Shannon-Hartley theorem for other bases

Question

The Shannon-Hartley theorem is given with terms referring to a binary signal.

What if a channel does not transmit via binary, but instead in another system, such as ternary or base-4?

Answer 1

The Shannon-Hartley theorem describes the capacity of an additive white Gaussian noise channel. The capacity-achieving distribution for this channel in Gaussian, too, so you can achieve this capacity only by transmitting a real-valued (not binary, ternary, etc.) signal.

The confusion arises because the capacity is measured in bits, i.e., you use $\log_2$ in the expression: This is as if a good costs 5 Dollars, and you ask how much it would cost in Euros. There is an exchange rate between these two prices, as there is an exchange rate between the measures if you take other logarithms. For example, you could state the Shannon-Hartley theorem as

$$ C=B\log_{256} \left(1+\frac{S}{N}\right) $$

if you want to measure the capacity in bytes, you could use $\log_3$, $\ln$ (for nats), whatever.