In the nutshell, the goal of Fourier analysis and signal processing is to be able to express a signal in terms of basic sinusoidal components, in terms of sines and cosines. Now, before we tackle the mathematical details of such a transformation, it is important to ask ourselves why we choose the sinusoidal components as the building blocks of our transform. Now, remember that the signal is as we said, the description of a physical phenomenon evolving in time. It turns out that oscillations are really the fundamental pulse of nature, so to speak. These are some examples, if you think about your heart beating you see that it has a periodic pattern that repeats. A train for instance, would have an engine that makes wheels turn in a circular motion. Waves are another example of periodic ebb and flow that can be modeled in a sinusoidal fashion. And of course, musical instruments generates sound by vibrating at a certain fundamental frequency. It turns out that any sustainable dynamic system exhibits an oscillatory behavior. Intuitively, it’s rather easy to see that things that don’t move in circle will sooner or later hit a dead end. Take bombs, an explosion is a clearly non reversible phenomenon. It’s impossible to bring the energy back. And so, it’s something that ends very quickly and very disruptively. Rockets, they burn fuel in a one directional way and have to be jettisoned after they reach the orbit. And human beings evolved unfortunately, in just one direction towards old age. And it would be nice if we could circle back to infancy, but that’s a metaphysical digression that we will not take at this point. Oscillations are very easy to describe and parametrize. Let’s take the very simple case of a two dimensional circular motion. If you have a point on the plane and this point starts moving in circles, you can characterize this motion by taking its period, P, which is the number of seconds it takes for this point to complete a full revolution. Or equivalently by its frequency, f, which is just the reciprocal of the period and which indicates how many revolutions per second the point performs. If you put a coordinate systems around the center of this oscillation, then you can express the instantaneous coordinates of this point on the plane via the very familiar sine and cosine functions, the abscissa will be simply the cosine of ft and the ordinate will be the sine of ft, where f is the frequency of the oscillation. So we claim the sinusoids are the building blocks of nature. And indeed to drive the point home even more effectively, I will remind you that you, yourself in your body have two specialized sinusoidal detectors. The cochlea in your inner ear that detects air pressure sinusoids at frequencies from 20Hz to 20KHz. And you have rods and cones in your retina, in your eye, to detect electromagnetic sinusoids with frequency from 430THz to 790THz. This is the range of frequencies for light in the visible spectrum. Very briefly, the cochlea is a snail like structure inside your ear, but if you unroll it, it’s like a cone, like this, and along this cone, you have hundreds of hair cells. And each hair cell will react to a specific frequency in the sounds that you hear. So, in a sense your cochlea is performing an instantaneous Fourier transform as we will see shortly, of the sound waves that are hitting your ear drum. In the eye, you have three types of cones, one is sensitive to red, one is sensitive to green, and another is sensitive to blue. So three different frequencies in the light spectrum and different colors sensations are given by different amounts of energy captured by the three different types of cones as processed by the brain. So to go back to our original intuition, it seems that, for instance, humans analyze complex signals in terms of the frequency components, we can build instruments that resonate at given frequencies, musical instruments, tuning forks. So although, as we say, signals seem to be the evolution of a physical phenomenon in time and very often we describe them as such. There seems to be a different domain, the domain of frequencies, that is just as good for describing physical phenomena as the time domain. So the fundamental question is can we decompose any signal into sinusoidal elements? The answer is yes, and Fourier showed us how to do it exactly. So from the analysis point of view, what we will do is move from the time domain to the frequency domain. We will decompose a signal into different frequencies that constitute the signal and by doing so, we will be able to discover hidden signal properties. The converse of this process is going back from the frequency domain to the time domain to synthesize a signal from its frequency components. We can use that to generate signals that have a known frequency content, and we can use that in signal processing to fit signals to specific frequency regions.
