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Linear and Time-Invariant Systems

Module by: Don Johnson. E-mail the author

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Summary: Linear time-invariant (LTI) systems are the most important class of systems in communications. They ensure that a system acting on a signal can be modeled by the system acting individually on the component parts of the signal and summed.

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A signal's complexity is not related to how wiggly it is. Rather, a signal expert looks for ways of decomposing a given signal into a sum of simpler signals, which we term the signal decomposition. Though we will never compute a signal's complexity, it essentially equals the number of terms in its decomposition. In writing a signal as a sum of component signals, we can change the component signal's gain by multiplying it by a constant and by delaying it. More complicated decompositions could contain derivatives or integrals of simple signals. In short, signal decomposition amounts to thinking of the signal as the output of a linear system having simple signals as its inputs. We would build such a system, but envisioning the signal's components helps understand the signal's structure. Furthermore, you can readily compute a linear system's output to an input decomposed as a superposition of simple signals.

Example 1

As an example of signal complexity, we can express the pulse p Δ t p Δ t as a sum of delayed unit steps.

p Δ t=ut−ut−Δ p Δ t u t u t Δ (1)
Thus, the pulse is a more complex signal than the step. Be that as it may, the pulse is very useful to us.

Exercise 1

Express a square wave having period T T and amplitude A A as a superposition of delayed and amplitude-scaled pulses.

Solution

sqt=∑n=-∞∞-1nA p T/2 t−nT2 sq t n 1 n A p T/2 t n T 2

Because the sinusoid is a superposition of two complex exponentials, the sinusoid is more complex. We could not prevent ourselves from the pun in this statement. Clearly, the word "complex" is used in two different ways here. The complex exponential can also be written (using Euler's relation) as a sum of a sine and a cosine. We will discover that virtually every signal can be decomposed into a sum of complex exponentials, and that this decomposition is very useful. Thus, the complex exponential is more fundamental, and Euler's relation does not adequately reveal its complexity.

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