※ [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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※ 編輯: supsymmetry 來自: 220.172.19.247 (03/07 11:37)
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