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This is the second post in the blinkdagger signal processing series.

Photo from Shonk

What IS Fourier Analysis??

This is an introductory post that tries to answer that question in a simple manner.

Fourier Analysis is based on the premise that any arbitrary signal can be constructed using a bunch of sine and cosine waves. See this crazy signal? It looks more like a conglomeration of random points.

Believe it or not, we can recreate that signal using a lot of sine and cosine waves (more specfically, we would need an infinite number of waves to make this happen)! There’s a bunch of theory that goes behind this statement, but we’re not going to be discussing that here. But, you can go sunlightd.com or complextoreal.com for a great explanation.

Why Would We Want to Recreate an Arbitrary Signal Using Sinusoids?

Why not recreate the signal using square waves? Or triangle waves? There are several reasons why sinusoids are used. Sine and Cosine waves have particular characteristics that make them special. Most notably, their amplitude, frequency, and phase. Sinusoids have the special property wherein any sinusoidal input to a linear time-invariant system results in a sinusoidal output that differs only in amplitude and phase shift, while retaining the frequency and wave shape.

Sinusoids are the ONLY waveform to have this useful property, which makes Fourier analysis possible. But what exactly is a linear time-invariant system? (if you aren’t familiar with LTI systems, click here to get the blinkdagger explanation).

So Why is This Useful?

All signals inherently have characteristics such as frequency, phase, and amplitude. In applications such as signal processing, image processing, communications, these characteristics are vital and offer invaluable insight. In the time domain, it is difficult to ascertain these qualities. But in the frequency domain, it is much easier.

How do we go from the time domain to the frequency domain?

Photo from Spacemonkey

The process by which this is done is called the Fourier transform. By representing a signal as the sum of sinusoids, we are effectively representing that same signal in the frequency domain. The details on this process can take an entire chapter within a textbook, so we won’t dig too much into this. But you can go to Fourier Analysis for Beginners for more information. Once we transform the signal to the frequency domain, it is much easier to extract the pertinent information.

Okay, so it Sounds Somewhat Useful, but can You Give us a Practical Example?

Let’s take another look at the crazy signal I showed you earlier:

This is a signal of a building’s response during an earthquake. Now, looking at that curve, there isn’t very much useful information there. But lets take the fourier transform of that and see what we get (see graph below). Once we do this, we will begin to get a better idea on why fourier analysis is so valuable.

Okay, so I took the Fourier Transform of the Signal, now What?

Photo from heypaul

If you notice, there are large spikes at specific frequencies. What does this mean? Those are the resonant frequencies of the building. At those frequencies, the building will sustain the most damage. This is useful information to know, because builders can modify the building so that these resonant frequencies are mitigated. Using this kind of analysis, we can build structures that are able to withstand earthquakes. In areas where earthquakes are a real concern, this is of utmost importance.

What’s Next?

This post was more theoretical and is meant to provide a foundation to build upon. We would like to stress that these posts, while informative and helpful, are not meant to be a substitute. , but the next couple of posts will involve the use of Matlab.

Next post: How to use FFT