Recall the system with dual binary channel

whose state diagram is

❶

## Trellis diagram

Trellis diagrams are basically state diagrams stretched out in time. For example let us consider two states *S*_{0} and *S*_{1}. 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
*S*_{0}, 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
*S*_{1}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 *S*_{0} or *x̅* of *S*_{1} 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.**

## Channel capacity

Suppose

- a noise-less channel that is non-catastrophic,
- either states
*S*(*t*= 0) or*S*(*t*=*t*) is known, and_{final} - output sequence
*y̅*is also known.

*y̅*can be figured out. In other words, all information from

*x̅*arrives at the output

*y̅*and hence lossless information channel.

[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.**