課程名稱︰信號與系統 課程性質︰系必修 課程教師︰電機系統一教學 開課學院：電資學院 開課系所︰電機系 考試日期（年月日）︰2010.6.21 考試時限（分鐘）：120 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : Signals and Systems Final 9:10a.m. ~ 11:10a.m., June 21,Mon.,2010 #Closed book, but open 1 sheet (both sides, 2 pages) of personal notes of A4 size #Total score: 120 #Total 3 pages in one B4 sheet ------------------------------------------------------------------------------ 1.Consider a system H(jω) giving an output signal y(t) for an input signal x(t). (a)[3]Explain what the group delay T(ω) is for a frequency ω. (b)[4]The system is called distortionless if y(t) = Kx(t-t0). Describe the conditions for H(jω) to be destortionless. 2.[8]x(t) is a continuous-time signal, p(t) is the shape of a narrow pulse with duration τ < T, and y(t) is a pulse train modulated by the samples of x(t), y(t) = Σn=-∞~∞ x(nT)p(t-nT) Explain why and how x(t) can be recovered from y(t) under what kind of conditions. 3.x[n] is a descrete-time signal, xp[n] is the sampled version of x[n] with sampling period N, xp[n] = x[n] , if n is an integer multiple of N 0 , else while xb[n] is a decimated version of x[n], xb[n] = xp[nN] (a)[8]Derive the relationship between X(e^jω), Xp(e^jω), and Xb(e^jω), the discrete-time Fourier transform of x[n] and xb[n], and explain what these relationships mean in frequency domain. (b)[8]Explain wht and how it is possible to recover x[n] from xb[n] under what kind of conditions, how thes can be done in the time-domain and what that means in frequency domain. 4.x1(t), x2(t) are two signals with |X1(jω)| = 0, |ω| > ω1 amd |X2(jω)| = 0 , |ω| > ω2, Find the conditions for the sampling frequency ωs = 2π/T to avoid aliasing when the signal y(t) below is sampled: (a)[2]y(t) = x1(t) = bx2(t-t0) (b)[2]y(t) = d/dt x1(t) (c)[2]y(t) = x1(t)x2(t) (d)[2]y(t) = x2(t)cos(ω2t) 5.[10]An engineer has a continous-time system Hc(jω) or hc(jω), and he wishes to design a discrete-time version Hd(e^jΩ) of hd[n] of it using the existing system Hc(jω) or hc(t), with input xd[n] and output yd[n] both at sampling period T. He comes up with the following design: xd[n]→→A→→→→hc(t), H(jω)→→→→B→→yd[n] xc(t) yc(t) hd[n], Hd(e^jΩ) where xd[n] = xc(nT) and yd[n] = yc(nT). So the operator A includes conversion of xd[n] to an impulse train plus an ideal iow-pass filter and the operator B is sampling at a period of T. Now derive the relationship between Hc(jω) and Hd(e^jΩ). 6.[10]Explain what the frequency modulation (FM) and phase modulation (PM) are, what the instantaneous frequency is and how it can be used to analyze FM and PM, and what the relationship between FM and PM is in terms of differentiation. 7.A descrete-time signal x[n] is discrete-time modulated by a descrete-time carrier signal c[n], c[n] = Acos(ω, nT) to produce a modulated signal y[n] = x[n]c[n]. Assume X(e^jω) looks like the following. X(e^jω) ↑ /\ /\ /\ __/ \_________/ \_____________/ \__→ω -2π -ωM ωM 2π Plot the spectra C(e^jω) and Y(e^jω), and show the conditon(s) for ωc to aviod aliasing. 8.[5]Assume a system h(t) has a Laplace transform H(s) = (s^2-s+1)/(s^2+s+1), ROC = {s+Re{s} > -1/2}. Use pole=zero plot and geometric evaluation to find out |H(jω)| for all ω. Note: There are problems in the back. 9.When an input signal x(t) = e^(-3t)u(t) is applied to a linear time-invariant system, the output signal is y(t) = [e^(-t) - e^(-2t)]u(t). (a)[4]Find the system function H(s) and its region of convergence. (b)[4]Determine the causality and stability of the system. (c)[3]Write down teh differential equation characterzing the system (with initial rest condition). 10.Prove the following property of z-transform, and discuss possible changes of region of convergence, if any. X(z) is the z-transform of x[n] with region of convergence R. For each case, write down the corresponding properties for Laplace transform and discrete-time Fourier transform, if any. z (a)[9]z0^n x[n]←→X(z/z0) z (b)[9]x(k)[n]←→X(z^k), where x(k)[n] = x[n/k] , if n is a multiple of k 0 , else 11.[9]A system has a system function H(z) = (1-7/4 z^(-1)-1/2 z^(-2)) / (1+1/4 z^(-1)-1/8 z^(-2)). Draw the block diagram of the system in (a) direct form, (b) parallel form and (c) cascade form. 12.[12]An engineer is designing a system H(z) with the requirement that both H(z) and H^(-1) (z) have to be causal and stable, where H^(-1) (z) is the inverse system of H(z). Can you describe the condition(s) for the locations of the poles and zeros of H(z) in order to meet the above requirement? --

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