Final Winter 2002

ELG3120C Winter 2002

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ELG 3120C

SIGNAL AND SYSTEM ANALYSIS Final Exam – Winter 2002

Wednesday, 24 April 2002 Time: 14:00 – 17:00 Vanier Hall

Room: 231

Prof. Jianping Yao

Time allowed: 3 hours Plain calculator permitted Textbook and notes not allowed (close book exam) Attempt all the questions (100 marks)

Last name:

First name:

Student number:

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ELG3120C Winter 2002

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Question 1 (12 marks)

1.1 (5 marks) Consider an LTI system whose response to the signal x1 (t ) in Fig. 1 (a) is the signal y1 (t ) shown in Fig. 1 (b). Determine and sketch carefully the response of the system to the input deposit in Fig. 1 (c).

Figure 1

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n

1 1.2 (7 marks) A signal x[n] =   u[n ] is applied to an LTI system with impulse response h[n]  2 shown in Figure 2. Calculate the output y[n] . 3 h[n]

2 1

0

1

1

2

3

Figure 2

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n

ELG3120C Winter 2002

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Question 2 (12 marks) e − t , 0 ≤ t ≤ 1 2.1 (5 marks) Consider the signal x0 (t ) =  . Determine the Fourier transform of elsewhere 0, the signal shown in Figure 3.

Figure 3

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2.2 (7 marks) Suppose g (t ) = x (t ) cos t and the Fourier transform of g (t ) is 1, ω ≤ 2 G ( jω ) =  0, otherwise Determine x(t ) .

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ELG3120C Winter 2002

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Questions 3 (12 marks) A causal and stable LTI system S has the frequency response H ( jω ) =

jω + 4 6 − ω 2 + 5 jω

(a) (4 marks) Determine a differential equation relating the input x(t ) and output y (t ) of S. (b) (4 marks) Determine the impulse response h(t ) of S. (c) (4 marks) Calculate the output y (t ) when the input is x(t ) = e −4 t u (t ) − te −4 t u (t ) .

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ELG3120C Winter 2002

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Question 4 (12 marks) Consider a system consisting of two cascaded LTI systems with frequency responses H 1 ( e jω ) =

1 2 − e − jω , and H 2 (e jω ) = . 1 1 1 − e − jω + e − j 2ω 1 + e − jω 4 2

(a) (5 marks) Determine the impulse response h[n] of the overall system. (b) (2 marks) If the system is stable? Justify your answer. n

1 (c) (5 marks) If an input signal x[n] =   u[n] is applied to the input of the overall system, 3 calculate the output y[n] .

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ELG3120C Winter 2002

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Question 5 (12 marks) Consider a continuous-time LTI system implemented as an RLC circuit shown in Figure 4. The voltage source x(t ) is considered the input to the system. The voltage y (t ) across the capacitor is considered the system output.

Figure 4 (a) (4 marks) Find the differential equation governing the input x(t ) and output y (t ) of this system. (b) (4 marks) What is the impulse response of the system? (c) (4 marks) If the resistance R can be adjusted, determine the value of R required to make the system have no oscillation .

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ELG3120C Winter 2002

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Question 6 (10 marks) 6.1 (4 marks) Sketch the straight-line approximation of the Bode amplitude plot for the frequency response below:

H ( jω ) =

100(1 + jω ) (10 + jω )(100 + jω )

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ELG3120C Winter 2002

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6.2 (6 marks) An LTI system is described by the following first-order difference equation: y[n ] − ay[n − 1] = x[n ] . Determine the amplitude response H (e jω ) . If a = 0.6 , sketch the amplitude response H (e jω ) .

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ELG3120C Winter 2002

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Question 7 (10 marks) 7.1 (5 marks) Let x(t ) be a signal with Nyquist rate ω 0 . Determine the Nyquist rate for the signal: x(t ) cos(5ω 0 t ) .

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ELG3120C Winter 2002

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7.2 (5 marks) A signal x(t ) with Fourier transform X ( jω ) undergoes impulse-train sampling to generate

x p (t ) =

∞

∑ x(nT )δ (t − nT ) ,

where

T = 10 −4 s.

For

the

constraint

n = −∞

X ( jω ) ∗ X ( jω ) = 0 for ω > 15000π , does the sampling theorem guarantee that x(t ) can be recovered exactly from x p (t ) ? Justify your answer.

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Question 8 (20 marks)

s ( s − 2) , ( s + 2)( s + 3) (a) (3 marks) Sketch all possible regions of convergence (ROCs) of H (s ) on a zero-pole plot. (b) (7 marks) State which ROC gives rise to a causal system (i.e., h(t ) = 0 for t < 0 ), and compute its associated impulse response h(t ) .

8.1 (10 marks) Consider an LTI system with transfer function H ( s ) =

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ELG3120C Winter 2002

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8.2 (10 marks) For the LTI system below

(a) (3 marks) Find the differential equation of this system. (b) (2 marks) What is the frequency response H ( jω ) of this system? (c) (2 marks) Find the impulse response h(t ) of this system. (d) (3 marks) If a signal x(t ) = e −3t u (t ) is applied to the input of this system, what is the output response y (t ) ?

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