Fourier Matrix Research Articles

Range-spread target detection using the time-frequency feature based on sparse representation

ABSTRACTThe wideband radar transmitting the linear frequency modulation signal often processes its echoes by the stretched processing. This paper deals with the range-spread target detection in white complex Gaussian noise. Here, we propose a new detection method for the range-spread target based on sparse representation, which selects the time-frequency feature to realise the target detection. It can be simply described as follows: first, the sketched signal is reconstructed from its noisy measurements by basis pursuit de-noising (BPDN); scatterers on the target are determined by its reconstruction and used to calculate the Wigner distribution; for the target embedded in noise, the time-frequency feature in its power-density spectrum is compared with the decision threshold. Meanwhile, the median absolute deviation (MAD) is adopted to estimate the noise variance. The mainly novelties can be concluded as follows: the Fourier matrix is selected to sparsely represent the sketched signal; the sparsity is used to improve the SNR of the received echoes; the Wigner transform is utilised to acquire the time-frequency feature of the range-spread target. Both the optimisation theory and time-frequency representation are introduced to solve the target detection problem. Experimental results on the raw data show that the proposed detector outperforms the conventional methods.

Multisource encoding and decoding using the signal apparition technique

Signal apparition is a method for encoding sources in simultaneous multisource seismic acquisition and decoding the multisource response of the earth into its single-source responses. For [Formula: see text] sources, encoding is performed by applying periodic sequences of period [Formula: see text] to each of the sources along source lines. Decoding is achieved in the wavenumber domain for each frequency by solving an [Formula: see text] linear system of equations. The system’s matrix is the product of a Fourier matrix and an encoding matrix, the latter containing the information of the codes. The solution of the system is unique when the encoding matrix is invertible. When the encoding sequences consist of time delays applied to sources’ firing times, the determinant of the encoding matrix becomes a polynomial. A unique solution to decoding then exists if the roots of the polynomial avoid the unit circle. Periodic time-shift sequences for two, three, four, and six sources are discussed. A model example of simultaneous four-source data acquisition illustrates the performance of the encoding/decoding technique for the spatially nonaliased case.

Measurement matrix optimization based on incoherent unit norm tight frame

This paper considers the problem of measurement matrix optimization for compressed sensing (CS) in which the dictionary is assumed to be given, such that it leads to an effective sensing matrix. Due to important properties of equiangular tight frames (ETFs) to achieve Welch bound equality, the measurement matrix optimization based on ETF has received considerable attention and many algorithms have been proposed for this aim. These methods produce sensing matrix with low mutual coherence based on initializing the measurement matrix with random Gaussian ensembles. This paper, use incoherent unit norm tight frame (UNTF) as an important frame with the aim of low mutual coherence and proposes a new method to construction a measurement matrix of any dimension while measurement matrix initialized by partial Fourier matrix. Simulation results show that the obtained measurement matrix effectively reduces the mutual coherence of sensing matrix and has a fast convergence to Welch bound compared with other methods.

Compressed-sensing wavenumber-scanning interferometry

The Fourier transform (FT), the nonlinear least-squares algorithm (NLSA), and eigenvalue decomposition algorithm (EDA) are used to evaluate the phase field in depth-resolved wavenumber-scanning interferometry (DRWSI). However, because the wavenumber series of the laser's output is usually accompanied by nonlinearity and mode-hop, FT, NLSA, and EDA, which are only suitable for equidistant interference data, often lead to non-negligible phase errors. In this work, a compressed-sensing method for DRWSI (CS-DRWSI) is proposed to resolve this problem. By using the randomly spaced inverse Fourier matrix and solving the underdetermined equation in the wavenumber domain, CS-DRWSI determines the nonuniform sampling and spectral leakage of the interference spectrum. Furthermore, it can evaluate interference data without prior knowledge of the object. The experimental results show that CS-DRWSI improves the depth resolution and suppresses sidelobes. It can replace the FTas a standard algorithm for DRWSI.

Uniform Recovery Bounds for Structured Random Matrices in Corrupted Compressed Sensing

We study the problem of recovering an $s$-sparse signal $\mathbf{x}^{\star}\in\mathbb{C}^n$ from corrupted measurements $\mathbf{y} = \mathbf{A}\mathbf{x}^{\star}+\mathbf{z}^{\star}+\mathbf{w}$, where $\mathbf{z}^{\star}\in\mathbb{C}^m$ is a $k$-sparse corruption vector whose nonzero entries may be arbitrarily large and $\mathbf{w}\in\mathbb{C}^m$ is a dense noise with bounded energy. The aim is to exactly and stably recover the sparse signal with tractable optimization programs. In this paper, we prove the uniform recovery guarantee of this problem for two classes of structured sensing matrices. The first class can be expressed as the product of a unit-norm tight frame (UTF), a random diagonal matrix and a bounded columnwise orthonormal matrix (e.g., partial random circulant matrix). When the UTF is bounded (i.e. $\mu(\mathbf{U})\sim1/\sqrt{m}$), we prove that with high probability, one can recover an $s$-sparse signal exactly and stably by $l_1$ minimization programs even if the measurements are corrupted by a sparse vector, provided $m = \mathcal{O}(s \log^2 s \log^2 n)$ and the sparsity level $k$ of the corruption is a constant fraction of the total number of measurements. The second class considers randomly sub-sampled orthogonal matrix (e.g., random Fourier matrix). We prove the uniform recovery guarantee provided that the corruption is sparse on certain sparsifying domain. Numerous simulation results are also presented to verify and complement the theoretical results.

Representations of group rings and groups

An isomorphism between the group ring of a finite group and a ring of certain block diagonal matrices is established. It is shown that for any group ring matrix $A$ of $mathbb{C} G$ there exists a matrix $U$ (independent of $A$) such that $U^{-1}AU= diag(T_1,T_2,ldots, T_r)$ for block matrices $T_i$ of fixed size $s_i × s_i$ where $r$ is the number of conjugacy classes of $G$ and $s_i$ are the ranks of the group ring matrices of the primitive idempotents. Using the isomorphism of the group ring to the ring of group ring matrices followed by the mapping $Amapsto P^{-1}AP$ (fixed $P$) gives an isomorphism from the group ring to the ring of such block matrices. Specialising to the group elements gives a faithful representation of the group. Other representations of $G$ may be derived using the blocks in the images of the group elements. For a finite abelian group $Q$ an explicit matrix $P$ is given which diagonalises any group ring matrix of $mathbb{C}Q$. The characters of $Q$ and the character table of $Q$ may be read off directly from the rows of the diagonalising matrix $P$. This is a special case of the general block diagonalisation process but is arrived at independently. The case for cyclic groups is well-known: Circulant matrices are the group ring matrices of the cyclic group and the Fourier matrix diagonalises any circulant matrix. This has applications to signal processing.

Channel Estimation for Finite Scatterers Massive Multi-User MIMO System

In finite scattering propagation environment, when the number of scatterers is very small compared to the number of base station antennas and users in the cell and if the same scatterers are shared by all users, then the correlation among the users increases. Hence, the high-dimensional multi-user MIMO system is likely to have low-rank channel. To estimate the channel matrix, weighted nuclear norm minimization method is proposed in this paper. Iterative weighted singular value thresholding algorithm is used to solve the optimization problem. To recover the low-rank channel, a partial random Fourier matrix (PRFM) satisfying the restricted isometric property is adapted as the training matrix. The PRFM forces the iterative algorithm to converge in one iteration which reduces the huge computational complexity of the proposed channel estimation method. The mean square error and achievable sum rate are the criteria used to measure the performance of the proposed method. The results show that the proposed method outperforms the least square estimation method and the nuclear norm minimization method for various finite scatterers in different SNR levels.

Solving underdetermined systems with error-correcting codes

In an underdetermined system of equations $Ax=y$, where $A$ is an $m\times n$ matrix, only $u$ of the entries of $y$ with $u < m$ are known. Thus $E_jw$, called `measurements', are known for certain $j\in J \subset \{0,1,\ldots,m-1\}$ where $\{E_i, i=0,1,\ldots, m-1\}$ are the rows of $A$ and $|J|=u$. It is required, if possible, to solve the system uniquely when $x$ has at most $t$ non-zero entries with $u\geq 2t$. Here such systems are considered from an error-correcting coding point of view. The unknown $x$ can be shown to be the error vector of a code subject to certain conditions on the rows of the matrix $A$. This reduces the problem to finding a suitable decoding algorithm which then finds $x$. Decoding workable algorithms are shown to exist, from which the unknown $x$ may be determined, in cases where the known $2t$ values are evenly spaced (that is, when the elements of $J$ are in arithmetic progression) for classes of matrices satisfying certain row properties. These cases include Fourier $n\times n $ matrices where the arithmetic difference $k$ satisfies $\gcd(n,k)=1$, and classes of Vandermonde matrices $V(x_1,x_2,\ldots,x_n)$ (with $x_i\neq 0$) with arithmetic difference $k$ where the ratios $x_i/x_j$ for $i\neq j$ are not $k^{th}$ roots of unity. The decoding algorithm has complexity $O(nt)$ and in some cases, including the Fourier matrix cases, the complexity is $O(t^2)$. Matrices which have the property that the determinant of any square submatrix is non-zero are of particular interest. Randomly choosing rows of such matrices can then give $t$ error-correcting pairs to generate a `measuring' code $C^\perp=\{E_j | j\in J\}$ with a decoding algorithm which finds $x$. This has applications to signal processing and compressed sensing.

Maximal coherence in a generic basis

Since quantum coherence is an undoubted characteristic trait of quantum physics, the quantification and application of quantum coherence has been one of the long-standing central topics in quantum information science. Within the framework of a resource theory of quantum coherence proposed recently, a \textit{fiducial basis} should be pre-selected for characterizing the quantum coherence in specific circumstances, namely, the quantum coherence is a \textit{basis-dependent} quantity. Therefore, a natural question is raised: what are the maximum and minimum coherences contained in a certain quantum state with respect to a generic basis? While the minimum case is trivial, it is not so intuitive to verify in which basis the quantum coherence is maximal. Based on the coherence measure of relative entropy, we indicate the particular basis in which the quantum coherence is maximal for a given state, where the Fourier matrix (or more generally, \textit{complex Hadamard matrices}) plays a critical role in determining the basis. Intriguingly, though we can prove that the basis associated with the Fourier matrix is a stationary point for optimizing the $l_1$ norm of coherence, numerical simulation shows that it is not a global optimal choice.

Analytic deformations of group commuting squares and complex Hadamard matrices

Let G be a finite group and denote by CG the commuting square associated to G. The defect of the group G, given by the formula d(G)=∑g∈G|G|order(g), was introduced in [9] as an upper bound for the number of linearly independent directions in which CG can be continuously deformed in the class of commuting squares. In this paper we show that this bound is actually attained, by constructing d(G) analytic families of commuting squares containing CG.In the case G=Zn, the defect d(Zn) can be interpreted as the dimension of the enveloping tangent space of the real algebraic manifold of n×n complex Hadamard matrices, at the Fourier matrix Fn (in the sense of [14,1]). The dimension of the enveloping tangent space gives a natural upper bound on the number of continuous deformations of Fn by complex Hadamard matrices, of linearly independent directions of convergence. Our result shows that this bound is reached, which is rather surprising. In particular our construction yields new analytic families of complex Hadamard matrices stemming from Fn.In the last section of the paper we use a compactness argument to prove non-equivalence (i.e. non-isomorphism as commuting squares) for dephased versions of the families of Hadamard matrices constructed throughout the paper.

A fast algorithm for solving circulant tensor systems

A new definition for circulant tensors is given, which is a generalization of the one for circulant matrices. Furthermore, we define the generalized circulant tensors which can be diagonalized by the Fourier matrix F and/or . We also consider solving the circulant tensor systems by a fast algorithm based on the fast Fourier transform (FFT). Such algorithm is similar to that for circulant linear systems and it can also be performed in .

Matrix Fourier Transforms for Consistent Mathematical Models

We create a matrix integral transforms method; it allows us to describe analytically the consistent mathematical models. An explicit constructions for direct and inverse Fourier matrix transforms with discontinuous coefficients are established. We introduce special types of Fourier matrix transforms: matrix cosine transforms, matrix sine transforms, and matrix transforms with piecewise trigonometric kernels. The integral transforms of such kinds are used for problems solving of mathematical physics in homogeneous and piecewise homogeneous media. Analytical solution of iterated heat conduction equation is obtained. Stress produced in the elastic semi-infinite solid by pressure is obtained in the integral form.

Improvements in PLC transmissions via difference sets schemes with compressive sampling

The paper deals with the damaging effects produced by impulsive noise in PLC transmissions and proposes a cancellation scheme using difference-set-based compressive sensing.Authors focus on a popular narrow-band PLC solution to evaluate its performance: PRIME. A simulation environment based on Matlab was developed to analyze PRIME’s behavior in an impulsive noise environment. To model the transmission channel, Middleton’s Class-A noise model was used in conjunction with noise parameters reported in the literature for the narrow-banded spectrum. The performance is measured in terms of bit error rate versus signal to noise ratio.Although the use of compressive sensing to cancel impulsive noise in communications has already been proposed in other studies, this paper details a modification based on Partial Fourier Matrix indexing according to difference sets. Results from simulations report an almost complete cancellation of the impulsive noise effects. The analysis is done studying the achieved bit error rate as a function of the signal to noise ratio at the receiver.

Accelerated Split Bregman Method for Image Compressive Sensing Recovery under Sparse Representation

Compared with traditional patch-based sparse representation, recent studies have concluded that group-based sparse representation (GSR) can simultaneously enforce the intrinsic local sparsity and nonlocal self-similarity of images within a unified framework. This article investigates an accelerated split Bregman method (SBM) that is based on GSR which exploits image compressive sensing (CS). The computational efficiency of accelerated SBM for the measurement matrix of a partial Fourier matrix can be further improved by the introduction of a fast Fourier transform (FFT) to derive the enhanced algorithm. In addition, we provide convergence analysis for the proposed method. Experimental results demonstrate that accelerated SBM is potentially faster than some existing image CS reconstruction methods.

Spectral properties of small Hadamard matrices

We prove that if A and B are Hadamard matrices which are both of size 4×4 or 5×5 and in dephased form, then tr(A)=tr(B) implies that A and B have the same eigenvalues, including multiplicity. We calculate explicitly the spectrum for these matrices. We also extend these results to larger Hadamard matrices which are permutations of the Fourier matrix and calculate their spectral multiplicities.

High resolution sparse estimation of exponentially decaying N-dimensional signals

In this work, we consider the problem of high-resolution estimation of the parameters detailing an N-dimensional (N-D) signal consisting of an unknown number of exponentially decaying sinusoidal components. Since such signals are not sparse in an oversampled Fourier matrix, earlier approaches typically exploit large dictionary matrices that include not only a finely spaced frequency grid, but also a grid over the considered damping factors. Even in the 2-D case, the resulting dictionary is typically very large, resulting in a computationally cumbersome optimization problem. Here, we introduce a sparse modeling framework for N-dimensional exponentially damped sinusoids using the Kronecker structure inherent in the model. Furthermore, we introduce a novel dictionary learning approach that iteratively refines the estimate of the candidate frequency and damping coefficients for each component, thus allowing for smaller dictionaries, and for frequency and damping parameters that are not restricted to a grid. The performance of the proposed method is illustrated using simulated data, clearly showing the improved performance as compared to previous techniques.

Elliptic Curve Public Key Algorithm Based on Fourier Matrix

Elliptic Curve Public Key Algorithm Based on Fourier Matrix

Research of incoherence rotated chaotic measurement matrix in compressed sensing

Measurement matrix construction is the hot issue of compressed sensing. How to construct a measurement matrix of good performance and easy hardware implementation is the main research problem in compressed sensing. In this paper, we present a novel simple and efficient measurement matrix named Incoherence Rotated Chaotic (IRC) matrix. We take advantage of the well pseudorandom of chaotic sequence, introduce the concept of the incoherence factor and rotation, and adopt QR decomposition to obtain the IRC measurement matrix which is suited for sparse reconstruction. The IRC matrix satisfies the Restricted Isometry Property criterion in sparse reconstruction and has a smaller RIP ratio. Simulations demonstrate IRC matrix has better performance than Gaussian random matrix, Bernoulli random matrix, Fourier matrix and can efficiently work on both natural image and remote sensing image. The peak signal-to-noise ratios of reconstructed images using IRC matrix are improved at 1.5 dB to 2.5 dB at least.

Small circulant complex Hadamard matrices of Butson type

We study the circulant complex Hadamard matrices of order n whose entries are lth roots of unity. For n=l prime we prove that the only such matrix, up to equivalence, is the Fourier matrix, while for n=p+q,l=pq with p,q distinct primes there is no such matrix. We then provide a list of equivalence classes of such matrices, for small values of n,l.

Uniqueness of Butson Hadamard matrices of small degrees

Let BHn×n(m) be the set of n×n Butson Hadamard matrices where all the entries are m-th roots of unity. For H1,H2∈BHn×n(m), we say that H1 is equivalent to H2 if H1=PH2Q for some monomial matrices P and Q whose nonzero entries are m-th roots of unity. In the present paper we show by computer search that all the matrices in BH17×17(17) are equivalent to the Fourier matrix of degree 17. Furthermore we shall prove that, for a prime number p, a matrix in BHp×p(p) which is not equivalent to the Fourier matrix of degree p gives rise to a non-Desarguesian projective plane of order p.