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Course by: Mark A. Davenport. E-mail the author

# Eigenvectors of LSI Systems

Module by: Mark A. Davenport. E-mail the author

Suppose that hh is the impulse response of an LSI system. Consider an input x[n]=znx[n]=zn where zz is a complex number. What is the output of the system? Recall that x*h=h*xx*h=h*x. In this case, it is easier to use the formula:

y [ n ] = k = - h [ k ] x [ n - k ] = k = - h [ k ] z n - k = z n k = - h [ k ] z - k = x [ n ] H ( z ) y [ n ] = k = - h [ k ] x [ n - k ] = k = - h [ k ] z n - k = z n k = - h [ k ] z - k = x [ n ] H ( z )
(1)

where

H ( z ) = k = - h [ k ] z - k . H ( z ) = k = - h [ k ] z - k .
(2)

In the event that H(z)H(z) converges, we see that y[n]y[n] is just a re-scaled version of x[n]x[n]. Thus, x[n]x[n] is an eigenvector of the system HH, right? Not exactly, but almost... technically, since zn2(Z)zn2(Z) it isn't really an eigenvector. However, most DSP texts ignore this subtlety. The intuition provided by thinking of znzn as an eigenvector is worth the slight abuse of terminology.

Next time we will analyze the function H(z)H(z) in greater detail. H(z)H(z) is called the zz-transform of hh, and provides an extremely useful characterization of a discrete-time system.

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