ECE111_2008SPRING_HW4__[0]

# ECE111_2008SPRING_HW4__[0] - The Cooper Union Department of...

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Unformatted text preview: The Cooper Union Department of Electrical Engineering ECE111 Signal Processing & Systems Analysis Problem Set IV: Time Frequency Transform Domains- continued March 7, 2008 1. Use the method of partial fractions to &nd the (causal) inverse Laplace transforms of the following. In the case of complex poles, simplify the expression for the time domain signal so that it is explicitly real (i.e., use sin ( & ) or cos ( & ) , not e j ( & ) ). (a) 3 s 3 +8 ( s +1)( s +4)( s +5) (b) 2 s 2 +3 s +1 ( s +3) 2 ( s +5) (c) 2 s 2 +3 s +1 ( s 2 +6 s +10)( s +2) 2. Use the method of partial fractions to &nd the (causal) inverse z-transforms of the following. In the case of complex poles, simplify as usual. (a) 2 z 2 +3 z +4 (2 z ¡ 1)(4 z +3) (b) 8 z +5 (3 z ¡ 2)(2 z +1) (c) 2 z 2 +3 z +4 2 z 2 +2 z +1 3. Write an explicit formula for the trasnfer function of the system shown in Figure 3. 4. Write a di/erence equation, and draw a direct form II transposed realization for: H ( z ) = 3 z 3 + 5 z 2 ¡ 6 z + 1 2 z 3 + 6 z 2 ¡ 3 z ¡ 4 5. Given the following discrete-time transfer function: H ( z ) = ( z + 3) 2 ( z + 4) ( z ¡ 2) (3 z + 1) 2 (5 z + 2) Do NOT attempt to &nd h ( n ) in this problem! (a) Identify the poles and zeros (including 1 ) , with multiplicity. (b) Sketch the poles and zeros in the complex plane, and indicate all possible regions of convergence. (c) Identify which region of convergence, if any, associates H ( z ) with: 1. the transfer function of a direct form II &lter structure; 2. a stable system; 3. a signal whose Fourier transform is well-de&ned. (d) Consider z ¡ 2 H ( z ) . How do your answers to this problem change for this case? 1 6. An analog system has simple poles at & 2 ¡ j 5 and a triple pole at & 3 . List the general real time-domain form of each mode; for each mode, identify the frequency of oscillation (if present) and the time-constant in seconds. Also identify the dominant mode. 7. A digital system has simple poles at 1 2 ¡ j 1 2 and a triple pole at 1 = 3 . List the general real time-domain form of the unit step response (i.e., the output when the input is a unit step). Identify, for each mode, the frequency of oscillation (in rad) if present and the time constant (in samp & 1 ); also identify the natural response and the forced response. 8. Invoke the following code in MATLAB to design a bandpass analog elliptic &lter: fp = [12 e 3 ; 15 e 3]; ; fs = [10 e 3 ; 16 e 3]; rp = 1 : 5; rs = 30; [ n; wn ] = ellipord (2 ¢ fp; 2 ¢ fs; rp; rs; s ); [ b; a ] = ellip ( n; rp; rs; wn; s ); MATLAB returns the numerator and denominator polynomials in the vectors b; a , respectively. The s in ellipord and ellip specify analog design (as opposed to digital...
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## This note was uploaded on 02/27/2012 for the course CHEMISTRY/ CH/ECE/PH/ taught by Professor Faculty during the Spring '08 term at Cooper Union.

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ECE111_2008SPRING_HW4__[0] - The Cooper Union Department of...

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