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December 13, 2009

Fourier Analysis: The Art and Science of Finding The Needle in a Haystack

Every time I listen to the heavy metal band Pantera my wife would invariably wonder aloud why I listen to all this ‘noise’. True, many music lovers would rather refer bands like Pantera as quintessential noise than music; there are persons like me who can dissect the melody from the apparent chaos of runaway frequencies of guitars, drums and so on. I can even analyze and follow individual instruments over time. This is what Fourier is about, or stated otherwise, my ear & brain can be said to be doing a Fourier transform on the said musical piece.

a JPEG image of Joseph FourierJoseph Fourier, a French mathematician, realized that all periodic waves could be ‘synthesized’ by mixing sine waves of right frequency, amplitude and phase. For example, a square wave could be prepared by ‘adding’ the fundamental frequency (the lowest frequency; say 70Hz) with an infinite number of its odd harmonics (e.g. 210Hz, 350Hz, 490Hz and so on. Harmonics are multiples of the fundamental frequency.) This is Fourier synthesis. Similarly, you could break down a periodic signal in which the amplitude varies over time into one of a frequency versus time graph. This is Fourier analysis, and it can be seen that here we are actually ‘decomposing’ the ‘signal’ into its frequency spectrum, over time. The process of decomposing a function into its constituent frequencies is known as Fourier transform. You can have a ‘hands on experience’ at what a square wave ‘looks’ or ‘sounds’ like and how a periodic wave is decomposed into its constituent parts here. Do experiment on the sine, cosine, triangle wave and square wave functions as well and turn on the sound of your PC while you are at this site!

While Fourier originally devised this to solve the problem of heat propagation, the impact of Fourier analysis can now be felt in almost every field of science, instrumentation, entertainment and telecommunications, and even arts. Whenever you use your audio graphic equalizer to suit a piece of musical performance to your taste, you are doing a Fourier. Here you are boosting some particular audio frequencies while suppressing others, obtained by a Fourier analysis of the audio signal. You are assigning relative weights to the frequencies by sliding those sliders. Likewise, when you compress a picture (graphic) file using software such as JPEG, an inbuilt program does a Fourier transform,--> eliminates the weaker components from the analysis and--> then saves the information in a compact way.

In Nuclear Magnetic Resonance Imaging (NMRI), the emitted radio frequency is Fourier transformed to give frequency versus time, throwing valuable information about nuclear spins. Fourier analysis may also be employed to remove mains AC hum frequencies, in mobile telephony and many other situations.

One day, we may expect, that Fourier analysis may be used to pick up the ‘right frequency’ in the brain EEG waves and may put the study of ‘mirror neuron’ and ‘thought controlled devices’ into a whole new domain.

Last modified: May15, 2013
Reference:
Fourier analysis (Wikipedia),
Explained: The Discrete Fourier Transform

1 comment:

Peter Kaminsky said...

I like the way you have derived an analogy between Fourier Transform and music. I am a college sophomore with a dual major in Physics and Mathematics @ University of California, Santa Barbara. By the way, i came across these excellent physics flash cards. Its also a great initiative by the FunnelBrain team. Amazing!!!

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