Generalized Discrete Fourier Transform Research Articles

Cooley-Tukey FFT Algorithm based on GDFT for Phasor Estimation by PMU under Power Quality Disturbances

Due to the high penetration of renewable energy sources, such as photovoltaic panels and wind turbines, in addition to the use of different electric power supplies in the power grid, there are major disturbances in the forms of electric waves. These variations and disturbances must be monitored and controlled for the efficient management of transmission and distribution of electrical energy, safety, and electrical protection systems. Nowadays, Phasor Measurement Unit (PMU) technology is an essential tool to develop the supervision, protection, and control of the electrical power grid. PMUs measure the amplitude and angle of current and voltage waveforms on a Coordinated Universal Time (UTC) time scale and speedily measure the fundamental frequencies and their rates of change using fast and accurate estimation algorithms. This paper presents a phasor estimation using a Modified Cooley-Tukey Fast Fourier Transform algorithm based on the Generalized Discrete Fourier Transform (DFT) used in PMUs, using simulations in MATLAB. This algorithm was utilized to accelerate and simplify the computation of DFTs. To validate the performance under waveform disturbances, several tests with different waveforms and disturbances were simulated and interpreted according to the standard and compared with DFT.

Oversampled filter bank channelizer for cryogenic detectors.

We present the field-programmable gate array (FPGA) implementation of an oversampled generalized discrete Fourier transform (GDFT) filter bank (FB), optimized for its usage in the readout of cryogenic detectors, such as microwave kinetic inductance detectors. A polyphase structure is derived underlining the efficiency of complex-valued subband processing, raised by a GDFT modulation. For the latter, a fast Fourier algorithm realization may be used, yielding a highly efficient polyphase implementation for arbitrary integer decimation ratios. The FB is tested through a 16-channel GDFT channelizer. The implementation is based on the Xilinx Zynq family of FPGAs and aims to show the data reuse and flexibility offered by the GDFT channelizer structure. General design criteria are summarized for the 16-channel polyphase FB channelizer. The performance in hardware resource usage is also presented, taking into account one of the main objectives of the current physics experiments, aiming to have an extremely large readout format.

Improving Generalized Discrete Fourier Transform (GDFT) Filter Banks with Low-Complexity and Reconfigurable Hybrid Algorithm

With ever-increasing wireless network demands, low-complexity reconfigurable filter design is expected to continue to require research attention. Extracting and reconfiguring channels of choice from multi-standard receivers using a generalized discrete Fourier transform filter bank (GDFT-FB) is computationally intensive. In this work, a lower compexity algorithm is written for this transform. The design employs two different approaches: hybridization of the generalized discrete Fourier transform filter bank with frequency response masking and coefficient decimation method 1; and the improvement and implementation of the hybrid generalized discrete Fourier transform using a parallel distributed arithmetic-based residual number system (PDA-RNS) filter. The design is evaluated using MATLAB 2020a. Synthesis of area, resource utilization, delay, and power consumption was done on a Quartus 11 Altera 90 using the very high-speed integrated circuits (VHSIC) hardware description language. During MATLAB simulations, the proposed HGDFT algorithm attained a 66% reduction, in terms of number of multipliers, compared with existing algorithms. From co-simulation on the Quartus 11 Altera 90, optimization of the filter with PDA-RNS resulted in a 77% reduction in the number of occupied lookup table (LUT) slices, an 83% reduction in power consumption, and an 11% reduction in execution time, when compared with existing methods.

An efficient and light weight polynomial multiplication for ideal lattice-based cryptography

Ring-Learning With Errors (Ring-LWE) based cryptographic schemes such as signature, key exchange, and encryption require polynomial multiplication. This multiplication operation is the most time consuming and computationally rigorous process in Ring-LWE. In order to improve the efficiency of the Ring-LWE based schemes, most of the existing schemes use Fast Fourier Transform (FFT) based polynomial multiplication algorithm. It is known that Discrete Sine Transformation (DST) and Discrete Cosine Transformation (DCT) are faster than the FFT. The combination of DCT and DST is Discrete Trigonometric Transform (DTT). When we generalize DTT in terms of FFT form, it becomes Generalized Discrete Fourier Transform (GDFT). In this paper, we propose two new polynomial multiplication techniques using DTT and GDFT. When we apply circular convolution and skew-circular convolution on DTT or GDFT for the polynomial multiplication, it gives us wrong results. To overcome this issue, we use symmetric convolution operation on DTT and GDFT. We implemented and compared the proposed polynomial multiplication schemes with the current state-of-the-art schemes in terms of computation and communication costs. The implementation results show that the proposed schemes DTT and GDFT perform more efficiently as compared to current state-of-the-art schemes in terms of computation and communication costs.

Retraction Note to: Filter Bank Multicarrier Based Generalized Discrete Fourier Transform for Peak to Average Power Ratio Reduction

The Filter Bank Multicarrier (FBMC), one of the multi-carrier modulation technique are subjected to extreme intensities of Peak to Average Power Ratio (PAPR). Moreover, the FBMC is a firm user, like the waveform-centred framework in the unconventional wireless communication frameworks, such as the 5G telecommunication system. This system framework acts as an alternative to the Orthogonal Frequencies Divisions Multiplexing (OFDM), including the Discrete Fourier Transforms (DFT). Signal Spreading is applicable in the FBMC purposed for PAPR reductions, which exemplifies a significant computational increase in its complexities that does not necessitate an overhead of signal placed at the receiver end. In this proposed work recommends the application of the general DFT, which is considered as a substitute to the DFT-spreading purposed for the PAPR reductions. To further we have derived the conditions whereby the Generalized Discrete Fourier Transform (GDFT) can significantly apply the single carriers influenced of the DFT-spreading, which targets the reductions in the PAPR. Moreover, this paper proposes a developed algorithm centred on the GDFT that considers more PAPR reductions eliminating any further computational complexities. Therefore, the recommended algorithm has been formulated to utilized Side Information (SI) represents a single bit of every data segment. This developed algorithm is also applicable in more enhanced DFT-spreading frameworks aimed at comparing data. Based on the collected simulation findings, it is evident that varied number of sub-carriers GDFT-spreading, which considers the availability or the presence of the developed algorithm, indicates the recommendable framework signifying that the enhanced algorithms received in most instances is an additional PAPR amount reductions based on other DFT-spreading frameworks. These frameworks, including the developed GDFT indicate effective power spectral densities (PSD) compared to varied DFT-spreading frameworks.

Eigenvectors of Ordinary, Generalized, Centered and Offset Discrete Fourier Transforms Based on Lookup Table Methods: Efficiency and Approximation Uses

In this paper, we first propose a methodology to construct an eigenvector of the N × N discrete Fourier transform (DFT) matrix from any eigenvector of the ( k 2 N ) × ( k 2 N ) DFT matrix, where k is a positive integer. Specifically, using the procedure to down-sample by k any ( k 2 N )-point DFT eigenvector with zero as the initial index and then fold modulus N and add the resultant down-sampled ( kN )-point vector, we obtain an N -point DFT eigenvector. Computing an N -point DFT eigenvector using the proposed method is very efficient, which requires only N ( k -1) additions and no multiplications with a pre-stored lookup table of ( k 2 N )-point DFT eigenvectors. Similar efficient methods are also developed to compute N -point eigenvectors of other DFT-related transforms, including the generalized DFT (GDFT), centered DFT (CDFT) and the offset DFT (ODFT), from ( k 2 N )-point eigenvectors of their corresponding transforms or the DFT. As application examples, we apply the proposed methods to compute N -point DFT, GDFT, CDFT and ODFT eigenvectors, which are closer to samples of the continuous Hermite-Gaussian function (HGF) than existing N -point eigenvectors, from their corresponding ( k 2 N )-point eigenvectors. These improved N -point Hermite-Gaussian-like eigenvectors can be used to define fractional versions of the various N × N DFT transforms whose outputs are closer to samples of the continuous fractional Fourier transform than outputs of the corresponding existing fractional transforms. Finally, we perform computer experiments to verify the effectiveness and demonstrate applications of our proposed methods.

Personal Sound Zones by Subband Filtering and Time Domain Optimization

Personal Sound Zones (PSZ) systems aim to render independent sound signals to multiple listeners within a room by using arrays of loudspeakers. One of the algorithms used to provide PSZ is Weighted Pressure Matching (wPM), which computes the filters required to render a desired response in the listening zones while reducing the acoustic energy arriving to the quiet zones. This algorithm can be formulated in time and frequency domains. In general, the time-domain formulation (wPM-TD) can obtain good performance with shorter filters and delays than the frequency-domain formulation (wPM-FD). However, wPM-TD requires higher computation for obtaining the optimal filters. In this article, we propose a novel approach to the wPM algorithm named Weighted Pressure Matching with Subband Decomposition (wPM-SD), which formulates an independent time-domain optimization problem for each of the subbands of a Generalized Discrete Fourier Transform (GDFT) filter bank. Solving the optimization independently for each subband has two main advantages: 1) lower computational complexity than wPM-TD to compute the optimal filters; 2) higher versatility than the classic wPM algorithms, as it allows different configurations (sets of loudspeakers, filter lengths, etc.) in each subband. Moreover, filtering the input signals with a GDFT filter bank (as in wPM-SD) requires lower computational effort than broadband filtering (as in wPM-TD and wPM-FD), which is beneficial for practical PSZ systems. We present experimental evaluations showing that wPM-SD offers very similar performance to wPM-TD. In addition, two cases where the versatility of wPM-SD is beneficial for a PSZ system are presented and experimentally validated.

RETRACTED ARTICLE: Filter Bank Multicarrier Based Generalized Discrete Fourier Transform for Peak to Average Power Ratio Reduction

The Filter Bank Multicarrier (FBMC), one of the multi-carrier modulation technique are subjected to extreme intensities of Peak to Average Power Ratio (PAPR). Moreover, the FBMC is a firm user, like the waveform-centred framework in the unconventional wireless communication frameworks, such as the 5G telecommunication system. This system framework acts as an alternative to the Orthogonal Frequencies Divisions Multiplexing (OFDM), including the Discrete Fourier Transforms (DFT). Signal Spreading is applicable in the FBMC purposed for PAPR reductions, which exemplifies a significant computational increase in its complexities that does not necessitate an overhead of signal placed at the receiver end. In this proposed work recommends the application of the general DFT, which is considered as a substitute to the DFT-spreading purposed for the PAPR reductions. To further we have derived the conditions whereby the Generalized Discrete Fourier Transform (GDFT) can significantly apply the single carriers influenced of the DFT-spreading, which targets the reductions in the PAPR. Moreover, this paper proposes a developed algorithm centred on the GDFT that considers more PAPR reductions eliminating any further computational complexities. Therefore, the recommended algorithm has been formulated to utilized Side Information (SI) represents a single bit of every data segment. This developed algorithm is also applicable in more enhanced DFT-spreading frameworks aimed at comparing data. Based on the collected simulation findings, it is evident that varied number of sub-carriers GDFT-spreading, which considers the availability or the presence of the developed algorithm, indicates the recommendable framework signifying that the enhanced algorithms received in most instances is an additional PAPR amount reductions based on other DFT-spreading frameworks. These frameworks, including the developed GDFT indicate effective power spectral densities (PSD) compared to varied DFT-spreading frameworks.

A New Algorithm for Fast Generalized DFTs

We give an new arithmetic algorithm to compute the generalized Discrete Fourier Transform (DFT) over finite groups G . The new algorithm uses O (∣ G ∣ ω /2 + o (1) ) operations to compute the generalized DFT over finite groups of Lie type, including the linear, orthogonal, and symplectic families and their variants, as well as all finite simple groups of Lie type. Here ω is the exponent of matrix multiplication, so the exponent ω/2 is optimal if ω = 2. Previously, “exponent one” algorithms were known for supersolvable groups and the symmetric and alternating groups. No exponent one algorithms were known, even under the assumption ω = 2, for families of linear groups of fixed dimension, and indeed the previous best-known algorithm for SL 2 (F q ) had exponent 4/3 despite being the focus of significant effort. We unconditionally achieve exponent at most 1.19 for this group and exponent one if ω = 2. Our algorithm also yields an improved exponent for computing the generalized DFT over general finite groups G , which beats the longstanding previous best upper bound for any ω. In particular, assuming ω = 2, we achieve exponent √ 2, while the previous best was 3/2.

An algorithm for Morlet wavelet transform based on generalized discrete Fourier transform

Continuous wavelet transform (CWT) is a linear convolution of signal and wavelet function for a fixed scale. This paper studies the algorithm of CWT with Morlet wavelet as mother wavelet by using nonzero-padded linear convolution. The time domain filter, which is a non-causal filter, is the sample of wavelet function. By making generalized discrete Fourier transform (GDFT) and inverse transform for this filter, we can get a geometrically weighted periodic extension of the filter when evaluated outside its original support. From this extension of the time domain filter, we can get a causal filter. In this paper, GDFT-based algorithm for CWT, which has a more concise form than that of linear convolution proposed by Jorge Martinez, is constructed by using this causal filter. The analytic expression of the GDFT of this filter, which is essential for GDFT-based algorithm for CWT, is deduced in this paper. The numerical experiments show that the calculation results of GDFT-based algorithm are stable and reliable; the running speed of GDFT-based algorithm is faster than that of the other two algorithms studied in our previous work.

A general construction of Reed-Solomon codes based on generalized discrete Fourier transform

In this paper, we employ the concept of the Generalized Discrete Fourier Transform, which in turn relies on the Hasse derivative of polynomials, to give a general construction of Reed-Solomon codes over Galois fields of characteristic not necessarily co-prime with the length of the code. The constructed linear codes enjoy nice algebraic properties just as the classic one.

Generalized Discrete Fourier Transform for FBMC Peak to Average Power Ratio Reduction

Filter bank multicarrier (FBMC) is a strong candidate as a waveform based technique in advanced wireless communication systems (e.g. 5G), as an alternative to orthogonal frequency division multiplexing (OFDM). Like all multicarrier modulation techniques, FBMC suffers from high levels of a peak-to-average power ratio (PAPR). Discrete Fourier transform (DFT) spreading can be used in FBMC for PAPR reduction with a quite notable increase in computational complexity and without any need for side information (SI) overhead at the receiver. However, the achieved PAPR reduction is of a marginal amount, compared to the single carrier effect in the single carrier frequency division multiple access (SC-FDMA). This is due to the in-phase and quadrature phase (IQ) overlapping structure between offset quadrature amplitude modulation (OQAM) FBMC symbols. In this paper, we propose the use of generalized DFT (GDFT) spreading as an alternative to DFT spreading for PAPR reduction in systems employing FBMC. We derive the conditions at which the GDFT can totally make use of the single carrier effect of DFT spreading and hence reducing the PAPR. We also propose an enhancement algorithm that is utilized in the GDFT for further PAPR reduction without any additional complexity overhead. From the simulation results that are run at different values of subcarriers, it is shown that the GDFT spreading with the enhancement algorithm (enhanced GDFT) attain an extra amount of PAPR reduction over the other DFT spreading techniques. In addition, the GDFT spreading technique and the enhanced one also show better power spectral density (PSD) over the other DFT spreading techniques.

Fast and Flexible Selective Harmonic Extraction Methods Based on the Generalized Discrete Fourier Transform

Selective harmonic extraction plays an important role in power quality assessment, harmonic compensation, and so on. Among the various methods, discrete Fourier transform (DFT)-based algorithms has found wide and popular applications due to advantages like simplicity and excellent selectively filtering properties. However, DFT suffers from disadvantages like slow dynamics and sensitivity to frequency variations. In order to alleviate these drawbacks, an improved and extended DFT—generalized DFT (GDFT) is proposed. It is revealed in this paper that the conventional DFT can be viewed as a comb filter connected in series with complex resonators, and DFT relies on the mechanism of pole-zero cancellation for harmonic extraction. More importantly, the key reason behind the DFT slow transient responses is the large delay introduced by the comb filter. Therefore, this paper proposed to reconfigure the comb filter according to specific harmonic patterns of the input signal for improving the dynamic responses and system flexibility. The proposed GDFT not only maintains the advantages of simplicity and selectively filtering properties but also features fast transients, around 0.3 fundamental cycle for typical applications, which is much shorter than the one-cycle settling time of the conventional DFT. A phase-locked-loop block is also incorporated into the harmonic-detection system to deal with possibly large frequency variations in practical applications. Extensive tests are provided to validate effectiveness of the proposed method.

Fast Algorithm for Computing the Discrete Hartley Transform of Type-II

The generalized discrete Hartley transforms (GDHTs) have proved to be an efficient alternative to the generalized discrete Fourier transforms (GDFTs) for real-valued data applications. In this paper, the development of direct computation of radix-2 decimation-in-time (DIT) algorithm for the fast calculation of the GDHT of type-II (DHT-II) is presented. The mathematical analysis and the implementation of the developed algorithm are derived, showing that this algorithm possesses a regular structure and can be implemented in-place for efficient memory utilization.The performance of the proposed algorithm is analyzed and the computational complexity is calculated for different transform lengths. A comparison between this algorithm and existing DHT-II algorithms shows that it can be considered as a good compromise between the structural and computational complexities.

On commutativity of Discrete Fourier Transform

In this paper we have studied the commutative properties of general Discrete Fourier Transform (DFT) matrices Un. The problem is to characterize matrices An that commute with Un. We find complete solutions for An up to n=5 theoretically. We also provide a major result towards the complete solutions for general n. To find An which commutes with Un one needs to solve a system of n2 linear equations of n2 variables. We reduced this problem into solving two different systems of linear equations of more or less n2/4 many variables and same number of equations. To do this reduction we use the idea of symmetric, skew symmetric matrices as well as we consider the set of matrices as a vector space and use direct sum of subspaces.

Novel Tridiagonal Commuting Matrices for Types I, IV, V, VIII DCT and DST Matrices

In this letter, we first propose new nearly tridiagonal commuting matrices of discrete Fourier transform (DFT) matrix and generalized DFT (GDFT) matrix. Then, using the block diagonalizations technique of circular-centrosymmetric and centrosymmetric matrices, we derive novel tridiagonal commuting matrices for each discrete cosine transform (DCT) and discrete sine transform (DST) matrices of the types I, IV, V, and VIII from the commuting matrices of the DFT and GDFT based on the relationships in matrix forms among DFT, GDFT, and various types of DCT and DST. Moreover, the novel tridiagonal commuting matrices of various types of DCT and DST do not have multiple eigenvalues. Last, with these novel commuting matrices, we can easily determine an orthonormal set of Hermite-like eigenvectors for each of their corresponding DCT or DST matrix.

Implementation of Generalized Discrete Fourier Transform for CDMA on Real Time System

paper deals with simulation of Generalized Discrete Fourier Transform spreading codes used in modulation technique in Code Division Multiple Access on real time system using MATLAB environment. Binary sequences can be used as various spreading code in digital modulation technique. The spreading codes are used to increase the spread spectrum and proper utilization of the spread spectrum in cellular communication system. In CDMA for digital modulation technique various spreading codes like Gold codes, M-ary Binary sequences, Walsh codes and Constant modulus function set (GDFT) are used. These codes can be used in various applications like Orthogonal Frequency Division Multiple Access (OFDM), Discrete Multi-tone and Code Division Multiple Access. This paper proposes real time system model in which GDFT spreading code modulated with voice or speech as input analog signal. Performance of Generalized Discrete fourier transform with non linear phase is been determined by some factors. The factors are Bit-Error-Rate, Signal-to-Noise ratio and Mean Square Value of auto-correlation and Mean Square Value of Cross-Correlation responsible to maintain quality of services in digital modulation.

Fourier Analysis of Sequences over a Composition Algebra of the Real Number Field

To analyze the structure of a set of perfect sequences over a composition algebra of the real number field, transforms of a set of sequences similar to DFT (discrete Fourier transform) are introduced. Discrete cosine transform, discrete sine transform and generalized discrete Fourier transform (GDFT) of the sequences are defined and the fundamental properties of these transforms are proved. We show that GDFT is bijective and that there exists a relationship between these transforms and a convolution of sequences. Applying these properties to the set of perfect sequences, a parameterization theorem of such sequences is obtained.

A generalized Fourier domain: Signal processing framework and applications

In this paper, a signal processing framework in a generalized Fourier domain (GFD) is introduced. In this newly proposed domain, a parametric form of control on the periodic repetitions that occur due to sampling in the reciprocal domain is possible, without the need to increase the sampling rate. This characteristic and the connections of the generalized Fourier transform to analyticity and to the z-transform are investigated. Core properties of the generalized discrete Fourier transform (GDFT) such as a weighted circular correlation property and Parseval's relation are derived. We show the benefits of using the novel framework in a spatial-audio application, specifically the simulation of room impulse responses for auralization purposes in e.g. virtual reality systems.

Generalized Discrete Fourier Transform With Nonlinear Phase

Constant modulus transforms like discrete Fourier transform (DFT), Walsh transform, and Gold codes have been successfully used over several decades in several engineering applications, including discrete multi-tone (DMT), orthogonal frequency division multiplexing (OFDM) and code division multiple access (CDMA) communications systems. In this paper, we present a generalized framework for DFT called Generalized DFT (GDFT) with nonlinear phase by exploiting the phase space. We show that GDFT offers sizable correlation improvements over DFT, Walsh, Oppermann and Gold codes, leading to better performance in all multi-carrier communications scenarios investigated. We also highlight how known constant modulus orthogonal transforms are special solutions of the proposed GDFT framework. Moreover, we introduce practical design methods offering computationally efficient implementations of GDFT as enhancements to DFT. We conclude the paper with examples of communications applications where GDFT is shown to outperform DFT and other known constant modulus bases.