If several different defects are observed in measurements, additional steps are needed to construct the resulting Markov chain. The transition matrix of the system is then given by the Kronecker product of the individual defects (see [155]). For
independent defects, the resulting transition matrix is:
Note, that the size of the system matrix equals the product of the matrix sizes of the individual defects. This can be a serious problem in terms of computation time, because the number of different states quickly grows to impractical sizes (for four three-state defects, the resulting system would already have different states). The number of different levels of each defect
when changing its charge only depends on its electrostatic influence on the device (usually given in terms of
or
) and the number of states with a charge transfer.
Here, is the vector of different charge levels for defect
,
is the electrostatic influence of one charge,
is the number of states of the defect,
is the number of thermal states, and
is the absolute charge state of the defect in state
. Note that the levels for the thermal states correspond to the states they are tied to. That means that for each thermal state, a duplicate entry has to be created at the correct position in (6.27) so that the size of the set is again
.
As an example, for a three-state defect without thermal state, the levels for would be
. On the other hand, the defect depicted in Figure 6.4 would possess the
vector
.
To determine the corresponding levels for a system of defects according to equation (6.26), the Cartesian product of the sets defined in (6.27) can be used.
Since the resulting subsets in contain the corresponding levels of all defects for every state in (6.26), the final step is to sum over all the entries in each subset to obtain the vector
with the actual voltage levels for all the states in
(the indices here mark the
subset).
To clarify the construction of the levels in (6.27) – (6.29), an example of a system consisting of a two-state defect with
and the three-state defect from Figure 6.4 with
should be given.
Inserting of the defect into (6.28) gives
Performing the summation for every sub-vector in (6.30) delivers the final defect levels of the combined system:
In principle, the combined transition matrix and the combined levels derived above are sufficient to train a HMM to find the most likely combined system (see Section 6.7). However, the entries in the combined
transition matrix of the individual defects are not independent from each other. That means that two independent defects would converge towards one single defect with the combined number of states, which is of course unphysical (see Section 6.7.3). Another constraint is given by the levels of the defects depicted in (6.27) as they can only be multiples of the elementary charge for each state. As with the transition matrix for the combined system, the entries of
in (6.29) are also not independent of each other.
These two constraints require modifications for the parameter update in the HMM training which will be explained in Section 6.7.3.