Fast Approximate Joint Diagonalization Incorporating Weight Matrices

We propose a new low-complexity approximate joint diagonalization (AJD) algorithm, which incorporates nontrivial block-diagonal weight matrices into a weighted least-squares (WLS) AJD criterion. Often in blind source separation (BSS), when the sources are nearly separated, the optimal weight matrix for WLS-based AJD takes a (nearly) block-diagonal form. Based on this observation, we show how the new algorithm can be utilized in an iteratively reweighted separation scheme, thereby giving rise to fast implementation of asymptotically optimal BSS algorithms in various scenarios. In particular, we consider three specific (yet common) scenarios, involving stationary or block-stationary Gaussian sources, for which the optimal weight matrices can be readily estimated from the sample covariance matrices (which are also the target-matrices for the AJD). Comparative simulation results demonstrate the advantages in both speed and accuracy, as well as compliance with the theoretically predicted asymptotic optimality of the resulting BSS algorithms based on the weighted AJD, both on large scale problems with matrices of the size 100$,times,$100.      

基于Kronecker积的图像超分辨率快速算法

 本文提出了基于矩阵Kronecker积的图像超分辨率快速重构算法.基于观测模型的图像超分辨率重构算法是研究较多的方法,观测模型包含两个维数很大的降采样矩阵和模糊矩阵,这两个矩阵均可以表示为两个维数相对较低的矩阵的Kronecker积.因此图像降质可以分解为两个独立的过程,首先对行向降质,然后再对列向降质.根据这一观点,文章提出了一个与现有模型等价的新模型,并进一步证明用于克服逆向病态的正则化算子也可以作这样的分解.基于新的观测模型,文章提出了共轭梯度法来实现图像的超分辨率重构,与传统方法不同的是,本文算法直接使用矩阵而不是向量作为决策变量.文章给出了理论分析,实验结果证实新算法确实能显著的节省时间和存储空间开销.     

Tridiagonal Commuting Matrices and Fractionalizations of DCT and DST Matrices of Types I, IV, V, and VIII

In this paper, we first establish new relationships in matrix forms among discrete Fourier transform (DFT), generalized DFT (GDFT), and various types of discrete cosine transform (DCT) and discrete sine transform (DST) matrices. Two new independent tridiagonal commuting matrices for each of DCT and DST matrices of types I, IV, V, and VIII are then derived from the existing commuting matrices of DFT and GDFT. With these new commuting matrices, the orthonormal sets of Hermite-like eigenvectors for DCT and DST matrices can be determined and the discrete fractional cosine transform (DFRCT) and the discrete fractional sine transform (DFRST) are defined. The relationships among the discrete fractional Fourier transform (DFRFT), fractional GDFT, and various types of DFRCT and DFRST are developed to reduce computations for DFRFT and fractional GDFT.     

Quadratic optimization for simultaneous matrix diagonalization

Simultaneous diagonalization of a set of matrices is a technique that has numerous applications in statistical signal processing and multivariate statistics. Although objective functions in a least-squares sense can be easily formulated, their minimization is not trivial, because constraints and fourth-order terms are usually involved. Most known optimization algorithms are, therefore, subject to certain restrictions on the class of problems: orthogonal transformations, sets of symmetric, Hermitian or positive definite matrices, to name a few. In this paper, we present a new algorithm called QDIAG that splits the overall optimization problem into a sequence of simpler second order subproblems. There are no restrictions imposed on the transformation matrix, which may be nonorthogonal, indefinite, or even rectangular, and there are no restrictions regarding the symmetry and definiteness of the matrices to be diagonalized, except for one of them. We apply the new method to second-order blind source separation and show that the algorithm converges fast and reliably. It allows for an implementation with a complexity independent of the number of matrices and, therefore, is particularly suitable for problems dealing with large sets of matrices.     

Signal detection via spectral theory of large dimensional randommatrices

Results on the spectral behavior of random matrices as the dimension increases are applied to the problem of detecting the number of sources impinging on an array of sensors. A common strategy to solve this problem is to estimate the multiplicity of the smallest eigenvalue of the spatial covariance matrix R of the sensed data. Existing approaches rely on the closeness of the noise eigenvalues of sample covariance matrix to each other and, therefore, the sample size has to be quite large when the number of sources is large in order to obtain a good estimate. The theoretical analysis presented focuses on the splitting of the spectrum of sample covariance matrix into noise and signal eigenvalues. It is shown that when the number of sensors is large the number of signals can be estimated with a sample size considerably less than that required by previous approaches     

A Generalized Reverse Block Jacket Transform

Jacket matrices motivated by the center weight Hadamard matrices have played important roles in signal processing, communication, image compression, cryptography, etc. In this paper we propose a notation called block Jacket matrix which substitutes elements of the matrix into common matrices or even block matrices. Employing the well-known Pauli matrices which are very important in many subjects, block Jacket matrices with any size are investigated in detail, and some recursive relations for fast construction of the block Jacket matrices are obtained. Based on the general recursive relations, several special block Jacket matrices are constructed. To decompose high order block Jacket matrices, a fast decomposition algorithm for the factorable block Jacket matrices is suggested. After that some properties of the block Jacket matrices are investigated. Finally, several remarks are presented. These remarks are associated with comparisons between the Clifford algebra and the block Jacket matrices, generations of orthogonal and quasi-orthogonal sequences, and relations of the block Jacket matrices to the orthogonal transforms for signal processing. Since the Pauli matrices are actually infinitesimal generators of $SU(2)$ group, the proposed construction and decomposition algorithms for the block Jacket matrices are available in the signal processing, communication, quantum signal processing and information theory.      

Solution of large dense complex matrix equations using a fastFourier transform (FFT)-based wavelet-like methodology

Wavelet-like transformations have been used in the past to compress dense large matrices into a sparse system. However, they generally are implemented through a finite impulse response filter realized through the formulation of Daubechies (1992). A method is proposed to use a very high order filter (namely an ideal one) and use the computationally efficient fast Fourier transform (FFT) to carry out the multiresolution analysis. The goal here is to reduce the redundancy in the system and also guarantee that the wavelet coefficients drop off much faster. Hence, the efficiency of the new procedure becomes clear for very high order filters. The advantage of the FFT-based procedure utilizing ideal filters is that it can be computationally efficient and for very large matrices may yield a sparse matrix. However, this is achieved, as well known in the literature, at the expense of robustness, which may lead to a larger reconstruction error due to the presence of the Gibb's phenomenon. Numerical examples are presented to illustrate the efficiency of this procedure as conjectured in the literature     

Fast Analysis of RCS Over a Frequency Band Using Pre-Corrected FFT/AIM and Asymptotic Waveform Evaluation Technique

The pre-corrected fast Fourier transform (PFFT)/adaptive integral method (AIM) is combined with the asymptotic waveform evaluation (AWE) technique to present fast RCS calculation for arbitrarily shaped three-dimensional PEC objects over a frequency band. The electric field integral equation (EFIE) is used to formulate the problem and the method of moments (MoM) is employed to solve the integral equation. By using the AWE method, the unknown equivalent current is expanded into a Taylor series around a frequency in the desired frequency band. Then, instead of solving the equivalent current at each frequency point, it is only necessary to solve for the coefficients of the Taylor series (called “moments”) at each expansion point. Since the number of the expansion points is usually much smaller than that of the frequency points, the AWE can achieve fast frequency sweeping. To facilitate the analysis of large problems, in this paper, all the full matrices are stored in a sparse form and the PFFT/AIM method is employed to accelerate all the matrix-vector products on both sides of the matrix equation for the moments. Further, the incomplete LU preconditioner is used at each expansion point to improve the convergence behaviour of the matrix equation for the moments. The present method can deal with much larger problems than the conventional MoM-AWE method since the PFFT/AIM achieves considerable reduction in memory requirement and computation time. Numerical results will be presented to show the efficiency and capability of the method.      

Eigenvalue Distributions of Sums and Products of Large Random Matrices Via Incremental Matrix Expansions

This paper uses an incremental matrix expansion approach to derive asymptotic eigenvalue distributions (a.e.d.s) of sums and products of large random matrices. We show that the result can be derived directly as a consequence of two common assumptions, and matches the results obtained from using - and -transforms in free probability theory. We also give a direct derivation of the a.e.d. of the sum of certain random matrices which are not free. This is used to determine the asymptotic signal-to-interference-ratio of a multiuser code-division multiple-access (CDMA) system with a minimum mean-square error linear receiver.     

Fast subsampled-updating stabilized fast transversal filter (FSUSFTF) RLS algorithm for adaptive filtering

We present a new, doubly fast algorithm for recursive least-squares (RLS) adaptive filtering that uses displacement structure and subsampled-updating. The fast subsampled-updating stabilized fast transversal filter (FSU SFTF) algorithm is mathematically equivalent to the classical fast transversal filter (FTF) algorithm. The FTF algorithm exploits the shift invariance that is present in the RLS adaptation of an FIR filter. The FTF algorithm is in essence the application of a rotation matrix to a set of filters and in that respect resembles the Levinson (1947) algorithm. In the subsampled-updating approach, we accumulate the rotation matrices over some time interval before applying them to the filters. It turns out that the successive rotation matrices themselves can be obtained from a Schur-type algorithm that, once properly initialized, does not require inner products. The various convolutions that appear In the algorithm are done using the fast Fourier transform (FFT). The resulting algorithm is doubly fast since it exploits FTF and FFTs. The roundoff error propagation in the FSU SFTF algorithm is identical to that in the SFTF algorithm: a numerically stabilized version of the classical FTF algorithm. The roundoff error generation, on the other hand, seems somewhat smaller. For relatively long filters, the computational complexity of the new algorithm is smaller than that of the well-known LMS algorithm, rendering it especially suitable for applications such as acoustic echo cancellation     

QC－LDPC码的置换矩阵循环移位次数设计

 本文提出了一种循环移位次数的代数设计方法,该方法可用来构造基于置换矩阵的QC-LDPC码的稀疏奇偶校验矩阵H 。这个方法的基本思路是：将构造 q×t置换阵列 H矩阵的问题转化为构造 q×t下标矩阵 S(H)=[aij]的问题,然后根据Fosserier的充分必要条件,设计出能消除小围长（girth）的下标计算表达式 aij=f(q.t.n)。由该方法构造的H 矩阵能消除4环长,围长至少是6。     

Two methods for Toeplitz-plus-Hankel approximation to a datacovariance matrix

Recently, fast algorithms have been developed for computing the optimal linear least squares prediction filters for nonstationary random processes (fields) whose covariances have (block) Toeplitz-Hankel form. If the covariance of the random process (field) must be estimated from the data, the following problem is presented: given a data covariance matrix, computer from the available data, find the Toeplitz-plus-Hankel matrix closest to this matrix in some sense. The authors give two procedures for computing the Toeplitz-plus-Hankel matrix that minimizes the Hilbert-Schmidt norm of the difference between the two matrices. The first approach projects the data covariance matrix onto the subspace of Toeplitz-plus-Hankel matrices, for which basis functions can be computed using a Gram-Schmidt orthonormalization. The second approach projects onto the subspace of symmetric Toeplitz plus skew-persymmetric Hankel matrices, resulting in a much simpler algorithm. The extension to block Toeplitz-plus-Hankel data covariance matrix approximation is also addressed     

Eigenvalues and eigenvectors of generalized DFT, generalized DHT,DCT-IV and DST-IV matrices

In this paper, the eigenvalues and eigenvectors of the generalized discrete Fourier transform (GDFT), the generalized discrete Hartley transform (GDHT), the type-IV discrete cosine transform (DCT-IV), and the type-IV discrete sine transform (DST-IV) matrices are investigated in a unified framework. First, the eigenvalues and their multiplicities of the GDFT matrix are determined, and the theory of commuting matrices is applied to find the real, symmetric, orthogonal eigenvectors set that constitutes the discrete counterpart of Hermite Gaussian function. Then, the results of the GDFT matrix and the relationships among these four unitary transforms are used to find the eigenproperties of the GDHT, DCT-IV, and DST-IV matrices. Finally, the fractional versions of these four transforms are defined, and an image watermarking scheme is proposed to demonstrate the effectiveness of fractional transforms     

Constructing and decoding GWBE codes using Kronecker products

In this letter, we introduce a novel method for constructing large size Generalized Welch Bound Equality (GWBE) matrices. This method can also be used for the construction of large WBE matrices. The advantage of this method is its low complexity for constructing large size matrices and low computational complexity using Maximum Likelihood (ML) decoders for a subclass of these codes.     

对合Cauchy-Hadamard型MDS矩阵的构造

MDS矩阵和对合MDS矩阵在分组密码中有广泛应用。该文将考察同时是Hadamard矩阵和Cauchy矩阵的那些MDS矩阵,给出了这类矩阵的结构、构造方法和个数,从而得到了MDS矩阵一种新的构造方法。该文还证明了Cauchy-Hadamard型MDS矩阵都等效于对合的Cauchy-Hadamard型MDS矩阵,并给出了由Cauchy-Hadamard型MDS矩阵构造对合的Cauchy-Hadamard型MDS矩阵的方法。     

A method for computing large-scale two-dimensional transform without transposing data matrix

A fast algorithm is presented for a two-dimensional transform of a data matrix such as Fourier or Hadamard transforms. It can be applied to a matrix, which is too large for the main storage and is stored sequentially rowwise in an auxiliary storage. The size of the matrix may be of R^mrows and of arbitrary number of columns. Whenever R rows of working area are available in the main storage, the matrix is read out, processed, and stored back m times. This new algorithm does not require the presently used method of transposing the data matrix.     

Image compressed sensing based on wavelet transform in contourlet domain

Compressed sensing (CS) has been widely concerned and sparsity of a signal plays a crucial role in CS to exactly recover signals. Contourlet transform provides sparse representations for images, so an algorithm of CS reconstruction based on contourlet is considered. Meanwhile, taking into account the computation and the storage of large random measurement matrices in the CS framework, we are trying to introduce the wavelet transform into the contourlet domain to reduce the size of random measurement matrices. Several numerical experiments demonstrate that this idea is feasible. The proposed algorithm possesses the following advantages: reduced size of random measurement matrix and improved recovered performance.     

光腔的分数付里叶变换描述

运用矩阵光学技巧,球面镜光腔的往返一周光束变换矩阵可分解为两矩阵之积,它们各对应于一分数付里叶变换,因而光腔的振荡本质就是不断作用分数付里叶变换的过程。其稳定条件和束腰位置可用分数阶次描述,分数阶次可为复数。同时,给出了可能的拓展。     

A cutting-plane method based on redundant rows for improving fractional distance

Decoding performance of linear programming (LP) decoding is closely related to geometrical properties of a fundamental polytope: fractional distance, pseudo codeword, etc. In this paper, an idea of the cutting-plane method is employed to improve the fractional distance of a given binary parity-check matrix. The fractional distance is the minimum weight (with respect to l₁-distance) of nonzero vertices of the fundamental polytope. The cutting polytope is defined based on redundant rows of the parity-check matrix. The redundant rows are codewords of the dual code not yet appearing as rows in the paritycheck matrix. The cutting polytope plays a key role to eliminate unnecessary fractional vertices in the fundamental polytope. We propose a greedy algorithm and its efficient implementation based on the cutting-plane method. It has been confirmed that the fractional distance of some parity-check matrices are actually improved by using the algorithm.     

Modeling baluns with the method of moments

When using the method of moments (MoM) to model large arrays with large numbers of balanced feeds, the feed baluns can always be incorporated into the analysis by combining the multiport admittance matrix for the array with the admittance matrices of the baluns. This technique is straightforward but requires that the MoM equations be solved for a large number of right-hand sides. The paper shows how the effects of perfect baluns can be incorporated directly into the MoM equations, requiring only one right-hand side. This technique yields the exact results with a significant savings in computing resources