Eric M. Dowling

Efficient direction-finding methods employing forward/backward averaging

In this paper, we develop a general approach for reducing the computational complexity of any direction finding method implemented with forward/backward (FB) averaging. We develop simplified FB data matrices in a manner paralleling previous work related to centrohermitian (correlation) matrices. Based on these simplified data matrices, efficient construction and updating of the FB correlation matrix is developed. In addition, efficient FB FFT, FB beamspace, FB EVD updating, FB SVD, and FB SVD updating methods are derived. In most cases, FB-based direction-finding methods can be simplified so that the computational complexity is reduced below that of an analogous forward only implementation. Thus, effectively twice the amount of data is processed with less total computation. >

Lightwave lattice filters for optically multiplexed communication systems

It has proven desirable to use multistage etalons and resonators in lightwave communication systems. The design of these linear structures, however, is made difficult by the manner in which their transfer functions are nonlinear with respect to their composite reflection coefficients. If we interpret the etalons as discrete-time lattice filters, then z-transform techniques may be used to recursively synthesize filters with desirable properties. An algorithm is developed which can be used to design the arbitrary all-pole transfer functions in transmission, and the restricted class of pole-zero transfer functions in reflection, which are possible to implement with this architecture. We present some design examples such as notch, or channel-blocking, filters and flat-top bandpass, or channel-passing, filters which are appropriate for frequency-division multiple access and wavelength-division multiplexed communications systems. The theory predicts, and we show experimentally, how these structures may be used to discriminate, or route, signals based on their modulated or coded characteristics. >

Conjugate gradient eigenstructure tracking for adaptive spectral estimation

A conjugate gradient iteration is derived that converges to the set of r dominant/subdominant eigenpairs. This iteration is used to construct two eigenstructure tracking algorithms that track the r-dimensional dominant or subdominant subspaces of time-varying data or data-covariance matrices. The two eigenstructure tracking algorithms have update complexities O(m/sup 2/r) and the other O(mr/sup 2/), where m is the data dimension. The algorithms are customized to solve high resolution temporal and spatial frequency tracking problems. They are compared with existing techniques by tying into published simulation based performance tests. The algorithms demonstrate rapid convergence and tracking characteristics at a competitive cost. >

The Data Least Squares Problem and Channel Equalization

Using the constrained TLS method of Abatzoglou and Mendel, we develop a simple closed-form solution to the data least squares (DLS) problem, in which the error is assumed to lie in the data matrix only. Simulations demonstrate that DLS outperforms TLS and ordinary LS for certain types of deconvolution problems.

The constrained MUSIC problem

The MUSIC-based direction-of-arrival (DOA) method is generalized to include constraints involving known signal information. Projection operators are used to constrain the noise subspace to be orthogonal to a set of prespecified direction vectors. By incorporating known source directions, the estimation of unknown source directions can be significantly improved. Simulations are performed over a wide range of scenarios to demonstrate the usefulness of the approach. >

A TQR-iteration based adaptive SVD for real time angle and frequency tracking

The transposed VR (TQR) iteration is a square root version of the symmetric QR iteration. The TQR algorithm converges directly to the singular value decomposition (SVD) of a matrix and was originally derived to provide a means to identify and reduce the effects of outliers for robust SVD computation. The paper extends the TQR algorithm to incorporate complex data and weighted norms, formulates a TQR-iteration based adaptive SVD algorithm, develops a real time systolic architecture, and analyzes performance. The applications of high resolution angle and frequency tracking are developed and the updating scheme is so tailored. A deflation mechanism reduces both the computational complexity of the algorithm and the hardware complexity of the systolic architecture, making the method ideal for real time applications. Simulation results demonstrate the performance of the method and compare it to existing SVD tracking schemes. The results show that the method is exceptional in terms of performance to cost ratio and systolic implementation. >

Incorporating a priori information into MUSIC-algorithms and analysis

Abstract Constrained MUSIC and beamspace MUSIC are similar algorithms in that they both require a priori information about signal directions and they both involve linear transformations on the data. Constrained MUSIC uses precise information regarding the directions of a subset of the signal directions to improve the direction estimates for the remaining signals. Beamspace MUSIC uses approximate knowledge regarding all the signal directions to reduce computational complexity and improve breakdown properties. These two methods can be combined, resulting in constrained beamspace MUSIC. We also perform asymptotic 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 a 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 actually does not achieve its theoretically predicted advantage in those cases. While constrained MUSIC requires precise information about the known signal to improve performance when the unknown signal is very near, it can also offer performance advantages with only approximate knowledge if the unknown and known signals are not too close to each other.

Z-domain techniques in the analysis of Fabry–Perot étalons and multilayer structures

Z-domain and digital filter techniques are used to derive and interpret the transfer functions—transmission and reflection dependencies—for several etalonlike structures. Two-, three-, and four-mirror Fabry–Perot etalons and a seven-layer interference filter are analyzed. Although most of the examples have equally spaced mirrors, an example is also presented that illustrates the extension of the technique to structures with dissimilar lengths. Concepts such as group delay and state-variable descriptions may now be naturally applied to the analysis of these structures. The design of these structures may now benefit from algorithms developed for digital signal processing.

The equivalence of the total least squares and minimum norm methods (signal processing)

It is shown that the minimum norm solution is equivalent to the total least squares solution. It is noted that two versions of the total least squares solution exist, one based on the signal subspace and another based on the noise subspace. >

Adaptation dynamics of the spherical subspace tracker

L. Ljungs (1977) method for analyzing recursive stochastic algorithms is used to formulate a projection operator ordinary differential equation (ODE). The ODE describes the expected convergence dynamics of a noniterative spherical subspace tracker. The subspace ODE is a Riccati equation defined over the manifold of rank r projection matrices in C/sup nxn/. A Lyapunov function is defined that is shown to have global maximum and minimum at the signal and noise subspaces, respectively. By taking a derivative of the Lyapunov function along any trajectory, it is shown that the dynamics force all trajectories to converge to the signal subspace. If the sign of the derivative is changed, all trajectories will converge to the noise subspace. >