ECE 3084 QUIZ 2 SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING GEORGIA INSTITUTE OF TECHNOLOGY APRIL 2, Name:

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1 ECE 3084 QUIZ 2 SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING GEORGIA INSTITUTE OF TECHNOLOGY APRIL 2, 205 Name:. The quiz is closed book, except for one 2-sided sheet of handwritten notes. 2. Turn off your phone and put it away. No tablets/laptops/wifi/etc. Calculators are OK. 3. Final answers must be entered into the answer box. 4. Correct answers must be accompanied by concise justifications to receive full credit. 5. Do not attach additional sheets. If necessary, use the back of the previous page. Problem Points Score TOTAL: 00

2 PROBLEM. (20 points) Be sure to label both axes, i.e. label important frequencies and amplitudes! The left-hand side of the figure below shows a continuous-time sinc signal s( t ) = sin(2 t)/( t) being fed to an AM modulator with carrier frequency 40 Hz, resulting in a modulated signal x( t ): s( t ) = sin2 t x( t ) x( t ) w( t ) IDEAL y( t ) t LPF H(j ) sin(80 t) sin(80 t) y( t ) [This AM modulator differs from the usual because it multiplies by sin(. ) instead of cos(. ).] The right-hand side shows a demodulator consisting of another multiplication by sin(. ), resulting in the signal w( t ), followed by an ideal LPF with an unspecified gain and cutoff frequency. (a) Sketch X(j ), the Fourier transform of the modulated signal x( t ). (b) Sketch W(j ), the Fourier transform of the signal w( t ).. YES NO (c) Can the LPF be designed so that its output is the same as the modulator input, i.e. to achieve y( t )= s( t )? (d) If YES, sketch below the frequency response H(j ) of the LPF that results in y( t )= s( t ); If NO, explain below why this is impossible:

3 PROBLEM 2. (20 points) Suppose that the continuous-time sinc signal x( t ) = sin(202 t)/(202 t) is sampled at an unspecified sampling rate f s, and that the samples are immediately fed to an ideal D-to-C converter (with the same f s parameter), producing the continuous-time output signal y( t ), as shown below: x( t ) x[ n ] y( t ) IDEAL C-to-D CONVERTER IDEAL D-to-C CONVERTER (a) The zero-th sample is x[0] =. f s (b) In order for the D-to-C converter to reconstruct the original signal (i.e., to achieve y( t ) = x( t )), the sampling frequency must satisfy: f s > Hz. (c) In the space below, carefully sketch the output Fourier transform Y(j ) when f s = 200 Hz, carefully labeling important frequencies and amplitudes: Y(j ) (d) There are an infinite number of sampling frequencies f s such that the output signal will be y( t ) = sin( f s t)/( f s t). Name any three: f s = Hz OR f s = Hz OR f s =

4 PROBLEM 3. (20 points) (a) The Laplace transform of x( t ) = (t 2 )u( t 2) is X( s ) =. s + 3 (b) The inverse Laplace transform of X( s ) = is x( t ) = ( )u( t ). ss (c) Find the differential equation that relates + s the output y( t ) and input x( t ) of an LTI system whose transfer function is H( s ) = ss + (d) The impulse response h( t ) of the system from part (c) approaches the constant h( ) = in the limit as time grows large.

5 PROBLEM 4. (20 points) Subject to the initial conditions x(0 ) = x ( 0 ) =, the following differential equation: d x( t ) + 9x( t ) = 0, dt 2 has the following solution: where: x( t ) = (A + Bcos(Ct) + Dsin(Et ) + Ft)u( t ), A = B = C = D = E = F =

6 PROBLEM 5. (20 points) An LTI system (zero initial conditions) with input x( t ) and output y( t ) obeys the following differential equation: d 2 y( t ) = 2x( t ) 20y( t ) d y( t ). dt 2 dt (a) Circle one: The system is [ overdamped ] [ underdamped ] [critically damped ]? (b) Its d.c. gain is H 0 =. (c) Its natural frequency is n = (in rad/s). (d) Its damping ratio is =. (e) The impulse response of this system is h( t ) = Ae Bt sin(ct), where: A = B = C =

7 Table of Fourier Transform Pairs Signal Name Time-Domain: x(t) Frequency-Domain: X(jω) Right-sided exponential e at u(t) (a > 0) Left-sided exponential e bt u( t) (b > 0) Square pulse sinc function [u(t + T/2) u(t T/2)] sin(ω 0 t) πt a + jω b jω sin(ωt/2) ω/2 [u(ω + ω 0 ) u(ω ω 0 )] Impulse δ(t) Shifted impulse δ(t t 0 ) e jωt 0 Complex exponential e jω 0t 2πδ(ω ω 0 ) General cosine A cos(ω 0 t + ϕ) πae jϕ δ(ω ω 0 ) + πae jϕ δ(ω + ω 0 ) Cosine cos(ω 0 t) πδ(ω ω 0 ) + πδ(ω + ω 0 ) Sine sin(ω 0 t) jπδ(ω ω 0 ) + jπδ(ω + ω 0 ) General periodic signal a k e jkω 0t 2πa k δ(ω kω 0 ) Impulse train k= n= δ(t nt ) k= 2π T k= δ(ω 2πk/T ) Table of Fourier Transform Properties Property Name Time-Domain x(t) Frequency-Domain X(jω) Linearity ax (t) + bx 2 (t) ax (jω) + bx 2 (jω) Conjugation x (t) X ( jω) Time-Reversal x( t) X( jω) Scaling f(at) a X(j(ω/a)) Delay x(t t d ) e jωt d X(jω) Modulation x(t)e jω 0t X(j(ω ω 0 )) Modulation x(t) cos(ω 0 t) 2 X(j(ω ω 0)) + 2 X(j(ω + ω 0)) d k x(t) Differentiation dt k (jω) k X(jω) Convolution x(t) h(t) X(jω)H(jω) Multiplication x(t)p(t) 2π X(jω) P (jω) Date 4-Sept-203

8 Table of Laplace Transform Pairs Signal Name Time-Domain: x(t) Laplace-Domain: X(s) Impulse δ(t) Delayed Impulse δ(t t 0 ), t 0 0 e st 0 Step u(t) Rectangular Pulse u(t) u(t T ), T > 0 Ramp tu(t) Polynomial t k u(t), k 0 Exponential e at u(t) Polynomial Exponential t k e at u(t) Cosine cos(ω 0 t)u(t) Sine sin(ω 0 t)u(t) Exponential Cosine e at cos(ω 0 t)u(t) Exponential Sine e at sin(ω 0 t)u(t) s e st s s 2 k! s k+ s + a k! (s + a) k+ s s 2 + ω 2 0 ω 0 s 2 + ω0 2 s + a (s + a) 2 + ω 2 0 ω 0 (s + a) 2 + ω 2 0 Table of Laplace Transform Properties Property Name Time-Domain: x(t) Laplace-Domain: X(s) Linearity αx (t) + βx 2 (t) αx (s) + βx 2 (s) Right-Shift x(t t 0 ), t 0 0 e st 0 X(s) Time Scaling x(at), a > 0 ( s ) a X a First Derivative ẋ(t) = d x(t) sx(s) x(0) dt Second Derivative Integration ẍ(t) = d2 dt 2 x(t) t 0 x(τ)dτ s2 X(s) sx(0) ẋ(0) X(s) s Modulation x(t)e at X(s a) Convolution x(t) h(t) X(s)H(s) Final Value Theorem Initial Value Theorem lim x(t), (if limit exists) t lim x(t), (if limit exists) t 0 lim sx(s) s 0 lim sx(s) s ECE3084: October 6, 203