Fast Recursive Algorithms Research Articles

Antenna Beampattern Matched Optimum Coherent Detection in High-Resolution Mechanically Scanning Maritime Surveillance Radars

Mechanically scanning radars are still widely used in ocean guard and surveillance areas nowadays, owing to simple structure, light weight and low cost. The antenna rotation causes amplitude modulation of target returns along pulses by the antenna azimuth beampattern. The traditional coherent detection that ignores this modulation suffers from some performance loss. In this paper, an antenna beampattern matched optimum coherent detector and a dual-threshold detection scheme are proposed for high-resolution mechanically scanning maritime surveillance radars. Firstly, it is shown that the modulation of target returns is determined by the antenna azimuth beampattern, scan rate, and pulse repetition interval (PRI) and the antenna scanning also affect the temporal correlation of sea clutter. Secondly, the antenna beampattern matched optimum coherent detector is given in compound-Gaussian sea clutter with inverse Gamma distributed texture and the pulse scanning mode is used to deal with unknown target azimuth angle in the matched filter. Thirdly, to mitigate the high computational cost of the pulse scanning mode, a dual-threshold scheme is given, which searches all pulse-range cells by short-time integration and further confirms suspected cells whose test statistics are between the two lower and higher thresholds in the search by long-time integration. Moreover, two fast recursive clutter covariance matrix inversion algorithms along pulses and range cells are presented, which bring a great reduction in computational complexity. Finally, real measured radar data is used to verify the proposed detector and scheme and the experimental results show that it behaves better in detection of small targets than the traditional detectors.

Singular Signal Measurement Based on Bidirectional Recursive Complex-Valued Wavelet Algorithm

The normal operation fault of the power system is usually caused by a short-circuit fault. At this time, the system changes drastically from one state to another, accompanied by complex transient phenomena. Therefore, the measured signal contains a large number of transient components. How to effectively analyze such signals, extract their characteristics, and develop new protection devices has always been an important research field in power system protection technology. The protection of the power system is to achieve the purpose of correct action and elimination of faults by quickly detecting and locating faults. At present, the power signal analysis tools used in microcomputer protection include FFT, Kalman filter, and finite impulse response filter. They are efficient for the analysis of stationary signals, but have their limitations in analyzing nonstationary signals; especially it is difficult to identify nonlinear faults, such as the detection of high-impedance nonlinear short-circuit faults, which is a long-term unsolved problem in power systems. Based on wavelet transform, this paper selects complex-valued wavelet algorithm, analyzes a real-time recursive wavelet algorithm, and deduces the realization process of the algorithm in detail. The algorithm greatly reduces the computational complexity of the existing two-way recursive algorithm, can be used for real-time detection of fault signals in various fields of power system, and can be extended to realize other fast recursive algorithms of wavelet functions. Based on the sensitivity of complex-valued wavelet transform phase information to singularity, a method for real-time monitoring of power system fault mutation signals using the phase information of complex-valued wavelet fast recursion algorithm to assist amplitude information is proposed. The validity and practicability of this complex-valued wavelet and its real-time recursive algorithm for fault detection are demonstrated by an example.

Combined adaptive antenna array and adaptive channel equalizer, based on multichannel fast recursive least squares algorithms

An adaptive antenna array (AAA) is a sort of directional antennas, which is able to suppress the signals of the interference sources. The AAA works well if there is no multipath propagation of the information signal. A device, which removes the multipath effects, is an adaptive equalizer. However, the independent AAA and equalizer do not work well if the interferences and multipath exist simultaneously. The purpose of this paper is to present a new efficient solution of the task by using of the AAA, whose weights are shared with the weights of the fractionally-spaced feed-forward (FF) part of the equalizer. The array output signal is combined with that of the feed-backward (FB) part of the equalizer. The array and such distributed equalizer operate as an indivisible adaptive filter, providing good performance and ability to receive the multipath information signals in the presence of the signals of the external interference sources. Because the FF part of the equalizer operates at the up-sampled rate, which is a few times higher than the information symbols rate, but the FB filter operates at the symbol rate, the computationally efficient fast recursive least squares (RLS) algorithms cannot be used directly in the considered adaptive unit. To solve this problem, it is proposed to use polyphase representation of the FF part of the equalizer. In this case, the architecture of the adaptive unit is viewed as a multichannel adaptive filter with unequal number of weights in channels. The paper presents the architecture of the adaptive unit and the details of its equalizer part. The mathematical description of a multichannel fast adaptive filtering algorithm, called stabilized fast a posteriori sequential technique (FAEST), fitted for the weights calculation of the considered unit is also presented. The verification of the proposed adaptive unit has been carried out via simulation, using a linear array with sixteen omnidirectional antennas, which operates in the conditions of –30 dB signal-to-interference ratio (SIR), 10…30 dB signal-to-noise ratio (SNR), receives the phase shift keying (8-PSK) desired signal, passed through a two-rays communication channel with about –65 dB gaps in the channel amplitude-frequency response. In the steady-state, the proposed adaptive unit demonstrates about –100 dB radiation pattern gaps towards the interference sources and the transient response with less than 500 symbols duration. The simulation demonstrates the ability of the proposed unit simultaneously to steer the main lobe of antenna array radiation pattern towards the desired signal source without a priori search of its angular location and to remove the signals of the external interferences and intersymbol interference in the array output signal. Due to these properties, the proposed AAA/equalizer can be used in equipment of modern mobile communication systems.

Multiresolution GPC-Structured Control of a Single-Loop Cold-Flow Chemical Looping Testbed.

Chemical looping is a near-zero emission process for generating power from coal. It is based on a multi-phase gas-solid flow and has extremely challenging nonlinear, multi-scale dynamics with jumps, producing large dynamic model uncertainty, which renders traditional robust control techniques, such as linear parameter varying H∞ design, largely inapplicable. This process complexity is addressed in the present work through the temporal and the spatiotemporal multiresolution modeling along with the corresponding model-based control laws. Namely, the nonlinear autoregressive with exogenous input model structure, nonlinear in the wavelet basis, but linear in parameters, is used to identify the dominant temporal chemical looping process dynamics. The control inputs and the wavelet model parameters are calculated by optimizing a quadratic cost function using a gradient descent method. The respective identification and tracking error convergence of the proposed self-tuning identification and control schemes, the latter using the unconstrained generalized predictive control structure, is separately ascertained through the Lyapunov stability theorem. The rate constraint on the control signal in the temporal control law is then imposed and the control topology is augmented by an additional control loop with self-tuning deadbeat controller which uses the spatiotemporal wavelet riser dynamics representation. The novelty of this work is three-fold: (1) developing the self-tuning controller design methodology that consists in embedding the real-time tunable temporal highly nonlinear, but linearly parametrizable, multiresolution system representations into the classical rate-constrained generalized predictive quadratic optimal control structure, (2) augmenting the temporal multiresolution loop by a more complex spatiotemporal multiresolution self-tuning deadbeat control loop, and (3) demonstrating the effectiveness of the proposed methodology in producing fast recursive real-time algorithms for controlling highly uncertain nonlinear multiscale processes. The latter is shown through the data from the implemented temporal and augmented spatiotemporal solutions of a difficult chemical looping cold flow tracking control problem.

Fast and Robust Recursive Algorithmsfor Separable Nonnegative Matrix Factorization.

In this paper, we study the nonnegative matrix factorization problem under the separability assumption (that is, there exists a cone spanned by a small subset of the columns of the input nonnegative data matrix containing all columns), which is equivalent to the hyperspectral unmixing problem under the linear mixing model and the pure-pixel assumption. We present a family of fast recursive algorithms and prove they are robust under any small perturbations of the input data matrix. This family generalizes several existing hyperspectral unmixing algorithms and hence provides for the first time a theoretical justification of their better practical performance.

Fixed-Point Implementation of Fast QR-Decomposition Recursive Least-Squares Algorithms (FQRD-RLS): Stability Conditions and Quantization Errors Analysis

The fast QR-decomposition based recursive least-squares (FQRD-RLS) algorithms offer RLS-like convergence and misadjustment at a lower computational cost, and therefore are desirable for implementation on a fixed-point digital signal processor (DSP). Furthermore, the FQRD-RLS algorithms are derived from QR-decomposition based RLS algorithms that are well known for their numerical stability in finite precision. Hence, these algorithms are often assumed numerically stable, although there is no rigorous analysis addressing the stability of such algorithms in finite precision. In this paper, we derive the conditions that guarantee stability of the FQRD-RLS algorithms, and also derive mathematical expressions for the mean-squared quantization error (MSQE) of internal variables of the FQRD-RLS algorithms at steady state. The objective is to quantify the propagation error due to quantization effects. The derived MSQE expressions have been verified by comparisons with fixed-point computer simulations.

A Novel Computationally Efficient Algorithm for Multiuser Detection in Long-Code CDMA System

Computation of the Cholesky decomposition of the autocorrelation matrix is essential in developing zero-forcing, minimum mean square error (MMSE), and decision feedback detectors for a long code-division multiple-access (CDMA) system. However, computing the Cholesky decomposition of the time-varying correlation matrix requires many computer operations. This is because the computation of the filter weights is segment specific (varies from one symbol interval to next). In this paper, a novel and fast algorithm is developed to compute the Cholesky decomposition of the autocorrelation matrix for a long-code CDMA system. The smart update algorithms, to compute the solution (filter weights) of one segment from the solution of the previous segment, reduce the computational complexity, particularly for large-sized autocorrelation matrices. We exploit the fact that the channel matrix of a long-code CDMA system is block banded with a shift structure. Although the channel matrix is not Toeplitz for a long-code CDMA system, the shift structure of the matrix, along with its banded structure, allows the development of fast smart update recursive algorithms for data estimation. The relevance of this algorithm is discussed for channels with multipath delays extending over many symbol intervals. Simulations show that the proposed algorithm performs the same as competing multiuser detector algorithms for long-code CDMA at much reduced computational complexity.

Constant-Time Filtering Using Shiftable Kernels

It was recently demonstrated in [5] that the non-linear bilateral filter [14] can be efficiently implemented using a constant-time or O(1) algorithm. At the heart of this algorithm was the idea of approximating the Gaussian range kernel of the bilateral filter using trigonometric functions. In this letter, we explain how the idea in [5] can be extended to few other linear and non-linear filters [14, 17, 2]. While some of these filters have received a lot of attention in recent years, they are known to be computationally intensive. To extend the idea in [5], we identify a central property of trigonometric functions, called shiftability, that allows us to exploit the redundancy inherent in the filtering operations. In particular, using shiftable kernels, we show how certain complex filtering can be reduced to simply that of computing the moving sum of a stack of images. Each image in the stack is obtained through an elementary pointwise transform of the input image. This has a two-fold advantage. First, we can use fast recursive algorithms for computing the moving sum [15, 6], and, secondly, we can use parallel computation to further speed up the computation. We also show how shiftable kernels can also be used to approximate the (non-shiftable) Gaussian kernel that is ubiquitously used in image filtering.

Fast Algorithms in Slow and High Doppler Mobile Environments

High Doppler effects resulting from fast time varying dispersive channels give rise to the so-called intercarrier interference, whose accurate modeling for data recovery is paramount. We present a novel turbo estimation scheme based on a known linearized model of the time varying channel via its derivatives. The derivatives estimation is adaptive, in the sense that at each turbo estimation it incorporates information on previously estimated parameters. This is combined with a decision-directed scheme implemented at bit level with soft decision. We show that any pilot based scheme that is able to induce a Toeplitz structure in the channel correlation matrix, can make use of an existing class of so-called superfast algorithms for minimum mean-square error (MMSE) channel estimation. Given a structure for the vector of pilots and an upper bound for the channel delay spread, say N, we show that it is only necessary to store 2N coefficients per pilot structure in order to recover the entire channel parameters. We bring attention to the fact that either for static or slowly varying channels, any Toeplitz-like model can yield a superfast equalization method as well, and show how fast order recursive algorithms can be combined to a superfast receiver structure, in order to yield both efficient equalizer computation and equalization.

Implementation of nonsymmetric interior-point methods for linear optimization over sparse matrix cones

We describe an implementation of nonsymmetric interior-point methods for linear cone programs defined by two types of matrix cones: the cone of positive semidefinite matrices with a given chordal sparsity pattern and its dual cone, the cone of chordal sparse matrices that have a positive semidefinite completion. The implementation takes advantage of fast recursive algorithms for evaluating the function values and derivatives of the logarithmic barrier functions for these cones. We present experimental results of two implementations, one of which is based on an augmented system approach, and a comparison with publicly available interior-point solvers for semidefinite programming.

On Fast Recursive Algorithms for V-BLAST with Optimal Ordered SIC Detection

A fast recursive algorithm for the vertical Bell Laboratories layered space-time (V-BLAST) with the optimal ordered successive interference cancellation (SIC) detection has been proposed by Benesty-Huang-Chen and two improved algorithms have been also recently introduced by Szczecinski-Massicotte and Zhu-Lei-Chin independently. In this letter, our first contribution is to incorporate the existing improvements into the original fast recursive algorithm to provide the fastest known algorithm for the optimal ordered SIC detection of V-BLAST. Subsequently, we propose two new algorithms that result from one further improvement for the fast recursive algorithm. Compared with the fastest known algorithm built from the existing improvements, one new proposed algorithm has a speedup of 1.3 times in both the number of multiplications and the number of additions, and the other new proposed algorithm requires less intermediate variables and saves memories.

Image restoration with sliding sinusoidal transforms

A space-variant restoration technique with sliding sinusoidal transforms is presented. Minimum mean-square error estimators in the domains of sliding discrete sine and cosine transforms for image restoration are derived. To achieve image processing in real time, fast recursive algorithms for computing the sliding sinusoidal transforms are utilized. Computer simulation results using a real image are provided and compared with that of common restoration techniques.

Fast linear algebra is stable

In Demmel et al. (Numer. Math. 106(2), 199–224, 2007) we showed that a large class of fast recursive matrix multiplication algorithms is stable in a normwise sense, and that in fact if multiplication of n-by-n matrices can be done by any algorithm in O(n ω+η ) operations for any η > 0, then it can be done stably in O(n ω+η ) operations for any η > 0. Here we extend this result to show that essentially all standard linear algebra operations, including LU decomposition, QR decomposition, linear equation solving, matrix inversion, solving least squares problems, (generalized) eigenvalue problems and the singular value decomposition can also be done stably (in a normwise sense) in O(n ω+η ) operations.

Recursive and Fast Recursive Capon Spectral Estimators

The Capon algorithm, which was originally proposed for wavenumber estimation in array signal processing, has become a powerful tool for spectral analysis. Over several decades, a significant amount of research attention has been devoted to the estimation of the Capon spectrum. Most of the developed algorithms thus far, however, rely on the direct computation of the inverse of the input correlation (or covariance) matrix, which can be computationally very expensive particularly when the dimension of the matrix is large. This paper deals with fast and efficient algorithms in computing the Capon spectrum. Inspired from the recursive idea established in adaptive signal processing theory, we first derive a recursive Capon algorithm. This new algorithm does not require an explicit matrix inversion, and hence it is more efficient to implement than the direct-inverse approach. We then develop a fast version of the recursive algorithm based on techniques used in fast recursive least-squares adaptive algorithms. This new fast algorithm can further reduce the complexity of the recursive Capon algorithm by an order of magnitude. Although our focus is on the Capon spectral estimation, the ideas shown in this paper can also be generalized and applied to other applications. To illustrate this, we will show how to apply the recursive idea to the estimation of the magnitude squared coherence function, which plays an important role for problems like time-delay estimation, signal-to-noise ratio estimation, and doubletalk detection in echo cancellation.

An Efficient Recursive Procedure for Evaluating the Impedance Matrix of Linear and Planar Fractal Arrays

The self-similar geometrical properties of fractal arrays are exploited in this paper to develop fast recursive algorithms for efficient evaluation of the associated impedance matrices as well as driving point impedances. The methodology is demonstrated by considering two types of uniformly excited fractal arrays consisting of side-by-side half-wave dipole antenna elements. These examples include a triadic Cantor linear fractal array and a Sierpinski carpet planar fractal array. This class of self-similar antenna arrays become significantly large at higher order stages of growth and utilization of fractal analysis allows the impedance matrix, and hence the driving point impedances, to be obtained much more efficiently than would be possible using conventional analysis techniques.

Fast QR Algorithms Based on Backward Prediction Errors: A New Implementation and Its Finite Precision Performance

QR decomposition techniques are well known for their good numerical behavior and low complexity. Fast QRD recursive least squares adaptive algorithms benefit from these characteristics to offer robust and fast adaptive filters. This paper examines two different versions of the fast QR algorithm based on a priori backward prediction errors as well as two other corresponding versions of the fast QR algorithm based on a poste- riori backward prediction errors. The main matrix equations are presented with different versions derived from two distinct deployments of a particular matrix equation. From this study, a new algorithm is derived. The discussed algorithms are compared, and differences in computational complexity and in finite-precision behavior are shown. Ke yw ords: Adaptive system, fast RLS algorithm, QR decomposition.

Recursive algorithms for inner ellipsoidal approximation of convex polytopes

In this paper, fast recursive algorithms for the approximation of an n-dimensional convex polytope by means of an inscribed ellipsoid are presented. These algorithms consider at each step a single inequality describing the polytope and, under mild assumptions, they are guaranteed to converge in a finite number of steps. For their recursive nature, the proposed algorithms are better suited to treat a quite large number of constraints than standard off-line solutions, and have their natural application to problems where the set of constraints is iteratively updated, as on-line estimation problems, nonlinear convex optimization procedures and set membership identification.

Modeling of highly multimode waveguides for time-domain simulation

Optical interconnection technology on the printed circuit board level is a key technology for future microelectronic equipment. Its industrial application requires sophisticated processes and technologies for the manufacturing as well as the design. In this paper, a modeling approach for a static and a time-domain simulation of board-integrated highly multimode optical waveguides with manufacturing based rough surfaces is presented. The transfer behavior of the waveguides is determined by a numerical computation of the step response using a Monte Carlo based ray tracing procedure. The step response is piecewise approximated by exponential functions in order to obtain an analytical representation of the pulse response. This approach allows the application of very fast semi-analytical recursive convolution algorithms.

Fast recursive algorithms for 2-D discrete cosine transform

A new algorithm for computation of two-dimensional (2-D) type-III discrete cosine transform is presented. The algorithm is particularly suited to block size ( p 1∗2 m ) by ( p 2∗2 n ), where p 1 and p 2 are odd integers, and m and n are non-negative integers. It shows that the 2-D type-III DCT can be decomposed into cosine–cosine, cosine–sine, sine–cosine, sine–sine sequences, which can be further decomposed into similar sequences. The proposed algorithm provides the flexibility in choosing block size and has a simple indexing mapping scheme and a fairly regular computation structure. The algorithm also requires a smaller number of arithmetic operations for p 1= p 2=3.

Fast recursive algorithm for a class of structured matrices

In this paper, we continue the study of a class of structured matrices which may be treated as a generalization of the class of Hessenberg matrices. For this class fast recursive algorithms for solution of the corresponding linear systems are obtained. The implementation of algorithms is illustrated by numerical experiments.