mohatta2015.pdf

MORE MATH 65
In the later chapter on MMSE equalization, we consider addressing cochannel interference
by modeling it as some form of noise.
If the number of receive antennas is equal to the number of transmitters (N r —
N t ), then we can obtain zero-forcing solution by solving the set of equations
N,
y m «][>,s (¿) (m) (3-39)
for the s'*) (m) values. We can write these equations in the form
y = Hs, (3.40)
where we've collected all the symbols into one column vector. The zero-forcing
solution is then
β = Η-ν· (3.41)
Notice that we had just enough equations to solve for the symbols. If we have fewer
equations, zero-forcing is not possible.
If N r > N t , then we have more equations than unknowns (H is no longer a
square matrix). We could simply discard some extra equations, but this would
not be the best strategy. To reduce the number of equations, we introduce spatial
matched filtering for each symbol:
z = H H y. (3.42)
This allows us to collect signal energy and reduce the number of equations to the
number of unknowns. The symbol estimates are then given by
s = (Η'Ή)-^ = (Η'ΉΓ'Ην (3.43)
Is this the only choice when JV r > JV t ? Actually not. Recall that there is a
copy of s m in y m +i- Before we sacrificed this copy to avoid ISI from future symbol
blocks. However, if N r > 2N t , then we have enough equations that we could use
y m+ i as well.
3.4.2 MIMO/cochannel scenario
Recall that in the MIMO/cochannel scenario, the received sample vector corresponding
to a particular PMC can be modeled as
x μ HAs + n, (3.44)
where n is a vector of Gaussian r.v.s with zero-mean and covariance C n and s is the
set of symbols transmitted from different transmitters. We will assume C n — N tt I
and A = E S I. Let's initially assume N r = N t , so that H is a square matrix.
With ZF DFE, we first need to triangularize the problem. Using QR decomposition,
we can write H as QR, where Q is orthonormal (Q _1 = Q H ) and R is upper
triangular. Substituting H = QR into (3.44) and multiplying both sides by Q H
gives
r (= v/ËjRs + e, (3.45)