ECE 308 SIGNALS AND SYSTEMS SPRING 2013 Examination #2 14 March 2013

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1 ECE 308 SIGNALS AND SYSTEMS SPRING 2013 Examination #2 14 March 2013 Name: Instructions: The examination lasts for 75 minutes and is closed book, closed notes. No electronic devices are permitted, including but not limited to calculators, cellphones, and other handheld devices. (Any such items in the examination room must be off and put away, subject to a 20 point penalty for the first violation and a score of 0 on the exam for the second violation.) A table of properties of the Fourier transform is attached for your convenience, as is a brief table of Fourier transform pairs, and also a couple of trigonometric identities. There are five problems on the exam. Do all your work on the pages in this exam booklet. Do not unstaple these pages. Any unstapled or restapled pages will NOT be graded. An extra worksheet follows each problem, and attached at the back of the exam booklet is one more extra work page. You may write on the backs of the pages if you need to. Show your work and clearly indicate your final answers. Neatness and organization in your work is important and will influence your grade. Each problem is weighted toward the final total as shown below. Grades 1. (20 pts.) 2. (20 pts.) 3. (20 pts.) 4. (20 pts.) 5. (20 pts.) Total (100 pts.)

2 Some properties of the Fourier transform Linearity Time shift Time scaling Time reversal Multiplication by a power of t ax(t) + bv(t) ax(ω) + bx(ω) x(t c) X(ω)e jωc x(at) 1 a X ( ω a ), for a > 0 x( t) X( ω) t n x(t) j n dn dω n X(ω), n = 1, 2,... Multiplication by sinusoids x(t) e jω 0t X(ω ω 0 ), for ω 0 real x(t) cos(ω 0 t) 1 2 [X(ω + ω 0) + X(ω ω 0 )] x(t) sin(ω 0 t) j 2 [X(ω + ω 0) X(ω ω 0 )] Differentiation Integration Convolution d n dt n x(t) (jω) n X(ω), n = 1, 2,... t 1 x(λ)dλ X(ω) + πx(0)δ(ω) jω x(t) v(t) X(ω)V (ω) Multiplication x(t)v(t) 1 X(ω) V (ω) 2π Duality X(t) 2πx( ω)

3 Some Fourier transform pairs δ(t) 1 u(t) jω 1 + π δ(ω) e at u(t), a > 0 1 jω + a p T (t) T sinc ( ) T ω 2π Function definitions { 1, T/2 < t < T/2 p T (t) = 0, otherwise sinc(x) = sin(πx) πx Some trigonometric identities sin(α ± β) = sin(α) cos(β) ± cos(α) sin(β) cos(α ± β) = cos(α) cos(β) sin(α) sin(β) sin(α) cos(β) = 1 [sin(α + β) + sin(α β)] 2 cos(α) cos(β) = 1 [cos(α + β) + cos(α β)] 2 sin(α) sin(β) = 1 [cos(α β) cos(α + β)] 2

4 1. [20 points] Express the periodic signal x(t) shown below as an exponential Fourier series. x(t) t

5 EXTRA WORKSHEET for problem 1

6 2. [20 points] The Fourier series for x(t) t is x(t) = (2l 1) 2 π cos ((2l 1)πt) + ( 1) k 2 2 kπ sin(kπt). l=1 Note that the even part x e (t) and odd part x o (t) of x(t) = x e (t) + x o (t) are given by the following plots. x e (t) t Find Fourier series expansions for x e (t) and x o (t). k=1 1 x o (t) t 1

7 EXTRA WORKSHEET for problem 2

8 3. [20 points total] The signal x(t) has a Fourier transform given by (a) [10 points] Find Y (ω) when X(ω) = ω 2. y(t) = d [t x(t)] dt (b) [10 points] For this x(t), show whether or not it is true that [ ] [ ] d dx(t) dx(t) dt [t x(t)] = 1 3 dt dt

9 EXTRA WORKSHEET for problem 3

10 4. [20 points total] The Fourier transform X(ω) of x(t) is shown below. X(ω) ω Sketch V (ω) for the following functions defined in terms of x(t). (a) [10 points] (b) [10 points] v(t) = x(t) cos(5 t) v(t) = x(t) + 1 ( 2 x t + π ) + 1 ( 4 2 x t π ) 4

11 EXTRA WORKSHEET for problem 4

12 5. [20 points] The Fourier transforms for x 1 (t), x 2 (t), and x 3 (t) are shown in the following plots. X 1 (ω) t X 2 (ω) t X 3 (ω) t Let y(t) is constructed from x 1 (t), x 2 (t), and x 3 (t) from the following diagram, where ( ) t h(t) = 2π sinc e j 1 2 t 2π is the impulse response of the subsystems in the diagram. x 1 (t) h(t) x 2 (t) e jt h(t) + x 3 (t) e j2t h(t) + y(t) Sketch Y (ω) as a function of ω.

13 EXTRA WORKSHEET for problem 5

14 EXTRA WORKSHEET (clearly label which problem your work pertains to)

Communication Theory Summary of Important Definitions and Results

Signal and system theory Convolution of signals x(t) h(t) = y(t): Fourier Transform: Communication Theory Summary of Important Definitions and Results X(ω) = X(ω) = y(t) = X(ω) = j x(t) e jωt dt, 0 Properties