White noise with two-sided power spectral density N 0 /2 drives a second-order Butter worth filter with frequency response function magnitude |Hbu()| = Vi+(1s) /1

White noise with two-sided power spectral density N₀/2 drives a second-order Butter worth filter with frequency response function magnitude

where f₃ is its 3-dB cutoff frequency.

(a) What is the power spectral density of the filter€™s output?
(b) Show that the auto correlation function of the output is

Plot as a function of f₃Ï„. Use the integral given below:

(c) Does the output power obtained by taking lim_Ï„†’0 R₀(Ï„) check with that calculated using the equivalent noise bandwidth for a Butter worth filter as given by (7.115)?

|Hbu()| = Vi+(1s)" /1 + (f/f3)* exp (-/2r f3l7|) [Vzn falr| 1/4| R,(r) = nf3 No Vza.f31= %D 2 cos