mohatta2015.pdf

174 PRACTICAL CONSIDERATIONS
In practice, we have to estimate these quantities or related quantities. As these
parameters change with time, we need to adapt the equalizer over time, giving rise
to adaptive equalization.
To understand what options we have, let's assume we are building an MMSE
linear equalizer. One option is to estimate the channel response and noise power
and use them to compute the weights. We refer to such an approach as indirect
adaptation of the equalizer, because we adaptively estimate one set of parameters
(channel response and noise power) and then use them to calculate another set of
parameters (equalization weights).
How would we estimate the channel response and noise power? It depends on the
particular way signals are transmitted. Often known symbols are sent, referred to as
pilot symbols. These symbols may be clustered together into a synchronization word
or training sequence, which can occur at the beginning of set of data (preamble) or
in the middle of the data (midamble). The receiver searches for this known pattern.
Finding it gives us the packet or frame timing. We can then estimate the channel
response as follows. Suppose the known symbols are s«, si and s 2 and suppose we
have determined that we only need to estimate two channel coefficients, c and d.
We can estimate these using
c = (0.5)( Sl r 1 +s 2 r 2 ) (8.1)
d = (0.5)(sori+sir 2 ). (8.2)
These operations are correlations (weighted summations), as they correlate the
received samples to the known symbol values. Why does it work? Let's substitute
the models for r\ and r 2 and see what happens to c. Using the fact that s^ = 1,
we obtain
c = (0.5)[«i(csi+dsii+ ni) + s 2 (cs 2 +dsi+n 2 )] (8.3)
= c + 0.5d(siSo + s 2 si) + 0.5(«ini + s 2 n 2 ). (8.4)
The first term on the right is what we want, the true value c. The second term
is interference from the delayed path. It would be nice if this were zero. Because
we are transmitting known symbol values, we can select their values such that
siso + s 2 «i is zero! The third term is a noise term which has power (1/2)σ 2 . This
is half the noise power of a single received sample because we have used two received
values to estimate c.
In general, if we have N s +1 known symbols and use N s received values, the noise
on the channel estimate has power (1/N s )a 2 . Thus, while the channel estimate is
noisy, it can be made much less noisy than the received signal.
Once we have channel estimates, we can obtain a noise power estimate by subtracting
an estimate of the signal and estimating power. Specifically, we have
σ 2 = (0.5)[(n - cs! - ds () ) 2 + (r 2 - cs 2 - ¿ Sl ) 2 }. (8.5)
Thus, with indirect adaptation, we estimate the channel response and noise power
from the received samples and use these estimates to form the equalization weights.