PSD after filtering: The relation between a ,b and c and φ size 12{φ} {} which describe the noise components can be seen to be identical with that between X,Y and R and θ size 12{θ} {}. Hence pdf of c is Rayleigh and that of θ size 12{θ} {} is uniform. fck=ckPke−ck2/2Pkck0 size 12{f left (c rSub { size 8{k} } right )= { {c rSub { size 8{k} } } over {P rSub { size 8{k} } } } e rSup { size 8{ - c rSub { size 6{k} } rSup { size 6{2} } /2P rSub { size 6{k} } } } ~c rSub {k} size 12{ > = 0}} {}, fθk=12π−π≤θk≤π size 12{f left (θ rSub { size 8{k} } right )= { {1} over {2π} } ~ - π <= θ rSub { size 8{k} } <= π} {} Let a spectral component of noise be the input to a filter whose transfer function at frequency kΔf size 12{kΔf} {} is H kΔf = ∣ H kΔf ∣ e jϕk = ∣ H kΔf ∣ ∠ ϕ k size 12{H left (kΔf right )= lline H left (kΔf right ) rline e rSup { size 8{jϕk} } = lline H left (kΔf right ) rline ∠ϕ rSub { size 8{k} } } {} The output spectral component of noise is n ko t = ∣ H kΔf ∣ a k cos 2πkΔ ft + ϕ k + ∣ H kΔf ∣ b k sin 2πkΔ ft + ϕ k size 12{n rSub { size 8{ ital "ko"} } left (t right )= lline H left (kΔf right ) rline a rSub { size 8{k} } "cos" left (2πkΔ ital "ft"+ϕ rSub { size 8{k} } right )+ lline H left (kΔf right ) rline b rSub { size 8{k} } "sin" left (2πkΔ ital "ft"+ϕ rSub { size 8{k} } right )} {} The power associated with the input component is P ki = a k 2 ¯ + b k 2 ¯ 2 size 12{P rSub { size 8{ ital "ki"} } = { { {overline {a rSub { size 8{k} } rSup { size 8{2} } }} + {overline {b rSub { size 8{k} } rSup { size 8{2} } }} } over {2} } } {} As ∣HkΔf∣ size 12{ lline H left (kΔf right ) rline } {}is a deterministic function, ∣HkΔf∣ak2¯=∣HkΔf∣2ak2¯ size 12{ {overline { left [ lline H left (kΔf right ) rline a rSub { size 8{k} } right ] rSup { size 8{2} } }} = lline H left (kΔf right ) rline rSup { size 8{2} } {overline {a rSub { size 8{k} } rSup { size 8{2} } }} } {} Similarly for bk size 12{b rSub { size 8{k} } } {}, and thus the power associated with noise output is P ko = ∣ H kΔf ∣ 2 a k 2 ¯ + b k 2 ¯ 2 size 12{P rSub { size 8{ ital "ko"} } = lline H left (kΔf right ) rline 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} } } {} And the power spectral densities are related by G no f = ∣ H f ∣ 2 G ni f size 12{G rSub { size 8{ ital "no"} } left (f right )= lline H left (f right ) rline rSup { size 8{2} } G rSub { size 8{ ital "ni"} } left (f right )} {} Where the kΔf size 12{kΔf} {} has been replaced by f size 12{f} {} as a continuous variable as Δf size 12{Δf} {}tends to 0.

Superposition of noises: Noise can be represented as superposition of (orthogonal) harmonics of Δf size 12{Δf} {} therefore total power is the result of superposition of component powers. Consider Two processes n1 size 12{n rSub { size 8{1} } } {} and n2 size 12{n rSub { size 8{2} } } {} with overlapping spectral components. Power of the sum of n1 size 12{n rSub { size 8{1} } } {} and n2 size 12{n rSub { size 8{2} } } {} will be p1+p2+2En1n2 size 12{p rSub { size 8{1} } +p rSub { size 8{2} } +2E left [n rSub { size 8{1} } n rSub { size 8{2} } right ]} {} and since n1 size 12{n rSub { size 8{1} } } {} and n2 size 12{n rSub { size 8{2} } } {}are uncorrelated, the last term = 0. Then these noises also obey the superposition of powers rule.

Mixing of noise with a sinusoid If kth size 12{k rSup { size 8{ ital "th"} } } {} component of noise is mixed with a sinusoid n k t cos 2πf o t = a k 2 cos 2π kΔf + f o t + b k 2 sin 2π kΔf + f o t + a k 2 cos 2π kΔf − f o t + b k 2 sin 2π kΔf + f o t alignl { stack { size 12{n rSub { size 8{k} } left (t right )"cos"2πf rSub { size 8{o} } t= { {a rSub { size 8{k} } } over {2} } "cos"2π left (kΔf+f rSub { size 8{o} } right )t+ { {b rSub { size 8{k} } } over {2} } "sin"2π left (kΔf+f rSub { size 8{o} } right )t} {} # + { {a rSub { size 8{k} } } over {2} } "cos"2π left (kΔf - f rSub { size 8{o} } right )t+ { {b rSub { size 8{k} } } over {2} } "sin"2π left (kΔf+f rSub { size 8{o} } right )t {} } } {} Sum and difference frequency noise spectral components with 1/2 amplitude are generated and G n f + f o = G n f − f o = G n f 4 size 12{G rSub { size 8{n} } left (f+f rSub { size 8{o} } right )=G rSub { size 8{n} } left (f - f rSub { size 8{o} } right )= { {G rSub { size 8{n} } left (f right )} over {4} } } {} Considering power spectral components at kΔf size 12{kΔf} {} and lΔf size 12{lΔf} {}, let the mixing frequency be f0=k+lΔf size 12{f rSub { size 8{0} } = { size 8{1} } wideslash { size 8{2} } left (k+l right )Δf} {}. This will generate 2 difference frequency components at the same frequency: pΔf=f0−kΔf=lΔf−f0 size 12{pΔf=f rSub { size 8{0} } - kΔf=lΔf - f rSub { size 8{0} } } {} Then difference frequency components are n p1 t = a k 2 cos 2πpΔ ft − b k 2 sin 2πpΔ ft size 12{n rSub { size 8{p1} } left (t right )= { {a rSub { size 8{k} } } over {2} } "cos"2πpΔ ital "ft" - { {b rSub { size 8{k} } } over {2} } "sin"2πpΔ ital "ft"} {} n p2 t = a l 2 cos 2πpΔ ft + b l 2 sin 2πpΔ ft size 12{n rSub { size 8{p2} } left (t right )= { {a rSub { size 8{l} } } over {2} } "cos"2πpΔ ital "ft"+ { {b rSub { size 8{l} } } over {2} } "sin"2πpΔ ital "ft"} {} But as akal¯=akbl¯=bkal¯=bkbl¯=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} {}, We find Enp1tnp2t=0 size 12{E left [n rSub { size 8{p1} } left (t right )n rSub { size 8{p2} } left (t right ) right ]=0} {} and Enp1t+np2t2=Enp1t2+Enp2t2 size 12{E left lbrace left [n rSub { size 8{p1} } left (t right )+n rSub { size 8{p2} } left (t right ) right ] rSup { size 8{2} } right rbrace =E left lbrace left [n rSub { size 8{p1} } left (t right ) right ] rSup { size 8{2} } right rbrace +E left lbrace left [n rSub { size 8{p2} } left (t right ) right ] rSup { size 8{2} } right rbrace } {} Thus superposition of power applies even after shifting due to mixing.