mohatta2015.pdf

92 LINEAR EQUALIZATION
sense that its distribution depends on where you sample (aligned with a symbol vs.
inbetween two symbols). The interference is actually cyclostationary, in that the
distribution changes periodically over time (it is the same at times t and t + T).
It is possible to consider simpler models, particularly for symbols from transmitters
that aren't transmitting symbols of interest. These models include the
following.
White, stationary noise. Interference is folded into the AWGN term, increasing
the value of No. If this is applied to all ISI, we end up with matched filtering.
Colored, stationary noise. This model captures the fact that interference samples
are correlated in time, due to the bandlimited symbol waveform and the
medium response. If only the effect of the symbol waveform is modeled, the
difference between this model and the white noise model is typically small.
However, if the medium response is accounted for and the medium response is
highly dispersive, then the model can be significantly different from the white
noise model. In essence, we are replacing the sequence of Gaussian interfering
symbols with a white, stationary Gaussian noise. The symbol waveform and
medium color that noise.
4.4.7 Block and sub-block forms
So far, we have assumed we will detect symbols one at a time. At the other extreme,
we can detect a whole block of symbols all at once. Such an approach is called block
equalization. There is also something inbetween, in which we detect a sub-block
of symbols. The sub-block corresponds to the symbols taken from certain symbol
periods, certain PMCs, and certain transmit antennas. For example, they could
correspond to all symbols within a certain symbol period.
With both block and sub-block equalization, the received signal vectors to be
processed can be stacked into a vector r which can be modeled using (1.54), i.e.,
r \= HAs + n, (4.132)
where n is a vector of zero-mean, complex r.v.s with covariance C n . Unlike the
purely MIMO scenario, the symbols in s do not necessarily correspond to symbols
from different transmitters during the same symbol period. We would like H to
have more rows than columns, so symbols at the edge of the sub-block may be
folded into n.
To achieve pure zero-forcing, we need H to have at least as many rows as there
are columns. The decision variable vector is given by
z = (ΑΗ Η ΗΑ)ΆΗ Η Γ. (4.133)
Notice we need AH ff HA to be full rank. Observe that if H is square and full rank,
then (4.133) simplifies to
z = A _1 H _1 r. (4.134)
The solution in (4.133) has several interpretations. One is that it is a leastsquares
estimate of s, in that it minimizes the sum of the squares of the difference