mohatta2015.pdf

180 PRACTICAL CONSIDERATIONS
8.3.2 Time-varying channel and known symbol sequence
Next consider the case of a time-varying channel and assume we know all the
transmitted symbols. For simplicity, we assumed a symbol-spaced receiver and
consider a single path channel with time-varying coefficient g{qT). We can model
a vector of symbol-spaced received values r as
r = Sg + n, (8.15)
where S is a diagonal matrix with the known symbol values, g is a vector of the
channel coefficient value at different times, and n is a vector of noise values.
8.3.2.1 Filtering With the filtering approach, we use some or all of the received
samples to estimate the channel coefficient at each moment in time. Using all the
samples for each estimate, the vector of coefficient estimates is obtained by a matrix
multiplication, i.e.,
g = W f/ r, (8.16)
where different columns of matrix W correspond to filters for estimating the channel
coefficient at different times.
Similar to MMSE linear equalization, MMSE channel estimation (also known as
Wiener filtering) estimates the channel at time mT using
where
g(mï')=w H r, (8.17)
w = C; 1 p = (SC fl S H + C n )- 1 p (8.18)
C 9 = E{gg"} (8.19)
p = E{rg*(kT)}. (8.20)
The values for C g and p depend on how correlated the channel coefficient is from
one symbol period to the next.
The correlation between the channel coefficient at time mT and (m+d)T becomes
small as |ef| becomes large. As a result, it is reasonable to only use received values
in the vicinity of mT when estimating the channel at time mT. This gives rise to
a transversal Wiener filter. Except at the edges, the coefficients are the same for
different values of m.
A simpler transversal filtering approach is the moving average filter. With this
approach, the channel coefficient at time mT is estimated using
where the number of filter taps is 2N + 1.
N
g{mT) = Σ s*{m)r{m'T), (8.21)