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I have the following equation to solve for the coefficients

[itex]\sum\limits_{m,n=0}^N b_m b_n^* (x_{mn}+y_{mn}) e^{ ik (ux_{mn} + vy_{mn}) } = 0[/itex]

which must be satisfied for all

[itex]||b||^2=1[/itex],

and

[itex]x_{mn}=x_m-x_n \text{ and } y_{mn}=y_m-y_n[/itex].

This equation arises in an analysis of plane waves incident on an array of sensors; the question (not homework) is to find weights on the various sensors such that the weighted element voltages sum to create an omni-directional sensitivity power pattern. I am happy to provide the derivation of how the physical system leads to this equation if it would be helpful.

Arguing from physical intuition, the sole solution appears to me that all [itex]b_m=0[/itex] except for one (call it [itex]b_0[/itex], it doesn't matter which we choose) that has unity modulus, [itex]b_0=e^{i\phi}[/itex]. This works because [itex]x_{00}=y_{00}=0[/itex] and is equivalent physically to turning

[itex]\mathbf{b}\rm=\array{[1&0&&0&...&0]}^T[/itex]

where, again, the indexing is arbitrary so we can assign the active element to any element in the array.

Can anyone suggest a more rigorous argument?

**b**:[itex]\sum\limits_{m,n=0}^N b_m b_n^* (x_{mn}+y_{mn}) e^{ ik (ux_{mn} + vy_{mn}) } = 0[/itex]

which must be satisfied for all

*u*and*v*in the interval [-1,1]. Here*k*is a constant,**b**is a vector of length*N*with unit norm[itex]||b||^2=1[/itex],

and

**x**and**y**are vectors of constants with[itex]x_{mn}=x_m-x_n \text{ and } y_{mn}=y_m-y_n[/itex].

This equation arises in an analysis of plane waves incident on an array of sensors; the question (not homework) is to find weights on the various sensors such that the weighted element voltages sum to create an omni-directional sensitivity power pattern. I am happy to provide the derivation of how the physical system leads to this equation if it would be helpful.

Arguing from physical intuition, the sole solution appears to me that all [itex]b_m=0[/itex] except for one (call it [itex]b_0[/itex], it doesn't matter which we choose) that has unity modulus, [itex]b_0=e^{i\phi}[/itex]. This works because [itex]x_{00}=y_{00}=0[/itex] and is equivalent physically to turning

*M*-1 sensors off and receiving with just a single active sensor. Choosing the arbitray constant as [itex]\phi=0[/itex] results in[itex]\mathbf{b}\rm=\array{[1&0&&0&...&0]}^T[/itex]

where, again, the indexing is arbitrary so we can assign the active element to any element in the array.

Can anyone suggest a more rigorous argument?

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