms on a computer? • CTFT is continuous, so that’s out • Can we compute DTFT?

  5. DTFT Computation • Infinite length discrete time signals • DTFT defined on continuous frequency interval of length 2p; not computer friendly • What do we do? • Only deal with finite length signals • Recall: discrete time Fourier series has finitely many coefficients • So, we can use DTFS coefficients • More detail on this technique, called Discrete Fourier Transform, or DFT, later in the course • For now, just how to compute and plot “Fourier transform” in MATLAB • All the acronyms can be overwhelming, but we’ve seen almost all by now

  6. DTFT Computation (method 1) • Given finite length N signal, extend it to infinity by padding with zeros to make X[n] • DTFT X(ejw) has period 2p • Instead of storing X(ejw) for every possible w, we can store N evenly spaced values of w • Then, define X[k] = X(ej2pk/N) • The coefficients X[k] have period N • MATLAB command fft() computes X[k] • Side note: FFT (sorry, one more acronym: Fast Fourier Transform) is a popular algorithm that computes transforms quickly – not important now

  7. DTFT Computation (method 2) • Another perspective • Treat finite length N signal X[n] as a periodic (infinite length) signal Xp[n] with period N • Compute the DTFS of Xp[n], ak • Define X[k] = ak • Note: X[k] has period N, just like X[n] • These two approaches are equivalent • For now, think of DTFT/DFT in whichever way is easier for you (probably method 1)

  8. Example: DTFT of rectangular signal • Rectangular signal

  9. Example: DTFT of rectangular signal x=[zeros(1,10) ones(1,7) zeros(1,10)]; y=fft(x); figure; subplot(2,1,1); stem(x); subplot(2,1,2); plot(abs(y))

  10. Example: DTFT of rectangular signal

  11. Zero padding - finer scale in Fourier transform y2=fft(x,64); y3=fft(x,128); figure; subplot(3,1,1); plot(abs(y)) subplot(3,1,2); plot(abs(y2)) subplot(3,1,3); plot(abs(y3))

  12. fftshift() • By default, MATLAB’s fft() command outputs X(ej2pk/N) for 0 <= 2pk/N < 2p • To get the output into the standard range of –p to p, use the fftshift() command: my_xform = fftshift(fft(my_signal));

  13. Filters – FIR vs. IIR • A discrete time LTI filter can be expressed as a difference equation • If all a(i)’s are zero except a(1) we call it FIR, otherwise it’s IIR • MATLAB: z = filter(b,a,x)

  14. Filter example • Input: x[n] is the rectangular signal from before • The difference equation describing the filter is a=[1,-1]; b=1; z=filter(b,a,x) figure; subplot(2,1,1); stem(x); subplot(2,1,2) stem(z);

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