Numerical Analysis: A fast fourier transform algorithm for real-valued series

Abstract

A new procedure is presented for calculating the complex, discrete Fourier transform of real-valued time series. This procedure is described for an example where the number of points in the series is an integral power of two. This algorithm preserves the order and symmetry of the Cooley-Tukey fast Fourier transform algorithm while effecting the two-to-one reduction in computation and storage which can be achieved when the series is real. Also discussed are hardware and software implementations of the algorithm which perform only (N/4) log2 (N/2) complex multiply and add operations, and which require only N real storage locations in analyzing each N-point record.KEY WORDS AND PHRASES: fast Fourier transform, time series analysis, digital filtering, spectral analysis, real-tlme spectrum analyzers, Fourier analysis, discrete Fourier transform, digital spectrum analysis, Fourier analysis algorithm, Fourier synthesis algorithm CR CATEGORIES: 3.80, 3.81, 4.9, 5.49, 6.22 [10][11][12][13][14]; and improving the computational efficiency when the input series consists solely of real numbers [14,15].There have been two basic approaches to the evaluation of real-valued time series. The first approach makes use of the conventional complex FFT algorithm and depends upon forming an artificial N/2-term complex record from each N-term real record [15]. A more direct approach has recently been proposed by Edson [16] in which he suggests specializing the complex FFT algorithm by eliminating those computations which lead to redundant results. The algorithm discussed in this paper extends this latter approach not only to decrease the required computation but also to preserve the order and symmetry which made the original complex Cooley-Tukey algorithm so convenient to implement in both hardware and software. Redundancy in the Cooley-Tukey AlgorithmAn interpretation of the A~ values of the Cooley-Tukey algorithm for i = 0, 1, 2,-.., m, where N = 2 m, has been noted by Cooley [17] and described in detail by Shively [7] and others [18]. This interpretation describes the A i values at each stage as being sets of unnormalized Fourier coefficients formed from interleaved sets of samples. Although this concept can be conveniently generalized to apply when N = rlr2.., r,~, an example ~vill be carried through for N = 2% In this case the original samples (i.e. the A0 values) can be thought of as N distinct one-term Fourier series representations of the dc (direct current) value of the time function. In Figure 1 704

Low-complexity Concurrent Error Detection for convolution with Fast Fourier Transforms

“…Thus, the total computational complexity of the CED checks is 6N real multiplications and 14N-4 real additions per output block. In the case of real input sequences, the computational complexity of the overall convolution can be reduced by noting that the Discrete Fourier Transform (DFT) of a real sequence is symmetric [12]. If this optimization is employed, the proposed CED method must be augmented by adding DMR to calculation of the imaginary part of the element-wise multiplication, i.…”

A Pipelined FFT Architecture for Real-Valued Signals

“…This is the basic idea of algorithms proposed for splitradix [11], [26], radix-2 [27], [30] and high radices [28]. These algorithms are, however, not valid for the DIF (Decimation In Frequency) decomposition of the FFT because it is not possible to apply the property (3) at each stage.…”

“…Both in the specific algorithms for the computation of the RFFT and in the CFFT-based ones, the output samples are provided in different orders [27], which are different from the bit-reversal one of the CFFT [35]. The sorting of the outputs of the RFFT is also a problem not solved in the literature so far.…”

Symmetric FFTs

“…Therefore, only half of the transforms in columns 5, 6, and 7 need to be computed. The resulting algorithm was published by Bergland [1], who credits Edson as its originator. It requires a little less than half the operations of the complex FFT.…”