If I have a time-varying signal containing a sinewave at some frequency f, and if I want to detect that signal, I could try generating a bunch of sinewaves that

If I have a time-varying signal containing a sinewave at some frequency f, and if I want to detect that signal, I could try generating a bunch of sinewaves that were changing at different rates and multiply each by the original signal of interest over time. If one of the sinewaves was close enough in frequency to the signal of interest, the sequential multiplication in the time domain would leave a nonzero result. I could call that a spectral line that identified the original signal. For those test frequencies not matched to the incoming signal, the sequential multiplications would produce a zero result. It's sort of like tuning a radio, trying to detect a narrow incoming signal -- the signal is only detectable at a specific receiver frequency. The difference is the DFT tunes a spectrum of frequencies simultaneously, and presents its results as a set of spectral lines covering a specified frequency domain. All the players in this story are complex numbers. This allows an unambiguous multiplication result when the signal is compared to various test frequencies -- for a monotonic frequency signal, the result is zero except at the frequency of interest. The FFT is a refinement of the DFT that minimizes the amount of computation, but its results are identical to the DFT. One of the more interesting aspects of Fourier analysis is the role of the Euler formula: http://en.wikipedia.org/wiki/Euler's_formula http://en.wikipedia.org/wiki/Euler's_formula Understanding Euler's formula given one a profound insight into how Fourier and inverse Fourier transforms work. It turns out that an inverse Fourier transform is accomplished by simply changing the sign of a subexpression: http://arachnoid.com/sage/fourier.html#Fourier_Transform http://arachnoid.com/sage/fourier.html#Fourier_Transform The same sign change determines whether a DFT converts from time to frequency domain or the reverse: http://arachnoid.com/sage/fourier.html#Discrete_Fourier_Transform http://arachnoid.com/sage/fourier.html#Discrete_Fourier_Tran... HTH