Waves and Fourier sums

INVERTIBLE SLOW FT PROGRAM

Typically, signals are real valued. But the programs in this chapter are for complex-valued signals. In order to use these programs, copy the real-valued signal into a complex array, where the signal goes into the real part of the complex numbers; the imaginary parts are then automatically set to zero.

There is no universally correct choice of scale factor in Fourier transform: choice of scale is a matter of convenience. Equations (8) and (9) mimic the $Z$-transform, so their scaling factors are convenient for the convolution theorem--that a product in the frequency domain is a convolution in the time domain. Obviously, the scaling factors of equations (8) and (9) will need to be interchanged for the complementary theorem that a convolution in the frequency domain is a product in the time domain. I like to use a scale factor that keeps the sums of squares the same in the time domain as in the frequency domain. Since I almost never need the scale factor, it simplifies life to omit it from the subroutine argument list.

Waves and Fourier sums