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A study of quadratic processing for passive receiving arrays Turner, Ross Maclean


This work is an investigation and development of theory for the quadratic processing of arrays with Gaussian signals and noise. The quadratic processor is characterized by a weighting matrix, q, in the quadratic form, x[sup t]qx, where x is a column vector with its i[sup th] component the filtered output of the i[sup th] receiving element. The study is restricted to linear arrays of equally spaced elements. The conditions are established whereby such a quadratic processor has a specified directive pattern. The directive properties of the quadratic processor are determined and it is shown that the optimum pattern has side lobes which are 3 dB lower than those of the power pattern of the optimum Dolph-Chebyshev array. The versatility of the quadratic processor for pattern synthesis is demonstrated. It is shown that specification of the directive pattern of the processor is not sufficient to specify the elements of the weighting matrix, q, and these additional degrees of freedom can be used to reduce the output noise variance. The performance of the quadratic processor in the presence of noise is studied for a point source model for the background noise. The resulting signal- to-noise ratio is shown to depend on the q matrix through two performance functions, P[sub V](q) and P[sub A](q). It is shown that √ P[sub V](q) and P[sub A](q) become inversely proportional to array directive gain when q is chosen so that the quadratic processor reduces to the square-law-detected array. However, for the general quadratic processor, it is shown that these performance functions are not proportional to array directive gain but may be considered as ail extension to the concept of array gain. Two methods for the reduction of the output noise variance are studied, based on a minimization of P[sub V](q) and P[sub A](q) subject to the constraint of specified pattern function and resulting in solution q matrices, q[sub V] and q[sub A]. An approximate method for the minimization of the minimum detectable signal is given based on an interpolation between q[sub V] and q[sub A]. Numerical results are presented showing how the performance functions vary with array element spacing, steering angle, array size, bandwidth and pattern type. Comparisons are made between the quadratic processor and conventional processors on the basis of the performance functions and signal-to-noise ratio. A q matrix which is maximally uniform subject to the pattern constraints is shown to maximize the signal-to-receivernoise ratio and to maximize the signal-to-background noise ratio for a half-wavelength spaced narrow-band broadside array and to provide performance nearly as good as the optimum quadratic processor for element spacings greater than half-a wavelength. Similar results are established for the endfire array for element spacings greater than or equal to a quarter of a wavelength.

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