Showing papers on "Prime-factor FFT algorithm published in 1975"

A new algorithm in spectral analysis and band-limited extrapolation

Abstract: If only a segment of a function f (t) is given, then its Fourier spectrum F(\omega) is estimated either as the transform of the product of f(t) with a time-limited window w(t) , or by certain techniques based on various a priori assumptions. In the following, a new algorithm is proposed for computing the transform of a band-limited function. The algorithm is a simple iteration involving only the fast Fourier transform (FFT). The effect of noise and the error due to aliasing are determined and it is shown that they can be controlled by early termination of the iteration. The proposed method can also be used to extrapolate bandlimited functions.

The use of finite fields to compute convolutions

Abstract: A transform is defined in the Galois field of q^2 elements GF(q^2) , a finite field analogous to the field of complex numbers, when q is a prime such that (--1) is not a quadratic residue. It is shown that the action of this transform over GF(q^2) is equivalent to the discrete Fourier transform of a sequence of complex integers of finite dynamic range. If q is a Mersenne prime, one can utilize the fast Fourier transform (FFT) algorithm to yield a fast convolution without the usual roundoff problem of complex numbers.

Quantization errors in the fast Fourier transform

Abstract: When a fast Fourier transform (FFT) is implemented on a digital machine, quantization errors will arise due to finite word lengths in the digital system. The magnitudes and characteristics of these errors must be known if an FFT is to be designed with the minimum word lengths needed for acceptable performance. Two forms of FFT quantization, coefficient rounding and floating point arithmetic quantization, are analyzed in this paper. A theory is presented from which several new results can be obtained. The error characteristics of FFT's using exact and truncated values for the coefficients 1 and -j are found to be roughly equivalent. The accuracy of the theory is tested by computer simulations. Using the models introduced in this paper, new and accurate models can be derived to model quantization errors in high-speed convolution filters.

A Parallel Radix-4 Fast Fourier Transform Computer

Abstract: The organization and functional design of a parallel radix-4 fast Fourier transform (FFT) computer for real-time signal processing of wide-band signals is introduced.

Modular system for performing the discrete fourier transform via the chirp-Z transform

Abstract: Two kinds of apparatus for combining N 2 chirp-Z transform (CZT) modu of length N 1 to perform a discrete Fourier transform (DFT) of length N 1 N 2 . The first method uses an auxiliary parallel-input, parallel-output, DFT device of size N 2 and allows the transform of size N 1 N 2 to be performed in the same time as is required for a single CZT module to perform a size N 1 transform. The second method uses an auxiliary parallel-input, serial-output, DFT device of size N 2 . If the second method is implemented entirely in a single technology, such as with charge-coupled devices (CCDs), it performs the size N 1 N 2 transform in N 2 times the amount of time required for a single CZT module to perform a size N 1 transform. If N 2 is a composite number, say N 2 = M 1 M 2 , the second method also permits the same hardware to perform M 1 simultaneous transforms of length N 1 M 2 .

Roundoff error in fast Fourier transforms

Abstract: The finite word length used in the computer causes round-off error in the calculation of Fourier coefficients. When the fast Fourier transform method is used, the statistical mean-square error has been previously determined [3] for the case of the decimation-infrequency algorithm. This letter treats the same problem for the decimation-in-time algorithm.