K mitra and hence the transient response decays to

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Unformatted text preview: lication of the input x[n] • In practice, excitation x[n] to a discrete-time system is usually a right-sided sequence applied at some sample index n = no • We develop the expression for the output for such an input Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Response to a Causal Exponential Sequence • Without any loss of generality, assume x[ n] = 0 for n < 0 • From the input-output relation y[n] = ∑∞= −∞ h[k ] x[n − k ] k we observe that for an input x[n] = e jωnµ[n] the output is given by ⎞ ⎛n y[n] = ⎜ ∑ h[k ] e jω( n −k ) ⎟ µ[n] ⎟ ⎜ 58 ⎝ k =0 ⎠ Copyright © 2005, S. K. Mitra Response to a Causal Exponential Sequence ⎛n • Or, y[n] = ⎜ ∑ h[k ] e− jωk ⎜ ⎝ k =0 59 Response to a Causal Exponential Sequence ⎞ ⎟ e jωnµ[n] ⎟ ⎠ • The output for n < 0 is y[n] = 0 • The output for n ≥ 0 is given by ⎛n ⎞ y[n] = ⎜ ∑ h[ k ] e− jωk ⎟ e jωn ⎜ ⎟ ⎠ ⎝ k =0 ∞ ⎞ jωn ⎛ ∞ ⎞ ⎛ = ⎜ ∑ h[k ] e − jωk ⎟ e − ⎜ ∑ h[k ] e− jωk ⎟ e jωn ⎜ ⎟ ⎜ ⎟ ⎝ k =0 ⎝ k = n +1 Copyright © 2005,⎠ K. Mitra ⎠ S. • Or, ⎛∞ y[n] = H (e jω ) e jωn − ⎜...
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