System Level Modeling and Optimization of the LTE Downlink

4. Extensions to the L2S ModelThe MSE is calculated based on the actual transmitted signal (x 0 ) and the estimatedreceive symbols (ˆx 0 ) as{MSE = E (ˆx 0 − x 0 ) (ˆx 0 − x 0 ) H} , (4.10)The estimated receive symbol vector ˆx 0 is, as previously shown in Equation (3.5),obtained as()N∑intˆx 0 = Gy = G H 0 x 0 + n + H i x i , (4.11)where the ZF receive filter G is calculated as) −1G =(ĤĤ 0 Ĥ 0 ĤH 0 , (4.12)Ĥ 0 = H 0 + E, (4.13)e ij ∼ CN ( 0, σe) 2 . (4.14)i=1The estimated channel (Ĥ 0 ) is modeled as the actual channel plus an error matrixE whose entries (e ij ) are modeled as complex-normal with mean power σ 2 e [118].Applying a Taylor series expansion at E = 0, i.e., assuming a small channel estimationnoise variance σ 2 e [119], the following MSE expression is obtained:MSE = E{(ˆx 0 − x 0 ) (ˆx 0 − x 0 ) H}{} ( ) H≈ ˜H −10 E EW 0 x 0 x H 0 W0 H E H ˜H −10{ } ( ) {H+ ˜H −10 E nn H ˜H −10 + ˜H −10 E EW 0 ˜H ( ) } H ( −10 nnH ˜H −10 WH0 E H ˜H −10N int∑+i=0[˜H −10 E {˜H i x i x H i ˜H H i} ( ) H˜H −10{+ ˜H −10 E EW 0 ˜H −1 ˜H( ) } H ( ) ] HH0 i x i x i˜H H i ˜H −10 W H E H ˜H −10(( (H= σeσ 2 x 2 0+ σv 2 + σvσ 2 eTr2 )H −1)) (0 H 0 ˜H H ˜H) −10 0+I∑[ (˜H −10 σx 2 iH i W i Wi H H H ii=1where H denotes the channel matrix,(+ σx 2 iσeTr2 H i H H i( )HH −1) ( ) )] H0 H 0 ˜H −10 ,(4.15)˜H the effective channel matrix (i.e., theprecoder-and-channel-matrix combination HW), and E the channel estimation errormatrix. The full derivation of Equation (4.15) can be found in Appendix C.) H59