System Level Modeling and Optimization of the LTE Downlink

C. Taylor Expansion of the ZF MSEAdditionally, as each non-zero element of W has a power equally distributed overall the non-zero elements of W, each element of W has an average power of1Ẽ = EW,ẽ i,j ∼ CN(0, σ 2 e√NtxThe following notation is employed for the model parameters:ν),ˆ˜H = ˜H + Ẽ.σ 2 x = Average power allocated for transmission over all transmit antennas.σ 2 v = Average received noise power per antenna.σ 2 e = Average power of the elements of E ∼ CN ( 0, σ 2 e).σ 2 s = Average power of the transmitted symbols on each layer.ν N TX,(C.5)ν = Number of spatial layers being employed. i.e., ν symbols are being transmitted.H = Channel matrix ∈ N rx×Ntx .W = Precoding matrix ∈ R Ntx×ν .˜H = Effective channel matrix.˜H = HW ∈ C Nrx×ν .Denoting as H 0 the channel between the transmitter and receiver and as H i thechannel for each of the I interferers, where i = 1, 2, . . . , I, the receiver filter G ZF isexpressed asG ZF =( ˆ˜HH 0ˆ˜H0) −1ˆ˜HH 0 =( (˜H0 + Ẽ) H (˜H0 + Ẽ) ) −1 (˜H0 + Ẽ) H, (C.6)while the difference between the receive symbol vector ŝ ZF = G ZF(˜H0 s 0 + v + ∑ Ii=1 ˜H i s i)and the transmitted symbol vector s isŝ ZF − s = G ZF(−EWs 0 + v +)I∑˜H i s ii=1(−1= − ((H 0 + E) W) H (H 0 + E) W)((H0 + E) W) H EW} {{ }(++) −1((H 0 + E) W) H (H 0 + E) W ((H0 + E) W) H v} {{ }NI∑i=0Ds 0(C.7)(−1((H 0 + E) W) H (H 0 + E) W)((H0 + E) W) H ˜Hi} {{ }I i =N ˜H is i , (C.8)where we separate the expression into a signal part (D), a noise part (N), and aninterference part (I i ).94