Parametric Beamformers
One of the major pros of the spectral based beamformers is their relatively lighter computational complexity, but they may not give accurate DOA estimation if the signals are correlated or coherent. An alternative approach is the parametric beamformers, also known as Maximum Likelihood (ML) beamformers. One example of maximum likelihood method commonly used in engineering is least squares method. In the least square approach, a quadratic penalty function is used. To get the minimum value (or least square) of the quadratic penalty function (or objective function), take its derivative (which is linear) and let it equal zero, and solve linear equations.
In ML Beamformers, quadratic penalty functions are used to the spatial covariance matrix and the signal/noise model. One example of ML beamformer penalty function is:
where N is the number of snapshots and is the Euclidean norm. It can be seen from Eq. (4) that minimizing the penalty function of Eq. (9) is to make the noise term as much as possible, or the signal model as close to the sample covariance matrix as much as possible. In oether words, the Maximum Likelihood beamformer is to find the DOA, the independent variable of vector V, so that the penalty function such as Eq. (9) is minimized. In practice, the penalty functions used look different, depending on the signal/noise model employed, but they are the same in essence. For this reason, there are two major categories of maximum likelihood beamformers: Deterministic ML beamformers and Stochastic ML beamformers, corresponding to the deterministic noise model and the stochastic noise model, respectively.
Another factor to change the former of the penalty equation is the consideration of simplifying the minimization of the penalty function. Although the derivative of a quadratic function is linear, matrix operations and trigonometric operations make it highly non-linear. In order to simplify the optimization algorithm, logrithmic operation and probability density function (PDF) of the observations may be used in some ML beamformers.
In ML beamformer, the optimizing problem becomes finding the roots of the equation of zeroing the derivative of the penalty function. Because the equation is highly non-linear, usually a numerical searching approach, such as Newton-Raphson method, is used. The Newton-Raphson method is an ierative root search method as given by
The search starts from an initial guess . If the Newton-Raphson search method is employed in minimizing the penalty function in beamforming, the resulting beamformer is called Newton ML beamformer. Several best-known ML beamformers are described below without giving their formulas due to the complexity of expression.
- Deterministic Maximum Likelihood Beamformer
In Deterministic Maximum Likelihood Beamformer (DML), the noise is modeled as a stationary Gaussian white random processes while the signal waveform as deterministic (but arbitrary) and unknown.
- Stochastic Maximum Likelihood Beamformer
In Stochastic Maximum Likelihood Beamformer (SML), the noise are modeled as a stationary Gaussian white random processes (the same as in DML) whereas the signal waveform as Gaussian random processes.
- Method of Direction Estimation
Method of Direction Estimation (MODE) is subspace maximum likelihood beamformer, just as MUSIC is the subspace spectral based beamformer. Subspace ML beamforming is obtained by eigen-decomposition of the sample covariance matrix.
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