Jean Baptiste Fourier once had a radical idea. He attempted to explain mathematically that the overshoots and undershoots in an oscillating square wave were the result of missing sine/cosine frequencies. However, it was rejected during his time by another renowned mathematician who argued that it was impossible to create a perfect waveform from such sinusoids(which is definitely true, since an infinite number of sinusoids would have to satisfy such), and thus, his theory that every wave can be decomposed into numerous sine/cosine components has been stored on the shelf for decades until its importance was fully recognized and accepted.

It is hard to imagine how any wave can be interpreted as a sum of many sine and cosine waves at different frequencies. In fact, many other modern studies define any waveform as a sum of any other different waveforms at different frequencies, may it be an exponential waveform or whatever. This concept could perhaps be better understood if we imagine the waveform as a piece of art, the empty canvas as the entire possible spectrum of sine and cosine frequencies (which extends to infinity), and the amplitudes of these sine and cosine frequencies as the paint colors. Say we relate a certain waveform, let us name it waveform A, to a certain piece of art, say, the Mona Lisa. The Mona Lisa's colors on the pallet gives it its distinguishing features, just like how the amplitudes of the different sine/cosine frequencies give waveform A its distinguishing features. There are some spots on the Mona Lisa that have no color (say white); these corresponds to an amplitude approaching zero of a sine/cosine frequency. The Mona Lisa, when zoomed in/zoomed out at, will appear pixelated. This resolution is the same as the number of sine/cosine frequencies that actually compose waveform A. The more the sine/cosine frequencies summed, the higher the resolution of waveform A. Of course, if we sum an infinite number of sine/cosine frequencies, we get perfect resolution and a perfect form we desire for waveform A.

Also, to get a more intuitive grasp of the Fourier, we can view it as the convolution of your desired signal with a real-complex exponential function.

With such a magnificent tool at our disposal, we can paint any kind of art we like, or any waveform we choose to generate, as long as we sum enough sine/cosine waveforms to satisfy the required resolution of the waveform. But wait, there's more! Remember I said above that at points where the Mona Lisa has no color (say white), the amplitudes of the respective sine/cosine frequencies become close to zero. If that is the case, then we can actually identify the frequencies where the amplitudes of the sine/cosine frequencies are at a maximum, i.e. which parts of the painting that have color/which frequencies are crucial to the formation of waveform A. Thus, we would know what frequencies are important to forming waveform A and use it to determine which mediums/communications channel waveform A can pass through (since all communications channels only allow a certain range of frequencies to pass through it, like a bandpass filter only cruder). To be able to know the essential frequencies of a waveform is what makes the fourier transform popular and powerful.

The fourier transform took time to gain popularity, but when it did, it found application on almost every field of study that required signal analysis. The fourier transform was useful, but it still had a major flaw. Its algorithm is too slow to keep up with the demands of its applications. Thus, in the 1960s, the fast fourier transform was introduced, which uses 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 fast fourier transform had a big-o notation of n*log(n) which is certainly a lot smaller in value than n^2 at larger values of n. (big-o notation is a means of expressing the speed of an algorithm)

When digital signal processing became mainstream, the fourier transform was also adopted, now in its digital form called the discrete fourier transform. Its not very different from the original fourier transform, the algorithm is very much the same the change being it dealing with sampled values in discrete quantities.

But what if we are given a number of sinusoids and wish to obtain an unknown waveform from it? Then we go to the inverse operation of the fourier transform, simply termed the inverse fourier transform. Mathematically, it is a simple summation of the sinusoids, and finds less application than its counterpart.

Without fourier analysis, a lot of technologies would never have emerged, and many applications in communications would never have been made possible. The next time a beautiful painting is in display, let us remember Jean Baptiste Fourier and how his ingeniuty managed to yield something of significant beauty as well.