mohatta2015.pdf

THE MATH 181
With recursive channel tracking, we track the channel coefficient by updating it as
we go along in time.
For example, with exponential "filtering" (also known as an alpha tracker), we
update the coefficient estimate using
g((m + l)T) = ag(mT) + (1 - a)r*{mT)s{mT), (8.23)
where parameter a is between 0 and 1. Another example is LMS tracking, mentioned
earlier, which would use
g((m + 1)T) = g(mT) + μs(m)(τ·(m7 , ) - g*(mT)s{mT))*, (8.24)
where parameter μ is a step size controlling the rate of tracking.
All channel tracking algorithms are based on a model of how the channel coefficient
changes over time. The LMS tracker is implicitly based on a first-order
model, the random walk model, i.e.,
g{(m + 1)T) = g(mT) + a g e(mT), (8.25)
where e(mT) is a sequence of uncorrelated, complex Gaussian random values with
unity variance. More advanced tracking algorithms have been developed by assuming
more advanced models, such as second-order models. These advanced algorithms
are particularly useful when the channel changes rapidly.
A classic model used in signal processing is the Kaiman filter model. It has the
form
x(m + 1) = F(m)x(m) 4- G(m)e(m) (8.26)
r*{mT) = h H (m)x(m) + n*(mT). (8.27)
where x(m) is an internal state vector. Notice that the random walk model is a
special case for which x(m) = g(mT), F(m) = 1, G(m) = 1, and h(m) = s(m). In
more sophisticated channel models, the state vector x(m) includes other quantities,
such as the derivative the channel coefficient.
The Kaiman filter is an MMSE recursive approach for estimating the state vector.
It has the update equations
x(m+l) = F(m)x(m)+k(m)(r(mT)-5*(mr)s(m2'))* (8.28)
P(m + 1) =
„/ N Λ,, ^ P m h m h " m p m \
F(m) P(ro) - , „/ .„/ ,, ) ; , F w v ; (m)
V h H y
{m)P{m)h(m) + alJ '
+ G(m)G H (m), (8.29)
where
Irfmï
k(m) =
F(m)P(m)h(m)
h«(m)P(m)h(m)+ ff r
The vector k(m) is referred to as the Kaiman gain vector.
(8·30)
8.3.3 Time-varying channel and partially known symbol sequence
Sometimes the transmitter sends periodic clusters of pilot symbols, with unknown
traffic symbols in between. One approach is to still use only the pilot symbols.