Complete Communications Engineering

ULA Signal To Noise Plus Interference Ratio Improvements With Delay and Sum Beamformer. The rule of thumb for delay and sum beamformers is that a doubling of the number of microphones leads to a 3dB gain. This holds iff there is no interfering signal which is correlated across all microphones. We derive the expected signal to noise plus interference ratio (SNIR) gains for ULA microphones. Consider a far field source impinging N ULA microphones through an anechoic medium 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(w) e^{\left(-jw \frac{(i-1) d}{c} \sin{\beta} \right)} + n_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(w) is the directional interfering signal, \beta is the DOA of the interfering signal and n_i(w) is zero mean i.i.d noise with variance \sigma_n^2.

The input SNIR per frequency bin w, denoted iSNIR(w) is given as:

iSNIR = \frac{|s(w)|^2}{|v(w)|^2 + \sigma_n^2 }

After the delay and sum beamformer, the output becomes:

x(w) = s(w) + v(w) \frac{1}{N} \frac{\sin{\left(w N \frac{d}{2c}( \sin{\theta} - \sin{\beta})\right)}}{\sin{\left(w \frac{d}{2c}( \sin{\theta} - \sin{\beta})\right)}} e^{\left(jw \frac{N-1}{2} \frac{d}{c}( \sin{\theta} - \sin{\beta}) \right)}+ \frac{1}{N} \sum\limits_{n=0}^{N-1} n_{i+1} (w) e^{\left(jw n \frac{d}{c}\sin{\theta} \right)}

The output SNIR per frequency bin $w$, denoted $oSNIR(w)$ is given as

oSNIR = \frac{|s(w)|^2}{\frac{\left |v(w) \right|^2}{N^2} \left| \frac{1}{N} \frac{\sin{\left(w N \frac{d}{2c}( \sin{\theta} - \sin{\beta})\right)}}{\sin{\left(w \frac{d}{2c}( \sin{\theta} - \sin{\beta})\right)}}\right| + \frac{1}{N} \sigma_n^2}

The SNR improvement, SNIRI then becomes:

SNRI = \frac{oSNR}{iSNR} = \frac{N^2 (\alpha(w) +1)}{\alpha(w) \left |\frac{\sin{\left(w N \frac{d}{2c}( \sin{\theta} - \sin{\beta})\right)}}{\sin{\left(w \frac{d}{2c}( \sin{\theta} - \sin{\beta})\right)}} \right|^2 + N}

where \alpha(w) = \frac{|v(w)|^2}{\sigma_n^2}.

A sample expected SNIRI at a frequency of 4kHz is shown in Figure 2 below d = 50mm using 4 microphones for \alpha =0 and \alpha = 100 or 20dB. It should be noted that \alpha =0 reduces to conventional N fold improvement as shown on the plot. The desired direction is 90^{\circ}.

SNRI (dB) for dierent DOA(degrees) and d

Figure 2: SNRI (dB) for different DOA(degrees) and $latx d$

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