By J.E. Hudson
This publication develops the strategies underlying the layout of adaptive arrays from first ideas and is directed at study employees and architects whose mathematical heritage calls for refurbishment within the particular recommendations that have collected round the box, usually to the obscuration of the straightforward uncomplicated ideas.
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2. Minimum norm property of projections The use of projections to achieve orthogonality is more than just a convenient mathematical procedure; it is also a minimum norm solution in the following sense. If a vector is required with a certain amplitude gain g for a desired signal (with vector Sx say) and zero response for up to N - 1 other sources with vectors S m , m * 2 , . . , M then the vector defined by eqns. 25) which has been normalised by multiplying by scalar g is the vector having these properties with the smallest norm, a statement proved in Appendix HI.
3 Directional pattern of 3-element array with nulls at 45° and 180° Near-field nulls It is not necessary to assume plane waves in the definition of the source coefficients smn and an array will steer nulls to local points in space if the coefficients are set up with the appropriate spherical spreading terms. Consider the situation shown in Fig. 4 where a two-element array with^/ spacing is required to receive distant source A in the end-fire direction in the presence of interference from source B which lies at £/ outside the array in the same direction.
The broadside lobe is smaller in the projection solution having an amplitude of about 4 6 as against 7 in the matrix inversion method and the norm of the vector in eqn. 28) is 3-62 as compared to 4-24 for eqn. 16), so that in these two important aspects the projection solution is much superior. It is not much better when the maximum modulus in the weighting vector is Fig. 5 Directional response for weighting vector obtained by projection 46 Vector and matrix techniques compared yielding 2*54 instead of 2*87 so that if the coefficients were implemented using attenuators there would be much the same loss in signal level when the largest coefficient modulus was scaled down to unity, the projection solution being only about 1 dB better.