HW 6

• 10-1

• 10-2 (Note: second delta[n-2] term should be delta[n-3])

• 10-3

• 10-4

• 10-6

• 10-7

• 10-12

• 10-15

• The purpose of this problem is to derive or recall and use formulas to compute the frequency vector for a true scaled frequency axis in a spectral plot. Both even and odd length cases are considered.

  Let $x_c(t)=\cos (2\pi 220t)$$x_c(t) = \cos(2\pi 220 t)$ be sampled at $F_s=8000$$F_s=8000$ samples/second leading to $x[n]$$x[n]$. By windowing, extract a length $L=100$$L=100$ set of samples.

  a) Use the FFT function in Matlab to compute the $N=100$$N=100$ point DFT. Plot the magnitude of the DFT on a frequency axis scaled to show frequencies in Hertz with DC (zero frequency) in the center of the frequency axis. Turn in your plot and Matlab code.

  b) Use the FFT function in Matlab to compute the $N=101$$N=101$ point DFT. Plot the magnitude of the DFT on a frequency axis scaled to show frequencies in Hertz with DC (zero frequency) in the center of the frequency axis. Turn in your plot and Matlab code.

• This problem explores the use of the FFT for performing linear convolution of two sequences.

  Let $x[n]=1,n=0,1,2,\cdots ,9$$x[n] = 1, n=0, 1, 2, \cdots, 9$ be a length $L=10$$L=10$ sequence and let $h[n]=1,n=0,1,2,3,4$$h[n] = 1, n=0, 1, 2, 3, 4$ be a length $M=5$$M=5$ sequence.

  a) Using Matlab's conv function, compute the linear convolution $y[n]=x[n]\ast h[n]$$y[n] = x[n] \ast h[n]$. b) Let $K$$K$ be the length of the convolution result $y[n]$$y[n]$. For general $L$$L$ and $M$$M$, what is the length $K$$K$ of $y[n]$$y[n]$ in terms of $L$$L$ and $M$$M$? c) For the specific values of $L$$L$ and $M$$M$ in this problem, what is the length $K$$K$ of $y[n]$$y[n]$? d) The convolution property of the DFT may be stated as

  $$\begin{array}{cc}x[n]& \stackrel{\text{DFT}}{⟷}X[k]\\ h[n]& ⟷H[k]\\ y[n]& ⟷Y[k]\\ y[n]=x[n]{⊛}_{N}h[n]& ⟷Y[k]=X[k]H[k]\end{array}$$

  where $x[n]{⊛}_{N}h[n]$$x[n] \circledast_N h[n]$ denotes $N$$N$-point circular convolution

  $$y[n]=x[n]{⊛}_{N}h[n]=\sum _{k=0}^{N-1}x[k]h[(n-k{)}_N],$$

  where $(n-k{)}_N$$(n-k)_N$ denotes $n-k$$n-k$ modulo $N$$N$. What condition on $N$$N$ guarantees that the circular convolution is equal to the linear convolution? e) For the values of $L$$L$ and $M$$M$ in this problem, what is the minimum value of $N$$N$ for linear-circular convolution equivalence? f) Using Matlab code, show how to use the FFT to perform fast convolution of $x[n]$$x[n]$ and $h[n]$$h[n]$. Rather than using the minimum value of $N$$N$. Let the transform length be the smallest power of 2 greater than the minimum $N$$N$. What is this $N$$N$? Turn in your code and a plot of $y[n]$$y[n]$. Make sure that the result computed using the FFT matches the result obtained using conv.