Fourier Transforms and the Fast Fourier Transform ( FFT ) Algorithm

and the inverse Fourier transform is f (x) = 1 2π ∫ ∞ −∞ F(ω)e dω Recall that i = √−1 and eiθ = cos θ+ i sin θ. Think of it as a transformation into a different set of basis functions. The Fourier transform uses complex exponentials (sinusoids) of various frequencies as its basis functions. (Other transforms, such as Z, Laplace, Cosine, Wavelet, and Hartley, use different basis functions). A Fourier transform pair is often written f (x)↔ F(ω), or F ( f (x)) = F(ω) where F is the Fourier transform operator. If f (x) is thought of as a signal (i.e. input data) then we call F(ω) the signal’s spectrum. If f is thought of as the impulse response of a filter (which operates on input data to produce output data) then we call F the filter’s frequency response. (Occasionally the line between what’s signal and what’s filter becomes blurry).