Skip to content Skip to navigation

Connexions

You are here: Home » Content » Filtering of noise - 2

Navigation

Recently Viewed

This feature requires Javascript to be enabled.

Filtering of noise - 2

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

User rating (How does the rating system work?)
Ratings

Ratings allow you to judge the quality of modules. If other users have ranked the module then its average rating is displayed below. Ratings are calculated on a scale from one star (Poor) to five stars (Excellent).

How to rate a module

Hover over the star that corresponds to the rating you wish to assign. Click on the star to add your rating. Your rating should be based on the quality of the content. You must have an account and be logged in to rate content.

:
(0 ratings)

Summary: this module considers the effect on power spectrum of noise after ffiltering

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

PSD after filtering:

The relation between a ,b and c and φφ size 12{φ} {} which describe the noise components can be seen to be identical with that between X,Y and R and θθ size 12{θ} {}.

Hence pdf of c is Rayleigh and that of θθ size 12{θ} {} is uniform.

fck=ckPke−ck2/2Pkck0fck=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≤πfθk=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 kΔfkΔf size 12{kΔf} {} is

H kΔf = ∣ H kΔf ∣ e jϕk = ∣ H kΔf ∣ ∠ ϕ k 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} } } {} (1)

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 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 )} {} (2)

The power associated with the input component is

P ki = a k 2 ¯ + b k 2 ¯ 2 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} } } {} (3)

As ∣HkΔf∣∣HkΔf∣ size 12{ lline H left (kΔf right ) rline } {}is a deterministic function, ∣HkΔf∣ak2¯=∣HkΔf∣2ak2¯∣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 bkbk 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 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} } } {} (4)

And the power spectral densities are related by

G no f = ∣ H f ∣ 2 G ni f 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 )} {} (5)

Where the kΔfkΔf size 12{kΔf} {} has been replaced by ff size 12{f} {} as a continuous variable as ΔfΔf size 12{Δf} {}tends to 0.

Superposition of noises:

 Noise can be represented as superposition of (orthogonal) harmonics of ΔfΔf size 12{Δf} {} therefore total power is the result of superposition of component powers.

Consider Two processes n1n1 size 12{n rSub { size 8{1} } } {} and n2n2 size 12{n rSub { size 8{2} } } {} with overlapping spectral components.

Power of the sum of n1n1 size 12{n rSub { size 8{1} } } {} and n2n2 size 12{n rSub { size 8{2} } } {} will be p1+p2+2En1n2p1+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 n1n1 size 12{n rSub { size 8{1} } } {} and n2n2 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 kthkth 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 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 {} } } {} (6)

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 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} } } {} (7)

Considering power spectral components at kΔfkΔf size 12{kΔf} {} and lΔflΔf size 12{lΔf} {}, let the mixing frequency be f0=k+lΔff0=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−f0pΔf=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 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"} {} (8)
n p2 t = a l 2 cos 2πpΔ ft + b l 2 sin 2πpΔ 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"} {} (9)

But as akal¯=akbl¯=bkal¯=bkbl¯=0akal¯=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=0Enp1tnp2t=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+Enp2t2Enp1t+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.

Content actions

Give Feedback:

E-mail the module author | Rate module ( How does the rating system work?)

Rating system

Ratings

Ratings allow you to judge the quality of modules. If other users have ranked the module then its average rating is displayed below. Ratings are calculated on a scale from one star (Poor) to five stars (Excellent).

How to rate a module

Hover over the star that corresponds to the rating you wish to assign. Click on the star to add your rating. Your rating should be based on the quality of the content. You must have an account and be logged in to rate content.

(0 ratings)

Download:

Module PDF

Add module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections directly in Connexions. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need a Connexions account to use 'My Favorites'.

| A lens (?)

Definition of a lens

Lenses

A lens is a custom view of Connexions content. You can think of it as a fancy kind of list that will let you see Connexions through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to Connexions materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual Connexions member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks