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2.3: Signal Decomposition

Learning Objectives

  • Often times, signals can be decomposed into a superposition of two or more simpler signals.
  • In these cases, linearity can be exploited to make the processing of such signals much simpler.

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.

Example \(\PageIndex{1}\):

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

\[p\Delta (t) = u(t)-u(t-\Delta )\]

Thus, the pulse is a more complex signal than the step. Be that as it may, the pulse is very useful to us.

 

Exercise \(\PageIndex{1}\)

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

Solution

\[sq(t)= \sum_{n=-\infty }^{\infty }(-1)^{n}Ap_{\frac{T}{2}}\left ( t-n\tfrac{T}{2} \right )\]

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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