Narrowband Representation When passed through a narrowband filter noise can also be represented in terms of quadrature components n ( t ) = n c t cos 2πf o t − n s t sin 2πf o t size 12{n \( t \) =n rSub { size 8{c} } left (t right )"cos"2πf rSub { size 8{o} } t - n rSub { size 8{s} } left (t right )"sin"2πf rSub { size 8{o} } t} {} Where fo size 12{f rSub { size 8{o} } } {} corresponds to k=K size 12{k=K} {} and lies in the centre of the band. Letting fo=KΔf size 12{f rSub { size 8{o} } =KΔf} {}and using 2πfot−2πKΔft=0 size 12{2πf rSub { size 8{o} } t - 2πKΔ ital "ft"=0} {} We add the above to the arguments in the equation 1. n t = lim Δf → 0 ∑ k = 1 ∞ a k cos 2π f 0 + k − K Δf t + b k sin 2π f 0 + k − K Δf t size 12{n left (t right )= {"lim"} cSub { size 8{Δf rightarrow 0} } Sum cSub { size 8{k=1} } cSup { size 8{ infinity } } { left lbrace a rSub { size 8{k} } "cos"2π left [f rSub { size 8{0} } + left (k - K right )Δf right ]t+b rSub { size 8{k} } "sin"2π left [f rSub { size 8{0} } + left (k - K right )Δf right ]t right rbrace } } {} Using identities for the sine and cosine of sum of two angles, we get the equation in terms of nc size 12{n rSub { size 8{c} } } {} and ns size 12{n rSub { size 8{s} } } {}, where n c t = lim Δf → 0 ∑ k = 1 ∞ a k cos 2π k − K Δ ft + b k sin 2π k − K Δ ft size 12{n rSub { size 8{c} } left (t right )= {"lim"} cSub { size 8{Δf rightarrow 0} } Sum cSub { size 8{k=1} } cSup { size 8{ infinity } } { left [a rSub { size 8{k} } "cos"2π left (k - K right )Δ ital "ft"+b rSub { size 8{k} } "sin"2π left (k - K right )Δ ital "ft" right ]} } {} n s t = lim Δf → 0 ∑ k = 1 ∞ a k sin 2π k − K Δ ft − b k cos 2π k − K Δ ft size 12{n rSub { size 8{s} } left (t right )= {"lim"} cSub { size 8{Δf rightarrow 0} } Sum cSub { size 8{k=1} } cSup { size 8{ infinity } } { left [a rSub { size 8{k} } "sin"2π left (k - K right )Δ ital "ft" - b rSub { size 8{k} } "cos"2π left (k - K right )Δ ital "ft" right ]} } {} nc size 12{n rSub { size 8{c} } } {} and ns size 12{n rSub { size 8{s} } } {} are stationary random processes represented as superposition of components. It can be shown that nc size 12{n rSub { size 8{c} } } {} and ns size 12{n rSub { size 8{s} } } {}are Gaussian, zero mean equal variance uncorrelated variables.

Significance: n(t) size 12{n \( t \) } {}of frequency kΔf size 12{kΔf} {}gives rise to nc size 12{n rSub { size 8{c} } } {} and ns size 12{n rSub { size 8{s} } } {} of frequency f−fo size 12{f - f rSub { size 8{o} } } {}within band -B/2 to B/2 and thus change insignificantly during one fo size 12{f rSub { size 8{o} } } {}cycle. nc size 12{n rSub { size 8{c} } } {} and ns size 12{n rSub { size 8{s} } } {} represent amplitude variations of two slowly changing quadrature phasor components, the complete phasor is

rt=nc2t+ns2t12 size 12{r left (t right )= left [n rSub { size 8{c} } rSup { size 8{2} } left (t right )+n rSub { size 8{s} } rSup { size 8{2} } left (t right ) right ] rSup { size 8{ { {1} over {2} } } } } {} θ t = tan − 1 n s t / n c t size 12{θ left (t right )="tan" rSup { size 8{ - 1} } left [ {n rSub { size 8{s} } left (t right )} slash {n rSub { size 8{c} } left (t right )} right ]} {} The endpoint of r wanders randomly with passage of time.

PSD of orthogonal noise Select spectral components corresponding to k=K+λ size 12{k=K+λ} {} and k=K−λ size 12{k=K - λ} {} where λ size 12{λ} {} is an integer. k=K size 12{k=K} {}corresponds to frequency fo size 12{f rSub { size 8{o} } } {}, hence selected components correspond to fo+λΔf size 12{f rSub { size 8{o} } +λΔf} {} and fo−λΔf size 12{f rSub { size 8{o} } - λΔf} {} , and these frequencies generate 4 power spectral lines.

Select from nc(t) size 12{n rSub { size 8{c} } \( t \) } {} that part Δnc(t) size 12{Δn rSub { size 8{c} } \( t \) } {}corresponding to the frequencies we have selected above Δnct=aK−λcos2πλΔft−bK−λsin2πλΔft+aK+λcos2πλΔft+bK+λsin2πλΔft size 12{Δn rSub { size 8{c} } left (t right )=a rSub { size 8{K - λ} } "cos"2 ital "πλ"Δ ital "ft" - b rSub { size 8{K - λ} } "sin"2 ital "πλ"Δ ital "ft"+a rSub { size 8{K+λ} } "cos"2 ital "πλ"Δ ital "ft"+b rSub { size 8{K+λ} } "sin"2 ital "πλ"Δ ital "ft"} {} All 4 terms are at same frequency and represent uncorrelated processes. Then the power Pλ size 12{P rSub { size 8{λ} } } {} of Δnc(t) size 12{Δn rSub { size 8{c} } \( t \) } {} is the ensemble average of Δnc(t)2 size 12{ left [Δn rSub { size 8{c} } \( t \) right ] rSup { size 8{2} } } {} and this may be calculated at any time t1 size 12{t rSub { size 8{1} } } {}. Choosing t1 size 12{t rSub { size 8{1} } } {} such that λΔft1 size 12{λΔ ital "ft" rSub { size 8{1} } } {} is an integer, Δn c t = a K − λ + a K + λ size 12{Δn rSub { size 8{c} } left (t right )=a rSub { size 8{K - λ} } +a rSub { size 8{K+λ} } } {} P λ = E Δn c t 1 2 = E a K − λ + a K + λ 2 = a K − λ 2 ¯ + a K + λ 2 ¯ size 12{P rSub { size 8{λ} } =E left lbrace left [Δn rSub { size 8{c} } left (t rSub { size 8{1} } right ) right ] rSup { size 8{2} } right rbrace =E left [ left (a rSub { size 8{K - λ} } +a rSub { size 8{K+λ} } right ) rSup { size 8{2} } right ]= {overline {a rSub { size 8{ {} rSub { size 6{K - λ} } } } rSup {2} }} size 12{+ {overline {a rSub { {} rSub { size 6{K+λ} } } rSup {2} }} }} {} We then find P λ = 2G nc λΔf Δf = 2G n K − λ Δf Δf = 2G n K + λ Δf Δf size 12{P rSub { size 8{λ} } =2G rSub { size 8{ ital "nc"} } left (λΔf right )Δf=2G rSub { size 8{n} } left [ left (K - λ right )Δf right ]Δf=2G rSub { size 8{n} } left [ left (K+λ right )Δf right ]Δf} {} So that G nc λΔf = G n K − λ Δf = G n K + λ Δf size 12{G rSub { size 8{ ital "nc"} } left (λΔf right )=G rSub { size 8{n} } left [ left (K - λ right )Δf right ]=G rSub { size 8{n} } left [ left (K+λ right )Δf right ]} {} For continuous frequency variable, this becomes G nc f = G n f 0 − f + G n f 0 + f size 12{G rSub { size 8{ ital "nc"} } left (f right )=G rSub { size 8{n} } left (f rSub { size 8{0} } - f right )+G rSub { size 8{n} } left (f rSub { size 8{0} } +f right )} {} Similar equation also results for Gns size 12{G rSub { size 8{ ital "ns"} } } {}The psd's of the orthogonal components are shown below, and are obtained by shifting the +ve and -ve parts of plot of Gn size 12{G rSub { size 8{n} } } {}from +fo size 12{+f rSub { size 8{o} } } {}and −fo size 12{ - f rSub { size 8{o} } } {}to x=0 size 12{x=0} {} and adding the displaced plots. Thus the power of n(t) size 12{n \( t \) } {}and the orthogonal components are equal

Example: For white noise filtered by a rectangular BPF centered at fo size 12{f rSub { size 8{o} } } {} with BW=B size 12{ ital "BW"=B} {}, G n ( f ) = η 2 f o − B 2 ≤ ∣ f ∣ ≤ f o + B 2 , G n ( f ) = 0 elsewhere alignl { stack { size 12{G rSub { size 8{n} } \( f \) = { {η} over {2} } ~f rSub { size 8{o} } - { {B} over {2} } <= lline f rline <= f rSub { size 8{o} } + { {B} over {2} } ,~} {} # G rSub { size 8{n} } \( f \) =0~ ital "elsewhere" {} } } {} Hence Gn(fo+f)=Gn(fo−f) size 12{G rSub { size 8{n} } \( f rSub { size 8{o} } +f \) =G rSub { size 8{n} } \( f rSub { size 8{o} } - f \) } {} and G nc ( f ) = G ns ( f ) = G n ( f o − f ) + G n ( f o + f ) = η 2 + η 2 = η ∣ f ∣ ≤ B 2 size 12{G rSub { size 8{ ital "nc"} } \( f \) =G rSub { size 8{ ital "ns"} } \( f \) =G rSub { size 8{n} } \( f rSub { size 8{o} } - f \) +G rSub { size 8{n} } \( f rSub { size 8{o} } +f \) = { {η} over {2} } + { {η} over {2} } =η~~ lline f rline <= { {B} over {2} } } {} and are twice the magnitude of Gn(fo+f) size 12{G rSub { size 8{n} } \( f rSub { size 8{o} } +f \) } {}. The power of the orthogonal components is: σ nc 2 = σ ns 2 = ∫ − B / 2 B / 2 G nc ( f ) df = ηB size 12{σ rSub { size 8{ ital "nc"} } rSup { size 8{2} } =σ rSub { size 8{ ital "ns"} } rSup { size 8{2} } = Int rSub { size 8{ - {B} slash {2} } } rSup { size 8{ {B} slash {2} } } {G rSub { size 8{ ital "nc"} } } \( f \) ital "df"=ηB} {} And that of n(t) size 12{n \( t \) } {} is σ n 2 = ∫ − fo − B / 2 − fo + B / 2 G n ( f ) df + ∫ fo − B / 2 fo + B / 2 G n ( f ) df = 2 η 2 B = ηB size 12{σ rSub { size 8{n} } rSup { size 8{2} } = Int rSub { size 8{ - ital "fo" - {B} slash {2} } } rSup { size 8{ - ital "fo"+ {B} slash {2} } } {G rSub { size 8{n} } } \( f \) ital "df"+ Int rSub { size 8{ ital "fo" - {B} slash {2} } } rSup { size 8{ ital "fo"+ {B} slash {2} } } {G rSub { size 8{n} } } \( f \) ital "df"=2 { {η} over {2} } B=ηB} {}