# Variance of a signal

How to calculate the variance of noise samples modeled as follows:

$$n_a(t)$$ is a Gaussian zero-mean white noise process with (two-sided) power spectral density $$\frac{N_0}{2}$$. $$n_a(t)$$ is passed through an anti-alisasing filter to get the output $$n_f(t)$$. How to calculate the variance of the noise samples $$n[j]$$ in terms of $$N_0$$ and $$B$$, where $$n[j]$$=$$n_f(jT_s)$$ and $$T_s$$ is the sampling period.

• What is $B{}{}{}$? – Dilip Sarwate Nov 30 '19 at 23:58
• B is the Bandwidth. – Rashmika Dec 1 '19 at 1:09

How to calculate the variance of the noise samples $$n[j]$$ in terms of $$N_0$$ and $$B$$, where $$n[j]$$=$$n_f(jT_s)$$ and $$T_s$$ is the sampling period?
Do you know how to calculate the variance of the process $$\{n_f(t) \colon -\infty < t < \infty\}$$? No? Hint: it is the area under the power spectral density curve of $$\{n_f(t)\colon -\infty < t < \infty\}$$ provided that you are not using radian frequency but are using Hertzian frequency to express the PSD. Well, if you know the process variance, then be assured that each random variable $$n_f(t)$$ in the process has the same variance as the process variance. A more interesting question for you to ponder might be: "Are $$n[k]$$ and $$n[\ell]$$ independent random variables or correlated random variables?" which will bring in that nuisance parameter $$T_s$$ into play.
• @Ben Sampling is of $n_f(t)$ which has finite variance, not of the white noise $n_a(t)$. See the question Variance of White Gaussian noise for more details. – Dilip Sarwate Dec 2 '19 at 3:14