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Originally Posted by Rickertsen2
I have been Puzzled by FTs and FFTs etc for a while too. I don't have much of a math background. I am currently taking precal. I have learned a bit on my own. I am reasonably comfortable with integrals, derivitaves, limits etc. Is there any hope in me trying to understand this stuff with my level of education? I am looking to understand HOW/WHY they work. There is plenty of information on how to implement them, but but not how they work.
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Fourier Transforms are very interesting, and they seem almost magical to most people that don't fully understand the math behind them. Once you realize what mathematical concept is behind them, they are actually quite easy to understand.
The basis of the Fourier Transform is simple Linear Algebra. If you don't know anything about linear algebra, I apologize, but I'll try to make this as simple to understand as possible.
There are two simple principles of Linear Algebra that the Fourier Transform uses:
1) Any vector in n-dimensional space can be written as a linear combination of n linearly independent vectors. For example, in 3-dimensional space, let vector X = 1i + 0j + 0k; let Y = 0i + 1j + 0k; let Z = 0i + 0j + 1k. Then, the vector W = 3i + 4j + 5k can be written as: W = 3*X + 4*Y + 5*Z (i.e., W can be written as a linear combination of the 3 linearly independent vectors X,Y, and Z).
2) Any vector can be projected onto another vector that is not orthogonal (perpendicular) to that vector. This is like saying that any vector will cast a shadow on another vector. For example, assume that there is a vector along the ground pointing due north. Then assume you have another vector that is also pointing due north, but also a little bit up in the air (in other words, the vector is pointing diagonally north and up). Now assume you have a light bulb directly above where the two vectors intersect. The diagonal vector would cast a shadow onto the vector on the ground. This shadow is called the "vector projection onto" the vector on the ground. In the example in part 1) above, If we projected vector W onto vector X, the projection would be 3i+0j+0k (or 3X). You can also say that "the X component of W is 3".
Okay, now you ask, "what on earth does this have to do with Fourier Transforms". I guess I forgot to mention one other principle of linear algebra: a function can act exactly like a vector, and a set of functions can act exactly like a set of vectors.
What does this mean? It means that any function can be written as a linear combination of other linearly independent functions (i.e., principle (1) above).
Okay, so now we know that we can write any function using a bunch of sine waves. How do we know what you multiply each of the waves by? That is simple: use principle (2). You simply project your desired function onto each of the sine waves to get the component of each sine wave for your desired function.
Of course, this sounds fairly simple and the details are a little more difficult, but not much, really. Once you have a little linear algebra under your belt, making projections of vectors onto other vectors is fairly easy. Making projections of functions onto other functions is just a simple extension. Once you understand these principles, Fourier Transforms are not that magical or mysterious, but just make a lot of sense. Also, one you understand these principles, there isn't much need to memorize the Fourier Transform formulas.