Ch3Handouts_2

# K mitra k copyright 2005 s k mitra the frequency

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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 ©...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online