## Fourier Transform

"How do we split sound into frequencies? Our ears do it by mechanical means, mathematicians do it using Fourier transforms, and computers do it using FFT."  Citation

Let us consider a time-dependent signal.

Our heart-beat, our brain waves.

Sound like music beats, an antenna signal.

How did your favorite stock do at the stock exchange today? How did that stock do the last six months?

Is the climate really deteriorating? How did temperature average change in the last few years?

Are we in danger because of sea level rise? How did this change recently?

All these are time-dependent signals.

Let us take 6 sound waves with the patterns that are shown in the following picture in blue.

These are sine or cosine wave patterns. We could have used random waves but in this case we have taken a specific “series” and in particular a “harmonic series”; that is we have chosen a specific  frequency f and then we have taken its multiples, namely the 2f, 3f, 4f, 5f and 6f. Each member of a harmonic series is termed “harmonic”. The first is called fundamental frequency or first harmonic and those who follow, 2nd harmonic, 3rd harmonic etc.

(In the mechanical wave sections, we had associated frequency to the ups and downs of the green bead in the unit of time. As we see in the figure, higher frequencies means more ups and downs in the unit of time.)

What happens if we try to sum up the 6 frequencies?

We can do this graphically. In every time point of the midline (horizontal green line) in the figure that follows we can design on a paper the “vertical distance” of each waveform from the midline and then sum the distances by keeping in mind their directionality (if they have opposite directionality it is like adding up a positive number and a negative number – you actually substract).

Here is my effort to demonstrate the approach graphically for the five frequencies that I could distinguish on the Wikipedia figure. For instance, I chose a point on the midline, as represented by the black asterisk, and I drew a dark green line for the first member, a yellow for the second, a brown for the third, a purple for the fourth and a light green for the fifth (unfortunately I could not distinguish the sixth). In the yellow rectangle I have put together all of these. Next to it I have summed these up. The result is the red line. If you do this for all the points of the midline, you will get the red curve of the Wikipedia figure.

What if someone gives us a time-dependent signal like that of the red curve? Could we analyse it to its components? This is done using the procedure termed Fourier transform.

Amazingly, all waveforms can be deconstructed to sine and cosine components. This is done using the above analysis.

Legend of wikipedia figure:

Function s(x) (in red) is a sum of six sine functions of different amplitudes and harmonically related frequencies. Their summation is called a Fourier series. The Fourier transform, S(f) (in blue), which depicts amplitude vs frequency, reveals the 6 frequencies and their amplitudes.

In conclusion the analysis will provide us with a chart showing the frequencies that compose the time-dependent signal and their amplitudes.

Let us try to understand how the analysis works.

We had mentioned in the mechanical wave section that a green bead going up and down in an oscillatory manner could be associated to an angle "running" on a circle.

Let us consider an animation of the sine function

In the animation of the link http://www.mathopenref.com/triggraphsine.html

Notice how the red point below moves up and down on the green vertical axis. By taking this trace in time we obtain the red curve.

The following short video illustrates the Fourier analysis