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Simple, single-channel noise attenuation and dereverberation is often not sufficient for acceptable quality, especially in very noisy environments, such as in a car or on a busy sidewalk. To reach better quality, we can then add more microphones. The benefit of added microphones includes at least:

• Time-of-arrival differences between sources enables the use of beamforming filters, which use such time (phase) differences to separate between sources.
• Intensity level differences between sources; In for example a mobile phone, we can have forward and backward facing microphones such that the backward facing microphone is used primarily for estimating background noise, while the forward facing microphone records the desired speech signal. By using the background-noise estimate from the backward facing microphone, we can then use noise attenuation on the forward facing microphone to gain better quality.
• Each microphone will feature some level of sensor-noise, that is, the hardware itself causes some inaccuracies to the signal. Sensor-noises are for most parts independent across microphones such that with each additional microphone we can better separate desired sources from noise.

The most-frequently discussed approach is to use microphone arrays, typically in either a linear configuration, where microphones are equi-spaced on a straight line, or in a circular configuration, where microphones are equi-spaced on a circle. The benefits include that a linear configuration makes analytical analysis easier, whereas a circular array can have an almost uniform response in all directions.

## Delay-and-sum beamforming

As an introduction to beamforming consider a linear array of K microphones with input signals xk(t), where k and t are the microphone and time indices. We assume that the desired source is sufficiently far away that we can approximate it with a plane wave. Then the signal will arrive at the microphones at different times

Mathinline
\Delta t_{x_k}

and we can calculate time time difference of arrival (TDOA) of each microphone

Mathinline
t_{x_k} = \Delta t_{x_k}-\Delta t_{x_1}

where we used microphone k=1 as a reference point. The delayed signals thus have

Mathinline
x_k(t) = x(t-\Delta t_{x_k}) = x\left(t-\Delta t_{x_k}+\Delta t_{x_1} - \Delta t_{x_1}\right) = x_1(t-t_{x_k}).

Similarly, for the noise sources we have

Mathinline
y_k(t) = t_1(t-t_{y_k}).

If the desired and noise sources appear at different angles, then their corresponding delays will be different. Moreover, if we add a signal z(t) with itself at a random offset δ, then the summation is destructive, that is, smaller than the original

Mathinline
\frac12\left|z(t)+z(t+\delta)\right| \leq \left|z(t)\right|

. Addition without an offset is obviously constructive, such that we can form the delay and sum estimate as

Mathdisplay
\hat x(t) = \frac 1K \sum_{k=1}^K x_k(t-t_{x_k}).

In this summation, all signals approaching from the same direction as the desired source will be additive (constructive) and other directions will be (more or less) destructive.

Trivial as it is, the delay-and-sum should however be treated as a pedagogical example only. It does not ideally amplify the desired source nor attenuate the noise source, and it is sensitive to errors in the TDOAs.