This question already has an answer here:

In the Shannon Hartley theorem, if N = 0 but B is limited, the theorem tells us that C is infinite. Why is this intuitively true?

If bandwidth is limited, I can only post so many pulses per second over the channel.


marked as duplicate by hotpaw2, jojek Apr 5 '17 at 9:24

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.


Why is this intuitively true?

It's a purely hypothetical mathematical formula result. So it doesn't even need to be intuitive, at all.

It's very normal for humans to look for intuition in abstract things, but an observable noise-free channel doesn't exist, so meh.

But if you insist: in a noise free channel, a single symbol could take one of $N$ values, with $N$ becoming arbitrarily large, because, due to the absence of noise, you could tell arbitrarily fine voltage differences apart. With that, a single (equally probable) symbol $x$ would have Shannon info $I = -\log_2 p_x = -\log_2 \frac1N = -(\log_2 1 - \log_2 N) = \log N\rightarrow \infty$ information.

EDIT: I saw you just got the same answer not an hour ago:

But in terms of pure math, there is no ($\aleph_0$) finite limit on the amount of information that a single real number (one sample) can contain.


Not the answer you're looking for? Browse other questions tagged or ask your own question.