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# Fourier Series introduction

Fourier series are a powerful tool that can help us break down complex signals into their constituent parts. By using some basic mathematics, we can deconstruct signals into simple sine waves, making them much easier to understand and analyze. Created by Sal Khan.

## Want to join the conversation?

• Can you do a series on Fourier Transform and its application too? saw your Laplace Series and absolutely loved it.
Fourier Transform would be even more awesome!!
• Here's a youtube channel on signals and systems, which has videos on the fourier series, transform and applications:
• Are there any videos on the fourier transform?
• As of this writing there is nothing yet on Khan Academy for the Fourier Transform.
• Will we ever have practice exercises for this material available on Khan Academy? I've used this website to study a host of topics from my differential equations class but none of those have practice material exercises available.
• What is the relation of Fourier Series (specially this video) with the concept of the Fourier transform?
• Hello Brando,

The Fourier Series is a shorthand mathematical description of a waveform. In this video we see that a square wave may be defined as the sum of an infinite number of sinusoids.

The Fourier transform is a machine (algorithm). It takes a waveform and decomposes it into a series of waveforms. If you fed a pure sinusoid into a Fourier transform you would get an output that describes a single sinusoid. If you fed a square wave into a Fourier transform you would get an output that could be described as by a Fourier series. Which is to say, a square wave is composed of an infinite sum of sinusoids.

Regards,

APD
• What is the baseline constant?
• Sal calls the first term, a_0, of the Fourier series the "baseline constant". This is a nickname for the term. In electronics it is also called the "DC offset". The a_0 term is not multiplied by a sine or cosine, so it is just a constant numbered added to the result.

If a_0 = 1, f(t) shifts upwards on the y axis. If a_0 = -1 then f(t) shifts down.

If you set a_0 equal to the average value of the square wave (half way between the maximum and minimum value) this shifts the square wave up until the minimum value falls right on the time axis. This is what he shows in the drawing. This means the constant a_0 controls where the "baseline" of the function ends up, hence the nickname, baseline constant.
• why we are using the Fourier series? whats the
application?
• The Fourier series teaches you about the frequency content of waveforms that are not sine waves. When you see a square(-ish) wave you know it is contains frequency information at the odd harmonics of its fundamental frequency, 3rd, 5th, 7th harmonic, etc.

We understand a lot of circuits based on how they behave vs. frequency. These are called "filters", composed of R, L, and C. For example, in a digital circuit you drive square waves into wires and load gates. Those wires and gates act like filters. If you want the square wave to still look square-ish at the other end of the wire, that filter better allow at least the third and fifth harmonic of the square wave to pass through.
• why is it 2pi instead of just 2?
• Sines and cosines have periods of 2pi, which means they repeat after intervals of 2pi. You could, however, have a sinusoidal like sin((pi*x)/2), which would provide a period of 2 instead of 2pi.
• how come the function has som many values(straight line) at 0, 2Pi, 4Pi. Isn't it supposed to be a single point instead of line
• How are waves related to square representation?
(1 vote)
• The theory of Fourier Series says any periodic function (square, triangle, ramp, anything periodic) can be exactly reproduced by a sum of weighted sines and cosines. It often takes an infinite number of sines and cosines, but the match is exact. Pretty remarkable.

This theory is why we focus so intensely on understanding the math of sine and cosine. Because we know we can create any other periodic signal by adding up different-sized sines and cosines.