why use complex numbers when representing periodic signals?

Question

a large class of periodic signals can be defined with sinusoidals. but many texts introduce these and then use a representation of periodic signals that has sinusoidals with real and imaginary parts - like here: http://www2.hawaii.edu/~gurdal/EE315/class2.pdf and here: http://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/lecture-notes/MITRES_6_007S11_lec02.pdf

why are complex numbers used to represent these periodic signals? explanations like the one linked above use Euler's rule to define complex periodic signals in terms of sinusoidals, but these still have imaginary parts. why is this used? What is the intuitive usefulness of using complex numbers here versus just sticking to sinusoidals that take real numbers as arguments?

Answer 1

It's not a matter of intuition^*, it's a matter of expedience.

My professor in Electromagnetism and Waves used to like to say, "Now let's complexify that." What he meant was, "At this point I'm going to rewrite this using $e^{ix}$-like formulas so that we can complete this lesson this morning rather than having to stretch it over two or three class periods."

You don't use a calculator (or computer) to compute square roots because it gives you a better intuition about square roots. You use the calculator because it takes so much longer to extract a square root with pencil and paper.

The difference is not quite that dramatic (perhaps) when you change real sinusoidals to complex exponentials, but it was dramatic enough to make that professor's phrase memorable to me decades later. It was kind of like dropping some sinusoidal functions into the hopper of a big mathematical machine, then you turn the crank and answers come dropping out the bottom.

The examples you linked to don't really seem to do anything particularly useful with the complex exponentials. They seem to just be introducing the idea that you can make a correspondence between sinusoids and complex exponentials. The wonderful mathematical machine is presumably waiting to be brought out in a later lesson.

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^*Or maybe it is a matter of intuition. As noted in a comment below, if you make certain associations between features of sinusoids and features of complex numbers, you can develop a good intuition for dealing with phase and amplitude.

Answer 2

Because complex numbers are a compact way to encode the basic trigonometric functions $\sin$ and $\cos,$ which are the basic sinusoids.

Answer 3

The fundamental reason of using complex representation is that when the signal is processed by a linear system (governed by a linear differential equation), the complex signal undergoes a simple linear transformation with complex coefficients: refer for instance to the concept of Impedance.

Answer 4

The complex exponentials do exist in the trigonometric form of Fourier series as well, they are just hidden within the sines/cosines and you do not have to deal with them as long as you are working with real-valued signals.

Apart from simplifying algebraic work, the complex exponential form requires to solve a single integral instead of two for the series coefficients.

Several results about the relation between the evenness and oddity of functions and similar properties of the corresponding Fourier series are much easier to establish with complex Fourier coefficients (as opposed to two real ones)