The FFT in the 21st Century: Eigenspace Processing
This book was undertaken to provide a text and reference on the theory and practice of the FFT and its common usage. This book is organized in only four chapters, and is intended as a tutorial on the use of the FFf and its trade space. The trade space of the FFT is the parameters in its usage and the relationships between them - the sampie rate, the total number of points or the interval over which processing occurs in a single FFf, the selectivity of tuning to a given frequency over signals out-of-band, and the bandwidth over which a signal appears. The examples given in this text are in FORTRAN 9512003. FORTRAN 2003 was frozen as a standard while this work was in progress. The listings given here are intended as an aid in understanding the FFT and associated algorithms such as spectral window weightings, with the goal of making the best of them more accessible to the reader. The code I use here provides a simple bridge between the material in the text and implementation in FORTRAN 2003, C++, Java, MATLAB ©, and other modem languages. The examples are sufficiently simple to be translated into older languages such as C and FORTRAN 77 if desired.
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100 Sidelobe level algorithm antenna aperture array efficiency back into data bandwidth and array Bit m-i Bit-reverse pairs butterflies c=cos(thet Chebychev 2 window Chebychev polynomials complex exponential computed conjugate Continuous Dolph-Chebychev continuous windows cosine windows data arrays data points dB Figure defined DIF FFT dimensions Dirac delta function Dirichlet kernel Dolph-Chebychev window end do end end function end if end equiripple Fast Fourier Transform Fourier matrix Fourier series Fourier transform frequency domain frequency response given by Equation implementation implicit input data integral inverse J. K. Beard Kaiser window main lobe Main Lobe Region N-Dimensional Power nbara noise bandwidth one-dimensional output data paragraph Parseval's Theorem peak performance point DFT problem radix real(kind(1dO s=sin(thet scratch storage self-reordering shown in Figure sidelobe height Split Chebychev subprogram subroutine subscripts summation Table toggles bits trade space transform pair twiddle factors Two-Dimensional Chebychev Two-Dimensional Dolph-Chebychev Two-Dimensional Taylor Window weighting function zero