ECE 30 Division 3, all 2007 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 4 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 9 pages. The last four 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. Anything else is strictly forbidden. Itemized Scores Problem : Problem 2a): Problem 2b): Problem 2c): Problem 3: Problem 4: Name: Email: Signature:
(0 pts). State the sampling theorem. (You may use your own words but be precise!) 2. Consider the signal x(t) = cos(3πt) + cos(5πt). (5 pts) a) Is x(t) band-limited? If so, what is the Nyquist rate for this signal (Justify your answers.) 2
(0 pts) b) Suppose the signal x(t) is modulated by the carrier signal c(t) = e 0jπt. Sketch the graph of the modulated signal in the frequency domain. Can one recover x(t) from the modulated signal? If you answered yes, draw the diagram of a system which could be used to demodulate the signal. If you answered no, explain why. (0 pts) c) Suppose the signal x(t) is modulated by the carrier signal c(t) = cos(0πt). Sketch the graph of the modulated signal in the frequency domain. Can one recover x(t) from the modulated signal? If you answered yes, draw the diagram of a system which could be used to demodulate the signal. If you answered no, explain why. 3
(20 pts) 3. DT processing of CT signals. A CT signal x(t) with Nyquist rate equal to 2000π is the input of a CT system consisting of a low-pass filter with gain 3 and cut-off frequency 500π. Draw the diagram of an equivalent system which would sample x(t), process the samples using a DT system, and reconstruct a CT signal from the processed samples. (Don t forget to specify an appropriate frequency for the sampling as well as the frequency response of the DT system.) Illustrate all the steps of the system by sketching (in the frequency domain) the graph of an input signal together with the graphs of the signals obtained at each of the intermediate steps of the system. 4
(5 pts) 4. Compute the Laplace transform of the following signal without using the table of Laplace transform pairs: x(t) = e 2t u(t) + e 5t u( t). 5
Table ourier Series of CT Periodic Signals with period T a k = T Some CT ourier 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) ourier 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) 6
CT ourier Transform.T. : X(ω) = Inverse.T.: x(t) = Z x(t)e jωt dt (8) Z X(ω)e jωt dω 2π (9) Properties of CT ourier Transform Let x(t) be a continuous-time signal and denote by X(ω) its ourier transform. Let y(t) be another continuous-time signal and denote by Y(ω) its ourier transform. Signal.T. Linearity: ax(t) + by(t) ax(ω) + by(ω) (0) Time Shifting: x(t t 0 ) e jωt 0 X(ω) () requency Shifting: e jω0t x(t) X(ω ω 0 ) (2) Time and requency 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 ourier Transform Pairs e jω 0t 2πδ(ω ω0 ) (7) sinwt πt δ(t) u(t + T ) u(t T ) X n= δ(t nt) 2πδ(ω) (8) u(ω + W) u(ω W) (9) (20) 2sin(ωT ) ω 2π T X k= (2) δ(ω 2πk T ) (22) 7
DT ourier Transform.T.:X(ω) = Inverse.T.: x[n] = X x[n]e jωn (23) n= Z X(ω)e jωn dω (24) 2π 2π Properties of DT ourier Transform Let x(t) be a signal and denote by X(ω) its ourier transform. Let y(t) be another signal and denote by Y(ω) its ourier transform. Signal.T. Linearity: ax[n] + by[n] ax(ω) + by(ω) (25) Time Shifting: x[n n 0 ] e jωn 0 X(ω) (26) requency 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 ourier Transform Pairs N X X a k e jk(2π N )n 2π a k δ(ω 2πk N ) (33) k=0 k= X 2π δ(ω 2πl) (34) δ[n] α n u[n], α < l= (35) (36) αe jω (37) (38) 8
Laplace Transform X(s) = Z x(t)e st dt (39) 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 (40) Time Shifting: x(t t 0 ) e st 0 X(s) R (4) Shifting in s: e s 0t x(t) X(s s 0 ) R + s 0 (42) Conjugation: x (t) X (s ) R (43) Time Scaling: x(at) s a X a ar (44) Convolution: x (t) x 2 (t) X (s)x 2 (s) At least R R 2 (45) Differentiation in Time: Differentiation in s: Integration : d x(t) dt sx(s) At least R (46) dx(s) tx(t) R (47) ds Z t x(τ)dτ X(s) At least R Re{s} > 0 (48) s Some Laplace Transform Pairs Signal LT ROC u(t) s Re{s} > 0 (49) u( t) Re{s} < 0 (50) s s u(t) cos(ω 0 t) s 2 + ω0 2 Re{s} > 0 (5) e αt u(t) s + α Re{s} > α (52) e αt u( t) s + α Re{s} < α (53) δ(t) all s (54) 9