Jean Baptiste Fourier once had a radical idea. He attempted to explain mathematically that overshoots and undershoots in an oscillating square wave were the resultant of high sine/cosine frequencies. This idea was rejected during his time though by another renowned mathematician who argued it was impossible to create perfect waveforms from such sinusoids. Thus, his theory that every wave can be decomposed into numerous sine/cosine components was shelved for decades until its importance was fully recognized and accepted - a story covered by Veritasium below.





It is hard to imagine how any wave can be interpreted as a sum of many sine and cosine waves at varying frequencies. In fact, some modern studies define any waveform as a sum of any other different waves at different frequencies, be it a sum of exponential functions or other function types. This concept could perhaps be better grasped intuitively if the waveform was likened to a piece of art, the empty canvas as the entire spectrum of sine and cosine frequencies extending to infinity, and amplitudes as the paint colors. 

Say a certain waveform A, is related to the Mona Lisa. The Mona Lisa's colors on the pallet gives it its distinguishing features, just like how amplitudes of different sine/cosine frequencies give waveform A its distinguishing features. Some spots on the Mona Lisa have no color (ex. white); corresponding to an amplitude approaching zero of a sine/cosine frequency. The Mona Lisa, when zoomed in/zoomed out  will appear pixelated. This resolution is similar as the number of sine/cosine frequencies that actually compose waveform A. More sine/cosine frequencies summed yield higher resolutions for waveform A. Of course, if the sum is an infinite number of sine/cosine frequencies, resolution becomes impeccable  and a perfect form is achieved for waveform A.

Another way to view the Fourier is as the convolution of your desired signal with a real-complex exponential function.



With such a magnificent tool at one's disposal, any kind of art can be painted, or any waveform chosen can be generated, as long as the sum of sine/cosine waveforms is enough to satisfy the required resolution of the waveform. 

But wait, there's more! Remember that at points where the Mona Lisa has no color (say white), the amplitudes of the respective sine/cosine frequencies approach zero. If that is the case, then peaks in the sine/cosine waveform can be identified, i.e. which constituents of the painting have color/which frequencies are crucial in waveform A's make-up. These frequencies can be used to determine which medium/communications channel waveform A is best suited to pass through (since all communications channels only allow a certain range of frequencies to pass through it, like a bandpass filter but cruder). The ability to know these essential frequencies of a waveform makes the Fourier transform popular and powerful.

The Fourier transform took time in gaining popularity, but when it did, found application on almost every field of study that required signal analysis. The Fourier transform was useful, but it still had a major flaw. The algorithm was too slow to keep up with demands from its applications. 

This was improved on in the 1960s, where the fast Fourier transform (FFT) was introduced, using an algorithm a lot faster than its predecessor, but not at all conditions. If the Fourier transform had a big-O notation of n^2, then the FFT had a big-O notation of n*log(n) which was certainly a lot smaller in value than n^2 at larger values of n. (big-O notation is a means of expressing algorithm speed)

When digital signal processing became mainstream, the Fourier transform was also adopted, now in its digital form called the discrete Fourier transform (DFT - not to be confused with Design for Test). It's not significantly different from the original Fourier transform, the domain being just sampled values in discrete quantities.

But what if there are a number of sinusoids and it is desired to obtain an unknown waveform from it? Then the inverse operation of the Fourier transform, simply coined as the inverse Fourier transform will suffice for the operation. Mathematically, it is simply a summation of sinusoids, and finds less application than its counterpart.

Without Fourier analysis, a lot of technologies may still be in the drawing board. Next time a beautiful painting is in display, it is worth remembering Jean Baptiste Fourier and how his ingenuity managed to provide something of similar beauty as well.