Fourier Transform Algorithms for Spectral Analysis Derived with MACSYMA

The general problem of computing the power spectrum from N samples of a signal x(t) by Fourier Transform (FT) methods is considered. It is commonly, and erroneously, believed that fast (FFT) methods apply efficiently only when N is a power of 2. Instead, I describe several techniques, using MACSYMA, applicable for computing transforms, correlation functions and power spectra that are faster than radix-2 FFTs. I also mention the possibility of a quasi-analytic B-spline method for computing transforms. For these problems, I will illustrate certain systematic techniques to generate optimized FORTRAN code from MACSYMA. One technique uses (1, exp(2πi/3)) as a basis for complex numbers instead of the more familiar (1, exp(πi/2)) to derive radix-3 and radix-6 FFTs without multiplications. A second technique rederives the Good algorithm for computing the discrete Fourier transform when N can be factored into relatively prime factors. Finally, I describe the automatic generation of complete FORTRAN programs within MACSYMA that implement the Good algorithm. Taken together, these automatic program generating examples illustrate several ideas important for further development of general automatic facilities for organizing, documenting, and producing run-time efficient large scale FORTRAN codes.