NAME: ht () 1 2π. Hj0 ( ) dω Find the value of BW for the system having the following impulse response.

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1 University of California at Berkeley Department of Electrical Engineering and Computer Sciences Professor J. M. Kahn, EECS 120, Fall 1998 Final Examination, Wednesday, December 16, 1998, 5-8 pm NAME: 1. he exam is closed-book. You are permitted to use three, two-sided pages of notes. No calculators are permitted. 2. Do all work in the space provided. If you need more room, use the back of previous page. 3. Indicate your answer clearly by circling it or drawing a box around it. Problem OAL Points Score Problem 1 (15 pts.) he equivalent noise bandwidth of a C LI system ht Hjω ( ) is defined as: BW = Hjω ( ) dω. Hj0 ( ) 2 Find the value of BW for the system having the following impulse response. ht () () F 2 ht () = 0 for t < 0andt> t 1

2 Problem 2 (25 pts.) he C LI system shown here can model many simple situations that produce echoes. Here, K and are real constants, and 0. + xt () Σ delay yt () + delay gain K (a) (10 pts.) Find an expression for the impulse response ht (). (b) (10 pts.) For what values of K is the system stable? Justify your answer. (c) (5 pts.) Let K = 1, and assume xt () = ut () ut ( 2 ). Sketch yt (). 2

3 Problem 3 (40 pts.) Consider a D system described by the difference equation: (a) yn [ ] + yn [ 1] 2y[ n 2] = xn [ ] + xn [ 1]. (5 pts.) Sketch a realization of this system using only two delay elements. (b) (5 pts.) Find the transfer function Hz ( ). (c) (5 pts.) Plot the poles and zeros and indicate the region of convergence of Hz ( ). Im{ z} Re{ z} (d) (5 pts.) Is the system BIBO stable? Justify your answer. 3

4 (e) (5 pts.) Find the impulse response hn [ ]. (f) (5 pts.) Let the input be xn [ ] = 3 n, n. Find yn [ ], n. (g) (10 pts.) Let the input be xn [ ] = 3 n un [ ], n. (If you are interested in the initial conditions, they are fully specified by our specification of the input signal.) Find yn [ ], n. Explain why, as n, yn [ ] doesn t agree with the result found in part (f). 4

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6 Problem 4 (45 pts.) In the diagram below, we process a C signal xt () using a C LI system h c () t L H c to obtain the output y c () t. (Here, the subscript c means that y c () t was processed using a C system). xt () h c () t L H c y c () t Alternatively, we can sample xt () at some appropriate rate 1 to obtain a D signal xn [ ], process this using a D LI system h d to obtain y d [ n], and perform reconstruction to obtain the C signal y d () t. (Here, the subscript d means that y d () t was processed using a D system). xn [ ] = xt () t = n h d [ n] Z y d [ n] xt () H d Reconstruction y d () t t = n System How to choose h d so that its effect is similar to that of h c () t L H c is a major topic in the study of digital signal processing. Here, we consider one technique, which makes reference to the diagrams below. y c () t xt () h c () t L H c t = n y c [ n] = y c t () t = n xt () t = n xn [ ] = xt () t = n h d y d [ n] We choose some typical input signal xt (), and then choose h d so that y d [ n] = y c [ n]. It should be emphasized that given the resulting h d, the condition y d [ n] = y c [ n] is satisfied only for that particular choice of xt (). In this problem, we choose xt () = ut (), which is a common choice, and we say that h d has been obtained from h c () t L H c by a step-invariant transformation. (We are not concerned by the fact that the unit step function is not strictly bandlimited.) 6

7 (a) (5 pts.) Consider a causal C system with impulse response h c () t. Show that the step-invariant h d are specified by the relations: un [ ]*h d [ n] = [ ut ()*h c () t ] and H d = ( 1 z 1 )Z{[ u()*h t c () t ] }. t = n t = n (b) (5 pts.) Consider the causal, first-order C system with transfer function: Sketch the poles and zeros of to be stable. Im{ s} H c a H c = s + a in the s-plane, and state the condition on a for this system Re{ s} 7

8 (c) (15 pts.) Using the results of part (a), find the step-invariant H d for the H c given in part (b). Sketch the poles and zeros of H d on the z-plane as a function of a and, and state the conditions on a and for the system described by H d to be stable. You may find it useful to define α = e a. Im{ z} Re{ z} (d) (5 pts.) Consider the causal, second-order C system with transfer function: s H c = s 2 + 2s Sketch the poles and zeros of H c in the s-plane, and state whether the system is stable. What type of system is this (e.g., integrator, highpass filter, etc.)? Im{ s} Re{ s} 8

9 (e) (15 pts.) Suppose = 0.02 s. Using the results of part (a), find the step-invariant H d for the H c given in part (d). Sketch the poles and zeros of H d on the z-plane, and state if the system described by H d is stable. You may want to define r = e 0.98 and θ = 10 = 0.2 rad 11. Im{ z} Re{ z} 9

10 Problem 5 (35 pts.) Let a C LI system be described by ht Hjω ( ). Suppose that Hjω ( ) is periodic in ω, i.e., Hjω ( ( + ω 0 )) = Hjω ( ), ω, for some ω 0. Describe ht () as precisely as you can, and show how to express ht () in terms of an integral of Hjω ( ) over one period, e.g., 0 ω< ω 0. Hint: it might help to first think about some specific example(s). () F 10

11 Problem 6 (40 pts.) his problem concerns sampling and reconstruction, i.e., conversion of C signals to D and back again. o simplify the mathematics, we will consider an equivalent all-c system. he signal xt () F Xjω ( ) is bandlimited to ω <. Samples are taken of xt () and of xt ()*h() t at a rate of 1/, half the rate that would be required to specify xt () if only xt () were sampled. his problem shows that for some ht () F Hjω ( ), it is possible to find G 1 ( jω) and G 2 ( jω) so that yt () = xt (). G 1 ( jω) xt () F Xjω ( ) pt () = δt ( n) Xjω ( ) 1 ht () F Hjω ( ) n = G 2 ( jω) + Σ + yt () F Yjω ( ) ω (a) (15 pts.) Find an expression for Yjω ( ). (b) (5 pts.) Find a simpler expression for Yjω ( ) that is valid for ω, i.e., including only those terms that contribute over that frequency range. 11

12 (c) (20 pts.) Show that if xt ()*h() t = dx dt and if G 1 ( jω) and G 2 ( jω) are as specified, then yt () = xt (). Hint: it is probably easiest to prove this using a series of carefully labeled sketches. Please be honest in answering the question. If you make an error in some intermediate step but miraculously obtain the desired answer, you will receive no credit G ( jω) ω G ( jω) 2 j j ω 12

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