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 zn∉ℓ2(Z)zn∉ℓ2(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.
