Uniform circular array directional noise SNR improvements with delay and sum beamformer. The use of a circular array topology for beamforming is on the rise mainly because it afford a $2\pi$ look direction for the array. Conventional derivations for the signal to noise ratio improvements using delay and sum beamformers indicate that you get $3dB$ gain for every doubling of the number of microphones being deployed. This however only holds for uncorrelated noise. In certain situations, a null is desired in one direction whilst simultaneously, a beam is desired in another direction. We derive the expected SNR gains for correlated noise on UCA microphones.Consider a far field source impinging N UCA microphones as shown in Figure 1: Figure 1: N UCA microphones

Suppose the signal at each microphone $i \in \{1, \cdots, N\}$ is given as $x_i(w) = s(w) e^{\left(-jw \frac{d}{c} \sin{\left((i-1)\psi - \theta\right)} \right)} + v(w) e^{\left(-jw \frac{d}{c} \sin{\left((i-1)\psi - \beta\right)} \right)}$

where $s(w)$ is the desired speech signal, $\theta$ is the direction of arrival (DOA) of the speech signal with respect to the normal to the axis joining all the microphones, $v_i(w)$ is the correlated noise such that $\mathbb{E}[v(w) v^*(w)] = \sigma_v^2$ and $\mathbb{E}[ s(w) e^{\left(-jw \frac{d}{c} \sin{\left((i-1)\psi - \theta\right)} \right)} v^*(w)] = 0, \forall i \in \{1, \cdots, N\}$. Further, $\beta$ is the direction of arrival (DOA) of the directional noise with respect to the normal to the axis joining all the microphones and
The input SNR per frequency bin $w$, denoted $iSNR(w)$ is given as: $iSNR = \frac{\mathbb{E}\left[|s(w)|^2 \right]}{\mathbb{E}\left[\left |v(w)\right|^2 \right]} =\frac{|s(w)|^2}{\sigma_v^2}$

where $\mathbb{E}[.]$ is the expectation operator.
After the delay and sum beamformer, the output becomes: $x(w) = s(w) + v(w) \frac{1}{N} \sum\limits_{n =0}^{N-1} e^{\left(2jw \frac{d}{c} \cos{\left(n\psi - \frac{1}{2}(\theta+\beta)\right)} \sin{(\frac{\beta - \theta}{2})} \right)}$

Without loss of generality, suppose $\beta = \theta + \pi$, then $x(w) = s(w) + v(w) \frac{1}{N} \sum\limits_{n =0}^{N-1} e^{\left(2jw \frac{d}{c} \sin{\left(n\psi - \theta\right)} \right)}$

The output SNR per frequency bin $w$, denoted $oSNR(w)$ is given as: $oSNR = \frac{|s(w)|^2}{\left | v(w) \frac{1}{N} \sum\limits_{n =0}^{N-1} e^{\left(2jw \frac{d}{c} \sin{\left(n\psi - \theta\right)} \right)} \right|^2}$

But $\frac{\sigma_v^2}{N} + \frac{\sigma_v^2}{N^2} \sum\limits_{n =0}^{N-1} \sum\limits_{m \neq n}^{N}e^{\left(2jw \frac{d}{c} (\cos{\left((\frac{n+m}{2})\psi - \theta \right)}) \sin{(\frac{n-m}{2}\psi)}\right)}$
This leads to an oSNR of: $oSNR = N \frac{|s(w)|^2 }{\sigma_v^2} \frac{1}{1 + \frac{1}{N} \sum\limits_{n =0}^{N-1} \sum\limits_{m \neq n}^{N}e^{\left(2jw \frac{d}{c} (\cos{\left((\frac{n+m}{2})\psi - \theta \right)}) \sin{(\frac{n-m}{2}\psi)}\right)}}$

The SNR improvement, SNRI then becomes $SNRI = \frac{oSNR}{iSNR} = \frac{N}{1 + \frac{1}{N} \sum\limits_{n =0}^{N-1} \sum\limits_{m \neq n}^{N}e^{\left(2jw \frac{d}{c} (\cos{\left((\frac{n+m}{2})\psi - \theta \right)}) \sin{(\frac{n-m}{2}\psi)}\right)}} \le N$

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