※ [本文轉錄自 Grad-ProbAsk 看板] 作者: supsymmetry (supsymmetry) 看板: Grad-ProbAsk 標題: [理工] discrete time signal processing Pro. 2.49 時間: Wed Mar 7 11:35:26 2007 Consider a discrete time LTI system with frequency response H(e^{jw}) and co rresponding impulse response h[n]: (a) We are first given the following three clues about the system: (i) The system is causal. (ii) H(e^{jw})=H^{*}(e^{-jw}) (iii) The DTFT of the sequence h[n+1] is real. Using the three clues, show that the system has an impulse response of finit e duration. (b) In addition to the preceding three clues, we are now given two more clue s: (iv) \int_{-\pi}^{\pi} H(e^{jw})dw=2 (v) H(e^{j\pi})=0. Is there enough information to identify6 the system uniquely? If so, determi ne the impulse response h[n]. If not, specify as much as you can about the s equence h[n]. My solution: (i) The system is causal. (ii) H(e^{jw})=H^{*}(e^{-jw}) (iii) The DTFT of the sequence h[n+1] is real. (iv) \int_{-\pi}^{\pi} H(e^{jw})dw=2 (v) H(e^{j\pi})=0. According to the conditions: (ii)=> h[n] is real (i)=> h[n]=0 when n<0. (iv)=> h[0]=2 (v)=>\sum_{n=0}^{\inf} (-1)^{n}h[n]=0 I conclude that h[0]=2,h[1]=-2,h[2]=2,h[3]=-2... But condition (iii) is troublesome, I deduce a controversial result with the last 2 conditions as follows: DTFT{h[n+1]} =\sum_{n=-\inf}^{\inf} h[n+1]e^{-jwn} =\sum_{n=-\inf}^{\inf} h[n+1]e^{-jw(n+1)}e^{jw} =H(e^{jw})e^{jw} So if DTFT{h[n+1]} is real, then H(e^{jw}) must be Ke^{-jw} where K is a rea l number.Hence h[n] must be K\delta(n-1). Except the deduction I can't conclude any other. Should we say that DTFT{h[n +1]} is real then h[n+1] is even? This seems not conforming to the formula o f DTFT and the property of real and property of conjugate symmetry. p.s. The solution of the question in the solutions manule seems incorrect. I down loaded a copy from Internet. --

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