Characterization of Noise: Understanding Different Types and Mathematical Description, Study notes of Statistics

Download Characterization of Noise: Understanding Different Types and Mathematical Description and more Study notes Statistics in PDF only on Docsity! Connexions module: m17196 1 Characterization of noise ∗ a.i. trivedi This work is produced by The Connexions Project and licensed under the Creative Commons Attribution License † Abstract Noise is described in terms of its power spectral density 1 Types of noise: • Thermal resistor noise: due to thermal agitation. Statistical uctuations. Always random, erratic and unpredictable. • Shot noise: uctuations 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 ampliers. 2 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 dierent 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. rectier. 3 Mathematical characterization: Frequency domain description can be derived for periodic and pulse-like aperiodic phenomena. • Noise is not repetitive and innite 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 (s) T (t) = ∑∞ k=1 (akcos2πk∆ft + bksin2πk∆ft) = ∑∞ k=1 ckcos (2πk∆ft + θk) Also, c2k = a 2 k + b 2 k and θk = −tan−1 bk ak ∗Version 1.1: Jul 7, 2008 3:57 am GMT-5 †http://creativecommons.org/licenses/by/2.0/ http://cnx.org/content/m17196/1.1/ Connexions module: m17196 2 • Two sided power spectrum and mean power spectral density may be derived Power associated with each spectral term is c2k 2 = a2k 2 + b2k 2 (1) Figure 1 The power spectral density at k∆f is Gn (k∆f) ≡ Gn (−k∆f) ≡ c2k 4∆f = a2k + b 2 k 4∆f (2) And total power in ∆f at k∆f is Pk = 2Gn (k∆f) ∆f (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 specic waveform, and hence the a,b,c values are specic calculable values. In the general case we can treat these as random variables, replace them by the ensemble average values. • Let T tend to innity - and ∆f tend to 0. then the actual noise waveform results. n (t) = lim ∆f→0 ∞∑ k=1 (akcos2πk∆ft + bksin2πk∆ft) = lim ∆f→0 ∞∑ k=1 ckcos (2πk∆ft + θk) (4) • Power contribution from coecients is now replaced be mean square of the random coecients which vary with chosen T or ensemble member. http://cnx.org/content/m17196/1.1/