In this section we present the update equations used
for recursive estimation of the fading coefficients 
and the noise pdf parameters 
.
These equations are new,
they provide extremely good estimates with sufficient training data,
and they have interesting interpretations which will be discussed in
Section 4.1. If the fading coefficients are already available,
from a pilot signal for example, then fewer quantities need to be
estimated.
The observations of T training symbols as modeled by (3) are assumed to be available. The EM algorithm is an iterative procedure for obtaining maximum likelihood parameter estimates, and its application to mixture pdfs is reviewed in [9]. The number of terms L in the mixture pdf (4) can be estimated. However, we have taken the simpler approach of fixing L equal to 2, 3 or 4. This simplifies the processing, and in addition other studies [11, 12] have demonstrated that using L = 2, 3 or 4 frequently produces a good approximation for cases of interest.
The general structure of the
EM algorithm is as follows.
Beginning with current estimates 
of the parameters, new estimates 
are computed by processing the current estimates
along with the training data.
Explicit formulas for this processing are given below.
Then the new estimates are assigned to the current
estimates,
and the same training
data is processed again to improve the estimates.
This process is repeated until the change in parameter estimates
is small from one iteration to the next iteration.
The EM update equations to estimate the model parameters are listed below, where first
is defined for 
.
Then the parameter estimates are updated as follows:
for 
.
Initial values for the parameters are required for the first
iteration of the EM algorithm.
We have developed rules of thumb for selecting initial values.
Preliminary investigations of the EM algorithm for parameter
estimation followed by MAP detection
are very encouraging.
