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Journal articles on the topic 'Discrete Fourier transforms (DFTs)'

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Published: 4 June 2021

Last updated: 7 February 2022

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1

Mahmood, Faisal, Märt Toots, Lars-Göran Öfverstedt, and Ulf Skoglund. "Algorithm and Architecture Optimization for 2D Discrete Fourier Transforms with Simultaneous Edge Artifact Removal." International Journal of Reconfigurable Computing 2018 (August 6, 2018): 1–17. http://dx.doi.org/10.1155/2018/1403181.

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Two-dimensional discrete Fourier transform (DFT) is an extensively used and computationally intensive algorithm, with a plethora of applications. 2D images are, in general, nonperiodic but are assumed to be periodic while calculating their DFTs. This leads to cross-shaped artifacts in the frequency domain due to spectral leakage. These artifacts can have critical consequences if the DFTs are being used for further processing, specifically for biomedical applications. In this paper we present a novel FPGA-based solution to calculate 2D DFTs with simultaneous edge artifact removal for high-performance applications. Standard approaches for removing these artifacts, using apodization functions or mirroring, either involve removing critical frequencies or necessitate a surge in computation by significantly increasing the image size. We use a periodic plus smooth decomposition-based approach that was optimized to reduce DRAM access and to decrease 1D FFT invocations. 2D FFTs on FPGAs also suffer from the so-called “intermediate storage” or “memory wall” problem, which is due to limited on-chip memory, increasingly large image sizes, and strided column-wise external memory access. We propose a “tile-hopping” memory mapping scheme that significantly improves the bandwidth of the external memory for column-wise reads and can reduce the energy consumption up to 53%. We tested our proposed optimizations on a PXIe-based Xilinx Kintex 7 FPGA system communicating with a host PC, which gives us the advantage of further expanding the design for biomedical applications such as electron microscopy and tomography. We demonstrate that our proposed optimizations can lead to 2.8× reduced FPGA and DRAM energy consumption when calculating high-throughput 4096×4096 2D FFTs with simultaneous edge artifact removal. We also used our high-performance 2D FFT implementation to accelerate filtered back-projection for reconstructing tomographic data.

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2

He, Y., K. Hueske, J. Götze, and E. Coersmeier. "Matrix-Vector Based Fast Fourier Transformations on SDR Architectures." Advances in Radio Science 6 (May 26, 2008): 89–94. http://dx.doi.org/10.5194/ars-6-89-2008.

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Abstract. Today Discrete Fourier Transforms (DFTs) are applied in various radio standards based on OFDM (Orthogonal Frequency Division Multiplex). It is important to gain a fast computational speed for the DFT, which is usually achieved by using specialized Fast Fourier Transform (FFT) engines. However, in face of the Software Defined Radio (SDR) development, more general (parallel) processor architectures are often desirable, which are not tailored to FFT computations. Therefore, alternative approaches are required to reduce the complexity of the DFT. Starting from a matrix-vector based description of the FFT idea, we will present different factorizations of the DFT matrix, which allow a reduction of the complexity that lies between the original DFT and the minimum FFT complexity. The computational complexities of these factorizations and their suitability for implementation on different processor architectures are investigated.

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3

Yang, Y., M. Crimp, R. A. Tomlinson, and E. A. Patterson. "Quantitative measurement of plastic strain field at a fatigue crack tip." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 468, no. 2144 (March 14, 2012): 2399–415. http://dx.doi.org/10.1098/rspa.2011.0682.

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A novel approach is introduced to map the mesoscale plastic strain distribution resulting from heterogeneous plastic deformation in complex loading and component geometries, by applying the discrete Fourier transform (DFT) to backscattered electron (BSE) images of polycrystalline patches. These DFTs are then calibrated against the full width at half the maximum of the central peak of the DFTs collected from the same material tested under in situ scanning electron microscopy uniaxial tensile conditions, which indicates a close relationship with the global tensile strain. In this work, the technique is demonstrated by measuring the residual strain distribution and plastic zone size around a fatigue crack tip in a commercially pure titanium compact tension specimen, by collecting BSE images in a 15×15 array of 115 μm square images around the fatigue crack tip. The measurement results show good agreement with the plastic zone size and shape measured using thermoelastic stress analysis.

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4

Peng, Yaqiu, and Mingqi Li. "Discrete Fourier Transform-Based Block Faster-Than- Nyquist Transmission for 5G Wireless Communications." Applied Sciences 10, no. 4 (February 14, 2020): 1313. http://dx.doi.org/10.3390/app10041313.

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Faster-than-Nyquist (FTN) signaling is regarded as a potential candidate for improving data rate and spectral efficiency of 5G new radio (NR). However, complex detectors have to be utilized to eliminate the inter symbol interference (ISI) introduced by time-domain packing and the inter carrier interference (ICI) introduced by frequency-domain packing. Thus, the exploration of low complexity transceiver schemes and detectors is of great importance. In this paper, we consider a discrete Fourier transform (DFT) block transmission for multi-carrier FTN signaling, i.e., DBT-MC-FTN. With the aid of DFTs/IDFTs and frequency domain windowing, time- and frequency domain packing can be implemented flexibly and efficiently. At the receiver, the inherent ISI and ICI can be canceled via a soft successive interference cancellation (SIC) detector. The effectiveness of the detector is verified by the simulation over the additive white Gaussian noise channel and the fading channel. Furthermore, based on the characteristics of the efficient architecture of DFT-MC-FTN, two pilot-aided channel estimation schemes, i.e., time-division-multiplexing DBT-MC-FTN symbol-level pilot, and frequency-division-multiplexing subcarrier-level pilot within the DBT-MC-FTN symbol, respectively, are also derived. Numerical results show that the proposed channel estimation schemes can achieve high channel estimation accuracy.

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5

SAKK, ERIC. "ON THE COMPUTATION OF MOLECULAR SURFACE CORRELATIONS FOR PROTEIN DOCKING USING FOURIER TECHNIQUES." Journal of Bioinformatics and Computational Biology 05, no. 04 (August 2007): 915–35. http://dx.doi.org/10.1142/s0219720007002916.

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The computation of surface correlations using a variety of molecular models has been applied to the unbound protein docking problem. Because of the computational complexity involved in examining all possible molecular orientations, the fast Fourier transform (FFT) (a fast numerical implementation of the discrete Fourier transform (DFT)) is generally applied to minimize the number of calculations. This approach is rooted in the convolution theorem which allows one to inverse transform the product of two DFTs in order to perform the correlation calculation. However, such a DFT calculation results in a cyclic or "circular" correlation which, in general, does not lead to the same result as the linear correlation desired for the docking problem. In this work, we provide computational bounds for constructing molecular models used in the molecular surface correlation problem. The derived bounds are then shown to be consistent with various intuitive guidelines previously reported in the protein docking literature. Finally, these bounds are applied to different molecular models in order to investigate their effect on the correlation calculation.

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6

de Oliveira Neto, Jose R., and Juliano B. Lima. "Discrete Fractional Fourier Transforms Based on Closed-Form Hermite–Gaussian-Like DFT Eigenvectors." IEEE Transactions on Signal Processing 65, no. 23 (December 1, 2017): 6171–84. http://dx.doi.org/10.1109/tsp.2017.2750105.

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7

Han, Feng, Yao Lin Liu, Zhen Liu, and Hai Dong Zeng. "Comments on Errors of DFT Spectrum." Applied Mechanics and Materials 568-570 (June 2014): 189–92. http://dx.doi.org/10.4028/www.scientific.net/amm.568-570.189.

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Discrete Fourier transform (DFT/FFT) spectrums contain a variety of inherent errors in asynchronous sampling. Spectrum analysis with the accuracy above 10-3 are generally challenging issues. This work divides the DFT procedure into four signal transforms and exams six spectrum errors originated from these distortions. Besides the review of traditional errors, a so-called energy loss-gain (ELG) error is briefly introduced, which is proved to be a considerable error on the basis of Parseval's theorem. With the help of full error analysis mentioned here and the further development of analytical error estimators, it is expectable to obtain a DFT spectrum with a specified accuracy.

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8

BI, GUOAN, A. C. KOT, and ZIHOU MENG. "GENERALISED SPLIT-RADIX ALGORITHMS FOR DFT." Journal of Circuits, Systems and Computers 08, no. 03 (June 1998): 405–9. http://dx.doi.org/10.1142/s0218126698000201.

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This letter presents a general split-radix algorithm based on the decimation-in-time decomposition. It can flexibly compute the discrete Fourier transforms of length-q*2m where q is an odd integer. In comparison with other reported algorithms, our proposed one supports a wider range of sequence lengths, achieves a reduction of arithmetic operations and requires a simple computational structure.

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9

Wang, Bangbing, Bo Sun, Jiaxin Wang, Jamin Greenbaum, Jingxue Guo, Laura Lindzey, Xiangbin Cui, Duncan A. Young, Donald D. Blankenship, and Martin J. Siegert. "Removal of ‘strip noise’ in radio-echo sounding data using combined wavelet and 2-D DFT filtering." Annals of Glaciology 61, no. 81 (March 28, 2019): 124–34. http://dx.doi.org/10.1017/aog.2019.4.

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ABSTRACTRadio-echo sounding (RES) can be used to understand ice-sheet processes, englacial flow structures and bed properties, making it one of the most popular tools in glaciological exploration. However, RES data are often subject to ‘strip noise’, caused by internal instrument noise and interference, and/or external environmental interference, which can hamper measurement and interpretation. For example, strip noise can result in reduced power from the bed, affecting the quality of ice thickness measurements and the characterization of subglacial conditions. Here, we present a method for removing strip noise based on combined wavelet and two-dimensional (2-D) Fourier filtering. First, we implement discrete wavelet decomposition on RES data to obtain multi-scale wavelet components. Then, 2-D discrete Fourier transform (DFT) spectral analysis is performed on components containing the noise. In the Fourier domain, the 2-D DFT spectrum of strip noise keeps its linear features and can be removed with a ‘targeted masking’ operation. Finally, inverse wavelet transforms are performed on all wavelet components, including strip-removed components, to restore the data with enhanced fidelity. Model tests and field-data processing demonstrate the method removes strip noise well and, incidentally, can remove the strong first reflector from the ice surface, thus improving the general quality of radar data.

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10

Lai, Chao-Yuan, Tien-Yen Yang, and Chih-Wen Liu. "Performance of Implementing Smart DFT in Micro Phasor Measurement Unit." E3S Web of Conferences 69 (2018): 01005. http://dx.doi.org/10.1051/e3sconf/20186901005.

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The advance of the signal processing, this paper implement a more accurate digital measurement algorithm, Smart DFT(SDFT) in Micro Phasor Measurement Unit(μPMU), which is based on Discrete Fourier Transforms(DFT) to estimate frequency information from Taiwan power system. μPMU, the sensor we plan to acquire the information from the power system by using the signal of outlet voltage-level 110V in Taiwan, such as frequency, voltage and angle. The performance of SDFT implemented in μPMU represents a more precise frequency information when frequency fluctuation occurred just as the frequency of power system. We offer the results of simulations, stable frequency generated from waveform generator and real frequency from main electricity to compare SDFT with DFT method which implemented in μPMU.

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11

Brown, C. M., W. Dreyer, and W. H. Mu¨ller. "The Convergence of a DFT-Algorithm for Solution of Stress-Strain Problems in Composite Mechanics." Journal of Engineering Materials and Technology 125, no. 1 (December 31, 2002): 27–37. http://dx.doi.org/10.1115/1.1526859.

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This paper addresses the convergence characteristics of an iterative solution scheme of the Neumann-type useful for obtaining homogenized mechanical material properties from a representative volume element. The analysis is based on Eshelby’s idea of “equivalent inclusions” and, within the context of mechanical stress/strain analysis, allows modeling of elastically highly heterogeneous bodies with the aid of discrete Fourier transforms. Within the iterative scheme the proof of convergence depends critically upon the choice of an appropriate, auxiliary stiffness matrix, which also determines the speed of convergence. Mathematically speaking it is based on Banach’s fixpoint theorem and only results in sufficient convergence conditions. However, all cases of elastic heterogeneity that are of practical importance are covered and some evidence is provided that other choices of auxiliary stiffness may result in faster convergence even if this cannot explicitly be shown within the theoretical framework chosen.

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12

Deng, Zheng-Tao, H. John Caulfield, and Marius Schamschula. "Fractional discrete Fourier transforms." Optics Letters 21, no. 18 (September 15, 1996): 1430. http://dx.doi.org/10.1364/ol.21.001430.

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13

Belmont, M. R. "Taylor discrete Fourier transforms." IEE Proceedings F Radar and Signal Processing 136, no. 2 (1989): 101. http://dx.doi.org/10.1049/ip-f-2.1989.0018.

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14

Ariyanto, Yuri, Rizky Ardiansyah, and Bias Paris. "STEGANOGRAFI MENGGUNAKAN METODE DISCRETE FOURIER TRANSFORM (DFT)." Jurnal Informatika Polinema 4, no. 2 (February 1, 2018): 87. http://dx.doi.org/10.33795/jip.v4i2.151.

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Seiring dengan kemajuan teknologi, serangan terjadi pada industri photography di mana banyak penyalahgunaan foto yang memiliki hak cipta tanpa seijin pemilik foto tersebut. Karena itulah dibuat sebuah aplikasi yang berfungsi untuk menyisipkan watermark dengan menggunakan metode DFT (Discrete Fourier Transform). Metode tersebut adalah metode matematika yang sering digunakan dalam bidang elektronika dan komputer. Metode ini secara khusus digunakan untuk menyelesaikan masalah yang berhubungan dengan frekuensi, sehingga metode ini dapat digunakan dalam bidang citra digital. Metode ini diterapkan untuk melakukan penyisipan dan ekstraksi watermark pada citra penampung. Watermark tersebut disisipkan kedalam frekuensi domain pada gambar dan akan menghasilkan output citra ber-watermark atau embeded image. Hal ini adalah untuk mencegah penyalahgunaan hak cipta, namun watermark tersebut tidak nampak secara fisik. Hal ini dilakukan selain memberikan jaminan keamanan terhadap gambar, tapi juga tidak mengurangi estetika pada gambar tersebut. Analisa yang dilakukan adalah tingkat keberhasilan proses insertion dan extraction, serangan pada citra, uji kemiripan dengan pengujian NPCR (Number of Pixel of Change Rate), UACI (Unified Averaged Changed Intensity), dan PSNR (Peak Signal-to-Noise Ratio) pada proses insertion dan extraction. DFT disimpulkan aman terhadap serangan berupa cropping, resize, dan editing. Selain itu, dihasilkan nilai presentase perubahan yang rendah pada pengujian NPCR & UACI dan nilai yang tinggi pada pengujian PSNR.

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15

Reed, Michael J., Hung V. Nguyen, and Ronald E. Chambers. "Computing the Fourier transform in geophysics with the transform decomposition DFT." GEOPHYSICS 58, no. 11 (November 1993): 1707–9. http://dx.doi.org/10.1190/1.1443386.

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The Fourier transform and its computationally efficient discrete implementation, the fast Fourier transform (FFT), are omnipresent in geophysical processing. While a general implementation of the discrete Fourier transform (DFT) will take on the order [Formula: see text] operations to compute the transform of an N point sequence, the FFT algorithm accomplishes the DFT with an operation count proportional to [Formula: see text] When a large percentage of the output coefficients of the transform are not desired, or a majority of the inputs to the transform are zero, it is possible to further reduce the computation required to perform the DFT. Here, we review one possible approach to accomplishing this reduction and indicate its application to phase‐shift migration.

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16

Stade, Eric, and E. G. Layton. "Generalized discrete Fourier transforms: the discrete Fourier-Riccati-Bessel transform." Computer Physics Communications 85, no. 3 (March 1995): 336–70. http://dx.doi.org/10.1016/0010-4655(94)00124-k.

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17

Forrest, T. G., and Robert B. Suter. "The Discrete Fourier Transform (DFT) in Behavioural Analysis." Journal of Theoretical Biology 166, no. 4 (February 1994): 419–29. http://dx.doi.org/10.1006/jtbi.1994.1037.

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18

Atakishiyev, Natig M., Luis Edgar Vicent, and Kurt Bernardo Wolf. "Continuous vs. discrete fractional Fourier transforms." Journal of Computational and Applied Mathematics 107, no. 1 (July 1999): 73–95. http://dx.doi.org/10.1016/s0377-0427(99)00082-5.

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19

Soo-Chang Pei, Chien-Cheng Tseng, Min-Hung Yeh, and Jong-Jy Shyu. "Discrete fractional Hartley and Fourier transforms." IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing 45, no. 6 (June 1998): 665–75. http://dx.doi.org/10.1109/82.686685.

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20

Engström, Alexander. "Discrete Morse Functions from Fourier Transforms." Experimental Mathematics 18, no. 1 (January 2009): 45–53. http://dx.doi.org/10.1080/10586458.2009.10128886.

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21

Cooley, James W., and Vaclav Cizek. "Discrete Fourier Transforms and Their Applications." Mathematics of Computation 50, no. 182 (April 1988): 643. http://dx.doi.org/10.2307/2008635.

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22

Samadi, S., M. O. Ahmad, and M. N. S. Swamy. "Ramanujan sums and discrete Fourier transforms." IEEE Signal Processing Letters 12, no. 4 (April 2005): 293–96. http://dx.doi.org/10.1109/lsp.2005.843775.

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23

Hrivnák, Jiří, Mariia Myronova, and Jiří Patera. "Central Splitting of A2 Discrete Fourier–Weyl Transforms." Symmetry 12, no. 11 (November 4, 2020): 1828. http://dx.doi.org/10.3390/sym12111828.

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Two types of bivariate discrete weight lattice Fourier–Weyl transforms are related by the central splitting decomposition. The two-variable symmetric and antisymmetric Weyl orbit functions of the crystallographic reflection group A2 constitute the kernels of the considered transforms. The central splitting of any function carrying the data into a sum of components governed by the number of elements of the center of A2 is employed to reduce the original weight lattice Fourier–Weyl transform into the corresponding weight lattice splitting transforms. The weight lattice elements intersecting with one-third of the fundamental region of the affine Weyl group determine the point set of the splitting transforms. The unitary matrix decompositions of the normalized weight lattice Fourier–Weyl transforms are presented. The interpolating behavior and the unitary transform matrices of the weight lattice splitting Fourier–Weyl transforms are exemplified.

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24

Hekrdla, J. "Index transforms of multidimensional cyclic convolutions and discrete Fourier transforms." IEEE Transactions on Acoustics, Speech, and Signal Processing 34, no. 4 (August 1986): 996–97. http://dx.doi.org/10.1109/tassp.1986.1164893.

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25

Csuka, Barna, and Zsolt Kollár. "R–DFT-based Parameter Estimation for WiGig." Periodica Polytechnica Electrical Engineering and Computer Science 61, no. 2 (May 23, 2017): 224. http://dx.doi.org/10.3311/ppee.9737.

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In this paper we present parameter estimation methods for IEEE 802.11ad transmission to estimate the frequency offset value and channel impulse response. Furthermore a less known low complexity signal processing architecture – the Recursive Discrete Fourier Transform (R-DFT) – is applied which may improve the estimation results. The paper also discusses the R-DFT and its advantages compared to the conventional Fast Fourier Transform.

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26

Briggs, William L., and Van Emden Henson. "A table of analytical discrete Fourier transforms." Applied Numerical Mathematics 21, no. 4 (October 1996): 375–84. http://dx.doi.org/10.1016/0168-9274(96)00018-9.

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27

Thompson, W. J., and J. R. Macdonald. "Discrete and integral fourier transforms: analytical examples." Proceedings of the National Academy of Sciences 90, no. 15 (August 1, 1993): 6904–8. http://dx.doi.org/10.1073/pnas.90.15.6904.

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28

Saksida, Pavle. "On discrete and continuous nonlinear Fourier transforms." Journal of Physics: Conference Series 563 (November 26, 2014): 012025. http://dx.doi.org/10.1088/1742-6596/563/1/012025.

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29

Ruzzi, M. "Jacobi ϑ-functions and discrete Fourier transforms." Journal of Mathematical Physics 47, no. 6 (June 2006): 063507. http://dx.doi.org/10.1063/1.2209770.

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30

Prabhu, K., and H. Renganathan. "Optimum binary windows for discrete Fourier transforms." IEEE Transactions on Acoustics, Speech, and Signal Processing 34, no. 1 (February 1986): 216–20. http://dx.doi.org/10.1109/tassp.1986.1164790.

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31

Knezevic, Marko, Hamad F. Al-Harbi, and Surya R. Kalidindi. "Crystal plasticity simulations using discrete Fourier transforms." Acta Materialia 57, no. 6 (April 2009): 1777–84. http://dx.doi.org/10.1016/j.actamat.2008.12.017.

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32

Grigoryan, A. M. "New algorithms for calculating discrete fourier transforms." USSR Computational Mathematics and Mathematical Physics 26, no. 5 (January 1986): 84–88. http://dx.doi.org/10.1016/0041-5553(86)90044-3.

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33

Strawderman, Robert, William L. Briggs, and Van Emden Henson. "The DFT: An Owner's Manual for the Discrete Fourier Transform." Journal of the American Statistical Association 94, no. 445 (March 1999): 349. http://dx.doi.org/10.2307/2669724.

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34

Serbes, Ahmet, and Lutfiye Durak-Ata. "The discrete fractional Fourier transform based on the DFT matrix." Signal Processing 91, no. 3 (March 2011): 571–81. http://dx.doi.org/10.1016/j.sigpro.2010.05.007.

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35

Sottek, Roland, and Thiago Lobato. "High-resolution spectral analysis (HSA) vs. discrete fourier transform (DFT)." INTER-NOISE and NOISE-CON Congress and Conference Proceedings 263, no. 4 (August 1, 2021): 2555–66. http://dx.doi.org/10.3397/in-2021-2172.

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The Discrete Fourier Transform (DFT) is the standard technique for performing spectral analysis. It is used in the form of the well-known fast implementation (FFT) in almost all areas that deal with signal processing. However, the DFT algorithm has some limitations in terms of its resolution in time and frequency: the higher the time resolution, the lower the frequency resolution, and vice versa. The product of time (analysis duration) and analysis bandwidth (frequency resolution) is a constant. DFT results depend on the analysis window used (type and duration), although the physical signal properties do not change. The High-Resolution Spectral Analysis (HSA) method, published at the ASST '90, considers the window influence through spectral deconvolution and thus leads to a much lower time-bandwidth product, correlating better with human perception. Recently, variants of the HSA have been used for a psychoacoustic standard (roughness). Additionally, HSA is planned for a new model of fluctuation strength. This paper describes the improvements made to the HSA algorithm as well as its robustness against noise, and compares application results for both methods: HSA and DFT.

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Suresh, K., and T. V. Sreenivas. "Block Convolution Using Discrete Trigonometric Transforms and Discrete Fourier Transform." IEEE Signal Processing Letters 15 (2008): 469–72. http://dx.doi.org/10.1109/lsp.2008.923789.

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Ponomareva, Olga, and Aleksey Ponomarev. "Theoretical foundations of digital vector Fourier analysis of two-dimensional signals padded with zero samples." Information and Control Systems, no. 1 (March 3, 2021): 55–65. http://dx.doi.org/10.31799/1684-8853-2021-1-55-65.

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Introduction: The practice of using Fourier-processing of finite two-dimensional signals (including images), having confirmed its effectiveness, revealed a number of negative effects inherent in it. A well-known method of dealing with negative effects of Fourier-processing is padding signals with zeros. However, the use of this operation leads to the need to provide information control systems with additional memory and perform unproductive calculations. Purpose: To develop new discrete Fourier transforms for efficient and effective processing of two-dimensional signals padded with zero samples. Method: We have proposed a new method for splitting a rectangular discrete Fourier transform matrix into square matrices. The method is based on the application of the modulus comparability relation to order the rows (columns) of the Fourier matrix. Results: New discrete Fourier transforms with variable parameters were developed, being a generalization of the classical discrete Fourier transform. The article investigates the properties of Fourier transform bases with variable parameters. In respect to these transforms, the validity has been proved for the theorems of linearity, shift, correlation and Parseval's equality. In the digital spectral Fourier analysis, the concepts of a parametric shift of a two-dimensional signal, and a parametric periodicity of a two-dimensional signal have been introduced. We have estimated the reduction of the required memory size and the number of calculations when applying the proposed transforms, and compared them with the discrete Fourier transform. Practical relevance: The developed discrete Fourier transforms with variable parameters can significantly reduce the cost of Fourier processing of two-dimensional signals (including images) padded with zeros.

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Siu, Wan-Chi, and A. G. Constantinides. "Fast mersenne number transforms for the computation of discrete fourier transforms." Signal Processing 9, no. 2 (September 1985): 125–31. http://dx.doi.org/10.1016/0165-1684(85)90035-0.

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39

Brus, Adam, Jiří Hrivnák, and Lenka Motlochová. "Connecting (Anti)Symmetric Trigonometric Transforms to Dual-Root Lattice Fourier–Weyl Transforms." Symmetry 13, no. 1 (December 31, 2020): 61. http://dx.doi.org/10.3390/sym13010061.

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Explicit links of the multivariate discrete (anti)symmetric cosine and sine transforms with the generalized dual-root lattice Fourier–Weyl transforms are constructed. Exact identities between the (anti)symmetric trigonometric functions and Weyl orbit functions of the crystallographic root systems A1 and Cn are utilized to connect the kernels of the discrete transforms. The point and label sets of the 32 discrete (anti)symmetric trigonometric transforms are expressed as fragments of the rescaled dual root and weight lattices inside the closures of Weyl alcoves. A case-by-case analysis of the inherent extended Coxeter–Dynkin diagrams specifically relates the weight and normalization functions of the discrete transforms. The resulting unique coupling of the transforms is achieved by detailing a common form of the associated unitary transform matrices. The direct evaluation of the corresponding unitary transform matrices is exemplified for several cases of the bivariate transforms.

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40

Chapman, David. "Spreadsheet Demonstrations of Discrete and Fast Fourier Transforms." International Journal of Electrical Engineering & Education 30, no. 3 (July 1993): 211–15. http://dx.doi.org/10.1177/002072099303000303.

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Spreadsheet demonstration of discrete and fast Fourier transforms The creation of discrete and fast Fourier transforms on a spreadsheet is a simple and informative exercise which graphically illustrates the simplicity and symmetries of the FFT. Experiments with ‘what-if’ exercises on the resulting spreadsheets lead to insight into concepts of the digital time and frequency domains.

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Ricci, P. E., and G. Mastroianni. "ERROR ESTIMATES FOR A CLASS OF INTEGRAL AND DISCRETE TRANSFORMS." Studia Scientiarum Mathematicarum Hungarica 36, no. 3-4 (December 1, 2000): 291–306. http://dx.doi.org/10.1556/sscmath.36.2000.3-4.1.

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We consider a class of integral transforms which generalize the classical Fourier Trans- form.We erive some theoretical error bounds for the corresponding approximate iscrete transforms,inclu ing the Discrete Fourier Transform.

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Britanak, Vladimir, and K. R. Rao. "The fast generalized discrete Fourier transforms: A unified approach to the discrete sinusoidal transforms computation." Signal Processing 79, no. 2 (December 1999): 135–50. http://dx.doi.org/10.1016/s0165-1684(99)00088-2.

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Su, Xinhua, Ran Tao, and Xuejing Kang. "Analysis and comparison of discrete fractional fourier transforms." Signal Processing 160 (July 2019): 284–98. http://dx.doi.org/10.1016/j.sigpro.2019.01.019.

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MOSCA, MICHELE, and CHRISTOF ZALKA. "EXACT QUANTUM FOURIER TRANSFORMS AND DISCRETE LOGARITHM ALGORITHMS." International Journal of Quantum Information 02, no. 01 (March 2004): 91–100. http://dx.doi.org/10.1142/s0219749904000109.

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Abstract:

We show how the Quantum Fast Fourier Transform (QFFT) can be made exact for arbitrary orders (first showing it for large primes). Most quantum algorithms only need a good approximation of the quantum Fourier transform of order 2n to succeed with high probability, and this QFFT can in fact be done exactly. Kitaev1 showed how to approximate the Fourier transform for any order. Here we show how his construction can be made exact by using the technique known as "amplitude amplification". Although unlikely to be of any practical use, this construction allows one to make Shor's discrete logarithm quantum algorithm exact. Thus we have the first example of an exact non black box fast quantum algorithm, thereby giving more evidence that "quantum" need not be probabilistic. We also show that in a certain sense the family of circuits for the exact QFFT is uniform. Namely, the parameters of the gates can be approximated efficiently.

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Ersoy, O. K., and N. C. Hu. "Fast algorithms for the discrete Fourier preprocessing transforms." IEEE Transactions on Signal Processing 40, no. 4 (April 1992): 744–57. http://dx.doi.org/10.1109/78.127949.

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Zhongde Wang. "On computing the discrete Fourier and cosine transforms." IEEE Transactions on Acoustics, Speech, and Signal Processing 33, no. 5 (October 1985): 1341–44. http://dx.doi.org/10.1109/tassp.1985.1164710.

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PENNY, J. E. T., M. I. FRISWELL, and S. D. GARVEY. "DETECTING ALIASED FREQUENCY COMPONENTS IN DISCRETE FOURIER TRANSFORMS." Mechanical Systems and Signal Processing 17, no. 2 (March 2003): 473–81. http://dx.doi.org/10.1006/mssp.2001.1445.

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Cabras, G., V. Roberto, and G. Salemi. "UNIDFT: A package of optimized discrete fourier transforms." Computer Physics Communications 47, no. 1 (October 1987): 113–27. http://dx.doi.org/10.1016/0010-4655(87)90071-3.

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Soo-Chang Pei and Jian-Jiun Ding. "Closed-form discrete fractional and affine Fourier transforms." IEEE Transactions on Signal Processing 48, no. 5 (May 2000): 1338–53. http://dx.doi.org/10.1109/78.839981.

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Marhic, M. E. "Discrete Fourier transforms by single-mode star networks." Optics Letters 12, no. 1 (January 1, 1987): 63. http://dx.doi.org/10.1364/ol.12.000063.

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