mohatta2015.pdf

THE MATH 179
8.3.1 Time-invariant channel and training sequence
Consider the case in which the channel does not change with time (at least over the
data burst being demodulated), and we wish to estimate the channel using only a
set of contiguous pilot symbols. If we select received values that only depend on
the training sequence, then we can model a vector of them as
r = Sg + n, (8.12)
where matrix S depends on the training sequence symbols, the path delays, and
possibly the pulse shape, g is a vector of channel coefficients corresponding to
different path delays, and n is noise.
One approach, introduced earlier, is correlation channel estimation, in which
the correlation of the received signal to the training sequence at the path delay of
interest gives an estimate of the channel coefficient. Sometimes the first and last
few symbols of the training sequence are not used, so that the channel estimate is
not influenced by unknown, traffic symbols and the autocorrelation properties can
be made perfect (the channel coefficient of one path is not interfered by the channel
coefficient of another path).
Another common approach is least-squares channel estimation. The idea is to
find the channel coefficients that best predict the received samples corresponding to
the training sequence. Here "best" means that the sum of the magnitude-squares
of the differences between the received samples and the predicted samples is minimized.
The predicted samples correspond to convolving the pilot symbols with the
estimated channel. The result, mathematically, is
g = (S H S)^1S H r. (8.13)
The least-squares approach ensures that different paths don't interfere with one
another, even if the autocorrelation properties of the training sequence are not
perfect.
The least-squares approach is a special case of maximum likelihood channel estimation
in which the noise is assumed to be white. If noise or interference is modeled
as colored noise, then the noise covariance (if known or estimated) can be used to
improve channel estimation.
So far, we have implicitly treated the channel coefficients as unknown, deterministic
(nonrandom) quantities. Alternatively, we can treat them as random quantities
with a certain distribution. A popular approach is MMSE channel estimation,
which requires knowledge of the mean and covariance of the set of channel coefficients,
independent of the distribution of the coefficients. Assuming the channel
coefficients have zero mean and covariance C 9 , the MMSE estimate is given by
g = C 9 S H (SC 9 S" + C Il )- 1 r. (8.14)
If the coefficients are modeled with a Gaussian distribution, then MMSE channel
estimation corresponds to MAP channel estimation. MAP channel estimation is a
more general approach, as it allows for other distributions to be used.