System Level Modeling and Optimization of the LTE Downlink
System Level Modeling and Optimization of the LTE Downlink
System Level Modeling and Optimization of the LTE Downlink
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C. Taylor Expansion <strong>of</strong> <strong>the</strong> ZF MSEApplying a Taylor series expansion at E = 0, we obtain⎡ŝ ZF − s ≈ − ⎣ ∑ ∂D}}R{[E] nr,nt n r,n t ∂R{[E] nr,nt ∣E=0+ ∑ ∂D{ } ⎤}I [E] ⎦ nr,nt s 0n r,n t ∂I{[E] nr,nt ∣E=0⎡+ ⎣ ( ) −1˜HH 0 ˜H0 ˜HH 0 + ∑ ∂N}n r,n t ∂R{[E] nr,nt ∣E=0+ ∑ ∂N{ } ⎤}I [E] ⎦ nr,nt vn r,n t ∂I{[E] nr,nt ∣E=0⎡I∑+ ⎣ ( ) −1˜HH 0 ˜H0 ˜HH 0 ˜Hi + ∑ ∂I i}∣i=0n r,n t ∂R{[E] nr,nt + ∑ ∂I i{ } ⎤}∣ I [E] ⎦ nr,nt s i .n r,n t ∂I{[E] nr,nt ∣E=0}R{[E] nr,nt ∣E=0}R{[E] nr,nt (C.9)Extending <strong>the</strong> aforementioned solutions for <strong>the</strong> Taylor expansion in [118, 119, 142,143] to include interferers:ŝ ZF − s ≈ −(˜HH 0 ˜H0) −1˜HH 0 Ẽs 0 +(˜HH 0 ˜H0) −1 (˜HH 0 + ẼH) v−(˜HH 0 ˜H0) −1 (˜HH 0 Ẽ + ẼH ˜H0) (˜HH 0 ˜H0) −1˜HH 0 v+−I∑[ ( ) −1 (˜HH 0 ˜H0 ˜HH 0 + ẼH) ˜Hi s ii=1(˜HH 0 ˜H0) −1 (˜HH 0 Ẽ + ẼH ˜H0) (˜HH 0 ˜H0) −1˜HH 0 ˜Hi s i], (C.10)which expressed in shorter form,ŝ ZF − s = −−1 ˜H 0 Ẽs 0 +˜H−10v − ˜H−10−1Ẽ ˜H0 v + I∑i=1[ ˜H−1 0˜H i s i −]−1 −1 ˜H 0 Ẽ ˜H 0˜H i s i .(C.11)With <strong>the</strong> reasonable assumption that s i , s, <strong>and</strong> E are statistically independent, <strong>the</strong>95