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:

N element ULA Microphone Array

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\}


\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]}


\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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