# How is noise variance related to bandwidth?

I have to calculate asymptotic coding gain with soft decision decoding over an uncoded reference system. Lets say the coded system requires a bandwidth $k$ times higher than the bandwidth of the uncoded system. Does the variance of the white additive Gausian noise of the coded system is $k$ times the variance of the same noise in the uncoded sytem?

White noise has a flat power spectrum (i.e. its power spectral density is the same at all frequencies). For a white noise process with power spectral density $S_n(f) = \frac{N_0}{2}\text{ W/Hz}$, the noise power within a bandwidth of $B\text{ Hz}$ is:
$$P_n = BS_n(f) = \frac{N_0}{2}B\text{ (Watts)}$$
So, as you asked, for a white noise process, the total noise power over a particular bandwidth is proportional to the bandwidth. Therefore, if you increase a signal's bandwidth by a factor $k$, then the total noise power across the newly-expanded signal bandwidth also increases by the same factor $k$.