seasonder_computePowerMatrix: Compute Power Matrix

In SeaSondeR: Radial Metrics from SeaSonde HF-Radar Data

Compute Power Matrix

Description

This function calculates the power matrix based on the provided steering vector, eigenvalues, and eigenvectors. The computation differs depending on the number of columns in the steering vector matrix.

Usage

seasonder_computePowerMatrix(eig, a)

Arguments

eig	A list containing the eigenvalues and eigenvectors of a covariance matrix. The list should include: values: A numeric vector of eigenvalues. vectors: A matrix where each column is an eigenvector.
a	A complex matrix representing the steering vector(s). Each column corresponds to a direction of arrival.

Details

The function computes the power matrix using the following steps:

• If a has two columns:

  1. Select the first two eigenvalues and their corresponding eigenvectors.

  2. Compute the matrix G = a^* \cdot \text{eigVector}, where a^* is the conjugate transpose of a.

  3. Calculate the inverse of G and its conjugate transpose.

  4. Compute the power matrix P = G_{\text{inv}}^* \cdot \text{diag(eigValues)} \cdot G_{\text{inv}}.

• If a has one column:

  1. Select the first eigenvalue and its corresponding eigenvector.

  2. Follow similar steps as above with single-column operations.

If a has no columns, the function returns NULL.

Value

A complex matrix representing the power matrix, calculated based on the provided eigenvalues, eigenvectors, and steering vectors. If the number of columns in a is zero, the function returns NULL.

Mathematical Formula

For a steering vector matrix a, eigenvectors \text{eigVector}, and eigenvalues \text{eigValues}, the power matrix is calculated as:

P = G_{\text{inv}}^* \cdot \text{diag(eigValues)} \cdot G_{\text{inv}}

where: G = a^* \cdot \text{eigVector} and G_{\text{inv}} is the inverse of G.

References

• Paolo, T. de, Cook, T., & Terrill, E. (2007). Properties of HF RADAR Compact Antenna Arrays and Their Effect on the MUSIC Algorithm. OCEANS 2007, 1–10. doi:10.1109/oceans.2007.4449265.

SeaSondeR documentation built on June 8, 2025, 10:50 a.m.