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I need to study an array's performance in terms of interference. I have already analyzed the array's white noise gain which is the ratio of output and input SNR of the beamformer i.e. $\rm{SNR_o} / {SNR_i}$. This was based on defining array's output as $${\bf y}(t, \phi) = {\bf a}(\phi)s(t) + {\bf n}(t),$$ where $s(t)$ is the signal of interest, ${\bf a}(\phi)$ is the array manifold, and ${\bf n}(t)$ an isotropic AWGN that is spatio-temporally uncorrelated (and uncorrelated with $s(t)$). Note that ${\bf y}(t, \phi), {\bf a}(\phi), {\bf n}(t) \in \mathbb{R}^{M}$, $M$ being the number of sensors in the array.

If an interference signal is added, the array outputs $${\bf y}(t, \phi_s, \phi_v) = {\bf A}(\phi_s, \phi_v){\bf r}(t) + {\bf n}(t),$$ where ${\bf A}(\phi_s, \phi_v) := [{\bf a}(\phi_s) \; {\bf a}(\phi_v)]$, ${\bf r}(t) := [s(t)\; v(t)]^T$, $s(t)$ is the signal-of-interest and $v(t)$ is the interference signal. $\phi_s$ and $\phi_v$ are the angles of arrival of the signal of interest and interference, respectively.

The idea is to study how the beamformer improves the signal of interest over the interference.

The concept of signal-to-noise-plus-interference ratio ${\rm SNIR}$ is introduced.

Question 1: Is the ${\rm SNIR}$ the best choice of metric for this study?

At the beamformer's input, ${\rm SNIR}_i = P_s / (P_v + P_n)$, ${\rm SNR}_i = P_s / P_n$ and ${\rm SIR}_i = P_s / P_v$. The interference-to-noise ratio ${\rm INR}_i = P_v / P_n$, where $P_s, P_v, P_n$ are signal-of-interest power, interference signal power, and additive noise power, respectively.

${\rm SNIR}_i = \dfrac{P_s}{P_v (1 + {\rm INR}_i^{-1})} = \dfrac{{\rm SIR}_i}{1 + {\rm INR}_i^{-1}}$.

The beamformer weight is ${\bf w}$, and the beamformer outputs \begin{eqnarray} B &=& {\bf w}(\phi_L)^T ~ {\bf y}(t, \phi) \nonumber \\ %&=& %{\bf w}(\phi)^T {\bf A}(\phi){\bf x} \nonumber \\ %&=& %{\bf w}(\phi)^T \left[ {\bf a}(\phi_s)s(t) + {\bf a}(\phi_v)v(t) + {\bf n}(t) %\right] \nonumber \\ &=& {\bf w}(\phi_L)^T{\bf a}(\phi_s)s(t) + {\bf w}(\phi_L)^T {\bf a}(\phi_v)v(t) +{\bf w}(\phi)^T{\bf n}(t) \nonumber \end{eqnarray} where $\phi_L$ is the beamformer's look direction.

Assuming the signal of interest, interference, and noise are uncorrelated, the output signal-to-noise-plus-interference ratio \begin{eqnarray} {\rm SNIR}_o &=& \dfrac{ {\bf w}(\phi_L)^T {\bf a}(\phi_s) {\bf a}(\phi_s)^T {\bf w}(\phi_L) ~P_s }{ {\bf w}(\phi_L)^T {\bf a}(\phi_v) {\bf a}(\phi_v)^T {\bf w}(\phi_L)~ P_v + {\bf w}(\phi_L)^T {\bf w}(\phi_L) ~P_n } \nonumber \\ &=& \dfrac{ {\bf w}(\phi_L)^T {\bf a}(\phi_s) {\bf a_s}(\phi)^T {\bf w}(\phi_L) ~ {\rm SIR}_i }{ {\bf w}(\phi_L)^T {\bf a}(\phi_v) {\bf a}(\phi_v)^T {\bf w}(\phi_L) + {\bf w}(\phi_L)^T {\bf w}(\phi_L) ~{\rm INR}^{-1} } \end{eqnarray}

And the beamformer's ${\rm SNIR}$ gain \begin{eqnarray} \dfrac{{\rm SNIR}_o}{{\rm SNIR}_i} &=& \dfrac{ {\bf w}(\phi_L)^T {\bf a}(\phi_s) {\bf a}(\phi_s)^T {\bf w}(\phi_L) \left[1 + {\rm INR}^{-1}\right] }{ {\bf w}(\phi_L)^T {\bf a}(\phi_v) {\bf a}(\phi_v)^T {\bf w}(\phi_L) + {\bf w}(\phi_L)^T {\bf w}(\phi_L) ~{\rm INR}^{-1} } \end{eqnarray}

For the beamformer, $\underset{\phi}{\arg\max} ~{\bf w}(\phi_L)^T {\bf a}(\phi) = \phi_L$, where $||{\bf w}|| = 1$. Assume that $\phi_s = \phi_L$ while $\phi_v = \phi_L \pm \Delta_{\phi}$.

Question 2: At this point, what is the best way to show that this beamformer enhances the signal-of-interest over the interference signal?

Question 3: Is it acceptable to do this study using only signal-of-interest and interference while ignoring the additive noise? That way, I will have just ${\rm SIR}$ gain instead of ${\rm SNIR}$ gain to consider.

BoltzBooz
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  • an array manifold is complex. could you explain why yours is real. –  Mar 16 '18 at 06:14
  • An array manifold could be complex or real depending on the geometry of the array. For an array of spatially collocated directional sensors (i.e. inter-sensor spacing is too small compared to the wavelength), there is no time delay, hence no complex phase information, hence the array manifold is real. An example of such is the acoustic vector sensor http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=317869 – BoltzBooz Mar 16 '18 at 06:23
  • But I don't see the relevance of realness or complexness of the array manifold in my question. Even if this is a phased array, how can one show a certain proposed (data-independent) beamformer enhances signal of interest (arriving from the look direction) over an interference signal arriving from another direction (not the look direction). – BoltzBooz Mar 16 '18 at 06:27
  • Thanks for the Nahori paper, but Equation 3.1 is a narrowband complex model. It shows a complex manifold. Have a better example? In terms of relevance one can't give an informed answer if the question isn't fully understood. I think one of Baggeroer's students has a relatively recent thesis on optimal processing of vector sensor arrays. I would look there –  Mar 16 '18 at 07:31
  • Is this answer what you're looking for? https://dsp.stackexchange.com/questions/72298/how-to-derive-the-well-known-solution-to-unconstrained-array-gain/73181#73181 – Gillespie May 14 '22 at 16:24

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There's an article that defines some measures for blind source separation that I used to measure the output of a beamformer Performance Measurement in Blind Audio Source Separation. Vincent E., Gribonval R. and Févotte C. Transactions on Audio, Speech, and Language Processing, Vol. 14, No. 4, July 2006. They basically use orthogonal projections to decompose the output signal and use these projections to calculate Signal to Noise, Interference, Distorsion and Artifacts ratio, I know the paper is based on BSS, but it could be an starting point.

http://www.irisa.fr/metiss/demonstrations-fr/bssperf.html

Hkxs
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