Petter Wirfält

On Kronecker and Linearly Structured Covariance Matrix Estimation

The estimation of covariance matrices is an integral part of numerous signal processing applications. In many scenarios, there exists prior knowledge on the structure of the true covariance matrix; e.g., it might be known that the matrix is Toeplitz in addition to Hermitian. Given the available data and such prior structural knowledge, estimates using the known structure can be expected to be more accurate since more data per unknown parameter is available. In this work, we study the case when a covariance matrix is known to be the Kronecker product of two factor matrices, and in addition the factor matrices are Toeplitz. We devise a two-step estimator to accurately solve this problem: the first step is a maximum likelihood (ML) based closed form estimator, which has previously been shown to give asymptotically (in the number of samples) efficient estimates when the relevant factor matrices are Hermitian or persymmetric. The second step is a re-weighting of the estimates found in the first steps, such that the final estimate satisfies the desired Toeplitz structure. We derive the asymptotic distribution of the proposed two-step estimator and conclude that the estimator is asymptotically statistically efficient, and hence asymptotically ML. Through Monte Carlo simulations, we further show that the estimator converges to the relevant Cramér-Rao lower bound for fewer samples than existing methods.

Subspace-based frequency estimation utilizing prior information

In certain frequency estimation applications one or more of the underlying frequencies are known. For example, in rotary machines the known frequency may be a strong network frequency masking important closely spaced frequencies. Being able to include this information in the design of the estimator can be expected to improve the performance when estimating such closely spaced frequencies. We present a framework to include such prior information in a class of subspace-based estimators. Through Monte Carlo simulations and real-data applications we show the usefulness of our approach.

On Toeplitz and Kronecker structured covariance matrix estimation

A number of signal processing applications require the estimation of covariance matrices. Sometimes, the particular scenario or system imparts a certain theoretical structure on the matrices that are to be estimated. Using this knowledge allows the design of algorithms exploiting such structure, resulting in more robust and accurate estimators, especially for small samples. We study a scenario with a measured covariance matrix known to be the Kronecker product of two other, possibly structured, covariance matrices that are to be estimated. Examples of scenarios in which such a problem occurs are MIMO-communications and EEG measurements. When the matrices that are to be estimated are Toeplitz structured, we show our algorithms to be able to achieve the Cramér-Rao Lower Bound already at very small sample sizes.

Fast communication: Line spectrum estimation with probabilistic priors

For line spectrum estimation, we derive the maximum a posteriori probability estimator where prior knowledge of frequencies is modeled probabilistically. Since the spectrum is periodic, an appropriate distribution is the circular von Mises distribution that can parameterize the entire range of prior certainty of the frequencies. An efficient alternating projections method is used to solve the resulting optimization problem. The estimator is evaluated numerically and compared with other estimators and the Cramer-Rao bound.

ML Estimation of Covariance Matrices with Kronecker and Persymmetric Structure

Estimation of covariance matrices is often an integral part in many signal processing algorithms. In some applications, the covariance matrices can be assumed to have certain structure. Imposing this structure in the estimation typically leads to improved accuracy and robustness (e.g., to small sample effects). In MIMO communications or in signal modelling of EEG data the full covariance matrix can sometimes be modelled as the Kronecker product of two smaller covariance matrices. These smaller matrices may also be structured, e.g., being Toeplitz or at least persymmetric. In this paper we discuss a recently proposed closed form maximum likelihood (ML) based method for the estimation of the Kronecker factor matrices. We also extend the previously presented method to be able to impose the persymmetric constraint into the estimator. Numerical examples show that the mean square errors of the new estimator attains the Cramér-Rao bound even for very small sample sizes.

Optimal prior knowledge-based direction of arrival estimation

In certain applications involving direction of arrival (DOA) estimation the operator may have a-priori information on some of the DOAs. This information could refer to a target known to be present ...

Prior knowledge-based direction of arrival estimation

In a number of direction of arrival (DOA) estimation applications there exists prior knowledge about the sources whose bearings are to be determined. We study the case when this prior information concerns some of the source positions and their correlation state, which is a relevant case in, for example, RADAR scenarios where stationary objects exists in the regions of interest. Traditional DOA methods are not designed to exploit such information, and thus cannot obtain the highest theoretical accuracy. We present a method that can utilize in an asymptotically efficient manner both knowledge on some source positions and that the source signals are uncorrelated.

An esprit-based parameter estimator for spectroscopic data

The pulse spin-locking sequence is a common excitation sequence for magnetic resonance and nuclear quadrupole resonance signals, with the resulting measurement data being well modeled as a train of exponentially damped sinusoidals. In this paper, we derive an ESPRIT-based estimator for such signals, together with the corresponding Cramér-Rao lower bound. The proposed estimator is computationally efficient and only requires prior knowledge of the number of spectral lines, which is in general available in the considered applications. Numerical simulations indicate that the proposed method is close to statistically efficient, and that it offers an attractive approach for initialization of existing statistically efficient gradient or search based techniques.

Robust prior-based Direction of Arrival estimation

In certain Direction of Arrival (DOA) scenarios some of the sources are approximately known a priori. It is then desirable to be able to exploit this prior knowledge when estimating the DOAs of the unknown sources. In this paper we modify an estimator utilizing exact angular prior knowledge of some sources such that the estimator is able to exploit prior knowledge with some uncertainty. We derive the corresponding Cramér-Rao lower bound and present numerical results showing that the estimator can benefit from prior information, even when it is inaccurate.

Prior-exploiting direction-of-arrival algorithm for partially uncorrelated source signals

In certain direction-of-arrival (DOA) estimation scenarios some of the source directions are known to the operator even before measurements are acquired. It is then undesirable to use regular DOA-algorithms which waste data-samples estimating the known directions. Additionally, in some applications it is known that the signals emanating from the known directions are uncorrelated with those coming from the unknown directions. In this article we present a novel algorithm which exploits the combination of such prior knowledge in a manner more efficient (in terms of accuracy) than any algorithm known to the authors. Through numerical Monte-Carlo simulations we show the estimator to attain the theoretical accuracy bound for significantly lower signal-to-noise ratios than current state-of-the-art methods. Additionally we show the proposed algorithm to treat the stricter problem of entirely uncorrelated emitters better than current state of the art.