Skip to content Skip to navigation

Connexions

You are here: Home » Content » Characterization of noise

Navigation

Content Actions
  • Download module PDF
  • Add to ...
    Add the module to:
    • My Favorites
    • A lens
    • An external social bookmarking service
    • My Favorites (What is '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 (What is 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.

    • External bookmarks
  • E-mail the author

Recently Viewed

Characterization of noise

Module by: a.i. trivedi

Summary: Noise is described in terms of its power spectral density

Types of noise:

  • Thermal resistor noise: due to thermal agitation. Statistical fluctuations. Always random, erratic and unpredictable.
  • Shot noise: fluctuations in emission of electrons or crossing of junctions.
  • Additive channel noise: random disturbances added in the channel.
  • Multiplicative noise: created by non linear devices like diodes, mixers, power amplifiers.

Terminology:

  • Noise is a Random variable best described by statistics like mean, variance, pdf etc.
  • Stationary process: always independent of time of measurement - same statistics any time
  • Ensemble statistics: measured at the same time on an ensemble (group) of sources
  • Ensemble statistics may be different from statistics of any member of group measured over time.
  • Ergodic: If ensemble and time statistics are same. ergodic is always stationary but Stationary is not always ergodic.
  • Noise from natural phenomenon Gaussian, ergodic & stationary.
  • Nonlinear sources may not give Gaussian pdf e.g. rectifier. 

Mathematical characterization:

Frequency domain description can be derived for periodic and pulse-like aperiodic phenomena.

  • Noise is not repetitive and infinite in time. So not a periodic or aperiodic pulse type signal.
  • As an approximation take a sample of noise of interest from -T/2 to T/2 consider it repetitive.
  • Purely random so safely assume no DC exists. Now Fourier description applicable.

n T ( s ) t = ∑ k = 1 ∞ a k cos 2πkΔ ft + b k sin 2πkΔ ft = ∑ k = 1 ∞ c k cos 2πkΔ ft + θ k n T ( s ) t = ∑ k = 1 ∞ a k cos 2πkΔ ft + b k sin 2πkΔ ft = ∑ k = 1 ∞ c k cos 2πkΔ ft + θ k size 12{n rSub { size 8{T} } rSup { size 8{ \( s \) } } left (t right )= Sum cSub { size 8{k=1} } cSup { size 8{ infinity } } { left (a rSub { size 8{k} } "cos"2πkΔ ital "ft"+b rSub { size 8{k} } "sin"2πkΔ ital "ft" right )= Sum cSub { size 8{k=1} } cSup { size 8{ infinity } } {c rSub { size 8{k} } "cos" left (2πkΔ ital "ft"+θ rSub { size 8{k} } right )} } } {}

Also,

ck2=ak2+bk2ck2=ak2+bk2 size 12{c rSub { size 8{k} } rSup { size 8{2} } =a rSub { size 8{k} } rSup { size 8{2} } +b rSub { size 8{k} } rSup { size 8{2} } } {}  and θk=−tan−1bkakθk=−tan−1bkak size 12{θ rSub { size 8{k} } = - "tan" rSup { size 8{ - 1} } { {b rSub { size 8{k} } } over {a rSub { size 8{k} } } } } {}

  • Two sided power spectrum and mean power spectral density may be derived

Power associated with each spectral term is

c k 2 2 = a k 2 2 + b k 2 2 c k 2 2 = a k 2 2 + b k 2 2 size 12{ { {c rSub { size 8{k} } rSup { size 8{2} } } over {2} } = { {a rSub { size 8{k} } rSup { size 8{2} } } over {2} } + { {b rSub { size 8{k} } rSup { size 8{2} } } over {2} } } {} (1)
Figure 1
Figure 1 (graphics1.png)

The power spectral density at k∆f is

G n kΔf ≡ G n − kΔf ≡ c k 2 4Δf = a k 2 + b k 2 4Δf G n kΔf ≡ G n − kΔf ≡ c k 2 4Δf = a k 2 + b k 2 4Δf size 12{G rSub { size 8{n} } left (kΔf right ) equiv G rSub { size 8{n} } left ( - kΔf right ) equiv { {c rSub { size 8{k} } rSup { size 8{2} } } over {4Δf} } = { {a rSub { size 8{k} } rSup { size 8{2} } +b rSub { size 8{k} } rSup { size 8{2} } } over {4Δf} } } {} (2)

And total power in ∆f at k∆f is

P k = 2G n kΔf Δf P k = 2G n kΔf Δf size 12{P rSub { size 8{k} } =2G rSub { size 8{n} } left (kΔf right )Δf} {} (3)

Half of this power is associated with k∆f and other half with -k∆f

  • The power spectrum above is deterministic in the sense that it has been derived for a specific waveform, and hence the a,b,c values are specific calculable values. In the general case we can treat these as random variables, replace them by the ensemble average values.
  • Let T tend to infinity - and ∆f tend to 0. then the actual noise waveform results.
n t = lim Δf → 0 ∑ k = 1 ∞ a k cos 2πkΔ ft + b k sin 2πkΔ ft = lim Δf → 0 ∑ k = 1 ∞ c k cos 2πkΔ ft + θ k n t = lim Δf → 0 ∑ k = 1 ∞ a k cos 2πkΔ ft + b k sin 2πkΔ ft = lim Δf → 0 ∑ k = 1 ∞ c k cos 2πkΔ ft + θ k size 12{n 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"2πkΔ ital "ft"+b rSub { size 8{k} } "sin"2πkΔ ital "ft" right )= {"lim"} cSub { size 8{Δf rightarrow 0} } Sum cSub { size 8{k=1} } cSup { size 8{ infinity } } {c rSub { size 8{k} } "cos" left (2πkΔ ital "ft"+θ rSub { size 8{k} } right )} } } {} (4)
  • Power contribution from coefficients is now replaced be mean square of the random coefficients which vary with chosen T or ensemble member.

Power spectral density can now be defined from mean square coefficients

  • The Power spectral density then becomes
G n f = lim Δf → 0 c k 2 ¯ 4Δf = lim Δf → 0 a k 2 ¯ + b k 2 ¯ 4Δf G n f = lim Δf → 0 c k 2 ¯ 4Δf = lim Δf → 0 a k 2 ¯ + b k 2 ¯ 4Δf size 12{G rSub { size 8{n} } left (f right )= {"lim"} cSub { size 8{Δf rightarrow 0} } { { {overline {c rSub { size 8{k} } rSup { size 8{2} } }} } over {4Δf} } = {"lim"} cSub { size 8{Δf rightarrow 0} } { { {overline {a rSub { size 8{k} } rSup { size 8{2} } }} + {overline {b rSub { size 8{k} } rSup { size 8{2} } }} } over {4Δf} } } {} (5)
  • The power in the frequency range f1 to f2 is given by
P f 1 → f 2 = ∫ − f2 − f1 G n f df + ∫ f1 f2 G n f df = 2 ∫ f1 f2 G n f df P f 1 → f 2 = ∫ − f2 − f1 G n f df + ∫ f1 f2 G n f df = 2 ∫ f1 f2 G n f df size 12{P left (f rSub { size 8{1} } rightarrow f rSub { size 8{2} } right )= Int rSub { size 8{ - f2} } rSup { size 8{ - f1} } {G rSub { size 8{n} } left (f right ) ital "df"} + Int rSub { size 8{f1} } rSup { size 8{f2} } {G rSub { size 8{n} } left (f right ) ital "df"} =2 Int rSub { size 8{f1} } rSup { size 8{f2} } {G rSub { size 8{n} } left (f right ) ital "df"} } {} (6)
  • And total power PT is
P T = ∫ − ∞ ∞ G n f df = 2 ∫ 0 ∞ G n f df P T = ∫ − ∞ ∞ G n f df = 2 ∫ 0 ∞ G n f df size 12{P rSub { size 8{T} } = Int rSub { size 8{ - infinity } } rSup { size 8{ infinity } } {G rSub { size 8{n} } left (f right ) ital "df"} =2 Int rSub { size 8{0} } rSup { size 8{ infinity } } {G rSub { size 8{n} } left (f right ) ital "df"} } {} (7)

Comments, questions, feedback, criticisms?

Send feedback