Discrete Time Filtered White Noise

Theoretically, the variance of a filtered white noise should be: $$\sigma^2 = \int_{-\infty}^\infty \frac{N_0}{2}|H(f)|^2\,\mathrm df = \int_{-\infty}^\infty \text{PSD}_{\text{new}}(f)\,\mathrm df$$

So practically:

1. I should be able to calculate the variance from the PSD estimate of the filtered white noise?

2. What will be the correct way to convert this integral to a discrete sum?

I should be able to calculate the variance from the PSD estimate of the filtered white noise?

Yes. Since there's zero-measure sum at exactly 0 Hz, the expectation of the filter output $$h(t)$$ is 0, and thus the power of your signal is equal to its variance. And the integral over a power density is a power. So, yeah, if you integrate over the PSD for the whole frequency axis, you should get the signal power and thus the signal variance.

What will be the correct way to convert this integral to a discrete sum?

The Sampling Theory for time-domain signals requires the frequency domain representation of that same signal to have a finite support (i.e., the signal must be band-limited) in order for it to be able to correctly represent it by a finite sum over weighted sampling functions. In the same way, it requires a signal to be time-limited (i.e. do not exist outside a compact interval in the time domain) for you to be able replace that integral by a discrete sum.

So, the correct way would be:

1. make sure your signal is time-limited. White noise passed through a filter is decidedly not time-limited, so here's where things break in the strict sense, and you can't correctly represent this particular signal by a sum. In a practical sense, we go, ignore this fact and simply assume the signal stopped after a long enough amount of time, $$T_t$$.
2. You sample your PSD every $$\frac1{T_t}$$, $$\Phi[k] = \text{PSD}_{\text{new}}(k/T_T)$$. You normalize with a factor of $$\pi$$, because that's the energy reconstructing interpolating scaled sin f/f (Shannon-Whittaker formula).
• Thank you very much for your respone. Did you assume my system is analog? how can one sample a PSD? (sorry if i misunderstood you).. If my dealings are purely digital, i.e i generate wgn and pass it through a digital filter, i would only need to estimate the right PSD? with Welch's method or regular periodogram? and then integrate over it? Commented Oct 20, 2023 at 16:39