Digital Signal Processing Reference
In-Depth Information
Recall that a memoryless (static) system is one whose current value of output depends
on only the current value of input. A system with memory is called a dynamic system.
Expanding the convolution sum of (10.30), we see that for a memoryless system,
Á
y[n] =
+ x[n + 2]h[-2] + x[n + 1]h[-1]
Á
+ x[n]h[0] + x[n - 1]h[1] +
= h[0]x[n],
since only
x
[
n
] can contribute to the output
y
[
n
]. Then
h
[
n
] must be zero for
thus,
n Z 0;
h[n] = Kd[n],
where
K = h[0]
is a constant. An LTI system is memoryless if
and only if
h[n] = Kd[n].
A memoryless LTI system is then a pure gain, described
by
y[n] = Kx[n].
If the gain
K
is unity
(h[n] = d[n]),
the identity system results.
A system is invertible if its input can be determined from its output. An invertible sys-
tem (impulse response
h
[
n
]) cascaded with its inverse system (impulse response )
form the identity system, as shown in Figure 10.11(a). Hence, a discrete-time LTI sys-
tem with impulse response
h
[
n
] is invertible if there exists a function
h
i
[n]
h
i
[n]
such that
h[n]
*
h
i
[n] = d[n],
(10.31)
since the identity system has the impulse response
We do not present a procedure for finding the impulse response given
h
[
n
]. This problem can be solved with the use of the
z
-transform of Chapter 11.
A simple example of a noninvertible discrete-time LTI system is given in
Figure 10.11(b). The output is zero for
n
even; hence, the input cannot be deter-
mined from the output for
n
even.
d[n].
h
i
[n],
A discrete-time LTI system is causal if the current value of the output depends on
only the current value and past values of the input. This property can be expressed
as, for
n
1
any integer,
y[n
1
] = T(x[n]),
n F n
1
,
x
[
n
]
y
[
n
]
x
[
n
]
h
[
n
]
h
i
[
n
]
(a)
x
[
n
]
y
[
n
]
sin [
n
/2]
(b)
Figure 10.11
Illustrations of invertibility.
