I am trying to understated the the achievable bit rate in noiseless and noisy channels. Particularly, I would like to solve the following problem:
Let a real symbol $0\leq x_i < 1$ be transmitted every $1$ milliseconds through the following channels:
$I$: Noiseless channel such that the output $y_i=\alpha x_i$ where $\alpha$ is the attenuation.
$II$: Noisy channel such that $y_i= x_i + n_i$ where $n_i$ is the noise and $|n_i|<0.0000223$.
What are the achievable bit rates for (i) channel $I$ assuming $\alpha$ is known, (ii) channel $I$ assuming $\alpha$ is unknown (iii) channel $II$.
The rate is in bits/s and so we need to find how many symbols per second is transmitted and how many bits is needed per symbol. We are transmitting one real symbol each $1$ ms. Thus, we are transmitting $1000$ symbols/s. Now for the second part, I am not sure how many bits is needed per symbol. How do I find that out?
For channel $I$, I know that in noiseless channel we get an infinite rate, but that's only if we know $\alpha$. What is it that we do if we don't? Can we ever obtain it? and how?
For channel $II$, is the achievable bit rate same as the capacity? We have a bound for the noise magnitude and we can see that $|x_i|<1$, thus $\mathrm{SNR}=|x_i|^2/|n_i|^2$ and $C = B\log_2(1+\mathrm{SNR})$. I don't know how to proceed though.