Skip to content Skip to navigation

Connexions

You are here: Home » Content » Filtering of noise

Navigation

Recently Viewed

This feature requires Javascript to be enabled.
 

Filtering of noise

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

Summary: this module discusses the effects of filtering of noise

We will see the effects on the power spectrum when noise is passed through a filter.

Filtered output:

If h(t)h(t) size 12{h \( t \) } {} is the impulse response, filtered output no(t)no(t) size 12{n rSub { size 8{o} } \( t \) } {} obtained by convolution.

n o t = ∫ − ∞ ∞ n i τ h t − τ dτ = ∫ − ∞ t n i τ h t − τ dτ n o t = ∫ − ∞ ∞ n i τ h t − τ dτ = ∫ − ∞ t n i τ h t − τ dτ size 12{n rSub { size 8{o} } left (t right )= Int cSub { size 8{ - infinity } } cSup { size 8{ infinity } } {n rSub { size 8{i} } left (τ right )h left (t - τ right )dτ} = Int cSub { size 8{ - infinity } } cSup { size 8{t} } {n rSub { size 8{i} } left (τ right )h left (t - τ right )dτ} } {} (1)
  • Approximate continuous time by discrete interval.
n o t = lim Δτ → 0 ∑ k = − ∞ k = n i kΔτ h t − kΔτ Δτ n o t = lim Δτ → 0 ∑ k = − ∞ k = n i kΔτ h t − kΔτ Δτ size 12{n rSub { size 8{o} } left (t right )= {"lim"} cSub { size 8{Δτ rightarrow 0} } Sum cSub { size 8{k= - infinity } } cSup { size 8{k= { size 6{t} } wideslash { size 6{Δτ} } } } {n rSub { size 8{i} } left (kΔτ right )h left (t - kΔτ right )Δτ} } {} (2)
  • In the interval impulse response is deterministic and known, but nini size 12{n rSub { size 8{i} } } {} depends on sample chosen,
  • For white Gaussian noise nini size 12{n rSub { size 8{i} } } {} in interval k is different from and independent of that in interval l.
  • Thus output at any time t0t0 size 12{t rSub { size 8{0} } } {} is a linear superposition of independent Gaussian variables: hence it is Gaussian.
  • If we split filter H(f)H(f) size 12{H \( f \) } {} into H1(f)H1(f) size 12{H rSub { size 8{1} } \( f \) } {} and H2(f)H2(f) size 12{H rSub { size 8{2} } \( f \) } {}, output of H1(f)H1(f) size 12{H rSub { size 8{1} } \( f \) } {} is Gaussian but NOT white (due to filtering)
Figure 1
Figure 1 (graphics1.png)
  • Hence output nono size 12{n rSub { size 8{o} } } {} of H2(f)H2(f) size 12{H rSub { size 8{2} } \( f \) } {} is also non-white.
  • output no'no' size 12{n rSub { size 8{o} } rSup { size 8{'} } } {} of H1(f)H1(f) size 12{H rSub { size 8{1} } \( f \) } {} is non-white, thus no'no' size 12{n rSub { size 8{o} } rSup { size 8{'} } } {} in intervals k and l are dependent, but despite of this nono size 12{n rSub { size 8{o} } } {} is Gaussian.
  • Hence superposition of non-white dependent Gaussian variables also gives Gaussian output.

 Spectral components:

  • Let spectral component k be
n k t = a k cos 2πkΔ ft + b k sin 2πkΔ ft = c k cos 2πkΔ ft + θ k n k t = a k cos 2πkΔ ft + b k sin 2πkΔ ft = c k cos 2πkΔ ft + θ k size 12{n rSub { size 8{k} } left (t right )=a rSub { size 8{k} } "cos"2πkΔ ital "ft"+b rSub { size 8{k} } "sin"2πkΔ ital "ft"=c rSub { size 8{k} } "cos" left (2πkΔ ital "ft"+θ rSub { size 8{k} } right )} {} (3)
  • Then though coefficients are random over ensemble, they are deterministic for a specific case and depend on the particular sample chosen.
  • Hence spectral component is stationary but not ergodic.
  • Its normalized power is
P k = n k t 2 ¯ = a k 2 ¯ cos 2 2πkΔ ft + b k 2 ¯ sin 2 2πkΔ ft + 2a k b k ¯ cos 2πkΔ ft sin 2πkΔ ft P k = n k t 2 ¯ = a k 2 ¯ cos 2 2πkΔ ft + b k 2 ¯ sin 2 2πkΔ ft + 2a k b k ¯ cos 2πkΔ ft sin 2πkΔ ft size 12{P rSub { size 8{k} } = {overline { left [n rSub { size 8{k} } left (t right ) right ] rSup { size 8{2} } }} = {overline {a rSub { size 8{k} } rSup { size 8{2} } }} "cos" rSup { size 8{2} } 2πkΔ ital "ft"+ {overline {b rSub { size 8{k} } rSup { size 8{2} } }} "sin" rSup { size 8{2} } 2πkΔ ital "ft"+ {overline {2a rSub { size 8{k} } b rSub { size 8{k} } }} "cos"2πkΔ ital "ft""sin"2πkΔ ital "ft"} {} (4)
  • As nk(t)nk(t) size 12{n rSub { size 8{k} } \( t \) } {} is stationary, t can be any time. Choose t as t1t1 size 12{t rSub { size 8{1} } } {}when cosine is 1, sine is 0. Then
P k = a k 2 ¯ P k = a k 2 ¯ size 12{P rSub { size 8{k} } = {overline {a rSub { size 8{k} } rSup { size 8{2} } }} } {} (5)

and similarly

P k = b k 2 ¯ P k = b k 2 ¯ size 12{P rSub { size 8{k} } = {overline {b rSub { size 8{k} } rSup { size 8{2} } }} } {} (6)
P k = 2G n ( kΔf ) Δf = − 2G n ( kΔf ) Δf = a k 2 ¯ = b k 2 ¯ = a k 2 ¯ + b k 2 ¯ 2 = c k 2 ¯ 2 P k = 2G n ( kΔf ) Δf = − 2G n ( kΔf ) Δf = a k 2 ¯ = b k 2 ¯ = a k 2 ¯ + b k 2 ¯ 2 = c k 2 ¯ 2 size 12{P rSub { size 8{k} } =2G rSub { size 8{n} } \( kΔf \) Δf= - 2G rSub { size 8{n} } \( kΔf \) Δf= {overline {a rSub { size 8{k} } rSup { size 8{2} } }} = {overline {b rSub { size 8{k} } 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} } = { { {overline {c rSub { size 8{k} } rSup { size 8{2} } }} } over {2} } } {} (7)

Since

a k 2 ¯ = b k 2 ¯ a k 2 ¯ = b k 2 ¯ size 12{ {overline {a rSub { size 8{k} } rSup { size 8{2} } }} = {overline {b rSub { size 8{k} } rSup { size 8{2} } }} } {} (8)
  • Then pkpk size 12{p rSub { size 8{k} } } {} can be written as showing a constant term and a sin-cos time dependent term.
P k = a k 2 ¯ cos 2 2πkΔ ft + sin 2 2πkΔ ft + 2a k b k ¯ cos 2πkΔ ft sin 2πkΔ ft P k = a k 2 ¯ cos 2 2πkΔ ft + sin 2 2πkΔ ft + 2a k b k ¯ cos 2πkΔ ft sin 2πkΔ ft size 12{P rSub { size 8{k} } = {overline {a rSub { size 8{k} } rSup { size 8{2} } }} left ("cos" rSup { size 8{2} } 2πkΔ ital "ft"+"sin" rSup { size 8{2} } 2πkΔ ital "ft" right )+ {overline {2a rSub { size 8{k} } b rSub { size 8{k} } }} "cos"2πkΔ ital "ft""sin"2πkΔ ital "ft"} {} (9)

thus

P k = a k 2 ¯ + 2a k b k ¯ cos 2πkΔ ft sin 2πkΔf P k = a k 2 ¯ + 2a k b k ¯ cos 2πkΔ ft sin 2πkΔf size 12{P rSub { size 8{k} } = {overline {a rSub { size 8{k} } rSup { size 8{2} } }} + {overline {2a rSub { size 8{k} } b rSub { size 8{k} } }} "cos"2πkΔ ital "ft""sin"2πkΔf} {} (10)
  • But pkpk size 12{p rSub { size 8{k} } } {} is stationary and independent of time, so
a k b k ¯ = 0 a k b k ¯ = 0 size 12{ {overline {a rSub { size 8{k} } b rSub { size 8{k} } }} =0} {} (11)

Hence the coefficients are seen to be uncorrelated.

  • Also at t1t1 size 12{t rSub { size 8{1} } } {}, we have nk(t1)=aknk(t1)=ak size 12{n rSub { size 8{k} } \( t rSub { size 8{1} } \) =a rSub { size 8{k} } } {}. But nknk size 12{n rSub { size 8{k} } } {} is Gaussian as it can be considered output of a narrowband filter whose input is Gaussian.

Thus akak size 12{a rSub { size 8{k} } } {} and bkbk size 12{b rSub { size 8{k} } } {} are Gaussian.

  • Also akak size 12{a rSub { size 8{k} } } {} is output of filter in a non zero frequency interval k, it is not DC, thus mean value of ak=0ak=0 size 12{a rSub { size 8{k} } =0} {}. Hence coefficients are Zero-mean Gaussian.
  • Taking product of two samples nk(t)nk(t) size 12{n rSub { size 8{k} } \( t \) } {} and nl(t)nl(t) size 12{n rSub { size 8{l} } \( t \) } {} where
n k t = a k cos 2πkΔ ft + b k sin 2πkΔ ft n k t = a k cos 2πkΔ ft + b k sin 2πkΔ ft size 12{n rSub { size 8{k} } left (t right )=a rSub { size 8{k} } "cos"2πkΔ ital "ft"+b rSub { size 8{k} } "sin"2πkΔ ital "ft"} {} (12)
n l t = a l cos 2πlΔ ft + b l sin 2πlΔ ft n l t = a l cos 2πlΔ ft + b l sin 2πlΔ ft size 12{n rSub { size 8{l} } left (t right )=a rSub { size 8{l} } "cos"2πlΔ ital "ft"+b rSub { size 8{l} } "sin"2πlΔ ital "ft"} {} (13)

The ensemble averages of the result must be independent of time as each component is stationary, hence

a k a l ¯ = a k b l ¯ = b k a l ¯ = b k b l ¯ = 0 a k a l ¯ = a k b l ¯ = b k a l ¯ = b k b l ¯ = 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} {} (14)
  • Hence coefficients at a given frequency are uncorrelated, and also coefficients at different frequencies are uncorrelated.

Summary of noise characteristics:

Random, Gaussian, Ergodic, Linear superposition of random spectral components with coefficients which are Gaussian, zero-mean, with variances related to power spectral density and uncorrelated with each other and with other coefficients at different frequencies.

Content actions

Download module as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Add module to:

My Favorites (?)

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

| A lens I own (?)

Definition of a lens

Lenses

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

What is in a lens?

Lens makers point to 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 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