mohatta2015.pdf

THE MATH 79
and SINR becomes
w T h
SINR=- =-. (4.54)
1 - w'h
Thus, once we solve for the weights, computing SINR. is straightforward.
Why express things in matrix and vector form? Besides compactness, any results
we derive hold for more general cases. For example, suppose we weighted ten
received values with ten weights to form zi. The SINR expression we derived could
still be used, except that w and h would have 10 elements in each instead of 2.
4.2.4 General MIMO scenario
For the MIMO scenario, a full ZF solution is possible. It can be obtained by solving
the set of equations
r « Hs. (4.55)
This is also the solution to minimizing ISI (with a minimum value of zero). If
there are more receive antennas than transmit antennas (H has more rows than
columns), then one can first multiply both sides by H H .
The MMSE weights for detecting s m using w\{m)r\ + W2(m)r2 can be obtained
by solving the set of equations
where
Rw(m) = h(m), (4.56)
■o . i c 2 + d 2 + σ 2 de \ , ·,
R
= I de β 2 + / 2 +σ 2 I' W ^
w\(m
W2{m
h(l) = [ % ] , and h(2) = [ j } . (4.57)
The MSE for detecting s m is
w T (m)Rw(ro)-2w T (m)h(m) + l, (4.58)
and the SINR is
SINR= 1
w T ff h g.. (4.59)
1 — w' (m)h(m)
4.3 THE MATH
First, the MMSE solution for the weights is developed, assuming partial matched
filtering at the front end. A less common ML formulation is developed and shown
to be equivalent to the MMSE solution in terms of modem performance. The ML
formulation will become important in Chapter 7, when soft information is discussed.
Finally, other design criteria are briefly discussed.