EECE 340 Homework Problem Set 3, Summer 2011

EECE340, Summer 2011–Fadi N Karameh 1 AMERICAN UNIVERSITY OF BEIRUT Department of Electrical and Computer Engineering EECE340 Signals and Systems -Summer 2011 Prof Karameh Problem Set 3 Out: Thursday July 14, 2011 Due: Thurs July 21, 2011 Work individually and write your complete solutions on clean paper. Start early, no extensions will be allowed. Problem 1 Find the unit sample response h[n] for each of the system functions given below: (1) ˜H(z) = 1 (1 + 1 2z−1)(1 − 4z−1)(1 − 3z−1) ˜H(z) is stable system (2) ˜H(z) = z3 − 2z z − 2 , ˜H(z) is an anti-causal system (3) ˜H(z) = 1 (1 − 4z−1)(1 − 1 3z−1), R.O.C. 1 3 < |z| < 4 . Problem 2 For each of the following pairs of input sequences x[n] and unit sample responses h[n], find the output y[n] = x[n] ⋆ h[n] using convolution. Then verify the result using the Z-transform. a) x[n] = δ[n + 1] + 1 2δ[n] + δ[n − 1] + 3 2δ[n − 2], h[n] = δ[n + 1] − δ[n] + δ[n − 1]. b) x[n] = (−3)nu[−n − 1], h[n] = (1 2)nu[n + 2]. EECE340, Summer 2011–Fadi N Karameh 2 Problem 3 Consider the cascade interconnection of three LTI causal DT systems shown in figure 1. The unit sample response g[n] of the equivalent overall system is shown in the same figure. The unit sample response h2[n] is given by h2[n] = u[n] − u[n − 2] Find h1[n] at n = 2. Clearly show your work. Figure 1: EECE340, Summer 2011–Fadi N Karameh 3 Problem 4 Using CT convolution, compute the output of the system y(t) for the given pairs of input x(t) and impulse response h(t). Figure 2: System Block Diagram Problem 4 a Figure 3: System Block Diagram Problem 4 c a) x(t), h(t) as shown in figure 2. b) x(t) = e(−4+2t)u(2 − t) and h(t) = 4e−tu(t − 1). c) x(t) and h(t) as shown in figure 3. EECE340, Summer 2011–Fadi N Karameh 4 Problem 5 An CT LTI system has an input x(t), a system function H(s) and an output y(t). We are given the following: X(s) = s + 5 s − 4 x(t) = 0, t > 0 and the corresponding output y(t) = −2 3e4tu(−t) + 1 3e−2tu(t). a) Determine H(s) and its region of convergence. b) Determine h(t), the impulse response of the system. Problem 6 (a) Consider a CT LTI system described by the following differential equation d2y dt2 − 2dy(t) dt − 3y(t) = x(t) (a-1) Determine the impulse response h(t) if this system stable. (a-2) Determine the impulse response h(t) if this system is neither causal nor stable. (b) Consider a CT LTI system S with inputs x(t) and output y(t). If an input xo(t) = 2e−3tu(t − 1) is applied, the corresponding (unknown) output is yo(t). If an input x1(t) = dxo(t) dt , then the corresponding output is y1(t) = −3yo(t) + e−2(t−1)u(t − 1). Find the impulse response h(t) of this system. Problem 7 Determine the Continuous time signal corresponding to each of the following Fourier Transforms (a) X(w) is given by the magnitude and phase plots of figure 4 (b) X(w) =          w − 1, 1 ≤ w < 2; sign(w), 2 < |w| ≤ 3; w + 1, −2 ≤ w < −1; 0, else. EECE340, Summer 2011–Fadi N Karameh 5 Figure 4: Problem 7 (c) X(w) = e−2|w| (d) X(w) = � cos(4w), |w| < π 4; 0, else. (e) X(w) = 2sinw w(jw + 2). (f) X(w) = d dw � 4sin(4w)sin(2w) w � Problem 8 Compute the Fourier Transform for the following CT signals (a) x(t) = � 1 − t2, 0 < t < 1; 0, else. (b) x(t) = e−3|t| sin 2t (c) x(t) = �sin(t) πt � ∗ d dt ��sin(2t) π �� (d) x(t) = te−3|t−1| EECE340, Summer 2011–Fadi N Karameh 6 Problem 9 Most of the modulation and demodulation schemes we discussed so far used multipliers as basic elements. Multipliers are often difficult to implement in practice. Therefore, many practical systems use a nonlinear element. In this problem we will be using a squaring device, as shown in figure 5. Figure 5: Problem 9 Assume x(t) is a bandlimited signal such that X(w) = 0, |w| > wm. 1. Determine the bandpass filter properties (wl, wh and A) such that y(t) is a modulated version of x(t). That is, y(t) = x(t)coswct. 2. Specify the constraints, if any, on wc, wm.