oldfinal - Old Final Exam 1 Consider a linear and...

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Unformatted text preview: Old Final Exam 1. Consider a linear and time-invariant system with impulse response h(t) = u(t — 1). Find the system output if the input $(t) = u(t) — u(t — 2) + u(t-— 4) — u(t — 6). Do all computations in the time domain. 2.(a) Consider a discrete-time system With input signal and corresponding output signal = n$[n + 1]. Determine whether the system is (i) Linear (ii) Time-invariant (iii) BlBO stable In each case, justify your answer. 3. Find the continuous-time Fourier Transform of y(t) = $1(t)2:2(t), where 55105) = (2 k + 2) — u(t — 2)) and $2(t) = cos(10t). 4. Consider the following interconnection of discrete-time LTI systems. (a) Find the Discrete-time Fourier Transform (DTFT) of mm = and h2 = (b) Find H (63“), the overall frequency response of the system. (c) Find the overall impulse response h[n] from H(ejw). Data] —-> Imfn] ———~" WW] 5. Consider the continuous—time signal 3:60?) = 2%. We want to sample $c(t) to obtain the continuous and sampled signal :rp (t) = :1in wc(nT)6(t — nT), where T is the sampling period. (a) Plot Xc(jw). What is the largest frequency? (10) If T = 0.4, plot the Fourier transform Xp(jw) of 231,05). (c) What are the specifications of the low pass filter that can be used to retrieve :rc(t). 6. Consider the following interconnection of discrete-time LTI systems. (a) If h1[n] = and h2[n] = (%)”+1u[n + 2], find their z-transforms, H1(z) and Specify the ROC in each case. (b) Compute H (z), the Z—transform of the overall system. Sketch the pole-zero plot with the ROC for H (c) Does the Discrete-time Fourier Transform exist for this case? Justify your answer. DCEH] a) MBA] l—> [MD] ”—> 3‘91 7.(a) Plot the function y(t) = 33(1 * t/2) for 33(t) = [2 — t|. (b) Determine the Fourier series representation of the signal 33(15) = Sin(47r/3t) + cos(37r/3t). (c) A signal has the impulse train Spectrum X ( jw) pictured below. Comment on whether the signal is periodic, even/odd? Add a brief justification in each case. XCJ 00) ...
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