K mitra k copyright 2005 s k mitra the frequency

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Unformatted text preview: The Frequency Response • An important property of an LTI system is that for certain types of input signals, called eigen functions, the output signal is the input signal multiplied by a complex constant • We consider here one such eigen function as the input • Consider the LTI discrete-time system with an impulse response {h[n]} shown below 35 x [n ] y [n ] • Its input-output relationship in the timedomain is given by the convolution sum y[n] = 36 Copyright © 2005, S. K. Mitra h [n ] ∞ ∑ h[k ] x[n − k ] k = −∞ Copyright © 2005, S. K. Mitra 6 The Frequency Response The Frequency Response • If the input is of the form • Then we can write y[n] = H (e jω ) e jω n • Thus for a complex exponential input signal e jω n , the output of an LTI discrete-time system is also a complex exponential signal of the same frequency multiplied by a jω complex constant H (e ) • Thus e jω n is an eigen function of the system x[ n] = e jω n , − ∞ < n < ∞ then it follows that the output is given by ∞ ⎞ ⎛∞ y[n] = ∑ h[k ] e jω( n − k ) = ⎜ ∑ h[k ] e − jω k ⎟ e jω n ⎝ k = −∞ ⎠ k = −∞ • Let ∞ H (e jω ) = ∑ h[k ] e − jω k 37 k = −∞ 38 Copyright © 2005, S. K. Mitra Copyright ©...
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