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# lect23 - Wireless Communication Technologies Lectures 23...

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Wireless Communication Technologies Lectures 23 & 24 (April 22 & April 24) Instructor: Dr. Narayan Mandayam Summarized by: Varchaswi Rayaprolu ([email protected]) Optimum Linear Detectors Optimum linear detectors construct linear transformations of y to optimize one of the following criteria: 1.) Maximum asymptotic efficiency Consider y to be the normalized vector of MF outputs. Let k v be the linear transformation of user k. Then the decision rule is: h h T k j T k k j j j T k T k T k T k k v r v b A y v b RA v y v y v b + = + = = = ) ( ) ( ) sgn( 1 The asymptotic efficiency of the k th user is: - = k j j T k k j k T k k T k k k r v A A r v v R v v |} | , 0 { max 1 ) ( 2 h This is a non-linear optimization problem. There is no closed form solution. Lupus and Verdu proposed an algorithm to solve it. For the two user case, k = 2, without loss of generality, ] [ 2 1 2 1 2 1 2 2 1 1 1 2 1 | | 1 ) , , ( )} , , ( , 0 { max ) ( 1 x x x A A x A A x f where A A x f v x v T + + + - + = = = r r r r r h (I) The argument x * that maximizes f (.) is given by

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otherwise x A A if A A ; | | 1 ; ) sgn( * 1 2 1 2 r r r - = < - (II) The result says if the interferer is strong enough ( | | 2 1 r A A < ), the decorrelator maximizes asymptotic efficiency otherwise the received signal is correlated with ) ( ) sgn( ) ( 2 1 2 1 t s A A t s r - . The maximum asymptotic efficiency linear detector is a compromise between the decorrelator and the single-user matched filter. The asymptotic efficiency of this detector is obtained by substituting (II) in (I). otherwise v A A if A A A A ; 1 ) ( | | 1 ; | | 2 1 2 * 1 1 1 2 1 2 2 1 2 2 r h r r - = < - + Optimum Optimum Linear Single User Matched Filter
( Mandayam-Aazhang ) . exp )] ( [ min min k j b to respect with is ectation the where v R v r v b A r v A Q E P j k k T k k j j T k j j k T k k v v k v k k k + = s Infinitesimal Perturbation Analysis(IPA) is used to estimate the sensitivity of average probability of bit-error to different parameters. These estimates are shown to be unbiased. A stochastic gradient algorithm to solve this optimization problem is proposed and shown to converge almost surely when near – far resistance is strictly positive, that is, when eye is open. ( Refer to Appendix 1 ) 3.) Minimum mean square error ( Madhow- Honig ) The approach here is to turn linear multi-user detection problem into a linear estimation problem. Idea: Require MSE between k th bit and output of the linear transformation y v T k to be minimized. This approach does not minimize the bit error rate. For the k th user solve: k k T k k v v and K k for y v b E k = - ,....... 3 , 2 , 1 ; ] ) [( min 2 Combining the K equations, ]} ) )( [( { } { || || . exp ] || [|| min min 2 2 T M T k M y M b y M b E trace to equivalent is problem this xx trace x Since noise and bits to respect with is ectation the where y M b E k k k k - - = - × × Solution to this problem is given by 1 2

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lect23 - Wireless Communication Technologies Lectures 23...

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