EECS20N: Signals and Systems

# Time Invariance

Consider the set of signals whose domain is Time. Such signals are functions of time. This includes all audio signals, for example. Typically, we assume that Time = Reals. In other words, time has no beginning or end, and is a continuum. For such signals, we can define a system called a delay operator Dτ so that if x is a function of time, then Dτ (x) is a new function of time where
(Dτ (x))(t) = x(t − τ )

For example, suppose x is the red signal in the applet below. You can vary τ using the slider.

If you were able to run applets, you would have a Delay demo here.

Notice that positive values of τ result in positive delays, despite the subtraction in x(t − τ ).

Time delay is clearly related to phase for sinusoidal signals. Time delay and phase changes are equivalent, except for the fact that phase is measured in radians (or degrees) rather than in time. In addition, a phase change of θ is equivalent to a phase change of θ + n2π for any integer n. Phase applies to sinusoidal signals, whereas delay applies to any signal that is a function of time.

Consider the set of all systems that map functions of time into functions of time. Such systems are called time-domain systems. This includes all audio systems, for example. Suppose that H is a time-domain system. H is said to be time invariant if

H · Dτ = Dτ · H.

In other words, if for all t ∈ Time, y(t) = (H(x))(t), then it must also be true that

y(t − τ ) = (H(x))(t− τ ).

A time-invariant system is one whose behavior (its response to inputs) does not change with time.

Time invariance is a mathematical fiction. No man-made electronic system is time invariant in the strict sense. For one thing, such a system is turned on at some point in time. Clearly, its behavior before it is turned on is not the same as its behavior after it is turned on. However, it proves to be a very convenient mathematical fiction, and is a reasonable approximation for many systems if their behavior is constant over a relatively long period of time (relative to whatever phenomenon we are studying). For example, your audio amplifier is not a time-invariant system. Its behavior changes drastically when you turn it on or off, and changes less drastically when you raise or lower the volume. However, for the duration of a compact disc, if you leave the volume fixed, the system can be reasonably approximated as being time invariant.

Some systems have a similar property even though they operate on signals whose domain is not time. For example, the domain of an image is a region of a plane. The output of an image processing system may not depend significantly on where in the plane the input image is placed. Shifting the input image will only shift the output image. The generalized property, which applies in this image processing case, is called shift invariance.