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Video explaining Fourier Transform Used in Spectrum Analysis


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Natural phenomena we can relate to, like sound waves, heat transfer, weather patterns, and those of a more esoteric sort, like quantum particle interactions and cosmic objects moving through gravitational fields governed by general relativity, can all have their interactions approximated by classes of functions called orthogonal functions. 

The approximation is a sum of the functions of the class. Each function is multiplied by its own unique number, called the coefficients. The method of determining these coefficients from the observed data is called a transform.

The sine/cosine functions used in Fourier Transform form a class of orthogonal functions that are particularly useful when approximating phenomena that repeat in some fashion, but any class of orthogonal functions can be used to decompose the observed phenomena into a sum of the functions.

When deciding what class of orthogonal functions to use, at least two questions are asked:

1. How many functions do I need to get the approximation  to reasonably match the observation? This is the convergence question. The fewer number of functions one needs, the "better" in general. Although with the advent of readily available computing power to just about anyone who can afford a home computer, this had become less relevant. Let the computer use as many functions as it needs.

2. Do the functions of the class represent a solution to some mathematical model of the phenomenon? This is the question of relevance. Although any family of orthogonal functions can be used to model the phenomenon, a relevant family will almost always give some physical insight into what is happening, or suggest a useful way to manipulate the effects.

TLDR: There are a great many transforms that can be used to mathematically fit functions to observations. The Fourier transform is especially useful for vibration phenomena because it quickly converges to the observations, and the functions form a natural solution to simple models of vibration.

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I used to think the complex wave form we see when playing a violin note was just the summation of sine waves that were harmonic multiples of the fundamental frequency of the string's vibration.  The amplitudes of each of these harmonics were modified by the violin's body so the body acts as filter and violins differ with regard to how they act as filters for the string's vibrations.

The Fourier Transform goes backwards and shows the individual frequencies of these individual sine waves and their amplitudes which are actual physical happenings not just mathematical models.  An interesting exercise is to add sine waves together to see how the resultant sound changes and this is often done in classes on acoustics.

But now  when I look at a calculator keyboard or computer program I realize  I was morally wrong -- its not a bunch of sines it is actually a bunch of "sin"s.

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I'm kind of fascinated by this 'mathematical or informational  phenomena'.  Particularly, since the waves of compression through the air are essentially additive, there is a significant limit to our ability to distinguish between a complex pressure wave originating from multiple sources that add up to what we receive, versus a single source the directly creates the same pattern of pressure variation.    Hence, for essentially mathematical rather than physical reasons, we can't really distinguish between a saw tooth source and multiple sine waves sources that add up to the saw tooth.   The more I think about, that trippier that seems to me.   And then our internal mechanisms of hearing add to this.  Our ear essential carries out a physical Fourier Transform and produces nerve signals from a sound that amount to a Fourier analysis of the incoming sound, then our Brain puts that back together and we experiences what seems like a unified sound!  

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