K mitra it expresses an arbitrary input as a linear

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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 steady-state 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 discrete-time 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 discrete-time 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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