We will see the effects on the power spectrum when noise is passed through a filter.

Filtered output: If h(t) size 12{h \( t \) } {} is the impulse response, filtered output no(t) size 12{n rSub { size 8{o} } \( t \) } {} obtained by convolution. n o t = ∫ − ∞ ∞ n i τ h t − τ dτ = ∫ − ∞ t n i τ h t − τ dτ size 12{n rSub { size 8{o} } left (t right )= Int cSub { size 8{ - infinity } } cSup { size 8{ infinity } } {n rSub { size 8{i} } left (τ right )h left (t - τ right )dτ} = Int cSub { size 8{ - infinity } } cSup { size 8{t} } {n rSub { size 8{i} } left (τ right )h left (t - τ right )dτ} } {} Approximate continuous time by discrete interval. n o t = lim Δτ → 0 ∑ k = − ∞ k = n i kΔτ h t − kΔτ Δτ size 12{n rSub { size 8{o} } left (t right )= {"lim"} cSub { size 8{Δτ rightarrow 0} } Sum cSub { size 8{k= - infinity } } cSup { size 8{k= { size 6{t} } wideslash { size 6{Δτ} } } } {n rSub { size 8{i} } left (kΔτ right )h left (t - kΔτ right )Δτ} } {} In the interval impulse response is deterministic and known, but ni size 12{n rSub { size 8{i} } } {} depends on sample chosen, For white Gaussian noise ni size 12{n rSub { size 8{i} } } {} in interval k is different from and independent of that in interval l. Thus output at any time t0 size 12{t rSub { size 8{0} } } {} is a linear superposition of independent Gaussian variables: hence it is Gaussian. If we split filter H(f) size 12{H \( f \) } {} into H1(f) size 12{H rSub { size 8{1} } \( f \) } {} and H2(f) size 12{H rSub { size 8{2} } \( f \) } {}, output of H1(f) size 12{H rSub { size 8{1} } \( f \) } {} is Gaussian but NOT white (due to filtering)

Hence output no size 12{n rSub { size 8{o} } } {} of H2(f) size 12{H rSub { size 8{2} } \( f \) } {} is also non-white. output no' size 12{n rSub { size 8{o} } rSup { size 8{'} } } {} of H1(f) size 12{H rSub { size 8{1} } \( f \) } {} is non-white, thus no' size 12{n rSub { size 8{o} } rSup { size 8{'} } } {} in intervals k and l are dependent, but despite of this no size 12{n rSub { size 8{o} } } {} is Gaussian. Hence superposition of non-white dependent Gaussian variables also gives Gaussian output.

Spectral components: Let spectral component k be n k t = a k cos 2πkΔ ft + b k sin 2πkΔ ft = c k cos 2πkΔ ft + θ k size 12{n rSub { size 8{k} } left (t right )=a rSub { size 8{k} } "cos"2πkΔ ital "ft"+b rSub { size 8{k} } "sin"2πkΔ ital "ft"=c rSub { size 8{k} } "cos" left (2πkΔ ital "ft"+θ rSub { size 8{k} } right )} {} Then though coefficients are random over ensemble, they are deterministic for a specific case and depend on the particular sample chosen. Hence spectral component is stationary but not ergodic. Its normalized power is P k = n k t 2 ¯ = a k 2 ¯ cos 2 2πkΔ ft + b k 2 ¯ sin 2 2πkΔ ft + 2a k b k ¯ cos 2πkΔ ft sin 2πkΔ ft size 12{P rSub { size 8{k} } = {overline { left [n rSub { size 8{k} } left (t right ) right ] rSup { size 8{2} } }} = {overline {a rSub { size 8{k} } rSup { size 8{2} } }} "cos" rSup { size 8{2} } 2πkΔ ital "ft"+ {overline {b rSub { size 8{k} } rSup { size 8{2} } }} "sin" rSup { size 8{2} } 2πkΔ ital "ft"+ {overline {2a rSub { size 8{k} } b rSub { size 8{k} } }} "cos"2πkΔ ital "ft""sin"2πkΔ ital "ft"} {} As nk(t) size 12{n rSub { size 8{k} } \( t \) } {} is stationary, t can be any time. Choose t as t1 size 12{t rSub { size 8{1} } } {}when cosine is 1, sine is 0. Then P k = a k 2 ¯ size 12{P rSub { size 8{k} } = {overline {a rSub { size 8{k} } rSup { size 8{2} } }} } {} and similarly P k = b k 2 ¯ size 12{P rSub { size 8{k} } = {overline {b rSub { size 8{k} } rSup { size 8{2} } }} } {} P k = 2G n ( kΔf ) Δf = − 2G n ( kΔf ) Δf = a k 2 ¯ = b k 2 ¯ = a k 2 ¯ + b k 2 ¯ 2 = c k 2 ¯ 2 size 12{P rSub { size 8{k} } =2G rSub { size 8{n} } \( kΔf \) Δf= - 2G rSub { size 8{n} } \( kΔf \) Δf= {overline {a rSub { size 8{k} } rSup { size 8{2} } }} = {overline {b rSub { size 8{k} } rSup { size 8{2} } }} = { { {overline {a rSub { size 8{k} } rSup { size 8{2} } }} + {overline {b rSub { size 8{k} } rSup { size 8{2} } }} } over {2} } = { { {overline {c rSub { size 8{k} } rSup { size 8{2} } }} } over {2} } } {} Since a k 2 ¯ = b k 2 ¯ size 12{ {overline {a rSub { size 8{k} } rSup { size 8{2} } }} = {overline {b rSub { size 8{k} } rSup { size 8{2} } }} } {} Then pk size 12{p rSub { size 8{k} } } {} can be written as showing a constant term and a sin-cos time dependent term. P k = a k 2 ¯ cos 2 2πkΔ ft + sin 2 2πkΔ ft + 2a k b k ¯ cos 2πkΔ ft sin 2πkΔ ft size 12{P rSub { size 8{k} } = {overline {a rSub { size 8{k} } rSup { size 8{2} } }} left ("cos" rSup { size 8{2} } 2πkΔ ital "ft"+"sin" rSup { size 8{2} } 2πkΔ ital "ft" right )+ {overline {2a rSub { size 8{k} } b rSub { size 8{k} } }} "cos"2πkΔ ital "ft""sin"2πkΔ ital "ft"} {} thus P k = a k 2 ¯ + 2a k b k ¯ cos 2πkΔ ft sin 2πkΔf size 12{P rSub { size 8{k} } = {overline {a rSub { size 8{k} } rSup { size 8{2} } }} + {overline {2a rSub { size 8{k} } b rSub { size 8{k} } }} "cos"2πkΔ ital "ft""sin"2πkΔf} {} But pk size 12{p rSub { size 8{k} } } {} is stationary and independent of time, so a k b k ¯ = 0 size 12{ {overline {a rSub { size 8{k} } b rSub { size 8{k} } }} =0} {} Hence the coefficients are seen to be uncorrelated. Also at t1 size 12{t rSub { size 8{1} } } {}, we have nk(t1)=ak size 12{n rSub { size 8{k} } \( t rSub { size 8{1} } \) =a rSub { size 8{k} } } {}. But nk size 12{n rSub { size 8{k} } } {} is Gaussian as it can be considered output of a narrowband filter whose input is Gaussian. Thus ak size 12{a rSub { size 8{k} } } {} and bk size 12{b rSub { size 8{k} } } {} are Gaussian. Also ak size 12{a rSub { size 8{k} } } {} is output of filter in a non zero frequency interval k, it is not DC, thus mean value of ak=0 size 12{a rSub { size 8{k} } =0} {}. Hence coefficients are Zero-mean Gaussian. Taking product of two samples nk(t) size 12{n rSub { size 8{k} } \( t \) } {} and nl(t) size 12{n rSub { size 8{l} } \( t \) } {} where n k t = a k cos 2πkΔ ft + b k sin 2πkΔ ft size 12{n rSub { size 8{k} } left (t right )=a rSub { size 8{k} } "cos"2πkΔ ital "ft"+b rSub { size 8{k} } "sin"2πkΔ ital "ft"} {} n l t = a l cos 2πlΔ ft + b l sin 2πlΔ ft size 12{n rSub { size 8{l} } left (t right )=a rSub { size 8{l} } "cos"2πlΔ ital "ft"+b rSub { size 8{l} } "sin"2πlΔ ital "ft"} {} The ensemble averages of the result must be independent of time as each component is stationary, hence a k a l ¯ = a k b l ¯ = b k a l ¯ = b k b l ¯ = 0 size 12{ {overline {a rSub { size 8{k} } a rSub { size 8{l} } }} = {overline {a rSub { size 8{k} } b rSub { size 8{l} } }} = {overline {b rSub { size 8{k} } a rSub { size 8{l} } }} = {overline {b rSub { size 8{k} } b rSub { size 8{l} } }} =0} {} Hence coefficients at a given frequency are uncorrelated, and also coefficients at different frequencies are uncorrelated.

Summary of noise characteristics: Random, Gaussian, Ergodic, Linear superposition of random spectral components with coefficients which are Gaussian, zero-mean, with variances related to power spectral density and uncorrelated with each other and with other coefficients at different frequencies.