, ▁under water ▁ac oust ic ▁imag ing , ▁and ▁Chem ical ▁sensor ▁arrays ... etc . ▁ ▁General ▁model ▁and ▁problem ▁form ulation ▁Consider ▁a ▁system ▁that ▁consists ▁of ▁array ▁of ▁r ▁arbitrary ▁sens ors ▁that ▁have ▁arbitrary ▁locations ▁and ▁arbitrary ▁directions ▁( direction al ▁characteristics ) ▁which ▁receive ▁signals ▁that ▁generated ▁by ▁q ▁narrow ▁band ▁sources ▁of ▁known ▁center ▁frequency ▁ ω ▁and ▁locations ▁ θ 1 , ▁ θ 2 , ▁ θ 3 , ▁ θ 4 ▁... ▁ θ q . ▁since ▁the ▁signals ▁are ▁narrow ▁band ▁the ▁propag ation ▁delay ▁across ▁the ▁array ▁is ▁much ▁smaller ▁than ▁the ▁reci pro cal ▁of ▁the ▁signal ▁band width ▁and ▁it ▁follows ▁that ▁by ▁using ▁a ▁complex ▁en velop ▁representation ▁the ▁array ▁output ▁can ▁be ▁expressed ▁ ▁( by ▁the ▁sense ▁of ▁super position ) ▁as ▁: ▁ ▁Where : ▁▁ ▁is ▁the ▁vector ▁of ▁the ▁signals ▁received ▁by ▁the ▁array ▁sens ors , ▁▁ ▁is ▁the ▁signal ▁em itted ▁by ▁the ▁k th ▁source ▁as ▁received ▁at ▁the ▁frequency ▁sensor ▁ 1 ▁of ▁the ▁array , ▁▁ ▁is ▁the ▁ste ering ▁vector ▁of ▁the ▁array ▁toward ▁direction ▁( ), ▁▁ τ i ( θ k ): ▁is ▁the ▁propag ation ▁delay ▁between ▁the ▁first ▁and ▁the ▁it h ▁sensor ▁for ▁a ▁wave form ▁coming ▁from ▁direction ▁( θ k ), ▁▁▁ ▁is ▁the ▁noise ▁vector . ▁ ▁The ▁same ▁equation ▁can ▁be ▁also ▁expressed ▁in ▁the ▁form ▁of ▁vectors : ▁ ▁If ▁we ▁assume ▁now ▁that ▁M ▁snap sh