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Consider two continuous time functions and . In a particular signal processing application we are required to perform . In CE#2, you learned how to convolve two continuous time functions using the conv command in MATLAB. In this CE, let us follow the DFT approach to perform convolution.

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Sampling and Discrete Fourier Transform (100/100 points)

Consider two continuous time functions and . In a particular signal processing application we are required to perform . In CE#2, you learned how to convolve two continuous time functions using the conv command in MATLAB. In this CE, let us follow the DFT approach to perform convolution. In MATLAB, we can implement the DFT using a special version of the DFT called the fft (fast fourier transform).

  1. Create x[n] and y[n], sampled versions of x(t) and y(t) with a sampling period T = .001 seconds. Generate x[n] and y[n] in MATLAB using the onescommand. As an example, ones(1,101) generates an array of length 101 with all entries 1. Do not zero-pad your arrays (16 points)
  2. Compute the N-DFT coefficients of x[n] and y[n] in MATLAB. Call those Xrand Yr, respectively. Here is how you can do it:                                                                      >>Xr=fft(xn,N)            %xn is the array containing x[n]                                                Choose your N to be 9001. (16 points)
  3. Let Zr = Xr*Yr. Compute Zr in MATLAB (16 points)
  4. Now take the N-IFFT of Zr. Here is you can do it in MATLAB (16 points)                                   >>zn=ifft(Zr,N)
  5. Verify that zn is the convolution of xn and yn by doing the below:                    >>zn2 = T*conv(xn,yn);                                                                                                       Plot both z and z2 on the same plot(16 points).
  6. Recall from class that you need to appropriately zero pad your input sequences in order to perform linear convolution using the DFT. Why does the DFT approach work in this example for computing linear convolution? You were never explicitly asked to zero-pad your arrays, thus it is happening implicitly. How is this done? (HINT: Look at the commands you have used above) (20 points)
 
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