Lecture09 Summary
• Trellis diagrams are basically state diagrams stretched out in time. For example let us consider two states S0 and S1. At time t = 0 the states as nodes are depicted as ◯ on the other hand at time t = 1 the states as nodes are depicted as ⬤.
For example for various cases of relation
• If knowledge of (i.e, S(t = 0)) lets us know all possible output (generated) sequence , then this channel with memory is called non-catastrophic or conditionally catastrophic.
• If a channel does not incorporate a noise channel but there is still information loss, then such channels are called noise-less channels.
• Channel capacity can be used to determine maximum information rate for class of inofrmation channel which are
• a noise-less channel that is non-catastrophic,
• either states S(t = 0) or S(t = tfinal) is known, and
• output sequence is also known.
• The channel capacity or maximum information rate or maximum mutual information between X and Y for such class of channels with memory is where is the possible number of Trellis paths in t steps reaching all Sj such that .
• It is hard to compute the channel capacity by the approach of determining . An easier approach is to diagonalize D (). Thus and Cj are constants because they are derived from U−1, U and their constants.
• In general you do not need to compute all Cj, just Cm. The maximum channel capacity is where λm is the largest eigenvalue; .