Characterization of noise 1.1 2008/03/28 05:01:31.445 GMT-5 2008/07/07 03:57:44.819 GMT-5 ajay i. trivedi ai.trivedi@gmail.com ajay i. trivedi ai.trivedi@gmail.com noise spectral density 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 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+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−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 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} } } {}

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 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} } } {} And total power in ∆f at k∆f is P k = 2G n kΔf Δf size 12{P rSub { size 8{k} } =2G rSub { size 8{n} } left (kΔf right )Δf} {} 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 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 )} } } {} 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 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} } } {} 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 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"} } {} And total power PT is 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"} } {}