What's the point of differential beamforming?

Question

I recently got into differential beamforming, in particular into the work of Benesty and Cohen. It seems to me that differential beamforming is something that just "takes place" when microphones are close enough to each other, and therefore some mathematical simplifications can be done in order to express spatial patterns and beamforming weights. Also, spatial patterns will be more frequency independent (again due to the microphones' closeness). My question is: what is the "novelty" with respect to classic beamforming? I don't see new optimization problems, new methods to solve them, I just see another point of view for a beamforming particular case, where microphones are so close to each other that formulas can be simplified. Can someone tell me if this is the case or if I am missing something?

Answer 1

Introduction

To my (rather limited) knowledge, the main advantage that made Differential Microphone Arrays (DMA) popular is their ability to have a frequency invariant beampattern (up to some certain frequency of course). This is a very appealing feature, especially for fields like speech processing and hi-fidelity recordings.

A wee bit more info

As you may already know DMAs are mostly (if not exclusively) end-fire arrays and some assumptions are in place the main of which are:

1. The array inter-element distance is short compared to the wavelength.
2. The source is situated along the main axis of the array (which is considered to be the $\theta = 0^{o}$ - although as described below beam patterns can also be steered with differential beamformers).
3. In most formulations (if not all) the impinging waves are considered to be planar.

Please note that if assumption 1 does not hold the beampattern is not frequency invariant, but similar problem arise even for broadside beamformers (aliasing).

Similarly, assumption 3 is used extensively for broadside beamforming (although spherical wave approaches have been used and are very similar to the plane wave approximation).

Now, assumption 2 is most probably what allows for the use and modelling of DMAs in the way they are presented. In contrast to the broadside beamformers, having a non-moving (relative to the microphone arrangement) is something that allows to focus on using the configuration in the way it is used. Usually, beamformers, in their “traditional form” are used to steer the main beam, which generically means that the source is not going to be stationary (compared to the array). The implications of beam-steering are broadening of the beam when approaching the end-fire direction, grating lobes and some more, as you may already know. But alas, those are similar problems that arise with DMAs if you try to steer the beam.

There are indeed DMAs that can be steered. More information can be found in *"Steering Study of Linear Differential Microphone Arrays" by Jin et al. but please not all DMAs are steerable (for example 1st order DMAs are not based on this paper).

Differential beamformers

As you may already know, based on the work of mostly Benesty, Chen and Pan (some indicative articles are "On the Design of Target Beampatterns for Differential Microphone Arrays” by Pan et al., "Theoretical Analysis of Differential Microphone Array Beamforming and an Improved Solution” by Pan, Chen and Benesty and the textbook “Fundamentals of Differential Beamforming” by Benesty, Chen and Pan), probably the easiest way to design a Differential beamformer is by solving a linear system of $N + 1$ contraint equations, where $N$ is the number of array elements. This is of the form (taken from "Theoretical Analysis of Differential Microphone Array Beamforming and an Improved Solution” by Pan, Chen and Benesty)

$$ \mathbf{D} \left( \omega, \theta \right) \mathbf{h} \left( \omega \right) = \mathbf{i} \tag{1} \label{1} $$

where $\mathbf{D} \left( \omega, \theta \right)$ are the constraint equations in the array manifold $\mathbf{d}$ and is of the form

$$ \mathbf{D} \left( \omega, 0 \right) = \begin{bmatrix} \mathbf{d}^{H} \left( \omega, \theta_{1} \right) \\ \mathbf{d}^{H} \left( \omega, \theta_{2} \right) \\ \vdots \\ \mathbf{d}^{H} \left( \omega, \theta_{N} \right) \end{bmatrix} \tag{2} \label{2} $$

where $\left[ ~ \cdot \right]^{H}$ denotes Hermitian transposition. The vector $\mathbf{h}$ is the weight vector for the array (the weights of the element outputs) and $\mathbf{i}$ is usually of the form

$$ \mathbf{i} = \left[ 1, 0, 0, \ldots, 0 \right]^{T} \tag{3}, \label{3} $$

Then, equation \eqref{1} denotes that the beampattern at $\theta = 0^{o}$ must be unity (to ensure distortionless response) and the rest of the equations denote the nulls of the beam pattern. Solving equation \eqref{1} for $\mathbf{h}$ provides the weights for the needed beam pattern. In this formulation, there is no need to perform any simplifications in order to calculate those weights.

Comparison with broadside beamforming

The main “conceptual” difference between end-fire and broadside beamformers is that the former subtracts the output of its elements while the latter sum them. Of course, there’s more to that “than meets the eye”.

As already mentioned, one of the main attractions of the DMAs is their frequency-invariant beam pattern. It is important to note that the broadside beamformers cannot achieve that, no matter what the inter-element distance is. Below you can see the beam pattern of a first-order DMA with a "figure-of-eight" polar response for four frequencies (monochromatic calculations, not band-limited noise) on a wide range ($\sim 140 ~ Hz$ to $\sim 1 ~ kHz$). The array consists of two pressure sensors (omnidirectional sensors) and the inter-element distance is $0.05 ~ m$.

You can see that there are minimal differences in the polar response. You cannot achieve such a thing without using special techniques for broadside beamforming.

The trick here is that in the previous plot, four differential beamformers were designed, one for each frequency. If the beamformers were designed for a single frequency (only one $\mathbf{h}$ for all frequencies) a somewhat larger difference would be visible, but still very smaller than what you would achieve with a conventional Delay-and-Sum (DaS) beamformer.

Nevertheless, there are some suggestions that actually do for DaS beamformers something similar to what I presented for the above DMA. The most usual approach is to design a differential beamformer for each frequency range (even for a single FFT bin), getting as many weight vectors $\mathbf{h}$ as the frequency bands of interest and then combining them. You can find a similar approach to broadside/DaS beamformers in ”Beamforming for Broadband Constant Beamwidth Through FIR Filter and DSP Implementation” by Ma, Zhang and Vray.

Apart from the frequency invariant beam pattern, which as we can see can be “matched” with some extra effort for the DaS beamformers, another important feature of differential beamformers is that they achieve the highest Directivity Factor of all available beamformers. This is a highly sought feature when one seeks to minimise the effect of the sound environment on the signal (in essence this means to reduce the background noise as much as possible). This is why most super-directive beamformers are designed via a differential approach.

Conclusions

Differential beamformers do have their use and more often than not, their use cases are different to those of DaS/broadside beamformers. A good example of DMAs is RF antennas used for audio applications (such as antennas for wireless in-ears and microphones) where you really want to achieve a good reduction of RF interference over a (rather) large frequency band (otherwise you’d need to change the antenna every time you would tune the equipment at a different frequency).

As far as I am aware DMAs are also used in the hearing-aid industry due to some of the features mentioned above. Reducing the sensitivity towards the hearing aid's speaker is very important if you want to improve robustness against feedback (which is a huge problem in hearing aids).