Linear Time-Invariant (LTI) Systems

For time-domain systems, time-invariance is a useful (if fictional) property. For complex (or real) systems, linearity is a useful (if fictional) property. For complex (or real) time-domain systems, the combination of these properties is extremely useful. Linear time-invariant (LTI) systems turn out to be particularly simple with sinusoidal inputs. Given a sinusoid at the input, the output of the LTI system will be a sinusoid with the same frequency, although possibly a different phase and amplitude. Given an input that is described as a sum of sinusoids of certain frequencies, the output can be described as a sum of sinusoids with the same frequencies, although possibly with phase and amplitude changes.

A straightforward way to show that LTI systems have this property starts by considering complex exponentials. A complex exponential is a signal e Î [Time® Complex] where for all t Î Time,

e(t) = exp(jw t) = cos(wt) + j sin(wt).

Complex exponential functions have an interesting property that will prove useful to us: For all t and t Î Time,

e(t - t ) = exp(jw(t - t )) = exp(-jwt ) exp(jwt).

This represents the function Dt · e, and follows from the multiplication property of exponentials, which applies whether they are complex or not:

ab+c = abac.

In words, a delayed complex exponential is a scaled complex exponential, where the scaling constant, exp(-jwt ), is complex.

We will now show that if the input to an LTI system is exp(jwt), then the output will be H(w) exp(jwt), where H(w) is a constant (not a function of time) that depends on the frequency w of the complex exponential. In other words, the output is only a scaled version of the input.

When the output of a system is only a scaled version of the input, the input is called an eigenfunction, which comes from the German word for "same." The output is (almost) the same as the input. Complex exponentials are eigenfunctions of LTI systems, as we will now show. This is the single reason for the (somewhat obsessive) focus on complex exponentials in electrical engineering. This single property underlies much of the discipline of signal processing, and is used heavily in circuit analysis, communication systems, and control systems.

Given an LTI system H:[Time ® Complex] ® [Time ® Complex], let

x(t) = exp(jwt)

y = H(x).

So y represents the output if the input is e. Recall that time invariance implies that

H · Dt = Dt · H.

Thus, if the input is exp(jw(t - t )), the output will be y(t - t ). But if the input is exp(jw(t - t )) = exp(- jwt ) exp(jwt), a (complex) constant times exp(jwt), then by linearity, the output is exp(- jwt )y(t). Thus, for all t and t Î Time,

y(t - t ) = exp(- jwt )y(t).

In particular, this is true for t = 0, so for all t Î Time,

y( - t ) = exp(- jwt )y(0).

Letting t = - t , we note that this implies that for all t Î Time,

y( t) = exp(jwt)y(0).

Since y(0) is a constant (it does not depend on t, although it probably depends on w), this establishes that the output is a complex exponential, just like the input except that it is scaled by y(0).

Since y(0) in this case is a property of the system, and in general depends on w, we define

H(w ) = y(0)

when the input is exp(jwt). Using this notation, we write the output

y( t) = H(w ) exp(jwt)

when the input is exp(jwt).

Note that H(w) is a function of w Î Reals, the possible frequencies of the input complex exponential. The function H:Reals ® Complex, which we have defined as the output at time zero when the input is a complex exponential with a frequency in the domain Reals, is called the frequency response. It defines the response of the LTI system to a complex exponential input.