Question about Shannon's mathematics

[ScarTissue]

I'm trying to go through Shannon's paper "A Mathematical Theory of Communication" to improve my understanding of information theory.

In Part I (Discrete Noiseless Systems) Shannon states:

Suppose all sequences of the symbols S₁, . . . ,S_n are allowed and these symbols have durations t₁, . . . ,t_n. What is the channel capacity?
If N(t) represents the number of sequences of duration t we have

N(t) = N(t -t₁)+N(t -t₂)+


...+N(t -t_n):

The total number is equal to the sum of the numbers of sequences ending in S₁, S₂, . . . , S_n and these are N(t -t₁), N(t -t₂), . . . ,N(t -t_n), respectively.


So I can't understand how this sum is actually working. For example, if t₁=2s and t₂=4s, then the first term in the sum is the number of all sequences ending in S₁ as expected. However the second term is going to be the number of all sequences ending in either S₁,S₁ or S₂. So this means that some of the sequences ending in S₁ have been counted twice by this sum.

Am I missing something here? Or am I correct and the right hand side of the equation is going to be larger than the left?

 

[verty]

To get a sequence of duration t, we append some S_i to a sequence of duration t - t_i. N(t) is just the number of sequences of duration t, and is the sum of those with each S_i to be appended.

-- sorry, I see I answered the wrong question. The question was, is the sum correct?

 

[ScarTissue]

Yes, I understand what the terms mean (I think) but I don't see how the two sides of the equation are equal.

 

[verty]

We know Claude Shannon as one of the forefathers of the digital age. Someone with this much foresight would not easily make a mistake. Whatever he wrote there we must assume was intentional.

Therefore, look again. And focus too on what is being counted. We are counting only t-length sequences, not any shorter sequences.

In reference to: "So this means that some of the sequences ending in S1 have been counted twice by this sum."

PS. Sorry for biting your head off.

 

[ScarTissue]

We are counting only t-length sequences, not any shorter sequences.

Right. So if we have t₁=2 and t₂=4, any sequence ending in two S₁'s will have the same length as any sequence ending in one S₂. In such a case you count the S₁S₁ sequences twice.

I don't believe Shannon could have made a mistake in this paper, and I don't believe it could have gone unquestioned if he had. So really I'm just trying to understand why you don't count the sequences above twice, or if you do, why it doesn't matter.