0.1 Beamforming

Beamforming

The conventional frequency domain beamforming algorithm involves three components: a steering vector e , a weighting matrix W , and a vector Y which contains the frequency information about the signal received at each sensor. The steering vector is given its own section, and I will go into brief detail about the other two here.

Sensor data

The column vector Y is actually a collection of short-time Fourier Transforms. The signal received at each sensor at a given time t is multiplied by a window of duration D and then the Fourier Transform of that product is taken. Mathematically, this can be expressed as

This transform effectively represents the frequency components of the signal during various blocks of time, and is useful if the signal itself is finite and/or not uniform over all time. When performing a short-time Fourier Transform, it is best to pick a window that has a Fourier Transform which looks like a single narrow spike so that its width is narrow compared to the signal’s spectral variations – this allows the short-time Fourier Transform of a signal at one instant in time to best approximate the signal’s spectrum. The vector Y should actually be written as Y ( t, ω ) because it is dependent both on time and the frequency content of the signal, but often that notation is omitted for neatness. Each element Y­ n­ ­is the collection of short-time Fourier Transforms at sensor n , thus giving Y the form

The weighting matrix

The matrix W is diagonal; the only elements which are not zero are the sensor weights along its main diagonal. The actual content of those weights can vary by application. In a simple linear array like the one used in our particular example, all of the weights are simply one. However, in more interesting array configurations such as an array with sensors that were placed randomly or real-world sensors that have inherent noise components, the weighting matrix can become much more important.

Beamformer output

The Fourier Transform of the beamformer output can be written in the following way:

This is the inner product of e’ and WY , which is nothing more than a column vector of weighted short-time Fourier transforms. In order to get the beamformer output in the time domain, all that is necessary is taking an inverse short-time Fourier Transform of e’WY ( t, ω ).

Here is a visual representation of the entire beamforming process: