ECE 301. Division 2, Fall 2006 Instructor: Mimi Boutin Midterm Examination 3

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1 ECE 30 Division 2, Fall 2006 Instructor: Mimi Boutin Midterm Examination 3 Instructions:. Wait for the BEGIN signal before opening this booklet. In the meantime, read the instructions below and fill out the requested info. 2. You have 50 minutes to answer the 5 questions contained in this exam. When the end of the exam is announced, you must stop writing immediately. Anyone caught writing after the exam is over will get a grade of zero. 3. This booklet contains 4 pages. The last five pages contain a table of formulas and properties. You may tear out these pages once the exam begins. Each transform and each property is labeled with a number. To save time, you may use these numbers to specify which transform/property you are using when justifying your answer. In general, if you use a fact which is not contained in this table, you must explain why it is true in order to get full credit. The only exceptions are the properties of the ROC, which can be used without justification. 4. This is a closed book exam. The only personal items allowed are pens/pencils, erasers and something to drink. Scratch paper will be provided by the exam supervisors. Anything else is strictly forbidden. Name: Signature:

2 . Consider the following CT signals: sin (5πt) x(t) =, πt y(t) = x(t)e jωt, z(t) = x(t) cos (ω 2 t). (5 pts) a) Is the signal x(t) band limited? (Answer yes/no and justify.) If you answered yes, what is the signal s Nyquist rate? (5 pts) b) Assuming that ω > 0, is the signal y(t) band limited? (Answer yes/no and justify.) If you answered yes, what is the signal s Nyquist rate? (5 pts) c) Assuming that ω 2 > 0, is the signal z(t) band limited? (Answer yes/no and justify.) If you answered yes, what is the signal s Nyquist rate? 2

3 (0 pts) d) Can one recover x(t) from y(t)? (Answer yes/no. If you answer yes, explain how. If you answered no, explain why not.) (5 pts) e) Can one recover x(t) from z(t)? (Answer yes/no. If you answer yes, explain how. If you answered no, explain why not.) 3

4 (5 pts) f) Impulse-train sampling is used to obtain x p [n] = k= x(n)δ(n kn). If the sampling period is N = 2, will aliasing occur? Justify your answer. 4

5 (5 pts) g) Impulse-train sampling is used to obtain y p [n] = k= y(n)δ(n kn). If ω = 3π and the sampling period is N =, will aliasing occur? Justify 6 your answer. 5

6 (20 pts) 2. The following figure shows the overall system for filtering a CT signal using a DT filter. If X(ω) and H d (ω) are as shown below, with T = 20kHZ, sketch X p(ω), X d (ω), Y d (ω), Y p (ω), and Y(ω). 6

7 (5 pts) 3. Using the definition of the Laplace transform (i.e. do not simply take the answer from the table), compute the Laplace transform of x(t) = e 5t u( t) 7

8 (20 pts) 4. Using the definition of the Laplace transform (i.e. do not simply take the answer from the table), compute the inverse z-transform of X(z) = + z + z > 3. z, 3 8

9 (5 pts)5. A discrete-time LTI system as unit impulse response h[n] = n3 n u[ n ]. What is the output of the system when the input is x[n] = ( + j) n? (Leave your answer in unsimplified form.) 9

10 Table Fourier Series of CT Periodic Signals with period T a k = T Some CT Fourier series x(t) = a k e jk(2π T )t () k= T 0 x(t)e jk(2π T )t dt (2) Signal e jω 0t Periodic square wave a k a =, a k = 0 else. (3) {, t < T x(t) = 0 T < t < T 2 and x(t + T) = x(t) δ(t nt) n= sin kω 0 T kπ a k = T (4) (5) Fourier Series of DT Periodic Signals with period N x[n] = N k=0 a k e jk(2π N )n a k = N x[n]e jk(2π N )n N n=0 (6) (7) 0

11 CT Fourier Transform F.T. : X(ω) = Inverse F.T.: x(t) = Z x(t)e jωt dt (8) Z X(ω)e jωt dω 2π (9) Properties of CT Fourier Transform Let x(t) be a continuous-time signal and denote by X(ω) its Fourier transform. Let y(t) be another continuous-time signal and denote by Y(ω) its Fourier transform. Signal F.T. Linearity: ax(t) + by(t) ax(ω) + by(ω) (0) Time Shifting: x(t t 0 ) e jωt 0 X(ω) () Frequency Shifting: e jω0t x(t) X(ω ω 0 ) (2) Time and Frequency Scaling: x(at) ω a X a (3) Multiplication: x(t)y(t) X(ω) Y(ω) 2π (4) Convolution: x(t) y(t) X(ω)Y(ω) (5) Differentiation in Time: d x(t) dt jωx(ω) (6) Some CT Fourier Transform Pairs X n= e jω 0t F 2πδ(ω ω0 ) (7) sin Wt πt δ(t) u(t) δ(t nt) F 2πδ(ω) (8) F u(ω + W) u(ω W) (9) F (20) F + πδ(ω) (2) jω F 2π T X k= δ(ω 2πk T ) (22)

12 DT Fourier Transform F.T.:X(ω) = Inverse F.T.: x[n] = X x[n]e jωn (23) n= Z X(ω)e jωn dω (24) 2π 2π Properties of DT Fourier Transform Let x(t) be a signal and denote by X(ω) its Fourier transform. Let y(t) be another signal and denote by Y(ω) its Fourier transform. Signal F.T. Linearity: ax[n] + by[n] ax(ω) + by(ω) (25) Time Shifting: x[n n 0 ] e jωn 0 X(ω) (26) Frequency Shifting: e jω 0n x[n] X(ω ω 0 ) (27) Time Reversal: x[ n] X( ω) (28) j x[ n Time Exp.: x k [n] = k ], if k divides n X(ω) (29) 0, else. Multiplication: x[n]y[n] X(ω) Y(ω) 2π (30) Convolution: x[n] y[n] X(ω)Y(ω) (3) Differencing in Time: x[n] x[n ] ( e jω )X(ω) (32) Some DT Fourier Transform Pairs N X a k e jk( 2π X N )n F 2π a k δ(ω 2πk N ) (33) k=0 k= F X 2π δ(ω 2πl) (34) δ[n] u[n] l= (35) F (36) F X e jω + π δ(ω 2πk) (37) k= 2

13 Laplace Transform X(s) = Z x(t)e st dt (38) Properties of Laplace Transform Let x(t), x (t) and x 2 (t) be three CT signals and denote by X(s), X (s) and X 2 (s) their respective Laplace transform. Let R be the ROC of X(s), let R be the ROC of X (z) and let R 2 be the ROC of X 2 (s). Signal L.T. ROC Linearity: ax (t) + bx 2 (t) ax (s) + bx 2 (s) At least R R 2 (39) Time Shifting: x(t t 0 ) e st 0 X(s) R (40) Shifting in s: e s 0t x(t) X(s s 0 ) R + s 0 (4) Conjugation: x (t) X (s ) R (42) Time Scaling: x(at) s a X a ar (43) Convolution: x (t) x 2 (t) X (s)x 2 (s) At least R R 2 (44) Differentiation in Time: Differentiation in s: Integration : d x(t) dt sx(s) At least R (45) dx(s) tx(t) R (46) ds Z t x(τ)dτ X(s) At least R Re{s} > 0 (47) s Some Laplace Transform Pairs Signal LT ROC u(t) s Re{s} > 0 (48) u( t) Re{s} < 0 (49) s s u(t) cos(ω 0 t) s 2 + ω0 2 Re{s} > 0 (50) e αt u(t) s + α Re{s} > α (5) e αt u( t) s + α Re{s} < α (52) δ(t) all s (53) 3

14 z-transform X(z) = X n= x[n]z n (54) Properties of z-transform Let x[n], x [n] and x 2 [n] be three DT signals and denote by X(z), X (z) and X 2 (z) their respective z-transform. Let R be the ROC of X(z), let R be the ROC of X (z) and let R 2 be the ROC of X 2 (z). Signal z-t. ROC Linearity: ax [n] + bx 2 [n] ax (z) + bx 2 (z) At least R R 2 (55) Time Shifting: x[n n 0 ] z n 0 X(z) R, but perhaps adding/deleting z = 0 (56) Time Shifting: x[ n] X(z ) R (57) Scaling in z: e jω0n x[n] X(e jω 0 z) R (58) Conjugation: x (t) X (z ) R (59) Convolution: x [n] x 2 [n] X (z)x 2 (z) At least R R 2 (60) Some z-transform Pairs Signal LT ROC u[n] u( n ) α n u[n] α n u[ n ] nα n u[n] nα n u[ n ] z z > (6) z z < (62) αz z > α (63) αz z < α (64) αz ( αz ) 2 z > α (65) αz ( αz ) 2 z < α (66) δ[n] all z (67) 4