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Unformatted text preview: ∑ h[ k ] e− jωk
⎜
⎝ k = n+1 ⎞ jωn
⎟e
⎟
⎠ • The first term on the RHS is the same as
that obtained when the input is applied at
n = 0 to an initially relaxed system and is
the steadystate response:
60 ysr [ n] = H (e jω ) e jωn
Copyright © 2005, S. K. Mitra 10 Response to a Causal
Exponential Sequence Response to a Causal
Exponential Sequence • The second term on the RHS is called the
transient response:
⎛∞
⎞
ytr [ n] = − ⎜ ∑ h[k ] e − jωk ⎟ e jωn
⎜
⎟
⎝ k = n +1
⎠
• To determine the effect of the above term
on the total output response, we observe
ytr [n] =
61 ∞ ∞ ∞ k = n +1 k = n +1 k =0 • For a causal, stable LTI IIR discretetime
system, h[n] is absolutely summable
• As a result, the transient response ytr [n] is a
bounded sequence
• Moreover, as n → ∞ ,
∞
∑k =n+1 h[k ] → 0 ∑ h[k ] e− jω( k −n) ≤ ∑ h[k ] ≤ ∑ h[k ]
62 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Response to a Causal
Exponential Sequence The Concept of Filtering • For a causal FIR LTI discretetime system
with an impulse response h[n] of length
N + 1, h[n] = 0 for n > N
• Hence, ytr [n] = 0 for n > N − 1
• Here the out...
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 Fall '13

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