ULA Uncorrelated Noise SNR Improvements

Conventional derivations for the signal to noise ratio improvements using delay and sum beamformer is that you get $3dB$ gain for every doubling of the number of microphones being deployed. This holds $iff$ the noise is not directional or in other words uncorrelated. We derive the expected SNR gains for uncorrelated noise on ULA microphones.
Consider a far field source impinging N ULA microphones as shown in Figure 1:

Figure 1: N ULA microphones

Suppose the signal at each microphone $i \in \{1, \cdots, N\}$ is given as

$x_i(w) = s(w) e^{\left(-jw \frac{(i-1) d}{c} \sin{\theta} \right)} + v_i(w)$

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 uncorrelated noise such that

$\mathbb{E}[v_i(w) v_j^*(w)] = 0, i \neq j, \{i,j\} \in \{1, \cdots, N\}$

and

$\mathbb{E}[s(w) e^{\left(-jw \frac{(i-1) d}{c} \sin{\theta} \right)} v_j^*(w)] = 0, \forall {i,j} \in \{1, \cdots, N\}$ .

The input SNR per frequency bin $w$ , denoted $iSNR(w)$ is given as

$iSNR = \frac{\mathbb{E}\left[|s(w)|^2 \right]}{\mathbb{E}\left[\left |v_1(w)\right|^2 \right]}$

where $\mathbb{E}[.]$ is the expectation operator. After the delay and sum beamformer, the output becomes

$x(w) = s(w) + \frac{1}{N} \sum\limits_{n =0}^{N-1} v_{n+1}(w) e^{\left(jw n \frac{d}{c} \sin{\theta} \right)}$

The output SNR per frequency bin $w$ , denoted $oSNR(w)$ is given as

$oSNR = \frac{\mathbb{E}\left[|s(w)|^2 \right]}{\mathbb{E}\left[\left | \frac{1}{N} \sum\limits_{n =0}^{N-1} v_{n+1}(w) e^{\left(jw n \frac{d}{c} \sin{\theta} \right)} \right|^2 \right]}$

But

$\left| \frac{1}{N} \sum\limits_{n =0}^{N-1} v_{n+1}(w) e^{\left(jw n \frac{d}{c} \sin{\theta} \right)} \right|^2 =\frac{1}{N^2} \sum\limits_{n =0}^{N-1} |v_{n+1}(w)|^2 + \frac{1}{N^2} \sum\limits_{n =1}^{N} \sum\limits_{m \neq n}^{N} v_{n}(w) v_{m}^*(w) e^{\left(jw (n-m) \frac{d}{c} \sin{\theta} \right)}$

Since by assumption $\mathbb{E}[v_i(w) v_j^*(w)] = 0, i \neq j, \{i,j\} \in \{1, \cdots, N\}$

$\mathbb{E}\left[\left| \frac{1}{N} \sum\limits_{n =0}^{N-1} v_{n+1}(w) e^{\left(jw n \frac{d}{c} \sin{\theta} \right)} \right|^2\right] = \frac{\mathbb{E}[|v_1(w)|^2]}{N}$

This leads to an oSNR of

$oSNR = N \frac{\mathbb{E}\left[|s(w)|^2 \right]}{\mathbb{E}[|v_1(w)|^2]}$

The SNR improvement, SNRI then becomes

$SNRI = \frac{oSNR}{iSNR} = N = 2^{\frac{\log{N}}{\log{2}}}$

Thus, if $N$ is increased by a factor of $2$ , the SNRI increases by a factor of $10 \log{2} \approx 3dB$ .

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ULA uncorrelated noise SNR improvements with delay and sum beamformer