Showing papers on "Twiddle factor published in 1974"

Matrix representations for sorting and the fast Fourier transform

Abstract: The different versions of the fast Fourier transform (FFT) are described here for arbitrary base in terms of the matrix factors of the discrete Fourier transform matrix T_{N} . The Kronecker product notation and the ideal shuffle base r permutation operator form the basis for a unifying theory through which the various versions of the FFT can be viewed. The properties of the ideal shuffle base r permutation operator are used to arrive at FFT versions with such desirable properties as in-place computation or identical geometry from stage to stage. The FFT versions previously described in the literature are derived here. At the same time, algorithms for the sorting of FFT data in digit-reversed order are generated. These are explored and new sorting versions amenable to hardware implementation with sequential memory are presented. As an example of how the unifying theory is used, a number of FFT versions with identical geometry from stage to stage are derived. The hardware necessary for these algorithms is described for the base 4 case with N = 1024 data points.

Quadratic phase memory

Abstract: Twiddle factors are generated by a memory unit which is divided into two symmetrical sections; each section contains the magnitude words for one half of a quadrant. After the memories are addressed and read, the twiddle factor is assigned to either the sine or cosine by a pair of switching means, and then the proper sign is attached by a pair of sign generators. A clock circuit is provided to address the memory units and select the states of the switching networks and sign generators.

Roundoff error in multidimensional generalized discrete transforms

Abstract: The analysis of rounding error in the one-dimensional fast Fourier transform (FFT) is extended to a class of generalized orthogonal transforms [1] with a common fast algorithm similar to the FFT algorithm. This class includes the BInary FOurier REpresentation (BIFORE) transform (BT) [2], the complex BT (CBT) [3], and the discrete Fourier transform (DFT). Expressions for the mean square error (MSE) in the two-dimensional BT, CBT, and FFT are derived. In the case of white input data, the mean square error-to-signal ratio is derived for the multidimensional generalized transforms. The error-to-signal ratio for the one-dimensional FFT derived by Kaneko and Liu is modified with improvement. Some comparisons among BIFORE, DFT, and Haar transforms are also included. The theoretical results for the two-dimensional FFT and BIFORE have been verified experimentally. The experimental results are in good agreement with the theoretical results for lower order sequences, but deviate as the order increases due to the actual manner of rounding in the digital computer.