This paper studies the statistical behavior of the affine projection (AP) algorithm for /spl mu/=1 for Gaussian autoregressive inputs. This work extends the theoretical results of Rupp (1998) to the numerical evaluation of the MSE learning curves for the adaptive AP weights. The MSE learning behavior of the AP(P+1) algorithm with an AR(Q) input (Q/spl les/P) is shown to be the same as the NLMS algorithm (/spl mu/=1) with a white input with M-P unity eigenvalues and P zero eigenvalues and increased observation noise. Monte Carlo simulations are presented which support the theoretical results.

Linebarger et al. (see Proceedings of Thirty-First Asilomar Conference on Signals, Systems & Computers, Pacific Grove, California, USA, vol.1, p.123-27, 1997) introduced a new, optimal approach to low rank transform domain adaptive filtering is , using a least squares, matrix based framework. Further, Raghothaman (see Proceedings of 8th IEEE DSP Workshop, Utah, USA, 1998) provides a computationally efficient algorithm to solve the formulation of the problem proposed by Linebarger. In this paper, we examine an alternative method for applying an optimal low rank transform, within the framework derived by Linebarger, to convert an overdetermined, full rank system into a low rank system. In addition, we propose a computationally efficient algorithm for the implementation of our method, using the DCT as the unitary transformation. Finally, we evaluate the performance of our algorithm via simulation in an acoustic echo canceller application, and show that the performance of our method is superior to existing low rank methods, NLMS and affine projection.

A family of adaptive filtering algorithms for processing signals which have energy concentrated in a relatively small number of component subspaces in the spectral domain is introduced. The approach is based on transform domain signal decomposition and linear least squares filtering of the selected subset of transform domain signal components. The derivation is based on the linear least squares adaptive filtering framework introduced in previous work (1997). Fast convergence and computational efficiency are the main characteristics of the resulting algorithms. The method is applied to the problem of adaptive line enhancement comb filtering and DFT is used as a transform method. It is also shown that the resulting adaptive structure is capable of handling the case of non-coinciding frequencies. The performance of the algorithm is evaluated through a series of simulation experiments.

This paper analyzes an SVD-based low rank transform domain adaptive filtering algorithm and proves that it performs better than the normalized LMS. The method extracts an under-determined solution from an overdetermined least squares problem, using a part of the unitary transformation formed by the right singular vectors of the data matrix. The aim is to get as close to the solution of an overdetermined system as possible, using an under-determined system. Previous work based on the same framework, but with the DFT as the transformation, has shown considerable improvement in performance over conventional time domain methods like NLMS and affine projection. The analysis of the SVD-based variant helps us to understand the convergence behavior of the DFT-based low complexity method. We prove that the SVD-based method gives a lower residual than NLMS. Simulations confirm the theoretical results.

The subspace-based reduced rank and polynomial order (RRPO) methods estimate a reduced order linear prediction polynomial whose roots are the desired "signal roots". In this paper, we describe how to extend the RRPO methods to include constraints involving known signal information. Simulation results indicate that by incorporating known signal information such as source direction angle, the estimation of unknown source directions can be significantly improved, especially when the unknown source is weak, closely spaced and highly coherent with the known source.

We combine spherical subspace (SS) and eigen based updating methods with the affine projection (AP) method to produce a new family of fast SS-AP algorithms that offers additional tradeoffs between computation and adaptive filtering performance. Moreover, the implementation of SS-AP is less complicated than the fast RLS based AP algorithms. For certain applications, e.g., echo cancellation and equalization in digital subscriber loop (DSL) transceivers, SS-AP offers performance that is comparable to AP, but at computational costs that are less than the fast AP algorithms.

We present an empirical analysis of the fast affine projection (FAP) algorithm to be used in an acoustic echo cancellation application using a fixed-point DSP processor. We also introduce a modified FAP algorithm that was developed based on our FAP study. Out analysis study shows that the modified FAP algorithm is more robust and provides more consistent performance than the LMS algorithm. The new FAP algorithm is also numerically efficient and easy to implement with a fixed-point DSP processor.

Reduced rank linear predictive frequency and direction-of-arrival (DOA) estimation algorithms use the singular value decomposition (SVD) to produce a noise-cleaned linear prediction vector. These algorithms then root this vector to obtain a subset of roots, whose angles contain the desired frequency or DOA information. The roots closest to the unit circle are deemed to be the "signal roots". The rest of the roots are "extraneous". The extraneous roots are expensive to calculate. Further, a search must be done to discern the signal roots from the extraneous roots. Here, we present a reduced polynomial order linear prediction method that simplifies the rooting computation for applications where high-speed processing is critical.

This paper presents a new set of derivative constraints for the generalized sidelobe canceller (GSC) that can be used to reduce sensitivity to steering error. These constraints are designed to flatten the spatial null of the GSC blocking matrix so that for a small steering error, the desired signal is still blocked and the GSC does not experience signal cancellation. With this approach, the steered response of the GSC can be forced to locally approximate any realizable fixed-weight beampattern. A related set of constraints can be used with the eigenvector constraint calibrated GSC to control the steered response for use in the presence of array errors.

We perform an analysis of constrained and unconstrained MUSIC demonstrating that (asymptotically) improved subspace estimates always result from the use of constraints, and (asymptotically) the variance of constrained MUSIC is less than that of unconstrained MUSIC under either high coherence, large numbers of sensors, or high SNR conditions. As part of this analysis, we study the effects of coherence on MUSIC and derive best/worst case coherences in terms of the variance of MUSIC. We also demonstrate that those conditions where the variance of MUSIC is predicted to be less than that of constrained MUSIC generally correspond to conditions where MUSIC is in breakdown (and constrained MUSIC is not). So, unconstrained MUSIC does not achieve its predicted advantage in those cases.

The column-unitary matrix whose range space is closest, in the Frobenius norm metric, to the range of a given matrix and whose columns belong to the null space of another specified matrix is obtained. An application of the matrix optimization result derived herein to constrained beamformer design in array signal processing is briefly described.>

Difference bases are discussed and their relevance to sensor arrays is described. Several new analytical difference base structures that result in near optimal low-redundancy sensor arrays are introduced. Algorithms are also presented for efficiently obtaining sparse sensor arrays and/or difference bases. New bounds, related to arrays that have both redundancies and holes in their coarray, are presented. Some extensions to the idea of difference bases that may yield useful results for sensor array design are discussed.>

A nonlinear autoregressive (AR) and AR moving average (ARMA) approximation theory is developed for an important class of nonlinear systems, namely, feedback linearizable systems with polynomial nonlinearities. The focus is on nonlinear AR models (NAR) because (1) the NAR parameters can be estimated via linear equations, (2) NAR models can be used to approximate almost any nonlinear system, and (3) compact difference equation and state space forms exist for this class of models.>

Numerically stable closed form expressions for the solution of the total least squares (TLS) problem with linear equality constraints (LCTLS) are derived. A constrained subspace linear predictive frequency estimation technique called LCTLS-linear predictive (LCTLS-LP) is proposed. The method is suited for scenarios where the presence of a known frequency component makes it difficult to resolve a closely spaced component.>

An approach whereby conflict-free access of any constant stride can be made by selecting a storage scheme for each vector based on the accessing patterns used with that vector is considered. By factoring the stride into two components, one a power of 2 and the other relatively prime to 2, a storage scheme that allows conflict-free access to the vector using the specified stride can be synthesized. All such schemes are based on a variation of the row rotation mechanism proposed by P. Budnik and D. Kuck (ibid., vol.C-20. no.12, pp.1566-9, Dec. 1971). Each storage scheme is based on two parameters, one describing the type of rotation to perform and the other describing the amount of memory to be rotated as a single block. The performance of the memory under access strides other than the stride used to specify the storage scheme is also considered. Modeling these other strides represents a vector being accessed with multiple strides as well as situations when the stride cannot be determined prior to initializing the vector. Simulation results show that if a single buffer is added to each memory port, then the average performance of the dynamic scheme surpasses that of the interleaved scheme for arbitrary stride accesses.>

Systems theory and some canonical representations are introduced for a class of nonlinear systems. Techniques are devised to identify, synthesize, and model such systems and their signals. Nonlinear systems theory is introduced at a fundamental level. To identify a system from input and output (or just output) data, parameters are estimated for a canonical state-space or difference-equation representation, depending on which representation is most convenient for further analysis.>

Given a Volterra series representation of an input-output system, the authors present a result concerning conditions under which there exists a feedback linearizable realization of the nonlinear system. In addition, they develop a method to compute the unknown parameters in a canonical-form realization from the known Volterra kernels. A finite number of parameters can be calculated from an associated finite number of the kernels.>

The ability to resolve two closely spaced sources is a desirable quality for any direction-finding method. However, as sources become more closely spaced, any direction-finding method will eventually be unable to determine that there are in fact two sources. The point at which this breakdown occurs will differ among different algorithms. The authors discuss how resolution of direction-finding methods should be measured for two closely spaced sources. Probability densities are estimated for the predicted signal directions using three different high-resolution direction-finding methods.>

Previous investigations into data storage schemes have focused on finding a storage scheme that permits conflict-free access for a set of frequently encountered access patterns. This paper considers an alternative approach. Rather than forcing a single storage scheme to be used for all access patterns, conflict-free accesses of any constant stride can be made by selecting a storage scheme for each vector based on the accessing patterns used with that vector.

Averaging of the spatial correlation function over the spatial variable-spatial averaging- is performed on the estimated spatial correlation matrix prior to beamforming calculations. Generally speaking, this preprocessing operation reduces the intersignal coherence in the spatial correlation matrix and the variance of the spatial correlation function estimate. Two types of spatial averaging are considered: subaperture averaging and redundancy averaging. Subaperture averaging is shown to significantly reduce coherence only when the source propagation directions are widely spaced while variance reduction is accompanied by a reduction in apparent aperture. In contrast, the reduction of coherence resulting from redundancy averaging is relatively insensitive to source bearing separations. While the aperture is not affected, redundancy averaging may induce bias in the estimated source bearings.>

The fundamental signal model for narrowband direction finding - the propagation of several sinusoidal planar wavefronts in a medium containing an array of sensors with additive Gaussian noise present - is assumed implicitly in most high resolution beamforming algorithms. The "natural" parameters for this problem - angles of arrival, signal strengths, inter-signal coherences, and noise strength - specify entirely the statistic used by many algorithms, the spatial correlation matrixR. Combining the relevant parameters for a given situation in a parameter vector p, an estimate of the true parameter vector can be obtained as the solution of an optimization problem:\min{\hat{p}}\max{\min}\parallel\hat{R} - R(\hat{p})\parallelwhere\hat{R}is an estimate ofR. The minimizing\hat{p}yields direct estimates of the relevant parameters rather than extracting them from an intermediate quantity such as a beampattern. This parametric method is an unbiased estimator which is capable of resolving closely spaced, completely coherent sources at low signal to noise ratios and low time-bandwidth product.

A signal-processing model for the generation of point processes is derived. In the context of this model, the interval between successive events is treated as a time series. The input-output relationship of a system generating inter-event intervals is shown to be derivable from the intensity of the desired point process. Both stationary and non-stationary processes can be generated with this technique. The input-output characteristic of this system is shown to be nonlinear for most interesting cases.