# What is the point of Fourier Series and what is it used for?

• nabeel17

#### nabeel17

I recently had an asignment where i calculated the Fourier series coefficients for

f= 1+t for t= -1 to 0
f= 1-t for t=0-1 basically triangle looking.

And as i summed more and more coefficients my function started looking more like this triangle (which was really cool). My question is, what is the use of this Fourier expansion? Why not just use the original function instead of this Fourier expansion which looks so much more complicated?

That exercise was interesting and useful for you.
Looking at things in the time domain is sometimes more convenient and fruitful than looking in the frequency domain - and vice versa. If a transmission channel has a certain frequency response then looking at the spectrum of the signal can tell you the likely impairment as it goes through the channel. That exercise you did would show you how much bandwidth (i.e. which higher harmonics are needed) to produce the waveform you wanted to a precision that you could accept: the bandwidth needed to transmit your particular waveform.
Signal processing specialists are always hopping from one domain to another and back again.

Consider that your ears basically detect the Fourier transform of the sound, which is a pressure changing in time. Your ears naturally break it into frequency components which you perceive as pitches. Light is similar, with different frequencies corresponding to different frequencies of electromagnetic radiation (although your eyes can only pick up three different primary colors).

For light and sound, different frequencies are filtered or modified in different ways when passing through materials. For example, you could have a color filter that only passes light near 600nm. If you have a time domain signal, you have to do a Fourier transform to figure out what passes through the filter.

There are quite a few mathematical equations that are solved by sinusoidal functions. If you can express ANY function in terms of sinusoids (and the equations are linear) then you open the possibility for much more general solutions to these equations.

The Fourier coefficients can be plotted, giving you the frequency domain expression for the waveform. A spectrum analyzer.

Fourier transforms (and FFT, DFT, etc) transform waveforms between the time and frequency domain. Defining the frequency response of a filter in the frequency domain, for example, inverse transforming it into its time domain impulse response and then using the time domain values to implement an FIR filter (convolution) that approximates the initial frequency response.

Understanding frequency and time domain are central to Digital Signal Processing.

Note that the Fourier series in predicated on the assumption that the time domain waveform repeats forever, so cannot precisely express ANY function unless it repeats forever.

If you are interested in finding a very quiet submarine in a noisy ocean environment you'd rely on Fourier transforms, I guarantee you.

Or for finding starquakes on rapidly rotating pulsars.

1) It allows you to isolate different effects that happen at different time scales from a data set where all the different effects are happening simultaneously. Get for instance the data for the sky luminosity. If you Fourier transform that data set it will show a combination of daily variation with seasonal variation.
2) It allows you to solve equations that are hard to solve otherwise.