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推 yanshencun :推~ 03/21 18:48
分享下列命題的一個例子
對於一個 LTI System
如果 given an output signal y[n] corresponding to an input signal x[n],
則其 impulse response function h[n] 可以不唯一
--
Consider a DT system that generates the output signal
y[n] = 2( u[n-3] - u[n-5] ) = 2( d[n-3] + d[n-4] )
if the input signal is
x[n] = d[n-1] + d[n-2]
--
假設此系統是 LTI, 則輸入輸出會符合以下式子
h[n] ＊ x[n] = y[n]
=> h[n] ＊ ( d[n-1] + d[n-2] ) = 2( d[n-3] + d[n-4] )
=> h[n-1] + h[n-2] = 2( d[n-3] + d[n-4] )
=> ...
n = 2, h[1] + h[0] = 0
n = 3, h[2] + h[1] = 2
n = 4, h[3] + h[2] = 2
n = 5, h[4] + h[3] = 0
...
=> h[1] = h[3] = 2 - h[2], where h[2] is a constant, and
h[n] = h[1] (-1)^|n-1|, for n<1 or n>3
--
例如, ...
h[-1] = 1.5
h[ 0] = -1.5
h[ 1] = 1.5
h[ 2] = 0.5
h[ 3] = 1.5
h[ 4] = -1.5
h[ 5] = 1.5
...
就是一個能夠符合前述 i/p 與 o/p 的 impulse response function
而系統 y[n] = 2 x[n-2] with h[n] = 2 d[n-2] 只是一個取 h[2] = 2 的特例
如果非 time-invariant 或甚至非 linear 的話, 就更難決定一個系統為何了
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