Comments on, "A note on the computational complexity of the arithmetic Fourier transform" by N. Tepedelenlioglu

In the literature, there are various algorithms available for computing the discrete fractional Fourier transform (DFRFT). In this paper, all the existing methods are reviewed, classified into four categories, and subsequently compared to find out the best alternative from the view point of minimal computational error, computational complexity, transform features, and additional features like security. Subsequently, the correlation theorem of FRFT has been utilized to remove significantly the Doppler shift caused due to motion of receiver in the DSB-SC AM signal. Finally, the role of DFRFT has been investigated in the area of steganography.

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Effective Approach to Calculate Analysis Window in Infinite Discrete Gabor Transform

Complexity ◽

10.1155/2018/9039240 ◽

2018 ◽

Vol 2018 ◽

pp. 1-10 ◽

Cited By ~ 1

Author(s):

Rui Li ◽

Yong Huang ◽

Jia-Bao Liu

Keyword(s):

Fourier Transform ◽

Computational Complexity ◽

Linear Equation ◽

Discrete Fourier Transform ◽

Delta Function ◽

Gabor Transform ◽

Gabor Analysis ◽

Analysis Window ◽

Orthogonality Constraint ◽

Infinite Sequences

The long-periodic/infinite discrete Gabor transform (DGT) is more effective than the periodic/finite one in many applications. In this paper, a fast and effective approach is presented to efficiently compute the Gabor analysis window for arbitrary given synthesis window in DGT of long-periodic/infinite sequences, in which the new orthogonality constraint between analysis window and synthesis window in DGT for long-periodic/infinite sequences is derived and proved to be equivalent to the completeness condition of the long-periodic/infinite DGT. By using the property of delta function, the original orthogonality can be expressed as a certain number of linear equation sets in both the critical sampling case and the oversampling case, which can be fast and efficiently calculated by fast discrete Fourier transform (FFT). The computational complexity of the proposed approach is analyzed and compared with that of the existing canonical algorithms. The numerical results indicate that the proposed approach is efficient and fast for computing Gabor analysis window in both the critical sampling case and the oversampling case in comparison to existing algorithms.

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Beam-Doppler Unitary ESPRIT for Multitarget DOA Estimation

International Journal of Antennas and Propagation ◽

10.1155/2018/3568286 ◽

2018 ◽

Vol 2018 ◽

pp. 1-10 ◽

Cited By ~ 1

Author(s):

Cai Wen ◽

Yan Zhou ◽

Mingliang Tao ◽

Jianxin Wu ◽

Jinye Peng

Keyword(s):

Fourier Transform ◽

Computational Complexity ◽

Doa Estimation ◽

Critical Issue ◽

Radar System ◽

Estimation Accuracy ◽

Invariance Property ◽

Multitarget Tracking ◽

Numerical Examples ◽

System A

High-resolution direction of arrival (DOA) estimation is a critical issue for mainbeam multitarget tracking in ground-based or airborne early warning radar system. A beam-Doppler unitary ESPRIT (BD-UESPRIT) algorithm is proposed to deal with this problem. Firstly, multiple snapshots without spatial aperture loss are obtained by using the technique of time-smoothing. Then the conjugate centrosymmetric discrete Fourier transform (DFT) matrix is used to transform the extracted data into beam-Doppler domain. Finally, the rotational invariance property of the space-time beam is exploited to estimate DOA of the target. The DOA estimation accuracy is improved greatly because the proposed algorithm takes full advantage of temporal information of the signal. Furthermore, the computational complexity of the presented algorithm is reduced dramatically, because the degree of freedom after beam transformation is very small and most of the operations are implemented in real-number domain. Numerical examples are given to verify the effectiveness of the proposed algorithm.

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Fast τ-p transforms by chirp modulation

Geophysics ◽

10.1190/geo2018-0380.1 ◽

2019 ◽

Vol 84 (1) ◽

pp. A13-A17 ◽

Cited By ~ 1

Author(s):

Fredrik Andersson ◽

Johan Robertsson

Keyword(s):

Fourier Transform ◽

Computational Complexity ◽

Fourier Transforms ◽

Graphics Processing Unit ◽

Computational Cost ◽

Processing Unit ◽

Chirp Modulation ◽

Fourier Sums ◽

Graphics Processing ◽

Direct Implementation

We have developed simple, fast, and accurate algorithms for the linear Radon ([Formula: see text]-[Formula: see text]) transform and its inverse. The algorithms have an [Formula: see text] computational complexity in contrast to the [Formula: see text] cost of a direct implementation in 2D and an [Formula: see text] computational complexity compared to the [Formula: see text] cost of a direct implementation in 3D. The methods use Bluestein’s algorithm to evaluate discrete nonstandard Fourier sums, and they need, apart from the fast Fourier transform (FFT), only multiplication of chirp functions and their Fourier transforms. The computational cost and accuracy are thus reduced to that inherited by the FFT. Fully working algorithms can be implemented in a couple of lines of code. Moreover, we find that efficient graphics processing unit (GPU) implementations could achieve processing speeds of approximately [Formula: see text], implying that the algorithms are I/O bound rather than compute bound.

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Irregular Sampling of GPS Signal for Real-Time Implementation of Pipelined PFFT

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.519-520.969 ◽

2014 ◽

Vol 519-520 ◽

pp. 969-974 ◽

Cited By ~ 1

Author(s):

Jian Hua Huang ◽

Gao Yong Luo

Keyword(s):

Fourier Transform ◽

Computational Complexity ◽

Real Time ◽

High Speed ◽

Sampling Method ◽

Prime Factor ◽

Irregular Sampling ◽

Utilization Rate ◽

Sampled Data ◽

Simulation Results

We present a nonuniform method sampling to form the 1 105 points of GPS signal for real-time implementation of prime factor Fourier transform (PFFT) processor, which uses less resource with more utilization rate, and is more efficient and computationally faster than fast Fourier transform (FFT). The high-speed capturing method by irregular sampling for GPS quick positioning is developed to obtain the specified number of sampled data to suit the requirement of fast pipelined PFFT implementation. Simulation results show that the irregular sampling method is efficient in terms of computational complexity and achieves minimum operations of sampling for 1 105 samples based PFFT processing.

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Determining the Weights of A Fourier Series Neural Network on the Basis of the Multidimensional Discrete Fourier Transform

International Journal of Applied Mathematics and Computer Science ◽

10.2478/v10006-008-0033-8 ◽

2008 ◽

Vol 18 (3) ◽

pp. 369-375 ◽

Cited By ~ 3

Author(s):

Krzysztof Halawa

Keyword(s):

Neural Network ◽

Fourier Transform ◽

Fourier Series ◽

Computational Complexity ◽

Discrete Fourier Transform ◽

Dynamic Systems ◽

Low Computational Complexity

Determining the Weights of A Fourier Series Neural Network on the Basis of the Multidimensional Discrete Fourier TransformThis paper presents a method for training a Fourier series neural network on the basis of the multidimensional discrete Fourier transform. The proposed method is characterized by low computational complexity. The article shows how the method can be used for modelling dynamic systems.

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A note on the computational complexity of the arithmetic Fourier transform

IEEE Transactions on Acoustics Speech and Signal Processing ◽

10.1109/29.32291 ◽

1989 ◽

Vol 37 (7) ◽

pp. 1146-1147 ◽

Cited By ~ 10

Author(s):

N. Tepedelenlioglu

Keyword(s):

Fourier Transform ◽

Computational Complexity

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Inverse Kinematics of Discretely Actuated Ball-Joint Manipulators Using Workspace Density

Volume 5C: 39th Mechanisms and Robotics Conference ◽

10.1115/detc2015-46803 ◽

2015 ◽

Cited By ~ 1

Author(s):

Hui Dong ◽

Taosha Fan ◽

Zhijiang Du ◽

Gregory Chirikjian

Keyword(s):

Fourier Transform ◽

Computational Complexity ◽

Inverse Kinematics ◽

Solution Space ◽

Rotation Group ◽

Ball Joint ◽

End Effector ◽

Numerical Examples ◽

One Dimensional ◽

Elegant Form

We present a workspace-density-based (WSDB) method to solve the inverse kinematics of discretely actuated ball-joint manipulators. Intuitively speaking, workspace density measures the flexibility of a robotic manipulator when the end-effector is fixed at a certain pose or position. We use the SE(3) Fourier transform to derive the workspace density for ball-joint manipulators and show that the workspace density has a concise and elegant form. Then we show that the state for each joint is determined by maximizing the workspace density of subsequent sub-manipulators. We demonstrate our method with several numerical examples. In particular, we show that our method can provide a solution that approximately minimizes the deviation of the end-effector and its computational complexity is linear with respect to the number of joints. Hence our method is very efficient in solving the inverse kinematics of redundant discretely actuated ball-joint manipulators. In addition, we prove that the solution space of our method is reduced from the rotation group SO(3) to a one-dimensional interval.

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Peak-to-Average Power Ratio Reduction Method Based on Partial Transmit Sequence and Discrete Fourier Transform Spreading

Electronics ◽

10.3390/electronics10060642 ◽

2021 ◽

Vol 10 (6) ◽

pp. 642

Author(s):

Nahla Al Harthi ◽

Zhongfeng Zhang ◽

Daejin Kim ◽

Seungwon Choi

Keyword(s):

Fourier Transform ◽

Computational Complexity ◽

Discrete Fourier Transform ◽

Computer Simulations ◽

Orthogonal Frequency Division Multiplexing ◽

Complexity Analysis ◽

Average Power ◽

Power Ratio ◽

Partial Transmit Sequence ◽

Ber Performance

Recently, filter bank multicarrier with offset quadrature amplitude modulation (FBMC/OQAM) has received increasing attention from researchers, owing to its merits and superior spectral efficiency. High peak-to-average power ratio (PAPR) occurs in approximately all multicarrier systems, including FBMC/OQAM, and may cause bit-error-rate (BER) degradation if not appropriately handled. Conventional PAPR reduction methods for orthogonal frequency division multiplexing (OFDM), such as partial transmit sequence (PTS), selective mapping (SLM), and discrete Fourier transform (DFT) spreading, are ineffective in FBMC/OQAM because of the different structure of the symbols. This study proposes a novel method combining DFT spreading and PTS methods to reduce the PAPR of FBMC/OQAM systems with reasonable computational complexity. Numerical results obtained from various computer simulations show that the proposed method achieves a noticeable enhancement in the PAPR performance of the FBMC/OQAM signal compared to other existing methods without affecting the BER performance. Further, the computational complexity analysis and BER performance of the proposed method are presented in comparison to typical existing methods. From our computer simulations, the proposed method reduces the PAPR by approximately 32.8% compared to that of the conventional methods, and the BER performance is improved by 25% with a high-power amplifier effect.

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