FINAL EXAM EE -275 Digital Signal Processing & Filtering ...

**FINAL** **EXAM****EE** -**275** **Digital** **Signal** **Processing** & **Filtering**December 14, 2009, 8:00 am - 11:00 amNAMECLOSED BOOKDo all problems.Each problem 10 points.Please underline or box your answers.1.2.3.4.5.6.7.8.9.10.11.12.

Formulae:H(e jω ) =h[n] = 12πH(jΩ) =h(t) = 12π+∞∑n=−∞∫ +π∫ −π +∞∫ −∞ +∞˜x[n] = 1 N˜X[k] =∑x[n] = 1 N−∞∑k=n=N−1h[n]e −jωnH(e jω )e +jωn dωh(t)e −jΩt dtH(jΩ)e jΩt dω˜X[k]e +jk( 2π N )n˜x[n]e −jk( 2π N )n∑X[k]e +jk( 2π N )nk=0N−1∑X[k] = x[n]e −jk( 2π N )nn=0x[n] ↔ X[z]x[n − 1] ↔ z −1 X[z] + x[−1]2

1. (Linear & Circular convolution)Consider a 10-point signal x[n] and a 8-point signal h[n].(a)How long is the signal y[n] where y[n] = x[n] ∗ h[n].(b)Describe how you would find the linear convolution of part (a) using DFTs. (Please keep your answerbrief).3

2. (Linear phase)(a) Construct a causal 3-point filter h[n], symmetric about the center point. Determine H(e jω ) = |H(e jω )|e jθ(ω) .Give the phase angle θ(ω) and the corresponding delay. (You don’t have to be concerned with any180-degree phase shift in the cosine part.)(b) Construct a causal 3-point filter h[n] that is antisymmetric about the center point. Determine H(e jω ) =|H(e jω )|e jθ(ω) . Give the phase angle θ(ω) and the corresponding delay. (You don’t have to be concernedwith any 180-degree phase shift in the sine part.)(c) Consider signal x[n] = e jωon applied to the filter of (a) above. In the steady state, determine outputsignal y[n]. What is the effect of the linear phase of the filter?(d) Repeat (c) for the (so-called generalized) linear phase of (b).4

3. (Highpass, lowpass filters)If H 1 (z) is lowpass, show thatH 2 (z) = H 1 (−z)is a highpass filter.5

4. (All Pass)Consider the transfer functionH(z) = z − 2z − 1 2(a) Determine the frequency response of the filter, H(z)| z=e jω.In the language of highpass, lowpass etc., filters, how would you describe this filter?b) Consider (H(z)H(z −1 ))| z=e jω. For real h[n], what is this equal to? Explain.6

5. (IIR)Figure 1: Lowpass filter(a)For the lowpass filter shown above, show that the voltage transfer function H(s) is,H(s) =1/RCs + 1/RCYou can use any procedure you wish to determine the transfer function. (You have a voltage divider).(b) Now consider designing an impulse-invariant IIR filter from the anlog filter H(s). Find h(t). AssumeRe{s} > − 1RC .(c) Sample h(t) to get h(nT ) = h[n].(d) Find IIR H(z). How are the poles of H(s) mapped to the poles of H(z)?7

7. (FIR/Windowing)Figure 2: Low pass filter(a) For the ideal lowpass digital filter shown above, find the impulse response h[n]. Sketch the impulseresponse.(b) Now consider truncating the even signal h[n] to a (2M + 1)-point signal by windowing it with a rectangularwindow w R [n] = 1, −M ≤ n ≤ M such that the truncated signal h t [n] = h[n]w R [n].It can be easily shown that the spectrum of w R [n] isW R (e jω ) =Sinω(2M+1)2Sin ω 2Give a rough sketch of H t (e jω ), showing the frequencies at which it goes to zero.phenomena showing up here?How is Gibb’s(c) If the issues with the non-ideal FIR low pass filter H t (e jω ) are passband and stopband ripple and size oftransition bandwidth (all caused by W R (e jω )), speculate on the ideal frequency characteristics W I (e jω )that would be desirable for an optimal window.(d) How would you make the non-causal filter h t [n], a causal filter? How will its frequency characteristicchange?8

8. (Sampling)Consider a signal x(t) = Cos(Ω o t), Ω o = 2π100 that is ideally sampled by an impulse train with samplingfrequency Ω s = 2π40. Let the sampled signal be converted by an ideal D/A into an analog signal. Show thespectrum of the sampled signal and that of the reconstructed analog signal. What is the frequency of theanalog signal?9

9. (z-Transform)(a) Find the z-transform of,Give the region of convergence.{x[n] =1, n = 0, 10, otherwise.(b) Find the z-Transform of,Give the region of convergence.(c) Find the z-Transform of,Give the region of convergence.x[n] =x[n] =u[n]u[−n](d) Can you find the DTFT from the z-transform in (b)?10

10. (z-Transform and difference equations.)Consider the difference equation,y[n] − 0.9y[n − 1] = u[n]where the system is initially at rest, that is y[−1] = 0. Find the solution y[n] as follows:(a) by solving the problem in the time domain.(b) by solving the problem using z-Transforms.11

11. Consider an analog signal of frequency 10-hz, sampled at 1000hz. The purpose here is to show howk, ω, Ω, f are related.(a) Assume that you take a 500-point DFT of this signal. For each of the 4 axis in Figure A, show andlabel the smallest, largest and mid-points of the 4 variables and indicate how they correspond. Alsoshow the two points where a 10-hz signal would show up on all 4 axis.(b) Repeat, but now assume that you took a 1000-point DFT. Show your results in Figure B.Figure 3: Figure AFigure 4: Figure B12

12. (DFT)Consider the 8-point DFT of a signal x[n], n = 0, 1, . . . , 7 where the signal x[n] has a frequency “commensurate”with one of the “basis” function frequencies, as in x[n] = 4e j( 2π 8 k)n where k could be any integerk = 0, 1, . . . , 7.(a) Consider the situation where you have x[n], with k = 3. That is, x[n] = 4e j( 2π 8 3)n What would the DFTX[k] look like for k=0,1,. . . 7?(b) Now consider another signal y[n] = 4e j 2π 310 2 n that is, with frequency not commensurate with any of thebasis function frequencies. Describe qualitatively (very briefly) what the DFT would look like.(c) What techniques are used to alleviate the problem in (b)?13