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

Letting f o = KΔf size 12{f rSub { size 8{o} } =KΔf} {} and using 2πf o t 2πKΔ ft = 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 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 n c size 12{n rSub { size 8{c} } } {} and n s 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 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 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 ]} } {}

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

It can be shown that n c size 12{n rSub { size 8{c} } } {} and n s 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 n c size 12{n rSub { size 8{c} } } {} and n s size 12{n rSub { size 8{s} } } {} of frequency f f o size 12{f - f rSub { size 8{o} } } {} within band -B/2 to B/2 and thus change insignificantly during one f o size 12{f rSub { size 8{o} } } {} cycle. n c size 12{n rSub { size 8{c} } } {} and n s size 12{n rSub { size 8{s} } } {} represent amplitude variations of two slowly changing quadrature phasor components, the complete phasor is

  r t = n c 2 t + n s 2 t 1 2 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 f o size 12{f rSub { size 8{o} } } {} , hence selected components correspond to f o + λΔf size 12{f rSub { size 8{o} } +λΔf} {} and f o λΔf size 12{f rSub { size 8{o} } - λΔf} {} , and these frequencies generate 4 power spectral lines.

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

  Δn c t = a K λ cos 2 πλ Δ ft b K λ sin 2 πλ Δ ft + a K + λ cos 2 πλ Δ ft + b K + λ sin 2 πλ Δ ft 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 P λ size 12{P rSub { size 8{λ} } } {} of Δn c ( t ) size 12{Δn rSub { size 8{c} } \( t \) } {} is the ensemble average of Δn c ( t ) 2 size 12{ left [Δn rSub { size 8{c} } \( t \) right ] rSup { size 8{2} } } {} and this may be calculated at any time t 1 size 12{t rSub { size 8{1} } } {} . Choosing t 1 size 12{t rSub { size 8{1} } } {} such that λΔ ft 1 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 G ns 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 G n size 12{G rSub { size 8{n} } } {} from + f o size 12{+f rSub { size 8{o} } } {} and f o 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 f o 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 G n ( f o + f ) = G n ( f o 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 G n ( f o + 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} {}

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Source:  OpenStax, Noise in communications. OpenStax CNX. Jul 07, 2008 Download for free at http://cnx.org/content/col10549/1.1
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