Proceedings with Extended Abstracts (single PDF file) - Radio ...

2 Proposed AlgorithmReceived signal of a phased array is given byÝ Ï À (1)where and Ï are the complex input signal vector and the weight vector, respectively. Theoutput power is expressed in terms of the covariance matrix Ê ÜÜ asÈ ½ ¾ ݾ ℄ ½ ¾ Ï À À Ï ½ ¾ Ï À Ê ÜÜ Ï (2)Principle of DCMP algorithm(Takao et al., 1976) is to minimize the output power underthe constraintÏ À À £ (3)where is the desired direction vector, and À is the constraint. Here we further apply analternate conditionÏ À Ï Í (4)which forces that the norm of the weight should be less than a given value Í, which is set to besufficiently lower than the main lobe level, but not to affect the weight control of the sideloberegion. This second constraint assures that the entire main lobe pattern is not affected by theweight control. Here we call this algorithm as ‘DCMP Constrained Norm’ (DCMP-CN) incontrast to conventional DCMP.The principle of DCMP-CN is thus expressed asÑÒÏÈ ÓÙØ ½ ¾ Ï À Ê ÜÜ Ï×ÙØ ØÓ Ì Ï £ À ² Ï À Ï Í (6)This minimization problem with an equality constraint and an inequality condition is solved byusing penalty function method. The cost function is expressed asÉ ´Üµ ´Üµ· ´Ö½´ ´Üµµ ¾ ·ÑÖ·½(5)´ ´Üµµ ¾ µ (7)where ´Üµ is the function to be minimized, ´Üµ ¼ gives an equality constraint, and ´Üµ ¼ gives an inequality constraint. Here ´µ ÑÒ¼ ´ µ¾, Ö is thenumber of equality constraints, and ´Ñ Öµ is the number of inequality constraints.We choose an arbitrary increasing series of the penalty factor which vanishes to ½.For each , we minimize É ´Üµ with a non-linear unconstrained optimization algorithm toobtain Ü starting from Ü ½ . This procedure is iterated from Ü ¼ by increasing the penaltyfactor so that Ü converges to the allowed region.The cost function for the current case is given byÉ ´Ï µ ½ ¾ Ï À Ê ÜÜ Ï · Ï À À ¾ ·´Í Ï À Ï µ ¾ ℄ ½ ¾ Ï À Ê ÜÜ Ï · ´Ï À Àµ´ À Ï À £ µ·´Í Ï À Ï µ ¾ ℄ (8)The gradient of É ´Ï µ in terms of the weight vector Ï is given byÖ Û É ´Ï µÊ ÜÜ Ï · ¾´ À Ï À £ µ Ï ´Í Ï À Ï µ ℄ (9)437