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Mukhammad Solikhin, Yohanssen Pratama, Purnama Pasaribu, Josua Rumahorbo, and Bona Simanullang. "Analisis Watermarking Menggunakan Metode Discrete Cosine Transform (DCT) dan Discrete Fourier Transform (DFT)." Jurnal Sistem Cerdas 5, no. 3 (December 11, 2022): 155–70. http://dx.doi.org/10.37396/jsc.v5i3.192.
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Digital image watermarking is the insertion of watermarks into digital image media. Several types of watermarking methods used are Discrete Cosine Transform (DCT) and Discrete Fourier Transform (DFT). Both of these watermarking methods work in the frequency domain (transform). Digital image watermarking using the frequency domain is carried out on the frequency coefficient. This study used 30 digital image data as material for digital image watermaking analysis with 10 data each in binary, grayscale and color digital images in jpg, png and bmp formats. Digital images in the binary and grayscale domains are conversions from digital images in the true color (RGB) domain. Digital image watermarking includes three main processes, namely embedding the watermarked image on the original digital image, extracting the watermarked image and measuring the correlation between the two digital images. Correlation aims to measure two variables that have the same relationship. The technology used in this research work is MATLAB (Matrix Laboratory) as a high-performance programming language for computing in solving problems with solutions expressed in mathematical notation. The results of the discussion prove that the watermarking process in terms of color, for DCT, RGB is better and binary is better for DFT. And the watermaking process, in terms of the type of watermark inserted, for both DCT and DFT, a good watermark is an invisible watermark.
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Balsa, Jose. "Comparison of Image Compressions: Analog Transformations." Proceedings 54, no. 1 (August 21, 2020): 37. http://dx.doi.org/10.3390/proceedings2020054037.
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A comparison between the four most used transforms, the discrete Fourier transform (DFT), discrete cosine transform (DCT), the Walsh–Hadamard transform (WHT) and the Haar-wavelet transform (DWT), for the transmission of analog images, varying their compression and comparing their quality, is presented. Additionally, performance tests are done for different levels of white Gaussian additive noise.
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Somasekhar, B., Ch Mohana Krishna, and Y. Murty. "Investigations on wavelet and Fourier transform based channel estimation in MIMO-OFDM system." International Journal of Engineering & Technology 7, no. 2.21 (April 20, 2018): 228. http://dx.doi.org/10.14419/ijet.v7i2.21.12178.
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In this paper channel estimation methods for MIMO-OFDM system are investigated based on Fourier Transform and Wavelet Transform. The channel estimation algorithm based on Discrete Fourier Transform (DFT) cause energy leakage in multipath channel with non-sample-spaced time delays. Discrete Cosine Transform (DCT) based channel estimator can mitigate the drawback of Discrete Fourier Transform based channel estimator, when the non-sample spaced path delays are available in multipath fading channels. Wavelet based systems provide better spectral efficiency because of no cyclic prefix requirement, with narrow side lobes and also exhibit improved BER performance. Simulation results reveal that the DWT based transform outperforms the conventional DFT and DCT based channel estimator in terms of bit error rate and mean square error.
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Ito, Izumi. "A New Pseudo-Spectral Method Using the Discrete Cosine Transform." Journal of Imaging 6, no. 4 (March 28, 2020): 15. http://dx.doi.org/10.3390/jimaging6040015.
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The pseudo-spectral (PS) method on the basis of the Fourier transform is a numerical method for estimating derivatives. Generally, the discrete Fourier transform (DFT) is used when implementing the PS method. However, when the values on both sides of the sequences differ significantly, oscillatory approximations around both sides appear due to the periodicity resulting from the DFT. To address this problem, we propose a new PS method based on symmetric extension. We mathematically derive the proposed method using the discrete cosine transform (DCT) in the forward transform from the relation between DFT and DCT. DCT allows a sequence to function as a symmetrically extended sequence and estimates derivatives in the transformed domain. The superior performance of the proposed method is demonstrated through image interpolation. Potential applications of the proposed method are numerical simulations using the Fourier based PS method in many fields such as fluid dynamics, meteorology, and geophysics.
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Ramadhyagita, Irsya Luthfiah, Annisa Annisa, Faishal Kamindra, and Farhan Muhammad Rizky. "Kajian Discrete Fourier Transform untuk Menganalisis Sinyal Arbitrer." Mitra Pilar: Jurnal Pendidikan, Inovasi, dan Terapan Teknologi 1, no. 1 (June 30, 2022): 7–16. http://dx.doi.org/10.58797/pilar.0101.02.
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Abstract This study aims to examine the Discrete Fourier Transform using arbitrary signals. Signal processing is a mathematical operation performed on a signal to obtain the required information. In this case, a transformation occurs. The Fourier transform is a popular method often used to change the time form to the frequency form intended to facilitate analysis. The Fourier transform is divided into two parts: the Continuous Fourier and the Discrete Fourier. The program developed by the researcher, DFT, analyzes arbitrary signals created in two programs. The first program is a function of the DFT and the second is the main program used to generate DFT graphs. In this project, researchers will analyze arbitrary signals decomposed into sine waves. This program uses two main libraries, namely matplotlib, and NumPy. In this case, we have used the DFT method and plotted the results of the calculations performed. Now the researcher will analyze the process that the researcher did while using the DFT method and plot the results. Using DFT, arbitrary signals can be arranged into a series of sinusoids, each with a different frequency. The DFT method can arrange signals into a series of sinusoids with different frequencies. In the DFT amplitude spectrum, the signal frequency is shown as a vertical bar, with the height being the signal amplitude in the time domain. DFT can convert a sequence of equally spaced signals into information about the frequencies of all sine waves needed to sum the time domain signals. Abstrak Penelitian ini bertujuan untuk mengkaji Discrete Fourier Transform dengan menggunakan sinyal arbitrer. Pengolahan sinyal adalah suatu operasi matematik yang dilakukan terhadap suatu sinyal sehingga diperoleh suatu informasi yang dibutuhkan. Dalam hal ini terjadi suatu transformasi. Transformasi Fourier merupakan salah satu metode popular yang sering digunakan untuk mengubah bentuk waktu bentuk frekuensi yang ditujukan untuk mempermudah analisis. Transformasi Fourier dibagi menjadi 2 bagian yaitu Fourier Kontinu dan Fourier Diskret. Program yang dikembangkan oleh peneliti yaitu, DFT untuk menganalisis sinyal arbitrer yang dibuat dalam dua program. Program pertama adalah fungsi dari DFT dan program kedua adalah program utama yang digunakan untuk memunculkan grafik DFT. Dalam project ini peneliti akan menganalisis sinyal arbitrer yang sudah diuraikan menjadi gelombang sinus. Program ini menggunakan dua library utama, yaitu matplotlib dan numpy. Pada case kali ini, telah menggunakan metode DFT dan membuat hasil plot dari perhitungan yang dilakukan. Sekarang peneliti akan menganalisis proses yang peneliti lakukan selama menggunakan metode DFT dan membuat hasil plot. Sebelum membahas hasil perhitungan, mari kembali membahas metode DFT itu sendiri. Dengan menggunakan DFT, dapat disusun sinyal arbitrer menjadi serangkaian sinusoid dan masing-masing akan memiliki frekuensi yang berbeda.Metode DFT dapat menyusun sinyal menjadi serangkaian sinusoid yang memiliki frekuensi yang berbeda-beda. Dalam spektrum amplitude DFT, frekuensi sinyal ditampilkan sebagai batang vertikal dengan ketinggiannya adalah amplitude sinyal dalam domain waktu. DFT dapat mengubah urutan sinyal dengan jarak yang sama menjadi informasi tentang frekuensi semua gelombang sinus yang diperlukan untuk menjumlahkan sinyal domain waktu.
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Shi, Zhiping, Yupeng Zhang, Yong Guan, Liming Li, and Jie Zhang. "The Formalization of Discrete Fourier Transform in HOL." Mathematical Problems in Engineering 2015 (2015): 1–8. http://dx.doi.org/10.1155/2015/687152.
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Traditionally, Discrete Fourier Transform (DFT) is performed with numerical or symbolic computation, which cannot guarantee 100% accurate analysis which may be necessary for safety-critical applications. Machine theorem proving is one of the formal methods that perform accurate analysis with completeness to some extent. This paper proposes the formalization of DFT in a higher-order logic theorem prover named HOL. We propose the formal definition of DFT and verify the fundamental properties of DFT. Two case studies are presented to illustrate usefulness and correctness of the formalized DFT, including formal verifications of Fast Fourier Transform (FFT) and cosine frequency shift.
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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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Yao, Xueyang, and Natalie Baddour. "Discrete two dimensional Fourier transform in polar coordinates part II: numerical computation and approximation of the continuous transform." PeerJ Computer Science 6 (March 2, 2020): e257. http://dx.doi.org/10.7717/peerj-cs.257.
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The theory of the continuous two-dimensional (2D) Fourier Transform in polar coordinates has been recently developed but no discrete counterpart exists to date. In the first part of this two-paper series, we proposed and evaluated the theory of the 2D Discrete Fourier Transform (DFT) in polar coordinates. The theory of the actual manipulated quantities was shown, including the standard set of shift, modulation, multiplication, and convolution rules. In this second part of the series, we address the computational aspects of the 2D DFT in polar coordinates. Specifically, we demonstrate how the decomposition of the 2D DFT as a DFT, Discrete Hankel Transform and inverse DFT sequence can be exploited for coding. We also demonstrate how the proposed 2D DFT can be used to approximate the continuous forward and inverse Fourier transform in polar coordinates in the same manner that the 1D DFT can be used to approximate its continuous counterpart.
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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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Fischer, Jens. "Four Particular Cases of the Fourier Transform." Mathematics 6, no. 12 (December 18, 2018): 335. http://dx.doi.org/10.3390/math6120335.
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In previous studies we used Laurent Schwartz’ theory of distributions to rigorously introduce discretizations and periodizations on tempered distributions. These results are now used in this study to derive a validity statement for four interlinking formulas. They are variants of Poisson’s Summation Formula and connect four commonly defined Fourier transforms to one another, the integral Fourier transform, the Discrete-Time Fourier Transform (DTFT), the Discrete Fourier Transform (DFT) and the integral Fourier transform for periodic functions—used to analyze Fourier series. We prove that under certain conditions, these four Fourier transforms become particular cases of the Fourier transform in the tempered distributions sense. We first derive four interlinking formulas from four definitions of the Fourier transform pure symbolically. Then, using our previous results, we specify three conditions for the validity of these formulas in the tempered distributions sense.
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Atakishiyeva, Mesuma K., Natig M. Atakishiyev, and Juan Loreto-Hernández. "More on algebraic properties of the discrete Fourier transform raising and lowering operators." 4open 2 (2019): 2. http://dx.doi.org/10.1051/fopen/2018010.
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In the present work, we discuss some additional findings concerning algebraic properties of the N-dimensional discrete Fourier transform (DFT) raising and lowering difference operators, recently introduced in [Atakishiyeva MK, Atakishiyev NM (2015), J Phys: Conf Ser 597, 012012; Atakishiyeva MK, Atakishiyev NM (2016), Adv Dyn Syst Appl 11, 81–92]. In particular, we argue that the most authentic symmetrical form of discretization of the integral Fourier transform may be constructed as the discrete Fourier transforms based on the odd points N only, while in the discrete Fourier transforms on the even points N this symmetry is spontaneously broken. This heretofore undetected distinction between odd and even dimensions is shown to be intimately related with the newly revealed algebraic properties of the above-mentioned DFT raising and lowering difference operators and, of course, is very consistent with the well-known formula for the multiplicities of the eigenvalues, associated with the N-dimensional DFT. In addition, we propose a general approach to deriving the eigenvectors of the discrete number operators N(N), that avoids the above-mentioned pitfalls in the structure of each even-dimensional case N = 2L.
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Wang, Yulin, and Gengxin Zhang. "Compressed Wideband Spectrum Sensing Based on Discrete Cosine Transform." Scientific World Journal 2014 (2014): 1–5. http://dx.doi.org/10.1155/2014/464895.
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Discrete cosine transform (DCT) is a special type of transform which is widely used for compression of speech and image. However, its use for spectrum sensing has not yet received widespread attention. This paper aims to alleviate the sampling requirements of wideband spectrum sensing by utilizing the compressive sampling (CS) principle and exploiting the unique sparsity structure in the DCT domain. Compared with discrete Fourier transform (DFT), wideband communication signal has much sparser representation and easier implementation in DCT domain. Simulation result shows that the proposed DCT-CSS scheme outperforms the conventional DFT-CSS scheme in terms of MSE of reconstruction signal, detection probability, and computational complexity.
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Baddour. "Discrete Two-Dimensional Fourier Transform in Polar Coordinates Part I: Theory and Operational Rules." Mathematics 7, no. 8 (August 2, 2019): 698. http://dx.doi.org/10.3390/math7080698.
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The theory of the continuous two-dimensional (2D) Fourier transform in polar coordinates has been recently developed but no discrete counterpart exists to date. In this paper, we propose and evaluate the theory of the 2D discrete Fourier transform (DFT) in polar coordinates. This discrete theory is shown to arise from discretization schemes that have been previously employed with the 1D DFT and the discrete Hankel transform (DHT). The proposed transform possesses orthogonality properties, which leads to invertibility of the transform. In the first part of this two-part paper, the theory of the actual manipulated quantities is shown, including the standard set of shift, modulation, multiplication, and convolution rules. Parseval and modified Parseval relationships are shown, depending on which choice of kernel is used. Similar to its continuous counterpart, the 2D DFT in polar coordinates is shown to consist of a 1D DFT, DHT and 1D inverse DFT.
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Korvel, Grazina, Bozena Kostek, and Olga Kurasova. "Comparative Analysis of Various Transformation Techniques for Voiceless Consonants Modeling." International Journal of Computers Communications & Control 13, no. 5 (September 29, 2018): 853–64. http://dx.doi.org/10.15837/ijccc.2018.5.3310.
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In this paper, a comparison of various transformation techniques, namely Discrete Fourier Transform (DFT), Discrete Cosine Transform (DCT) and Discrete Walsh Hadamard Transform (DWHT) is performed in the context of their application to voiceless consonant modeling. Speech features based on these transformation techniques are extracted. These features are mean and derivative values of cepstrum coefficients, derived from each transformation. Fea-ture extraction is performed on the speech signal divided into short-time seg-ments. The kNN and Naive Bayes methods are used for phoneme classification. Experiments show that DFT and DCT give better classification accuracy than DWHT. The result of DFT was not significantly different from DCT, but it was for DWHT. The same tendency was revealed for DCT. It was checked with the usage of the ANOVA test that the difference between results obtained by DCT and DWHT is significant.
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Chechetkin, V. R., and V. V. Lobzin. "Detection of Large-Scale Noisy Multi-Periodic Patterns with Discrete Double Fourier Transform." Fluctuation and Noise Letters 19, no. 02 (November 20, 2019): 2050019. http://dx.doi.org/10.1142/s0219477520500194.
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In many processes, the variations in underlying characteristics can be approximated by noisy multi-periodic patterns. If large-scale patterns are superimposed by a noise with long-range correlations, the detection of multi-periodic patterns becomes especially challenging. To solve this problem, we developed a discrete double Fourier transform (DDFT). DDFT is based on the equidistance property of harmonics generated by multi-periodic patterns in the discrete Fourier transform (DFT) spectra. As the large-scale patterns generate long enough equidistant series, they can be detected by the iteration of the primary DFT. DDFT is defined as Fourier transform of intensity spectral harmonics or of their functions. It comprises widely used cepstrum transform as a particular case. We present also the relevant analytical criteria for the assessment of the statistical significance of peak harmonics in DDFT spectra in the presence of noise. DDFT technique was tested by extensive numerical simulations. The practical applications of the DDFT technique are illustrated by the analysis of variations in solar wind speed related to solar rotation and by the study of large-scale multi-periodic patterns in DNA sequences. The latter application can be considered as a generic example for the general spectral analysis of symbolic sequences. The results are compared with those obtained by the cepstrum transform. The mutual combination of DFT and DDFT provides an efficient technique to search for noisy large-scale multi-periodic patterns.
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Zhang, Tianhao. "Fourier Transform in Three Differences Spaces." Highlights in Science, Engineering and Technology 38 (March 16, 2023): 527–43. http://dx.doi.org/10.54097/hset.v38i.5878.
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This paper starts by introducing the application of the Fourier analysis and divides the Fourier analysis into four cases according to the difference between physical space and frequency space: discrete points on a circle discrete points on a circle, , , and . The four cases are then grouped into three parts: the discrete Fourier transform (DFT), the Fourier series, and the Fourier transform. For each part, this paper discusses its characteristics and basic theory, and selectively deals with the proof and application examples. This paper will promote some research related to Fourier analysis by analyzing Fourier analysis in three different spaces as a whole.
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Luque, Amalia, Jesús Gómez-Bellido, Alejandro Carrasco, and Julio Barbancho. "Exploiting the Symmetry of Integral Transforms for Featuring Anuran Calls." Symmetry 11, no. 3 (March 20, 2019): 405. http://dx.doi.org/10.3390/sym11030405.
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The application of machine learning techniques to sound signals requires the previous characterization of said signals. In many cases, their description is made using cepstral coefficients that represent the sound spectra. In this paper, the performance in obtaining cepstral coefficients by two integral transforms, Discrete Fourier Transform (DFT) and Discrete Cosine Transform (DCT), are compared in the context of processing anuran calls. Due to the symmetry of sound spectra, it is shown that DCT clearly outperforms DFT, and decreases the error representing the spectrum by more than 30%. Additionally, it is demonstrated that DCT-based cepstral coefficients are less correlated than their DFT-based counterparts, which leads to a significant advantage for DCT-based cepstral coefficients if these features are later used in classification algorithms. Since the DCT superiority is based on the symmetry of sound spectra and not on any intrinsic advantage of the algorithm, the conclusions of this research can definitely be extrapolated to include any sound signal.
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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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Wei, Shu Ang, Li Gao, Zhi Ye Sun, and Shi Jue Zheng. "A Mobile Data Minging Algorithm Based on Discrete Fourier Transform by Genetic Algorithm." Advanced Materials Research 108-111 (May 2010): 1452–57. http://dx.doi.org/10.4028/www.scientific.net/amr.108-111.1452.
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In the mobile environment, considering resource constraint, the frequent disconnect, synchronous data flow, the cost of communication, mobility and so on, combined with discrete Fourier transform (Discrete Fourier Transform, DFT) algorithm to facilitate time-domain and frequency domain conversion advantages as well as the genetic algorithm’s (Genetic Algorithm, GA) good global search capability ,this paper proposes a mobile data mining model which is based on the combination of Discrete Fourier Transform and Genetic Algorithm (DFTGA).
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Ponomareva, O. V., A. V. Ponomarev, and N. V. Smirnova. "Algorithms for Direct and Inverse Parametric Fast Fourier Transform." Informacionnye Tehnologii 28, no. 1 (January 17, 2022): 9–19. http://dx.doi.org/10.17587/it.28.9-19.
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Русский Main page New issue Archive of articles Editorial board For the authors Publishing house ABSTRACTS OF ARTICLES OF THE JOURNAL "INFORMATION TECHNOLOGIES". No. 1. Vol. 28. 2022 DOI: 10.17587/it.28.9-19 O. V. Ponomareva, Dr. Sc., Tech., Professor, A. V. Ponomarev, PhD, Econ., Associate Professor, Kalashnikov Izhevsk State Technical University, Izhevsk, 426069, Russian Federation, N. V. Smirnova, PhD, Tech., Associate Professor, Sevastopol State University, Sevastopol, 299053, Russian Federation Algorithms for Direct and Inverse Parametric Fast Fourier Transform Classical Fourier processing of finite information discrete signals (FID signals) is the most important method of digital analysis, modeling, optimization, improvement of control and decision making. The theoretical basis of classical Fourier processing of FID signals is the discrete Fourier transform (DFT). The practical basis of classical Fourier processing of FID signals is the Fast Fourier Transform (FFT). The practice of using classical Fourier processing of FID signals, having confirmed its effectiveness, revealed a number of negative effects inherent in this type of digital signal processing (DSP). The aliasing effect, scalloping effect, picket fence effect, significantly affect the effectiveness of analysis, modeling, optimization, improvement of management and decision making. To increase the efficiency of Fourier processing of FID signals, the authors of the paper have developed a generalization of DFT in the form of a parametric DFT (DFT-P). Since the direct application of parametric Fourier processing of FID signals (as well as the use of classical Fourier processing of FID signals) requires complex multiplications, fast procedures are required for the practical implementation of this type of FID signals. Purpose of the research is to develop algorithms for the fast parametric discrete Fourier transform (FFT-P). The work developed fast procedures for the implementation of DFT-P by time decimation. Parametric FFT-P with substitution (in place) and without substitution (no place) are proposed. The estimation of the efficiency of the FFT-P algorithms is given. The practical significance of the work is in the fact that developing algorithms for the parametric fast Fourier transform can reduce the computational costs of performing parametric discrete transformations by three or more orders of magnitude.
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Alzain, Mohammed. "Image Encryption Using Chaotic Cat Mapping in the Discrete Fourier Transform." INTERNATIONAL JOURNAL OF COMPUTERS & TECHNOLOGY 18 (November 29, 2018): 7389–97. http://dx.doi.org/10.24297/ijct.v18i0.7907.
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The paper presents an secure image using the two dimensional chaotic cat mapping (2D-CCM) in the Discrete Fourier Transform domain (DFT). The ciphering phase begins by applying the DFT on the plainimage to be encrypted and the resulted Fourier transformed image are scrambled using the 2D-CCM and finally an inverse DFT is applied to obtain the final encrypted image. The decryption phase applies a reverse procedure to get the original plainimage. A set of encryption test experiments are employed to inspect the proposed DFT based 2D-CCM image cryptosystem. The experimental results verified and confirmed the superiority of the proposed DFT based 2D-CCM image cryptosystem.
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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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Byun, Keun-Yung, Chun-Su Park, Jee-Young Sun, and Sung-Jea Ko. "Vector Radix 2 × 2 Sliding Fast Fourier Transform." Mathematical Problems in Engineering 2016 (2016): 1–8. http://dx.doi.org/10.1155/2016/2416286.
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The two-dimensional (2D) discrete Fourier transform (DFT) in the sliding window scenario has been successfully used for numerous applications requiring consecutive spectrum analysis of input signals. However, the results of conventional sliding DFT algorithms are potentially unstable because of the accumulated numerical errors caused by recursive strategy. In this letter, a stable 2D sliding fast Fourier transform (FFT) algorithm based on the vector radix (VR) 2 × 2 FFT is presented. In the VR-2 × 2 FFT algorithm, each 2D DFT bin is hierarchically decomposed into four sub-DFT bins until the size of the sub-DFT bins is reduced to 2 × 2; the output DFT bins are calculated using the linear combination of the sub-DFT bins. Because the sub-DFT bins for the overlapped input signals between the previous and current window are the same, the proposed algorithm reduces the computational complexity of the VR-2 × 2 FFT algorithm by reusing previously calculated sub-DFT bins in the sliding window scenario. Moreover, because the resultant DFT bins are identical to those of the VR-2 × 2 FFT algorithm, numerical errors do not arise; therefore, unconditional stability is guaranteed. Theoretical analysis shows that the proposed algorithm has the lowest computational requirements among the existing stable sliding DFT algorithms.
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BHATNAGAR, GAURAV, and BALASUBRAMANIAN RAMAN. "DISTRIBUTED MULTIRESOLUTION DISCRETE FOURIER TRANSFORM AND ITS APPLICATION TO WATERMARKING." International Journal of Wavelets, Multiresolution and Information Processing 08, no. 02 (March 2010): 225–41. http://dx.doi.org/10.1142/s021969131000347x.
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The Fourier transform is undoubtedly one of the most valuable and frequently used tools in signal processing and analysis but it has some limitations. In this paper, we rectify these limitations by proposing a newer version of Fourier transform, namely, Distributed Multiresolution Discrete Fourier Transform (D-MR-DFT) and its application in digital watermarking. The core idea of the proposed watermarking scheme is to decompose an image into four frequency sub-bands using D-MR-DFT and then singular values of every sub-band are modified with the singular values of the watermark. The experimental results show better visual imperceptibility and resiliency of the proposed scheme against intentional or unintentional variety of attacks.
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Varghese, Justin, Omer Bin Hussain, Saudia Subash, and Abdul Razak T. "An effective digital image watermarking scheme incorporating DCT, DFT and SVD transformations." PeerJ Computer Science 9 (July 10, 2023): e1427. http://dx.doi.org/10.7717/peerj-cs.1427.
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Image watermarking prevents copyright infringements by attaching visible/invisible watermark images as authentication identities in the owner’s documents. The article made analysis on the advantages of different transformations for choosing better combinations to make the watermark embedding process and observed that watermarking techniques incorporating discrete cosine transform (DCT) provide better resistance towards JPEG based potential attacks, discrete Fourier transform (DFT) has strong energy compaction with geometrical invariance properties to resist geometric attacks while singular value decomposition (SVD) provides stability, proportion invariance and rotation invariance properties and it provides strong resistance against noise based attacks. Considering these advantages of different transformations, the article presents a new non-blind watermarking algorithm by utilizing advantages of DFT, DCT and SVD transforms while attaching secret contents in cover images. The algorithm starts by applying DFT followed by onion peel decomposition (OPD) for decomposing Fourier domain carrier image to four frequency sub images. The scheme then applies DCT on the frequency bands and orders them in zigzag fashion to form four individual frequency arrays. In the final step of embedding process, it embeds four copies of watermark singular value contents in DFT domain with the carrier image singular value contents to produce the watermarked image. The experimental results based on subjective and objective metrics on various test images from standard image databases with different test conditions demarcate the stability of new algorithm in producing high quality watermarked images with fewer distortions even when the watermarked images are extremely distorted by potential attacks.
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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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Saatcilar, R., S. Ergintav, and N. Canitez. "The use of the Hartley transform in geophysical applications." GEOPHYSICS 55, no. 11 (November 1990): 1488–95. http://dx.doi.org/10.1190/1.1442796.
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The Hartley transform (HT) is an integral transform similar to the Fourier transform (FT). It has most of the characteristics of the FT. Several authors have shown that fast algorithms can be constructed for the fast Hartley transform (FHT) using the same structures as for the fast Fourier transform. However, the HT is a real transform and for this reason, since one complex multiplication requires four real multiplications, the discrete HT (DHT) is computationally faster than the discrete FT (DFT). Consequently, any process requiring the DFT (such as amplitude and phase spectra) can be performed faster by using the DHT. The general properties of the DHT are reviewed first, and then an attempt is made to use the FHT in some seismic data processing techniques such as one‐dimensional filtering, forward seismic modeling, and migration. The experiments show that the Hartley transform is two times faster than the Fourier transform.
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Liu, Yao Lin, Feng Han, Zhen Liu, and Min Chen Zhai. "Analysis of Energy Loss-Gain Error in Discrete Fourier Transform." Applied Mechanics and Materials 568-570 (June 2014): 172–75. http://dx.doi.org/10.4028/www.scientific.net/amm.568-570.172.
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In asynchronous sampling, discrete Fourier transform (DFT) spectrum involves errors. Scholars have done great investigations on the correction techniques of DFT spectrum, but the errors have not been completely eliminated all along. In this paper, spectrums were examined from the principle of conservation of energy. It is unnoticed before that the energy of the digital signal, which is the analysis object of DFT, isn't equal to that of the finite continuous signal truncated by rectangular window. Thus the energy of their spectrums are different according to the Parseval's theorem. The Energy Loss-Gain (ELG) error was introduced to express the energy difference between these two spectrums. The ELG error is zero if the observed continuous signal is truncated in integral multiple of half cycle and it is related to the cycle number and sampling number in one cycle. Analysis show that the ELG error decreases with the increment of these two parameters, which are helpful to the engineering.
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Bai, Yuchan, Jinyu Liu, Huihao Wu, Lei Fan, Renqing Li, and Jun Xing. "Sinusoidal Frequency Estimator by Using Interpolation of Four DTFT Spectral Lines." Journal of Physics: Conference Series 2564, no. 1 (August 1, 2023): 012016. http://dx.doi.org/10.1088/1742-6596/2564/1/012016.
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Abstract This paper proposes a sinusoid frequency estimator by interpolating four Discrete Time Fourier Transform (DTFT) samples. Firstly, Discrete Fourier Transform (DFT) is performed on the received sinusoid. Then we find the positional information of the maximal DFT bin and obtain the coarse estimation results. Next, the proposed method uses four symmetrical DTFT samples on both sides of the maximal DFT bin to interpolate the signal frequency and obtain accurate estimation results. Simulation experiment results illustrate that the proposed method’s accuracy is higher than that of the competitive methods. When the signal-to-noise ratio (SNR) increases from 0dB to 90dB, the presented estimator performs better than the competitive methods.
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Amhar, Fahmi, Endang Purnama Giri, Florence Elfriede Sinthauli Silalahi, Shelvie Nidya Neyman, Anggrahito, Dadan Ramdani, Danang Jaya, et al. "Ownership Protection on Digital Elevation Model (DEM) Using Transform-Based Watermarking." ISPRS International Journal of Geo-Information 11, no. 3 (March 16, 2022): 200. http://dx.doi.org/10.3390/ijgi11030200.
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This research aims to protect Digital Elevation Model (DEM) data from piracy or counterfeiting. An invisible watermark inserted into the data, which will not considerably change the data value, is necessary. The proposed method involves the use of the two-dimensional discrete cosine transform (2D DCT), a combination of 2D DCT and discrete wavelet transform (DWT), and two-dimensional discrete Fourier transform (2D DFT) in the frequency domain. The data used include a National DEM file downloaded from the geoportal of the Geospatial Information Agency (Badan Informasi Geospasial—BIG). Three files represent mountainous, lowland/urban, and coastal areas. An “attack” is also conducted on the watermarked DEM by cropping. The results indicate that the watermarked DEM is well recognized. The watermark can be read 100% for 2D DCT, while that for 2D DFT can be read 90.50%. The distortion value of the elevation data under the DCT technique demonstrates the smallest maximum value of 0.1 m compared with 4.5 and 1.1 m for 2D DFT and 2D DCT–DWT. Meanwhile, the height difference (Max Delta), the peak signal-to-noise ratio, and the root mean squared error (RMSE) are highest in mountainous, lowland, and coastal areas, respectively. Overall, the 2D DCT is also superior to the 2D DFT and the2D DCT–DWT. Although only one can recognize the nine watermarks inserted on each sheet, DEMs attacked by the cropping process can still be identified. However, this finding can sufficiently confirm that DEMs belong to BIG.
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Waqas, Ghulam Jilani, Ishtiaq Ahmad, Muhammad Kashif Samee, Muhammad Nasir Khan, and Ali Raza. "A hybrid OFDM–CDMA-based robust image watermarking technique." International Journal of Wavelets, Multiresolution and Information Processing 18, no. 06 (August 20, 2020): 2050043. http://dx.doi.org/10.1142/s0219691320500435.
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Digital watermarking is a process of embedding hidden information called watermark into different kinds of media objects. It uses basic modulation, multiplexing and transform techniques of communication for hiding information. Traditional techniques used are least significant bit (LSB) modification, discrete cosine transform (DCT), discrete wavelet transform (DWT), discrete Fourier transform (DFT), code division multiple access (CDMA) or a combination of these. Among these, CDMA is the most robust against different attacks except geometric attacks. This paper proposes a blind and highly robust watermarking technique by utilizing the basis of orthogonal frequency division multiplexing (OFDM) and CDMA communication system. In this scheme, the insertion process starts by taking DFT of host images, permuting the watermark bits in randomized manner and recording them in a seed as a key. Then PSK modulation follows inverse DFT (IDFT) that gives watermark information as OFDM symbols. These symbols are spread using spreading codes and then arithmetically added to the host image. Finally, scheme applies inverse DCT (IDCT) to get watermarked host images. The simulation results of the proposed scheme are compared with CDMA-based scheme in DCT domain. The results show that the robustness of the proposed scheme is higher than the existing scheme for non-geometric attacks.
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Kanatov, Ivan, Dmitry Kaplun, Denis Butusov, Viacheslav Gulvanskii, and Aleksander Sinitca. "One Technique to Enhance the Resolution of Discrete Fourier Transform." Electronics 8, no. 3 (March 18, 2019): 330. http://dx.doi.org/10.3390/electronics8030330.
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Discrete Fourier transform (DFT) is a common analysis tool in digital signal processing. This transform is well studied and its shortcomings are known as well. Various window functions (e.g., Hanning, Blackman, Kaiser) are often used to reduce sidelobes and to spread the spectrum. In this paper, we introduce a transformation that allows removing the sidelobes of the Fourier transform and increasing the resolution of the DFT without changing the time sample. The proposed method is based on signal phase analysis. We give the comparison of the proposed approach with known methods based on window functions. The advantages and disadvantages of the proposed technique are explicitly shown. We also give a set of examples illustrating the application of our technique in some practical applications, including engine vibration analysis and a short-range radar system.
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Grindlay, J. "On an application of a generalization of the discrete Fourier transform to short time series." Canadian Journal of Physics 79, no. 5 (May 1, 2001): 857–68. http://dx.doi.org/10.1139/p01-054.
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A generalization of the discrete Fourier transform (DFT) is discussed. This generalization or GDFT provides a smooth interpolation between the points of the DFT. The GDFT of a sinusoidal function in a finite time window is (a) described in detail and (b) shown to coincide (aside from a simple scaling constant) with the corresponding Fourier transform, provided that certain conditions are satisfied by the sinusoidal parameters. The sinusoidal GDFT is proposed as a tool to investigate, (independently of any Fourier transform connection) the sinusoidal nature of time series. The method is applied successfully to the case of a specific trajectory of the Hénon and Heiles model. PACS Nos.: 02.30, 05.45
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Lušin, Tomaž, and Dušan Agrež. "Estimation of the Amplitude Square Using the Interpolated Discrete Fourier Transform." Metrology and Measurement Systems 18, no. 4 (January 1, 2011): 583–96. http://dx.doi.org/10.2478/v10178-011-0056-6.
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Estimation of the Amplitude Square Using the Interpolated Discrete Fourier Transform To improve the estimation of active power, the possibility of estimating the amplitude square of a signal component using the interpolation of the squared amplitude discrete Fourier transform (DFT) coefficients is presented. As with an energy-based approach, the amplitude square can be estimated with the squared amplitude DFT coefficients around the component peak and a suitable interpolation algorithm. The use of the Hann window, for which the frequency spectrum is well known, and the three largest local amplitude DFT coefficients gives lower systematic errors in squared interpolated approach or in better interpolated squared approach than the energy-based approach, although the frequency has to be estimated in the first step. All investigated algorithms have almost the same noise propagation and the standard deviations are about two times larger than the Cramér-Rao lower bound.
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Frąckowiak, Andrzej, and Michał Ciałkowski. "Application of discrete Fourier transform to inverse heat conduction problem regularization." International Journal of Numerical Methods for Heat & Fluid Flow 28, no. 1 (January 2, 2018): 239–53. http://dx.doi.org/10.1108/hff-09-2017-0381.
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Purpose This paper aims to present the Cauchy problem for the Laplace’s equation for profiles of gas turbine blades with one and three cooling channels. The distribution of heat transfer coefficient and temperature on the outer boundary of the blade are known. On this basis, the temperature on inner surfaces of the blade (the walls of cooling channels) is determined. Design/methodology/approach Such posed inverse problem was solved using the finite element method in the domain of the discrete Fourier transform (DFT). Findings Calculations indicate that the regularization in the domain of the DFT enables obtaining a stable solution to the inverse problem. In the example under consideration, problems with reconstruction constant temperature, assumed on the outer boundary of the blade, in the vicinity of the trailing and leading edges occurred. Originality/value The application of DFT in connection with regularization is an original achievement presented in this study.
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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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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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Li, Xiaomin, Huali Wang, Wanghan Lv, and Haichao Luo. "Sensing Fractional Power Spectrum of Nonstationary Signals with Coprime Filter Banks." Complexity 2020 (July 17, 2020): 1–17. http://dx.doi.org/10.1155/2020/2608613.
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The coprime discrete Fourier transform (DFT) filter banks provide an effective scheme of spectral sensing for wide-sense stationary (WSS) signals in case that the sampling rate is far lower than the Nyquist sampling rate. And the resolution of the coprime DFT filter banks in the Fourier domain (FD) is 2π/MN, where M and N are coprime. In this work, a digital fractional Fourier transform- (DFrFT-) based coprime filter banks spectrum sensing method is suggested. Our proposed method has the same sampling principle as the coprime DFT filter banks but has some advantages compared to the coprime DFT filter banks. Firstly, the fractional power spectrum of the chirp-stationary signals which are nonstationary in the FD can be sensed effectively by the coprime DFrFT filter banks because of the linear time-invariant (LTI) property of the proposed system in discrete-time Fourier domain (DTFD), while the coprime DFT filter banks can only sense the power spectrum of the WSS signals. Secondly, the coprime DFrFT filter banks improve the resolution from 2π/MN to 2π sin α/MN by combining the fractional filter banks theory with the coprime theory. Simulation results confirm the obtained analytical results.
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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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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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Yasin, Mohd Yusuf. "Simplified approach to DFT computation for nonprogrammable scientific calculators." BIBECHANA 12 (December 8, 2014): 13–19. http://dx.doi.org/10.3126/bibechana.v12i0.11681.
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Fourier analysis is an important tool used as it is or it’s different variants in many fields of sciences and engineering. It’s importance is due to it’s simplicity with which it expands a given function in terms of circular or complex exponents. Further it is quite versatile to handle many functions of practical interest, specifically, the functions with several mathematical disabilities that are hard to be handled with tools like Taylor series. Discrete Fourier Transform (DFT) is a form of Fourier analysis where the discrete function and it’s transform are both of finite length. This processing requires lot many computations. Here in this work a simplified and non programmable calculator based scheme is presented with which one can easily determine the DFT of the given function by feeding in the DFT equation once and a few presses of the calculator keys. DOI: http://dx.doi.org/10.3126/bibechana.v12i0.11681BIBECHANA 12 (2015) 13-19
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Yang, Ying-Hui, Bing-Bing Zhang, Xiao-Li Wang, Shi-Jiao Geng, and Pei-Ying Chen. "Characterizing an Uncertainty Diagram and Kirkwood–Dirac Nonclassicality Based on Discrete Fourier Transform." Entropy 25, no. 7 (July 17, 2023): 1075. http://dx.doi.org/10.3390/e25071075.
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In this paper, we investigate an uncertainty diagram and Kirkwood–Dirac (KD) nonclassicality based on discrete Fourier transform (DFT) in a d-dimensional system. We first consider the uncertainty diagram of the DFT matrix, which is a transition matrix from basis A to basis B. Here, the bases A, B are not necessarily completely incompatible. We show that for the uncertainty diagram of the DFT matrix, there is no “hole” in the region of the (nA,nB) plane above and on the line nA+nB=d+1. Then, we present where the holes are in the region strictly below the line and above the hyperbola nAnB=d. Finally, we provide an alternative proof of the conjecture about KD nonclassicality based on DFT.
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Hinich, Melvin J., John Foster, and Phillip Wild. "DISCRETE FOURIER TRANSFORM FILTERS: CYCLE EXTRACTION AND GIBBS EFFECT CONSIDERATIONS." Macroeconomic Dynamics 13, no. 4 (September 2009): 523–34. http://dx.doi.org/10.1017/s1365100509080237.
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The purpose of this note is to analyze the capability of bandpass filters to extract a known periodicity. The specific bandpass filters considered are a conventional discrete Fourier transform (DFT) filter and the filter recently proposed by Iacobucci and Noullez. We employ simulation methods to investigate cycle extraction properties. We also examine the implications arising from the Gibbs effect in practical settings that typically confront applied macroeconomists
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Soto-Quiros, Pablo. "A Parallel Framework with Block Matrices of a Discrete Fourier Transform for Vector-Valued Discrete-Time Signals." Scientific World Journal 2015 (2015): 1–7. http://dx.doi.org/10.1155/2015/348517.
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This paper presents a parallel implementation of a kind of discrete Fourier transform (DFT): the vector-valued DFT. The vector-valued DFT is a novel tool to analyze the spectra of vector-valued discrete-time signals. This parallel implementation is developed in terms of a mathematical framework with a set of block matrix operations. These block matrix operations contribute to analysis, design, and implementation of parallel algorithms in multicore processors. In this work, an implementation and experimental investigation of the mathematical framework are performed using MATLAB with the Parallel Computing Toolbox. We found that there is advantage to use multicore processors and a parallel computing environment to minimize the high execution time. Additionally, speedup increases when the number of logical processors and length of the signal increase.
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Jain, Gourav, and Shaik Rafi Ahamed. "High Throughput Realization of a New Systolic Array based FFT using CORDIC." International Journal of Measurement Technologies and Instrumentation Engineering 2, no. 2 (April 2012): 53–59. http://dx.doi.org/10.4018/ijmtie.2012040105.
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In this paper, the authors propose a new systolic array for radix-2, N-point discrete Fourier Transform (DFT) computation based on CORDIC (CO-ordinate Rotation Digital Computer). Complex multiplication can be done by this in a rather simple and elegant way. A CORDIC based multiplier less DFT architecture is designed in order to improve the performance of the system. It is able to provide two transforms per each clock cycle. The proposed design is well suited for high speed DSP-applications.
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BI, GUOAN, and YANQIU CHEN. "FAST DHT ALGORITHMS FOR COMPOSITE SEQUENCE LENGTHS." Journal of Circuits, Systems and Computers 08, no. 03 (June 1998): 421–34. http://dx.doi.org/10.1142/s0218126698000225.
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This paper presents fast algorithms for the computation of discrete Hartley transform (DHT). When the sequence length N = p*q, where p and q are integers and relatively prime, the one dimensional DHT can be decomposed into p length-q DHT's and q length-p discrete Fourier transforms (DFT). Compared to other reported algorithms, the proposed one has a regular computational structure, provides flexibility for composite sequence lengths and achieves substantial savings on the required number of operations.
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Majewski, M., and L. B. Magalas. "Hilbert-Twin – A Novel Hilbert Transform-Based Method To Compute The Elastic Modulus In High-Resolution Mechanical Spectroscopy HRMS." Archives of Metallurgy and Materials 60, no. 2 (June 1, 2015): 1099–103. http://dx.doi.org/10.1515/amm-2015-0266.
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Abstract A novel Hilbert-twin (H-twin) method is introduced as an alternative method for the computation of the resonant frequency for exponentially damped free decays embedded in noise. We also present the comparison among the following methods used to compute the dynamic elastic modulus in solids: the parametric OMI (Optimization in Multiple Intervals), the Yoshida-Magalas (YM) interpolated discrete Fourier transform, the Hilbert-twin (H-twin), and discrete Fourier transform (DFT) methods. It is concluded that the OMI and YM methods are the best methods to compute the elastic modulus from discrete exponentially damped free-elastic decays embedded in unavoidable noise.
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Tereshchenko, Andrii, and Valeriy Zadiraka. "Implementation of Multidigit Multiplication Basing on Discrete Cosine and Sine Transforms." Cybernetics and Computer Technologies, no. 4 (December 30, 2021): 61–79. http://dx.doi.org/10.34229/2707-451x.21.4.7.
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Introduction. The emergence of new parallel computing systems such as multi-core processors, clusters, distributed systems, due to the solution of various applications in different spheres. Among such problems are the calculation of systems of linear algebraic equations with the number of unknown 33-35 million, the calculation of nuclear reactor shells, modeling of physical and chemical processes, aerodynamics, hydrodynamics, information security, and so on. This greatly expands the use of multidigit arithmetic, due to the fact that ignoring rounding errors leads to the fact that sometimes computer solutions are obtained that do not correspond to the physical content. Multidigit multiplication operation is an integral part of the exponentiation by module operation, the speed of which determines the speed of asymmetric cryptographic software and hardware complexes. This paper presents algorithms for implementing the multiplication operation of two N-digit numbers based on discrete cosine and sine transforms (DCT and DST) by separating the calculation for the real and imaginary parts of the DFT. Calculation of DCT and DST at the expense of additional bit shifts, additions and subtractions reduces the algorithm complexity to linear complexity by the number of integer multiplication operations. The purpose of the article is to reduce the number of multiplication operations to speed up the execution time of the multiplication operation of two N-bit numbers based on discrete transforms. Reduce the number of complex multiplication operations. Reduce the overall computational complexity and find a modification in which the calculation steps will correspond to DCT, DSP, IDCT and IDST. Use the coefficients to take into account the rounding errors to exclude multiplication operations on calculating DCT, DST, IDCT and IDST. Results. The relationship between DCT, DST and DFT of a real signal is considered, which allows to separate calculations for real and imaginary parts of DFT of real signals. The computational complexity is reduced almost twice at the expense of use of DFT properties of real signals. It is shown that after optimization steps of the algorithm calculation correspond to DCT, DST, IDCT and IDST. Using additional coefficients, which allow to take into account rounding errors at each step so that all calculations use integers. An analysis of the choice of word length in the calculation is given. For each algorithm, examples of calculation are given. Tables of dependence of the minimum lengths of the coefficients on the length of the multidigit number and the length of the digit (in bits) are given. Conclusions. Multiplication algorithms of two N-digit numbers based on discrete cosine and sine transforms (DCT and DST) are presented in this paper. Separating the calculation for the real and imaginary parts of the DFT allows to reduce the number of multiplication operations by 33%. The use of additional coefficients and calculation of DCT, DST, IDCT, IDST at the expense of bit shifts, additions and subtractions reduces the complexity of the multiplication algorithm of two N-digit numbers to linear complexity by the number of simple integer multiplication operations. Based on comparative analysis, it is shown that the proposed method of multiplication based on DCT and DST using integers begins to exceed the Karatsuba method by the number of 32-bit multiplication operations when multiplying numbers, starting with a length of 4096 bits. Keywords: multidigit multiplication, multidigit arithmetic, asymmetric cryptography, discrete cosine transform, discrete sine transform, discrete Fourier transform, fast algorithm for Fourier calculation.
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Liu, Zhanhong, Lei Fan, Jinyu Liu, Nian Liu, Jiyu Jin, and Jun Xing. "Accurate Frequency Estimator for Real Sinusoid Based on DFT." Electronics 11, no. 19 (September 24, 2022): 3042. http://dx.doi.org/10.3390/electronics11193042.
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An accurate frequency estimator for real sinusoid based on Discrete Fourier Transform (DFT) is proposed. The proposed estimator is based on the interpolation of the maximum DFT spectral line and two Discrete-Time Fourier Transform (DTFT) spectral lines and can operate with both the rectangular window and the maximum sidelobe decay (MSD) window. In the coarse estimation step, the proposed estimator with the MSD window is used. According to the value of the coarse estimate, the negative frequency spectral component is removed. In the fine estimation step, the proposed estimator with the rectangular window is utilized to achieve the Cramer–Rao lower bound (CRLB). Simulation results show that the performance of this algorithm is better than that of the AM algorithm, Candan algorithm, Djukanovic algorithm, and FDIAM algorithm.
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Li, Chen Wu, Jian Zhang, Qin Xie, and Xiao Hong Zhang. "Carrier Modulation Technology Based on Orthogonal Frequency Division Multiplexing." Advanced Materials Research 774-776 (September 2013): 1671–76. http://dx.doi.org/10.4028/www.scientific.net/amr.774-776.1671.
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This paper first analyzes the transmission characteristics of low-voltage power line channels with the focus on the study of carrier modulation technology regarding the power line communication part, then proposes the orthogonal frequency division multiplexing technology that serves for the digital communication of family network power line communication gateways, analyzes the OFDM system principle, actulizes OFDM modulation and demodulation through discrete Fourier transform (DFT) and inverse discrete Fourier transform (IDFT), and build the OFDM simulation model. Finally, a specific plan of using power lines as the family network transmission media is proposed.