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Originally Posted by Venkatesh
There is an excellent book, "Music, Physics, and Engineering", in which fourier analysis and basic transforms are explored. You don't need much beyond Precalc to understand their explanations.
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I too had a great interest in the magical Fourier transform when I was young(er). If you have an interest in pursuing the more hard-engineering fields, like EE & ME, I cannot stress enough how important it is to master the ideas behind the FT. As an EE myself, I can use these ideas to solve some really tough problems just by thinking about what's happening in "Fourier space". As an example, I'm working on a new class of spacecraft avionics that use distributed computational nodes that are interconnected via a gigabit-class network. Trying to figure out why I'm getting transmission errors between those nodes by examining the network signals in the time domain is next to impossible. But, if you can imagine what's happening in Fourier or frequency space, the problem just might become easier to solve. There are also instruments that will do this for you as well, like a spectrum analyzer (like the one that comes with winamp). More sophisticated instruments, like a network analyzer, can also show you what's happening in other cool spaces like modulation and phase space.
You don't need sophisticated instruments to investigate the magic that is the Fourier transform, you can do it with a copy of Matlab, Mathematica, Igor Pro or even Excel (yes, Excel!). Start by creating a table that represents a series of samples of your waveform. Using a discrete time version of the Fourier transform, known as the Fast Fourier Transform (FFT), you can convert this waveform into a table of its frequency components. Start with something simple like a pure sine wave and work yourself up to more complex waveforms like a triangle wave. A really cool waveform to use is a square wave. By modifying the resulting frequency components of the square wave and then converting back to time space, you can see how the square wave can be distorted by attenuating the higher frequency components. Square wave fidelity is important in high speed digital signals, like the network I mentioned above. For the more adventurous, you can also write the c code that does the above. Try googling on something like "
FFT example source code" for some examples.
-Kevin