An algorithm for computing the mixed radix fast Fourier transform

Singleton, Richard C.

doi:10.1109/tau.1969.1162042

Cited by 475 publications

(132 citation statements)

References 8 publications

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“…The hierarchical classification tree in Figure 1 shows that the well-known Fast Fourier Transform (FFT) [3] and Welch Periodogram methods fall into the category of non-parametric methods. These methods transform a timedomain signal into a frequency-domain function where the signal is decomposed into a set of oscillatory components with a magnitude and phase.…”

Section: ) Non-parametric Methodsmentioning

confidence: 99%

Comparative review of methods for stability monitoring in electrical power systems and vibrating structures

Thambirajah

Barocio

Thornhill

2010

IET Gener. Transm. Distrib.

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Section: ) Non-parametric Methodsmentioning

confidence: 99%

Comparative review of methods for stability monitoring in electrical power systems and vibrating structures

Thambirajah

Barocio

Thornhill

2010

IET Gener. Transm. Distrib.

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“…The DFT can be calculated using many Fast Fourier Transform (FFT) algorithms. More detailed information about FFT can be found in [54][55][56]. In this case, we used the FFT algorithm in the IDL (Interactive Data Language) software.…”

Section: Discrete Fourier Transform Of a Time Seriesmentioning

confidence: 99%

Modeling and Reconstruction of Time Series of Passive Microwave Data by Discrete Fourier Transform Guided Filtering and Harmonic Analysis

Shang

Menenti

2016

Remote Sensing

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Daily time series of microwave radiometer data obtained in one-orbit direction are full of observation gaps due to satellite configuration and errors from spatial sampling. Such time series carry information about the surface signal including surface emittance and vegetation attenuation, and the atmospheric signal including atmosphere emittance and atmospheric attenuation. To extract the surface signal from this noisy time series, the Time Series Analysis Procedure (TSAP) was developed, based on the properties of the Discrete Fourier Transform (DFT). TSAP includes two stages: (1) identify the spectral features of observation gaps and errors and remove them with a modified boxcar filter; and (2) identify the spectral features of the surface signal and reconstruct it with the Harmonic Analysis of Time Series (HANTS) algorithm. Polarization Difference Brightness Temperature (PDBT) at 37 GHz data were used to illustrate the problems, to explain the implementation of TSAP and to validate this method, due to the PDBT sensitivity to the water content both at the land surface and in the atmosphere. We carried out a case study on a limited heterogeneous crop land and lake area, where the power spectrum of the PDBT time series showed that the harmonic components associated with observation gaps and errors have periods ≤8 days. After applying the modified boxcar filter with a length of 10 days, the RMSD between raw and filtered time series was above 11 K, mainly related to the power reduction in the frequency range associated with observation gaps and errors. Noise reduction is beneficial when applying PDBT observations to monitor wet areas and open water, since the PDBT range between dryland and open water is about 20 K. The spectral features of the atmospheric signal can be revealed by time series analysis of rain-gauge data, since the PDBT at 37 GHz is mainly attenuated by hydrometeors that yield precipitation. Thus, the spectral features of the surface signal were identified in the PDBT time series with the help of the rain-gauge data. HANTS reconstructed the upper envelope of the signal, i.e., correcting for atmospheric influence, while retaining the spectral features of the surface signal. To evaluate the impact of TSAP on retrieval accuracy, the fraction of Water Saturated Surface (WSS) in the region of Poyang Lake was retrieved with 37 GHz observations. The retrievals were evaluated against estimations of the lake area obtained with MODerate-resolution Imaging Spectroradiometer (MODIS) and Advanced Synthetic Aperture Radar (ASAR) data. The Relative RMSE on WSS was 39.5% with unfiltered data and 23% after applying TSAP, i.e., using the estimated surface signal only.

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“…Direct computation is suggested by many writers and implemented in published algorithms such as that of Singleton [18]. Another way of linking together shorter transformations, suggested by Good [20], also falls under the Type 1 ideas.…”

Section: The Fast Fourier Transformmentioning

confidence: 99%

Average running time of the fast Fourier transform

Diaconis

1980

Journal of Algorithms

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We compare several algorithms for computing the discrete Fourier transform of n numbers. The number of "operations" of the original Cooley-Tukey algorithm is approximately 2n A(n), where A(n) is the sum of the prime divisors of n. We show that the average number of operations satisfies (l/x)Z,,,2nThe average is not a good indication of the number of operations. For example, it is shown that for about half of the integers n less than x, the number of "operations" is less than n i 61. A similar analysis is given for Good's algorithm and for two algorithms that compute the discrete Fourier transform in O(n log n) operations: the chirp-z transform and the mixed-radix algorithm that computes the transform of a series of prime length p in O(p log p) operations.

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An algorithm for computing the mixed radix fast Fourier transform

Cited by 475 publications

References 8 publications

Comparative review of methods for stability monitoring in electrical power systems and vibrating structures

Comparative review of methods for stability monitoring in electrical power systems and vibrating structures

Modeling and Reconstruction of Time Series of Passive Microwave Data by Discrete Fourier Transform Guided Filtering and Harmonic Analysis

Average running time of the fast Fourier transform

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