Recall the system with dual binary channel
whose state diagram is
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 input x and output y when the input over output relation is , the Trellis diagram is
However when is the Trellis diagram is
Hence the combined diagram for the two cases of relation is complicated. However to avoid complication a clockwise arc is used such that it indicates
- for the first state S0, the first output arrow (labelled 1) encountered by the arc is for output y = −2 and the second output arrow (labelled 2) encountered is for output y = 0.
- And similarly the reverse for S1 state.
Non-catastrophic channel and Noise-less channel
Recall that the duo-binary channel contains catastrophic sequence, i.e, where two or more x̅ produce the same y̅ thus making x̅ ambiguous. For example if we have
Since the channel has memory, that is, the sequence x̅ has a particular relationship with the observed sequence y̅, if the sequence x̅ of the initial state (either x̅ of S0 or x̅ of S1 as the initial state) is known, there is NO information loss.
Recall the definition that a channel with memory is catastrophic if there is same output sequence y̅ for two for more different sequences x̅.
But if knowledge of x̅ (i.e, S(t = 0)) lets us know all possible output (generated) sequence y̅, 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.❹
- a noise-less channel that is non-catastrophic,
- either states S(t = 0) or S(t = tfinal) is known, and
- output sequence y̅ is also known.
[this statement is true because these channels satisfy corollary-1 pg.1288 Application of Set-Membership Techniques to Symbol-by-Symbol Decoding for Binary Data-Transmission Systems R.B Wells]
For such class of information channels maximum information rate can be determined from channel capacity.❺