Sensor Array - Spectral Based Beamforming

Spectral Based Beamforming

The delay and sum beamforming is a time domain approach. It is simple to implement, but it gives poor DOA estimation if SNR is low: if the signal is contaminated by very strong noise, there may be practical difficulty in implementing the delay-and-sum beamforming algorithm. The solution is frequency domain approaches of beamforming. The Fourier transform is used to transform the time domain signal vector to frequency domain signal vector. After the Fourier transform, the time delays between adjacent sensors becomes phase shifts. Thus, the array output vector at any time t can be denoted as, where stands for the signal received by the first sensor. All frequency domain beamforming algorithms use the spatial covariance matrix which is represented by . This M by M matrix plays a key role in all frequency domain beamforming algorithms because it carries the spatial and spectral information of the signals impinging the sensor array. Assuming zero-mean Gaussian white noise, the frequency signal-noise snapshot model of the spatial covariance matrix is given by

where is the variance of the white noise, I is the identity matrix and V is the array manifold vector: . This model is of central use in frequency domain beamforming algorithms.

Spectral base beamforming approaches include conventional (Barlett) beamformer, capon Beamformer and MUSIC beamformer.

• Barlett Beamformer:

The Barlett beamformer is a natural extension of the conventional spectral analysis (spectrogram) to the sensor array. Its spectral power is represented by

The angle that maximize this power is an estimation of the angle of arrival.

• Capon Beamformer

The Capon beamforming algorithm is also known as MVDR (maximum variance distortionless response) approach. Its power is given by

Though achieving fairly better resolution than the Barlett approach, Capon algorithm is computationally intensive due to full rank matrix inversion.

• MUSIC Beamformer

MUSIC (MUltiple SIgnal Classification) beamforming algorithm is derived from the Capon algorithm by decomposing the covariance matrix as given by Eq. (4) for both the signal part and the noise part. The eigen-decomposition of is represented by

The MUSIC uses the noise sub-space of the spatial covariance matrix in the denominator of the Capon algorithm:

Therefore MUSIC beamformer is also known as subspace beamformer. Compared Capon beamformer, it gives much better DOA estimation while avoiding matrix inversion. Computational intensity is reduced significantly if the number of sensors (M) is large.