1.6. TIMEINVARIANT SYSTEMS
A timeinvariant system is one where a time
delay (or shift) in the input sequence causes a equivalent time
delay in the system's output sequence. Keeping in mind that n is just an indexing variable we use to
keep track of our input and output samples, let's say a system
provides an output y(n) given an input of x(n),
or
For a system to be time invariant, with a
shifted version of the original x(n)
input applied, x'(n), the following applies:
Equation 124
where k is some
integer representing k sample
period time delays. For a system to be time invariant, expression
(124) must hold true for
any integer value of k and any
input sequence.
1.6.1 Example of a TimeInvariant
System
Let's look at a simple example of time
invariance illustrated in Figure 19. Assume that our initial x(n)
input is a unityamplitude 1Hz sinewave sequence with a y(n)
output, as shown in Figure
19(b). Consider a different input sequence x'(n),
where
Equation 125
Figure 19. Timeinvariant system
inputtooutput relationships: (a) system block diagram where y(n) =
–x(n)/2; (b) system input and output with a
1Hz sinewave applied; (c) system input and output when a 1Hz
sinewave, delayed by four samples, is applied. When x'(n) =
x(n+4), then, y'(n) = y(n+4).
Equation
(125) tells us that the input sequence x'(n)
is equal to sequence x(n) shifted four samples to the left,
that is, x'(0) = x(4), x'(1) = x(5), x'(2) = x(6), and so on, as shown on the left of
Figure 19(c). The
discrete system is time invariant because the y'(n)
output sequence is equal to the y(n)
sequence shifted to the left by four samples, or y'(n) =
y(n+4). We can see that y'(0) = y(4), y'(1) = y(5), y'(2) = y(6), and so on, as shown in Figure 19(c). For
timeinvariant systems, the y time
shift is equal to the x time
shift.
Some authors succumb to the urge to define a
timeinvariant system as one whose parameters do not change with
time. That definition is incomplete and can get us in trouble if
we're not careful. We'll just stick with the formal definition that
a timeinvariant system is one where a time shift in an input
sequence results in an equal time shift in the output sequence. By
the way, timeinvariant systems in the literature are often called
shiftinvariant systems.^{[]}
^{[]} An example of a
discrete process that's not timeinvariant is the downsampling, or
decimation, process described in Chapter 10.
