Capon Beamformer#

Implementation of the Capon beamformer, also referred to as Minimum Variance Distortionless Response (MVDR).

The Capon[1] beamformer estimates the power $$\hat{P}$$ received from a direction $$(\theta, \phi)$$, where $$\theta$$ is the zenith and $$\phi$$ is the azimuth angle of interest in spherical coordinates, respectively. Let $$\mathbf{X} \in \mathbb{C}^{N \times T}$$ be the the matrix of $$T$$ time-discrete samples acquired by an antenna arrary featuring $$N$$ antennas and

$\mathbf{R}^{-1} = \left( \mathbf{X}\mathbf{X}^{\mathsf{H}} + \lambda \mathbb{I} \right)^{-1}$

be the respective inverse sample correlation matrix loaded by a factor $$\lambda \in \mathbb{R}_{+}$$. The antenna array’s response towards a source within its far field emitting a signal of small relative bandwidth is $$\mathbf{a}(\theta, \phi) \in \mathbb{C}^{N}$$. Then, the Capon beamformer’s spatial power response is defined as

$\hat{P}_{\mathrm{Capon}}(\theta, \phi) = \frac{1}{\mathbf{a}^{\mathsf{H}}(\theta, \phi) \mathbf{R}^{-1} \mathbf{a}(\theta, \phi)}$

with

$\mathbf{w}(\theta, \phi) = \frac{\mathbf{R}^{-1} \mathbf{a}(\theta, \phi)}{\mathbf{a}^{\mathsf{H}}(\theta, \phi) \mathbf{R}^{-1} \mathbf{a}(\theta, \phi)} \in \mathbb{C}^{N}$

being the beamforming weights to steer the sensor array’s receive characteristics towards direction $$(\theta, \phi)$$, so that

$\mathcal{B}\lbrace \mathbf{X} \rbrace = \mathbf{w}^\mathsf{H}(\theta, \phi) \mathbf{X}$

is the implemented beamforming equation.

Parameters:

loading (float, optional) – Diagonal covariance loading coefficient $$\lambda$$. Defaults to zero.

_decode(samples, carrier_frequency, angles, array)[source]#

Decode signal streams for receive beamforming.

This method is called as a subroutine during receive() and probe().

Parameters:
• samples (np.ndarray) – Signal samples, first dimension being the number of signal streams $$N$$, second the number of samples $$T$$.

• carrier_frequency (float) – The assumed carrier central frequency of the samples $$f_\mathrm{c}$$.

• angles (numpy.ndarray) – Spherical coordinate system angles of arrival in radians. A three-dimensional numpy array with the first dimension representing the number of angles, the second dimension of magnitude number of focus points $$F$$, and the third dimension containing the azimuth and zenith angle in radians, respectively.

• array (AntennaArray) – The assumed antenna array.

Return type:

ndarray

Returns:

Stream samples of the focused signal towards all focus points. A three-dimensional numpy array with the first dimension representing the number of focus points, the second dimension the number of returned streams and the third dimension the amount of samples.

Required for robust matrix inversion in the case of rank-deficient sample covariances.

Returns:

Diagonal loading coefficient $$\lambda$$.

Raises:

Number of required receive focus points.

If this is $$1$$, the beamformer is considered to be a single focus point beamformer and receive_focus will return a single focus point. Otherwise, the beamformer is considered a multi focus point beamformer and receive_focus will return a Sequence of focus points.

Returns: Number of focus points.

Number of input streams required by this beamformer.

Dimension $$N$$ of the input sample matrix $$\mathbf{X} \in \mathbb{C}^{N \times T}$$.

Returns:

Number of input streams $$N$$.

Dimension $$M$$ of the output sample matrix $$\mathbf{Y} \in \mathbb{C}^{M \times T}$$.
Number of output streams $$M$$.