Maximum likelihood detection of MIMO system using second order ...

A¨+'WWWM+=W CM6MWby C¢' - A: - ; renaming , we see that such an Y' K¡ £¢< :¥¤ © ¤L:Y MSat most equalsQ¡ 1 CC (9)In order to minimize the last expression, we convert MAX-CUT problem into following quadratic problemThis minimum is approximately% as shown in [4].7790"¡§¦SK¡ £¢ K¡ £$013254¦678/8¥9 A ; A4. SIGNAL MODELWe consider BPSK case. We assume a discrete time block fadingmultiple antenna channel model D with transmit and receiveantennas, also we assume that the receiver has perfect channelknowledge. The received signal at an instant isA ¡IC ; A; A non-convex quadratic constraint problem is given byK¡ %¢& K¡ %$6 N A (17)R¨HW+,+' 1+.-/-/- + 8 (18)W1 + = ' 1 + ¨3 +4£C 1 + = ' 1 + ¨3 +4£; C AAwhere ¨ .is a known channel matrix, and£ . '©¨ $" £ +(10)>is the i.i.d. zero mean white Gaussian noise with variance . Thefading coefficients are i.i.d. Gaussian with zero mean and unitvariance. Under the aforementioned assumptions, the ML criterionrequires us find$ . > :to C¨minimizesML detection.$ > :, where $ : 0 C 1 + 1 5, which. Now we develop SOCP method for the5. SOCP ML DECODERThe problem can be written (after neglecting constant term) aswhere and N '©¨' C¨6¡$ '$ . Since [ W < : Y : b$A$"KNMON A$ +(11)+: 'D$ : + < +(12)$are the diagonal entries in J, can be absorbed in the vector . ThusNwithout loss of generality, we can assume that all diagonal entriesof are zeros. The above equation is equivalent towhere Ais of the formA6¡$ ' 8=F © [ W < : Y : b Y © ¥ N NA: CAAC (13)(14)Since this cost function is symmetric, C : ' 1 need not bemaintained explicitly. It can be shown that the MAX-CUT problemis equivalent to ML detection with BPSK [9]. As mentionedearlier that the MAX-CUT problem is NP-hard, therefore we willuse some relaxation scheme to find near optimum solution. In thisrespect, note that ; ' 1PC1! +; ' 1#" +;W1 + +;1 (15)RA ; A; 1 + A ¡IC ; A; C 1 (16) ; where is all zero vector except 1 position= at . MaximizingAA implies minimizing M subject toC AA where ' H > ¡¤£¦ + N A('H "*) AH "01?2;4¨678/8¥9set of real symmetric matrices. The above problem can be writtenasN5464¦P R¡ 'H "*) AH "$.234= . , ) H . and R H . . > ¡¤£6is the P8 + R¡>= (23)then the right hand side of the above equation alsoimplies the left hand side. A convex quadratic constraint can nowR¡4¡ C ( A ¡ (24)P8 +which are convex quadratic constraints. If' > ¡¤£¦ be easily transformed into second order cone constraints. To dofor P8 this, we decomposeD'D Afirst , where =.(>&?and 7is symmetric and positive semidefinite. The constraint(25)' RF£87U¡ . Such a decomposition is always possible, asis equivalent toIt is known that anyR¡ A (26)A A R¡ (27)W +F£!A@+CB/.+(28)9ZE.