Signal Eigenvalues Research Articles

Adaptive rank estimation for spherical subspace trackers

We develop a rank tracking method for spherical subspace trackers. The method adaptively sets a threshold based on the averaged noise eigenvalue. The signal eigenvalues are then compared with the threshold to reach a decision on the subspace rank. The threshold itself is chosen to balance the error probabilities due to rank underfitting and overfitting. Simulation results show that our method performs as well as (and, for very low SNRs, even better than) information theoretic criteria.

Signal detection via spectral theory of large dimensional random matrices

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. >

Characterizing attractors using local intrinsic dimensions calculated by singular-value decomposition and information-theoretic criteria.

An algorithm to estimate the average local intrinsic dimension (〈${\mathit{d}}_{\mathrm{LID}}$〉) of an attractor using signal versus noise separation methods based on information-theoretic criteria is explored in this work. Using noisy sample data the 〈${\mathit{d}}_{\mathrm{LID}}$〉 is computed from an eigenanalysis of local attractor regions, indicating the local orthogonal directions along which the data are clustered. The 〈${\mathit{d}}_{\mathrm{LID}}$〉 algorithm requires the separation of signal eigenvalues, i.e., the dominant eigenvalues, from the noise eigenvalues for which thresholding mechanisms based on two information-theoretic criteria are used. Singular-value decomposition is used to calculate the eigenvalues as well as to determine the rank of the local-phase-space data matrix. The two specific information-theoretic criteria which we consider to separate signal and noise are the Akaike information criterion (AIC) [H. Akaike, IEEE Trans. Auto. Control 19, 716 (1974)] and the minimum description length (MDL) of Rissanen [Automatica 14, 465 (1978)] and Schwarz [Ann. Stat. 6, 461 (1978)]. Several test cases are presented demonstrating the use of the AIC and MDL to calculate 〈${\mathit{d}}_{\mathrm{LID}}$〉. Results are then compared to the correlation dimension as computed by the Grassberger-Procaccia method [Physica D9, 189 (1983)]. The AIC and MDL separation techniques are found to be inappropriate at higher signal-to-noise ratios (SNR), above \ensuremath{\sim}12 dB, and the MDL produces results in good overall agreement with the accepted correlation dimension in the range of SNR between 5 and 12 dB. Two of the primary benefits of the 〈${\mathit{d}}_{\mathrm{LID}}$〉 technique are that it will yield reasonable and fairly accurate estimates of the dimension of attractors and that it can probe small length scales on the attractor even for moderate SNR. Other thresholding schemes are being pursued for SNR's above 12 dB.

The λ²-LMS algorithm for separating sinusoids using the intermediate-converged adaptive filter solution

A new procedure is proposed for automatically separating sinusoids of different, but unknown, frequencies, based only on the fact that they differ in amplitude. The procedure involves taking the intermediate solution from the LMS adaptive solution rather than the converged steady-state solution. In addition, a new algorithm which converges at a rate proportional to the squares of the signal eigenvalues is described and gives superior performance when taking the intermediate adaptive solution.