3.1 Complex fourier series

In an earlier module , we showed that a square wave could be expressed as a superposition of pulses. As useful asthis decomposition was in this example, it does not generalize well to other periodic signals:How can a superposition of pulses equal a smooth signal like a sinusoid?Because of the importance of sinusoids to linear systems, you might wonder whether they could be added together to represent alarge number of periodic signals. You would be right and in good company as well. Euler and Gauss in particular worried about this problem, and Jean Baptiste Fourier got the credit even though tough mathematical issues were notsettled until later. They worked on what is now known as the Fourier series : representing any periodic signal as a superposition of sinusoids.

But the Fourier series goes well beyond being another signal decomposition method.Rather, the Fourier series begins our journey to appreciate how a signal can be described in either the time-domain or the frequency-domain with no compromise. Let $s(t)$ be a periodic signal with period $T$ . We want to show that periodic signals, even those that haveconstant-valued segments like a square wave, can be expressed as sum of harmonically related sine waves: sinusoids having frequencies that are integer multiples of the fundamental frequency . Because the signal has period $T$ , the fundamental frequency is $\frac{1}{T}$ . The complex Fourier series expresses the signal as a superposition ofcomplex exponentials having frequencies $\frac{k}{T}$ , $k=\{\text{…}, -1, 0, 1, \text{…}\}$ .

To find the Fourier coefficients, we note the orthogonality property