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| − | 2. If x(t) is <math> | + | 2. If x(t) is <math> \frac{1/8}*sinc(\frac{t-15}{7}) </math> find <math> \chi(f) </math> and plot <math> |\chi(f)| </math> |
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| − | 4. Find the DFT of <math> x[n] = e^{j | + | 4. Find the DFT of <math> x[n] = e^{j \frac{\pi}{2} n} * cos( (\pi/5) n) </math> |
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| − | 5. Find the N-Point inverse DFT x[n] of <math> e^{j | + | 5. Find the N-Point inverse DFT x[n] of <math> e^{j \frac{\pi}{3} k} </math> |
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| − | 10. Consider a discrete space system defined by <math> g[m,n] = h[m,n] ** f[m,n] </math> with <math> h[m] = | + | 10. Consider a discrete space system defined by <math> g[m,n] = h[m,n] ** f[m,n] </math> with <math> h[m] = \frac{1}{25} * \begin{bmatrix} |
1 & 3 & 1 \\ | 1 & 3 & 1 \\ | ||
3 & 9 & 3\\ | 3 & 9 & 3\\ | ||
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0 & 1 & 1 & 1 & 0 \\ | 0 & 1 & 1 & 1 & 0 \\ | ||
0 & 0 & 0 & 0 & 0 \\ \end{bmatrix}</math> | 0 & 0 & 0 & 0 & 0 \\ \end{bmatrix}</math> | ||
| + | |||
| + | == SOLUTION == | ||
| + | |||
| + | |||
| + | 1. <math> \chi_d(w) = 400 rep_2\pi ((\delta((400/(2\pi) -250) + \delta((400/(2\pi) +250)) /2) </math> | ||
| + | However, aliasing takes place and the deltas experience a shift to + and - 3\pi/4 as shown: | ||
| + | |||
| + | |||
| + | |||
| + | |||
| + | |||
| + | 2. By known | ||
Revision as of 21:27, 23 April 2017
Sample Final Exam
ECE 438
Fall 2016
Instructor: Prof. Mireille Boutin
All questions below are derived from the homework of Prof. Mireille Boutin, all rights reserved by Prof. Mireille Boutin.
1. Compute and Sketch the graph of the DTFT of $ x_d[n] = cos(2 \pi 250 (1/400) ) $
2. If x(t) is $ \frac{1/8}*sinc(\frac{t-15}{7}) $ find $ \chi(f) $ and plot $ |\chi(f)| $
3. A continuous time signal is such that $ \chi(f) $ is 0 when |f| > 3 KHz. You would like a cutoff at 2KHz and a gain of 5.
a. A sample exists with a sampling rate of 9000 samples/sec. Can you process this signal to make a band limited interpolation? If no, explain why. If yes, explain how.
b. If this sample is downsampled by a factor of 3, can you still make the reconstruction?
4. Find the DFT of $ x[n] = e^{j \frac{\pi}{2} n} * cos( (\pi/5) n) $
5. Find the N-Point inverse DFT x[n] of $ e^{j \frac{\pi}{3} k} $
6. Compare the total number of operations for a N= 64 point DFT via summation formula, decimation by 2, and radix DFT. Which makes more sense for N=64. How about for N = 65?
7. Find the Z transform of $ x[n] = 2^{n} * u[n-10] $
8. Find the frequency response in two different ways of $ y[n] + 2y[n-1] + 7y[n-3] = x[n] + 3x[n-2] $ Is this an implicit or an explicit system? What does this tell us about the plot on the Z plane?
9. The sampling rate of a vocal recording is 10KHz. There exists a large formant at 2KHz, medium sized formants at 3 KHz and 1.5 KHz, and a small formant at 4KHz. Below, plot the poles of the frequency response of this vocal recording.
10. Consider a discrete space system defined by $ g[m,n] = h[m,n] ** f[m,n] $ with $ h[m] = \frac{1}{25} * \begin{bmatrix} 1 & 3 & 1 \\ 3 & 9 & 3\\ 1 & 3 & 1 \end{bmatrix} $ What is the response to the following input (assume 0-boundry condition):
$ \begin{bmatrix} 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ \end{bmatrix} $
SOLUTION
1. $ \chi_d(w) = 400 rep_2\pi ((\delta((400/(2\pi) -250) + \delta((400/(2\pi) +250)) /2) $ However, aliasing takes place and the deltas experience a shift to + and - 3\pi/4 as shown:
2. By known
