Discrete Fourier Transform Matrix Research Articles

Digit classification of ghost imaging based on similarity measures

In this paper, a pattern recognition scheme based on ghost imaging (GI) system is proposed, which is inspired by the fact that the one-dimensional vectors obtained by the GI system can be considered as an alternative representation of the original object, and these vectors will be able to greatly preserve the original object features when appropriate illumination patterns are adopted. The characteristics of GI is combined with discrete Fourier transform (DFT) to generate illumination patterns based on the DFT matrix, which are used to illuminate objects. The obtained bucket detector measurements are utilized to perform similarity metrics and output classification results. The scheme can classify objects with no need of reconstruction and neural network, thus reducing the running time. The experimental and numerical simulation results demonstrate that the proposed scheme requires less computation but has high recognition accuracy, and is also robust to environment noise. In addition, the recognition accuracy of the proposed scheme can also be well maintained when the sampling rate is 50%.

Fast, Algebraic Multivariate Multipoint Evaluation in Small Characteristic and Applications

Multipoint evaluation is the computational task of evaluating a polynomial given as a list of coefficients at a given set of inputs. Besides being a natural and fundamental question in computer algebra on its own, fast algorithms for this problem are also closely related to fast algorithms for other natural algebraic questions such as polynomial factorization and modular composition. And while nearly linear time algorithms have been known for the univariate instance of multipoint evaluation for close to five decades due to a work of Borodin and Moenck [ 7 ], fast algorithms for the multivariate version have been much harder to come by. In a significant improvement to the state-of-the-art for this problem, Umans [ 25 ] and Kedlaya & Umans [ 16 ] gave nearly linear time algorithms for this problem over field of small characteristic and over all finite fields, respectively, provided that the number of variables n is at most \(d^{o(1)}\) where the degree of the input polynomial in every variable is less than d . They also stated the question of designing fast algorithms for the large variable case (i.e., \(n \notin d^{o(1)}\) ) as an open problem. In this work, we show that there is a deterministic algorithm for multivariate multipoint evaluation over a field \(\mathbb {F}_{q}\) of characteristic p , which evaluates an n -variate polynomial of degree less than d in each variable on N inputs in time \(\begin{equation*} \left((N + d^n)^{1 + o(1)}\text{poly}(\log q, d, n, p)\right), \end{equation*}\) provided that p is at most d o (1) , and q is at most (exp (exp (exp (...(exp ( d ))))), where the height of this tower of exponentials is fixed. When the number of variables is large (e.g., n ∉ d o (1) ), this is the first nearly linear time algorithm for this problem over any (large enough) field. Our algorithm is based on elementary algebraic ideas, and this algebraic structure naturally leads to the following two independently interesting applications: — We show that there is an algebraic data structure for univariate polynomial evaluation with nearly linear space complexity and sublinear time complexity over finite fields of small characteristic and quasipolynomially bounded size. This provides a counterexample to a conjecture of Miltersen [ 21 ] who conjectured that over small finite fields, any algebraic data structure for polynomial evaluation using polynomial space must have linear query complexity. — We also show that over finite fields of small characteristic and quasipolynomially bounded size, Vandermonde matrices are not rigid enough to yield size-depth tradeoffs for linear circuits via the current quantitative bounds in Valiant’s program [ 26 ]. More precisely, for every fixed prime p , we show that for every constant ɛ > 0, and large enough n , the rank of any \(n \times n\) Vandermonde matrix V over the field \(\mathbb {F}_{p^a}\) can be reduced to ( n /exp (Ω (poly(ɛ)log 0.53 n ))) by changing at most n Θ (ɛ) entries in every row of V , provided a ≤ poly(log n ). Prior to this work, similar upper bounds on rigidity were known only for special Vandermonde matrices. For instance, the Discrete Fourier Transform matrices and Vandermonde matrices with generators in a geometric progression [ 9 ].

Smart Radio Environments with Intelligent Reflecting Surfaces for 6G Sub-Terahertz-Band Communications

Technology for sixth-generation (6G) mobile communication system is now being widely studied. A sub-Terahertz band is expected to play a great role in 6G to enable extremely high data-rate transmission. This paper has two goals. (1) Introduction of 6G concept and propagation characteristics of sub-Terahertz-band radio waves. (2) Performance evaluation of intelligent reflecting surfaces (IRSs) based on beamforming in a sub-Terahertz band for smart radio environments (SREs). We briefly review research on SREs with reconfigurable intelligent surfaces (RISs), and describe requirements and key features of 6G with a sub-Terahertz band. After that, we explain propagation characteristics of sub-Terahertz band radio waves. Important feature is that the number of multipath components is small in a sub-Terahertz band in indoor office environments. This leads to an IRS control method based on beamforming because the number of radio waves out of the optimum beam is very small and power that is not used for transmission from the IRS to user equipment (UE) is little in the environments. We use beams generated by a Butler matrix or a DFT matrix. In simulations, we compare the received power at a UE with that of the upper bound value. Simulation results show that the proposed method reveals good performance in the sense that the received power is not so lower than the upper bound value.

Characterizing an Uncertainty Diagram and Kirkwood-Dirac Nonclassicality Based on Discrete Fourier Transform.

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.

Transceiver design for an uplink asynchronous NOMA using MSK-type signals

The time offset is intentionally introduced into non-orthogonal multiple access (NOMA) to decrease inter-user interference and enhance performance. Recently, minimum shift keying type (MSK-type) signals suitable for power-efficient nonlinear amplifiers have been considered into the uplink asynchronous NOMA (ANOMA), but it is only for the capacity analysis. In this paper, the transceiver design for an uplink ANOMA using MSK-type signals is our concern. Following the maximum likelihood (ML) criterion under successive interference cancellation (SIC), we first propose a Viterbi algorithm (VA)-based detector. By the eigenvalue decomposition (ED) of Hermitian-Toeplitz matrices, an ED-based transceiver scheme is also given. Further, considering the asymptotic behavior of banded-Toeplitz matrices and the limitations of SIC-based transceivers, we propose a DFT matrix diagonalization (DFTD)-based transceiver scheme for joint ML (JML) detection. Finally, the complexity analysis of several detectors is provided. Numerical results reveal that the maximum sum throughput can be achieved by setting half the bit interval, and the time offset can bring a performance gain of 4 dB or 7.5 dB for SIC-based transceivers and 8 dB for JML-based transceivers. The relationship between bit-error-rate (BER) performance and different channel differences, and modulation formats is also shown by simulation.

Detection of Downlink Asynchronous NOMA With MSK-Type Signal

In this letter, motivated by the superiority of continuous phase modulation (CPM) signals, a binary CPM with modulation index 0.5 is considered into downlink symbol-asynchronous non-orthogonal multiple access (ANOMA) to enhance power and spectral efficiency, especially for resource-limited wireless communications. According to the maximum likelihood (ML) criterion, a Viterbi-based block-by-block detector (VB-BBD) is first proposed. Considering the asymptotics of Toeplitz matrices, a DFT matrix diagonalization-based symbol-by-symbol detector (DFTD-SSD) is also given. Finally, the detection complexity analysis is provided. Numerical results reveal the optimal time offsets with different modulation formats for a two-user system, and show that time asynchrony can bring about 3dB and 6dB performance gains at a bit error rate (BER) of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-4</sup> . The relationship between BER performance and different power levels, frequency pulses, response lengths is also investigated.

On the Derivation of Winograd-Type DFT Algorithms for Input Sequences Whose Length Is a Power of Two

Winograd’s algorithms are an effective tool for calculating the discrete Fourier transform (DFT). These algorithms described in well-known articles are traditionally represented either with the help of sets of recurrent relations or with the help of products of sparse matrices obtained on the basis of various methods of the DFT matrix factorization. Unfortunately, in the mentioned papers, it is not shown how the described relations were obtained or how the presented factorizations were found. In this paper, we use a simple, understandable and fairly unified approach to the derivation of the Winograd-type DFT algorithms for the cases N = 8, N = 16 and N = 32. It is easy to verify that algorithms for other lengths of sequences that are powers of two can be synthesized in a similar way.

Minimizing Aliasing in Multiple Frequency Harmonic Balance Computations

The harmonic balance method has emerged as an efficient and accurate approach for computing periodic, as well as almost periodic, solutions to nonlinear ordinary differential equations. The accuracy of the harmonic balance method can however be negatively impacted by aliasing. Aliasing occurs because Fourier coefficients of nonlinear terms in the governing equations are approximated by a discrete Fourier transform (DFT). Understanding how aliasing occurs when the DFT is applied is therefore essential in improving the accuracy of the harmonic balance method. In this work, a new operator that describe the fold-back, i.e. aliasing, of unresolved frequencies onto the resolved ones is developed. The norm of this operator is then used as a metric for investigating how the time sampling should be performed to minimize aliasing. It is found that a time sampling which minimizes the condition number of the DFT matrix is the best choice in this regard, both for single and multiple frequency problems. These findings are also verified for the Duffing oscillator. Finally, a strategy for oversampling multiple frequency harmonic balance computations is developed and tested.

Application of Joint Raw Moments-Based Probabilistic Power Flow Analysis for Hybrid AC/VSC-MTDC Power Systems

A joint raw moments-based analytical method (JRMAM) is first proposed herein to deal with probabilistic power flow (PPF) of hybrid AC&#x002F;VSC-MTDC (Alternate Current&#x002F;Voltage Source Control-Multiple Terminal Direct Current) power systems with integration of the offshore wind farms. JRMAM is able to overcome the challenges of some existing combined methods in estimating the high order moments of target outputs, and the dramatic time consumption of Monte Carlo simulation method (MCSM) in performing the PPF analysis particularly due to the complication of its deterministic power flow (DPF) model. The proposed JRMAM has a further superiority on efficiency, even compared to the joint cumulants-based analytical method (JCAM). As a preliminary step, discrete Fourier transformation matrix (DFTM) method is applied to calculate the joint raw moments of random inputs which are required by conducting JRMAM, and the probability density function (PDF) of outputs can be estimated by means of the Gram-Charlier expansions (GCE) based on the results obtained from JRMAM. Two modified IEEE test cases connected with DC subsystems through the VSC devices are adopted to verify the effectiveness and the scalability of JRMAM, where the result of MCSM is treated as a reference, and the properties of the hybrid AC&#x002F;VSC-MTDC power system with these master-slave controlled VSCs are investigated for a further acceleration on PPF analysis.

How Exponentially Ill-Conditioned Are Contiguous Submatrices of the Fourier Matrix?

Linear systems involving contiguous submatrices of the discrete Fourier transform (DFT) matrix arise in many applications, such as Fourier extension, superresolution, and coherent diffraction imaging. We show that the condition number of any such $p\times q$ submatrix of the $N\times N$ DFT matrix is at least $ \exp \left( \frac{\pi}{2} \left[\min(p,q)- \frac{pq}{N}\right]\right)$, up to algebraic prefactors. That is, fixing the shape parameters $(\al,\bt):=(p/N,q/N)\in(0,1)^2$, the growth is $e^{\rho N}$ as $N\to\infty$, the exponential rate being $\rho = \frac{\pi}{2}[\min(\alpha,\beta)- \alpha\beta]$. Our proof uses the Kaiser--Bessel transform pair (of which we give a self-contained proof), plus estimates on sums over distorted sinc functions, to construct a localized trial vector whose DFT is also localized. We warm up with an elementary proof of the above but with half the rate, via a periodized Gaussian trial vector. Using low-rank approximation of the kernel $e^{ixt}$, we also prove another lower bound $(4/e\pi \al)^q$, up to algebraic prefactors, which is stronger than the above for small $\al$ and $\bt$. When combined, the bounds are within a factor of two of the empirical asymptotic rate, uniformly over $(0,1)^2$, and become sharp in certain regions. However, the results are not asymptotic: they apply to essentially all $N$, $p$, and $q$, and with all constants explicit.

Research Spotlights

Research Spotlights

The p-norm of circulant matrices via Fourier analysis

Abstract A recent work derived expressions for the induced p-norm of a special class of circulant matrices A(n, a, b) ∈ ℝ n × n , with the diagonal entries equal to a ∈ ℝ and the off-diagonal entries equal to b ≥ 0. We provide shorter proofs for all the results therein using Fourier analysis. The key observation is that a circulant matrix is diagonalized by a DFT matrix. The results comprise an exact expression for ǁAǁ p , 1 ≤ p ≤ ∞, where A = A(n, a, b), a ≥ 0 and for ǁAǁ2 where A = A(n, −a, b), a ≥ 0; for the other p-norms of A(n, −a, b), 2 &lt; p &lt; ∞, upper and lower bounds are derived.

Poisson Phase Retrieval in Very Low-count Regimes.

This paper discusses phase retrieval algorithms for maximum likelihood (ML) estimation from measurements following independent Poisson distributions in very low-count regimes, e.g., 0.25 photon per pixel. To maximize the log-likelihood of the Poisson ML model, we propose a modified Wirtinger flow (WF) algorithm using a step size based on the observed Fisher information. This approach eliminates all parameter tuning except the number of iterations. We also propose a novel curvature for majorize-minimize (MM) algorithms with a quadratic majorizer. We show theoretically that our proposed curvature is sharper than the curvature derived from the supremum of the second derivative of the Poisson ML cost function. We compare the proposed algorithms (WF, MM) with existing optimization methods, including WF using other step-size schemes, quasi-Newton methods such as LBFGS and alternating direction method of multipliers (ADMM) algorithms, under a variety of experimental settings. Simulation experiments with a random Gaussian matrix, a canonical DFT matrix, a masked DFT matrix and an empirical transmission matrix demonstrate the following. 1) As expected, algorithms based on the Poisson ML model consistently produce higher quality reconstructions than algorithms derived from Gaussian noise ML models when applied to low-count data. Furthermore, incorporating regularizers, such as corner-rounded anisotropic total variation (TV) that exploit the assumed properties of the latent image, can further improve the reconstruction quality. 2) For unregularized cases, our proposed WF algorithm with Fisher information for step size converges faster (in terms of cost function and PSNR vs. time) than other WF methods, e.g., WF with empirical step size, backtracking line search, and optimal step size for the Gaussian noise model; it also converges faster than the LBFGS quasi-Newton method. 3) In regularized cases, our proposed WF algorithm converges faster than WF with backtracking line search, LBFGS, MM and ADMM.

New flexible deterministic compressive measurement matrix based on finite Galois field

Nowadays, the deterministic construction of sensing matrices is a hot topic in compressed sensing. The coherence of the measurement matrix is an important research area in the design of deterministic compressed sensing. To solve this problem, this paper proposes a novel sparse deterministic measurement matrix, which basically accesses the optimal low coherence of the measurement matrix. Firstly, a class of sparse square matrix is constructed based on finite fields' arithmetic. Then, the Hadamard matrix, or (discrete Fourier transform) DFT matrix, is nestled into the square matrix to construct an asymptotically optimal deterministic measurement matrix. That is, the relevant column vectors have orthogonal characteristics. Using this feature, the measurement matrix can be further optimized to reduce its mutual coherence, almost achieving the lower bound of the coherence (Welch bound). The two types of deterministic measurement matrices proposed are sparse with low mutual coherence and flexible measurement sizes. So, the proposed deterministic measurement matrices require less memory and time for the recovery as well as reducing the complexity due to their sparse structure. The simulation results show that compared with the existing (several typical) random matrices, the proposed method can reduce the mutual coherence and computational complexity of the measurement matrix.

Universal Measurement Matrix Design for Sparse and Co-Sparse Signal Recovery

Compressed Sensing (CS) avails mutual coherence metric to choose the measurement matrix that is incoherent with dictionary matrix. Random measurement matrices are incoherent with any dictionary, but their highly uncertain elements necessitate large storage and make hardware realization difficult. In this paper deterministic matrices are employed which greatly reduce memory space and computational complexity. To avoid the randomness completely, deterministic sub-sampling is done by choosing rows deterministically rather than randomly, so that matrix can be regenerated during reconstruction without storing it. Also matrices are generated by orthonormalization, which makes them highly incoherent with any dictionary basis. Random matrices like Gaussian, Bernoulli, semi-deterministic matrices like Toeplitz, Circulant and full-deterministic matrices like DFT, DCT, FZC-Circulant are compared. DFT matrix is found to be effective in terms of recovery error and recovery time for all the cases of signal sparsity and is applicable for signals that are sparse in any basis, hence universal.

Time-and-Frequency Hybrid Multiplexing for Flexible Ambiguity Controls of DFT-coded MIMO OFDM Radar

A time-and-frequency hybrid multiplexing technique for a multiple-input multiple-output (MIMO) orthogonal frequency-division multiplexing (OFDM) radar is proposed. Discrete Fourier transform (DFT)-coded OFDM waveforms are introduced, which allow it to be free from range-dependent angle errors. These are readily used to provide both time-domain multiplexing and frequency-domain multiplexing in the MIMO OFDM radar. The DFT- coded frequency-domain multiplexing shortens the maximum unambiguous range, while the DFT- coded time-domain multiplexing lowers the maximum Doppler ambiguity. A hybrid of both domain multiplexing techniques can mitigate the respective limitations by adaptively selecting the proper DFT matrix size in each multiplexing domain. This allows to solve the intrinsic range and Doppler ambiguity problems of MIMO radars by controlling the hybrid ratio of the two kinds of DFT- code based multiplexing methods. Range-Doppler and range-angle maps of four examples with different hybrid multiplexing ratios are simulated with a MIMO OFDM radar numerical platform.

Quantum Fourier transform revisited

SummaryThe fast Fourier transform (FFT) is one of the most successful numerical algorithms of the 20th century and has found numerous applications in many branches of computational science and engineering. The FFT algorithm can be derived from a particular matrix decomposition of the discrete Fourier transform (DFT) matrix. In this paper, we show that the quantum Fourier transform (QFT) can be derived by further decomposing the diagonal factors of the FFT matrix decomposition into products of matrices with Kronecker product structure. We analyze the implication of this Kronecker product structure on the discrete Fourier transform of rank‐1 tensors on a classical computer. We also explain why such a structure can take advantage of an important quantum computer feature that enables the QFT algorithm to attain an exponential speedup on a quantum computer over the FFT algorithm on a classical computer. Further, the connection between the matrix decomposition of the DFT matrix and a quantum circuit is made. We also discuss a natural extension of a radix‐2 QFT decomposition to a radix‐d QFT decomposition. No prior knowledge of quantum computing is required to understand what is presented in this paper. Yet, we believe this paper may help readers to gain some rudimentary understanding of the nature of quantum computing from a matrix computation point of view.

Automatic Generation of High-Performance FFT Kernels on Arm and X86 CPUs

This article presents AutoFFT, a template-based code generation framework that can automatically generate high-performance FFT kernels for all natural-number radices. AutoFFT is based on the Cooley-Tukey FFT algorithm, which exploits the symmetric and periodic properties of the DFT matrix, as the outer parallelization framework. Because butterflies are the core operations of the Cooley-Tukey algorithm, we explore additional symmetric and periodic properties of the DFT matrix and formulate multiple optimized calculation templates to further reduce the number of floating-point operations for butterflies of arbitrary natural numbers. To fully exploit hardware resources, we encapsulate a series of optimizations in an assembly template optimizer. Given any DFT problem, AutoFFT automatically generates C FFT kernels using these calculation templates and converts them into efficient assembly kernels using the template optimizer. Through a series of experiments on Arm, Intel, and AMD processors, we show that AutoFFT-generated kernels can outperform those in Fastest Fourier Transform in the West (FFTW), the Arm Performance Libraries (ARMPL), and the Intel Math Kernel Library (MKL).

DFT‐based direction‐of‐arrival estimation using a nested minimum redundancy array

Performance of the discrete Fourier transform (DFT)-based direction-of-arrival (DoA) estimation method depends on the number of antennas. Increasing the number of antennas will improve the accuracy of DoA estimation. In this study, the authors propose a new DFT-based DoA estimation method using a nested minimum redundancy array (NMRA). By vectorisation of the covariance matrix of the received signal of an NMRA with low number of antennas, a virtual uniform linear array with a large number of antennas can be obtained. An initial estimation of DoAs can be derived by applying the normalised centralised DFT matrix to the observations of virtual uniform linear array. By minimisation of the difference between the virtual linear array observations and their estimated values, the accuracy of DoA estimation will be improved. Simulation results show that the proposed method has better performance in comparison with the conventional ones.

Construction of unimodular tight frames for compressed sensing using majorization-minimization

In this paper, we propose a method to construct uni-modular tight frames (UMTFs), which are tight frames with the additional constraint that every entry of the matrix has the same magnitude. UMTFs are useful in many applications, since multiplication of a UMTF by a vector can be implemented in polar coordinates using very low computational cost. Since normalized UMTFs are unit norm tight frames (UNTFs), and since a UNTF is a minimizer of the frame potential, we propose an algorithm to find UMTFs by minimizing the frame potential. We show that minimizing the frame potential is equivalent to minimizing the total coherence when the frame is unimodular. We use the majorization-minimization approach to propose a low complexity, iterative, fast-converging algorithm for minimizing the frame potential. We also extend our algorithm to the cases where the phase angles of the sensing matrix are required to belong to a given finite set of feasible angles, and to the case where the signal being sampled is sparse in an arbitrary, possibly non-canonical basis. We illustrate the utility of our proposed construction in the context of sparse signal recovery. Partial DFT matrices, obtained by randomly selected rows from the full DFT matrix, are UMTFs. However, they perform poorly when dealing with signals that admit a sparse representation in the wavelet, Fourier and discrete cosine transform domains. In such scenarios, we illustrate the superior performance of our construction compared to the partial DFT, complex Gaussian and Bernoulli random matrices through simulations. The proposed algorithm offers the same performance as the partial DFT matrix, and outperforms the complex Gaussian and Bernoulli random matrices, when the signal is sparse in the canonical basis.