Re: Simple question on Shannon's 1948 paper


Inf wrote:
I have (if you don't mind) one last question.

I assume N(t-t_i) = 0 if t-t_i <0.

My question is what to assume about N(0).
If t_1 is the smallest of the t_i's and I choose
t=t_1 then I would expect that N(t_1)=1.

Plugging this into the formula Shannon derived,
I get N(t_1)=N(0) +0 + 0 + ...

So I evidently need to take N(0)=1. Is this correct
and if so what is its interpretation?

Ah, yes, the base cases. I hadn't thought of that, but here's what I
can work out:

Remember that N(t) is "number of sequences of duration t."

Clearly there are no sequences of duration < 0, so N(t) = 0 for t < 0.

For t = 0, there is exactly one sequence, namely the empty sequence
(i.e., a sequence of length 0). Assuming t_1, t_2, ..., t_n > 0, which
is a reasonable assumption. So N(0) = 1, as you said.

So your example:
N(t_1) = | {set of sequences of duration t_1} |
= | { S1 } |
= | { "" + S1} |
...

N(0) = | {set of sequences of duration 0} |
= | { "" } |
= 1

Michael

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