# Given a noise-free channel with limited bandwidth, why is the channel capacity infinite? [duplicate]

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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

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## 1 Answer

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.