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Module by: a.i. trivedi. E-mail the author

Summary: This module describes the quadrature components of narrowband noise

## 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 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} {}
(1)

Where fofo size 12{f rSub { size 8{o} } } {} corresponds to k=Kk=K size 12{k=K} {} and lies in the centre of the band.

Letting fo=KΔffo=KΔf size 12{f rSub { size 8{o} } =KΔf} {}and using 2πfot2πKΔft=02πfot2πKΔft=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 f 0 + k K Δf t + b k sin f 0 + k K Δf t n t = lim Δf 0 k = 1 a k cos f 0 + k K Δf t + b k sin 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 } } {}
(2)

Using identities for the sine and cosine of sum of two angles, we get the equation in terms of ncnc size 12{n rSub { size 8{c} } } {} and nsns size 12{n rSub { size 8{s} } } {}, where

n c t = lim Δf 0 k = 1 a k cos k K Δ ft + b k sin k K Δ ft n c t = lim Δf 0 k = 1 a k cos k K Δ ft + b k sin 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 ]} } {}
(3)
n s t = lim Δf 0 k = 1 a k sin k K Δ ft b k cos k K Δ ft n s t = lim Δf 0 k = 1 a k sin k K Δ ft b k cos 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 ]} } {}
(4)

ncnc size 12{n rSub { size 8{c} } } {} and nsns size 12{n rSub { size 8{s} } } {} are stationary random processes represented as superposition of components.

It can be shown that ncnc size 12{n rSub { size 8{c} } } {} and nsns size 12{n rSub { size 8{s} } } {}are Gaussian, zero mean equal variance uncorrelated variables.

## Significance:

n(t)n(t) size 12{n $$t$$ } {}of frequency kΔfkΔf size 12{kΔf} {}gives rise to ncnc size 12{n rSub { size 8{c} } } {} and nsns size 12{n rSub { size 8{s} } } {} of frequency ffoffo size 12{f - f rSub { size 8{o} } } {}within band -B/2 to B/2 and thus change insignificantly during one fofo size 12{f rSub { size 8{o} } } {}cycle. ncnc size 12{n rSub { size 8{c} } } {} and nsns size 12{n rSub { size 8{s} } } {} represent amplitude variations of two slowly changing quadrature phasor components, the complete phasor is

rt=nc2t+ns2t12rt=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 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 ]} {}
(5)

The endpoint of r wanders randomly with passage of time.

## PSD of orthogonal noise

Select spectral components corresponding to k=K+λk=K+λ size 12{k=K+λ} {} and k=Kλk=Kλ size 12{k=K - λ} {} where λλ size 12{λ} {} is an integer. k=Kk=K size 12{k=K} {}corresponds to frequency fofo size 12{f rSub { size 8{o} } } {}, hence selected components correspond to fo+λΔffo+λΔf size 12{f rSub { size 8{o} } +λΔf} {} and foλΔffoλΔf size 12{f rSub { size 8{o} } - λΔf} {} , and these frequencies generate 4 power spectral lines.

Select from nc(t)nc(t) size 12{n rSub { size 8{c} } $$t$$ } {} that part Δnc(t)Δnc(t) size 12{Δn rSub { size 8{c} } $$t$$ } {}corresponding to the frequencies we have selected above

Δnct=aKλcos2πλΔftbKλsin2πλΔft+aK+λcos2πλΔft+bK+λsin2πλΔftΔnct=aKλcos2πλΔftbKλsin2πλΔft+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 PλPλ size 12{P rSub { size 8{λ} } } {} of Δnc(t)Δnc(t) size 12{Δn rSub { size 8{c} } $$t$$ } {} is the ensemble average of Δnc(t)2Δnc(t)2 size 12{ left [Δn rSub { size 8{c} } $$t$$ right ] rSup { size 8{2} } } {} and this may be calculated at any time t1t1 size 12{t rSub { size 8{1} } } {}. Choosing t1t1 size 12{t rSub { size 8{1} } } {} such that λΔft1λΔft1 size 12{λΔ ital "ft" rSub { size 8{1} } } {} is an integer,

Δn c t = a K λ + a K + λ 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} } } {}
(6)
P λ = E Δn c t 1 2 = E a K λ + a K + λ 2 = a K λ 2 ¯ + a K + λ 2 ¯ 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} }} }} {}
(7)

We then find

P λ = 2G nc λΔf Δf = 2G n K λ Δf Δf = 2G n K + λ Δf Δf 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} {}
(8)

So that

G nc λΔf = G n K λ Δf = G n K + λ Δf 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 ]} {}
(9)

For continuous frequency variable, this becomes

G nc f = G n f 0 f + G n f 0 + f 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 )} {}
(10)

Similar equation also results for GnsGns 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 GnGn size 12{G rSub { size 8{n} } } {}from +fo+fo size 12{+f rSub { size 8{o} } } {}and fofo size 12{ - f rSub { size 8{o} } } {}to x=0x=0 size 12{x=0} {} and adding the displaced plots.

Thus the power of n(t)n(t) size 12{n $$t$$ } {}and the orthogonal components are equal

Example: For white noise filtered by a rectangular BPF centered at fofo size 12{f rSub { size 8{o} } } {} with BW=BBW=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 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" {} } } {}
(11)

Hence Gn(fo+f)=Gn(fof)Gn(fo+f)=Gn(fof) 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 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} } } {}
(12)

and are twice the magnitude of Gn(fo+f)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 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} {}
(13)

And that of n(t)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 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} {}
(14)

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