Final Examination for EECS 120 at University of California, Berkeley - Fall 1997, Exams of Signals and Systems Theory

Download Final Examination for EECS 120 at University of California, Berkeley - Fall 1997 and more Exams Signals and Systems Theory in PDF only on Docsity! University of California at Berkeley Department of Electrical Engineering and Computer Sciences Professor J.M. Kahn, EECS 120, Fall, 1997 Final Examination, Wednesday, December 17, 1997, 5-8 pm NAME: 1. The 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 1 (75 pts.) A LTI discrete-time system with input x[n] and output y[n] has an impulse response: . A sketch of h[n] is shown here. (a) (5 pts.) Give a closed-form expression for the system transfer function H(z). Problem 1 2 3 4 5 6 TOTAL Points 75 30 30 25 30 10 200 Score h n[ ] 12 1 3 --–    n 1 2 --–    n– u n[ ]= 0 2 4 6 8 10 −2 −1 0 1 2 n h[ n] (b) (5 pts.) Find the poles and zeros of H(z), and sketch them on the z-plane below. (c) (5 pts.) Find the output y[n] when input is . (d) (5 pts.) Clearly the frequency response exists. Sketch the magnitude response , labeling the vertical and horizontal axes of your plot. Re(z) Im(z) 1 x n[ ] 3n n∀,= H e jΩ( ) H e jΩ( ) Ω H e jΩ( )2 The step response exhibits overshoot, which is troublesome in many applications. It is proposed to compensate for the overshoot by cascading with another system , as shown below. The system is described by the difference equation: . (h) (15 pts.) Find values of the constants a, b, and c such that the cascaded system (with input x[n] and output w[n]) has a step response that is simply a delayed unit step, . (i) (5 pts.) Sketch a realization of the system . g n[ ] x n[ ] h n[ ] y n[ ] g n[ ] w n[ ] g n[ ] w n[ ] ay n[ ] by n 1–[ ] cy n 2–[ ]+ += ws n[ ] u n 1–[ ]= g n[ ]5 Problem 2 (30 pts.) Consider the discrete-time system enclosed in the dashed box, which has input x[n] and output y[n]. Suppose that the input x[n] has the discrete-time Fourier transform shown below. (a) (15 pts.) Sketch , , and , which are the discrete-time Fourier trans- forms of v[n], w[n], and y[n], respectively. x n[ ] v n[ ] h n[ ] 1 2 --sinc n 2 --   = y n[ ] 1–( )n w n[ ] 1–( )n X e jΩ( ) Ω X e jΩ( ) 0 ππ– 2π2– π 3π3– π 4π4– π ...... 1 V e jΩ( ) W ejΩ( ) Y ejΩ( ) Ω V e jΩ( ) 0 ππ– 2π2– π 3π3– π 4π4– π ...... Ω W e jΩ( ) 0 ππ– 2π2– π 3π3– π 4π4– π ...... Ω Y e jΩ( ) 0 ππ– 2π2– π 3π3– π 4π4– π ......6 Evidently, the system enclosed in the dashed box is equivalent to a LTI system . (b) (5 pts.) Sketch the frequency response . (c) (10 pts.) Give an expression for the impulse response . g n[ ] G ejΩ( )↔ G e jΩ( ) Ω G e jΩ( ) 0 ππ– 2π2– π 3π3– π 4π4– π ...... g n[ ]7 Problem 4 (25 pts.) As studied in class, Parseval’s identity for the continuous-time Fourier trans- form states that for : . (a) (15 pts.) Consider and . Prove a generalization of the identity: . (b) (10 pts.) Let be a real signal, and let be its Hilbert transform. Use the result of part (a) to evaluate the integral: . x t( ) X ω( )↔ x∗ t( )x t( ) td ∞– ∞ ∫ 12π----- X∗ ω( )X ω( ) ωd ∞– ∞ ∫= x1 t( ) X1 ω( )↔ x2 t( ) X2 ω( )↔ x1∗ t( )x2 t( ) td ∞– ∞ ∫ 12π----- X1∗ ω( )X2 ω( ) ωd ∞– ∞ ∫= x t( ) x̂ t( ) x∗ t( )x̂ t( ) td ∞– ∞ ∫ 10 Problem 5 (30 pts.) Consider the following system. The signal is bandlimited to , It is multiplied by , the periodic signal shown. The product, , is passed through the ideal lowpass filter , which has cutoff frequency W and gain G. (a) (10 pts.) Since is periodic with period , it can be represented as a Fourier series: . Without explicitly calculating the Pn, find an expression for , the Fourier transform of , in terms of and the Pn. (b) (20 pts.) State the conditions on , W and G such that . Be as specific as possi- ble, replacing variables by specific numbers when possible. x t( ) y t( ) x t( )p t( )= H ω( ) p t( ) z t( ) X ω( ) ωmω– m ω 2–2 1 p(t) 4–4 t 0 H ω( ) WW– ω G1 x t( ) ω ωm≤ p t( ) y t( ) H ω( ) p t( ) T0 4= p t( ) Pne jnπt 2 --------- n ∞= ∞ ∑= Y ω( ) y t( ) X ω( ) ωm z t( ) x t( )=11 Problem 6 (10 pts.) A discrete-time LTI system has transfer function: . The input is . Assuming , find . H z( ) 1 1 z 17– z 19– –– ---------------------------------= x n[ ] anu n[ ]= y n[ ] 0 n 0<,= y 15[ ]12