Approximate and Numerical Techniques in Fourier Transform
In this chapter we present some approximate and numerical techniques dealing with calculating the Fourier transform, which are needed (a) when no exact solution is available, or (b) the exact solution is too cumbersome. The five presented methods include approximating the time signal by a Taylor series and then finding the corresponding transform; approximating the function by a series of pulses and figuring the transform of these scaled, time-shifted pulses; approximating the function by a series of unit step functions and figuring the transform of these scaled, time-shifted unit step functions; using the time differentiating property; or using brute force numerical integration. We apply all 5 methods on a test case and for each case examine the spectrum and corresponding inverse transform which would be the reconstructed time signal. We show the relevant knobs and how by using more of those we achieve better accuracy. We even supply a C-program snippet to illustrate how to deal with complex numbers and do frequency integration. The brute numerical approach is especially important as it can pretty much be applied for any practical Fourier transform calculations, or inverse transform thereof.