mohatta2015.pdf
signal processing from power amplifier operation control point of view
signal processing from power amplifier operation control point of view
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THE MATH 181<br />
With recursive channel tracking, we track the channel coefficient by updating it as<br />
we go along in time.<br />
For example, with exponential "filtering" (also known as an alpha tracker), we<br />
update the coefficient estimate using<br />
g((m + l)T) = ag(mT) + (1 - a)r*{mT)s{mT), (8.23)<br />
where parameter a is between 0 and 1. Another example is LMS tracking, mentioned<br />
earlier, which would use<br />
g((m + 1)T) = g(mT) + μs(m)(τ·(m7 , ) - g*(mT)s{mT))*, (8.24)<br />
where parameter μ is a step size controlling the rate of tracking.<br />
All channel tracking algorithms are based on a model of how the channel coefficient<br />
changes over time. The LMS tracker is implicitly based on a first-order<br />
model, the random walk model, i.e.,<br />
g{(m + 1)T) = g(mT) + a g e(mT), (8.25)<br />
where e(mT) is a sequence of uncorrelated, complex Gaussian random values with<br />
unity variance. More advanced tracking algorithms have been developed by assuming<br />
more advanced models, such as second-order models. These advanced algorithms<br />
are particularly useful when the channel changes rapidly.<br />
A classic model used in signal processing is the Kaiman filter model. It has the<br />
form<br />
x(m + 1) = F(m)x(m) 4- G(m)e(m) (8.26)<br />
r*{mT) = h H (m)x(m) + n*(mT). (8.27)<br />
where x(m) is an internal state vector. Notice that the random walk model is a<br />
special case for which x(m) = g(mT), F(m) = 1, G(m) = 1, and h(m) = s(m). In<br />
more sophisticated channel models, the state vector x(m) includes other quantities,<br />
such as the derivative the channel coefficient.<br />
The Kaiman filter is an MMSE recursive approach for estimating the state vector.<br />
It has the update equations<br />
x(m+l) = F(m)x(m)+k(m)(r(mT)-5*(mr)s(m2'))* (8.28)<br />
P(m + 1) =<br />
„/ N Λ,, ^ P m h m h " m p m \<br />
F(m) P(ro) - , „/ .„/ ,, ) ; , F w v ; (m)<br />
V h H y<br />
{m)P{m)h(m) + alJ '<br />
+ G(m)G H (m), (8.29)<br />
where<br />
Irfmï<br />
k(m) =<br />
F(m)P(m)h(m)<br />
h«(m)P(m)h(m)+ ff r<br />
The vector k(m) is referred to as the Kaiman gain vector.<br />
(8·30)<br />
8.3.3 Time-varying channel and partially known symbol sequence<br />
Sometimes the transmitter sends periodic clusters of pilot symbols, with unknown<br />
traffic symbols in between. One approach is to still use only the pilot symbols.