0.1 Characterization of noise

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.

$n_T^{(s)}\left(t\right)=\sum _{k=1}^{\infty }\left(a_k\text{cos}\mathrm{2\pi k\Delta }\text{ft}+b_k\text{sin}\mathrm{2\pi k\Delta }\text{ft}\right)=\sum _{k=1}^{\infty }{c}_k\text{cos}\left(\mathrm{2\pi k\Delta }\text{ft}+{\theta }_k\right)$

Also,

$c_k^2=a_k^2+b_k^2$  and ${\theta }_k=-{\text{tan}}^{-1}\frac{b_k}{a_k}$

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

Power associated with each spectral term is

The power spectral density at k∆f is

And total power in∆f at k∆f is

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.

• 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

• The power in the frequency range f1 to f2 is given by

• And total power PT is