S4-5.3 Stochastic Processes: Temporal & Spectral

Work through these

• The random process concept and classification of processes

  A stochastic process is a random variable that evolves in time, which is the model for every real signal carrying information or noise. Classification sorts them by whether time and amplitude are continuous or discrete.

  NPTEL: Probability and Stochastic Processes · Course

• Stationarity: distribution/density functions, wide-sense and strict-sense

  Stationarity says the statistics do not drift as time passes, and the wide-sense version asks only that the mean and autocorrelation behave. Almost every result later assumes it, so knowing what it does and does not require matters.

• Time averages and ergodicity

  Ergodicity says one long recording reveals the same statistics as many short ones taken across the ensemble. It is the assumption that makes measurement possible at all, since in practice you only ever have one recording.

• Autocorrelation function and its properties

  Autocorrelation measures how much a signal resembles a delayed copy of itself, and its properties encode almost everything about the process. It is one half of the central result of this unit.

• Cross-correlation and covariance functions; Gaussian random process

  Cross-correlation compares two different signals rather than one with itself, and covariance removes the means before comparing. The Gaussian process is included here because it is the case where these two functions describe it completely.

• Power density spectrum and its properties

  The power density spectrum says how a signal's power is distributed across frequency, which is what a spectrum analyser draws. Its properties follow from the fact that power cannot be negative.

• Relationship between power spectrum and autocorrelation

  The Wiener-Khinchin relationship says autocorrelation and power spectrum are a transform pair, so correlation in time and power in frequency are two views of one object. This is the central theorem of the unit.

• Cross-power density spectrum and its relation to cross-correlation

  The same pairing extended to two signals: cross-correlation in time corresponds to cross-power spectrum in frequency. It is the tool for questions about how two signals share content.