Random Processes
What is Random Processes?
A collection of random variables indexed by time or another parameter, describing a system's probabilistic evolution.
Key formula / rule: Mean Function
Key points
- Understand the definition and properties of random processes.
- Differentiate between stationary and non-stationary processes.
- Analyze the statistical behavior of signals using autocorrelation and power spectral density.
- Apply concepts of random processes to model real-world phenomena.
Common exam trap
Confusing a random variable with a random process.
Definitions
- Term
Random Process
- Meaning
A collection of random variables indexed by time or another parameter, describing a system's probabilistic evolution.
- Term
Stationary Process
- Meaning
A process whose statistical properties (like mean and variance) do not change over time.
- Term
Wide-Sense Stationary (WSS)
- Meaning
A process where the mean is constant and the autocorrelation depends only on the time difference (lag).
- Term
Ergodic Process
- Meaning
A stationary process where time averages are equal to ensemble averages.
- Term
White Noise
- Meaning
A random process with a constant power spectral density over all frequencies.
- Term
Autocorrelation Function
- Meaning
A function that measures the similarity between a signal and a delayed copy of itself as a function of the delay.
- Term
Power Spectral Density (PSD)
- Meaning
The distribution of the power of a signal or process over frequency.
Learning objectives
Understand the definition and properties of random processes.
Differentiate between stationary and non-stationary processes.
Analyze the statistical behavior of signals using autocorrelation and power spectral density.
Apply concepts of random processes to model real-world phenomena.
Formulae
- Name
Mean Function
- Note
Expected value of the random process at time t.
- Expression
μX(t) = E[X(t)]
- Name
Autocorrelation Function
- Note
Statistical relationship between values separated by time lag τ.
- Expression
RX(τ) = E[X(t)X(t+τ)]
- Name
Power Spectral Density (PSD)
- Note
Fourier Transform of the Autocorrelation Function for WSS processes.
- Expression
SX(f) = ∫_{-∞}^{∞} RX(τ)e-j2πfτ dτ
- Name
Inverse PSD
- Note
Inverse Fourier Transform to get Autocorrelation from PSD.
- Expression
RX(τ) = ∫_{-∞}^{∞} SX(f)ej2πfτ df
- Name
Wiener-Khinchin Theorem
- Note
RX(τ) ↔ SX(f)
- Expression
Relates Autocorrelation and Power Spectral Density for WSS processes.
Prerequisites
Probability and Random Variables
Basic Calculus
Fourier Series and Transforms
Linear Systems
Common mistakes
Confusing a random variable with a random process.
Assuming stationarity or ergodicity without justification.
Incorrectly calculating autocorrelation or power spectral density.
Misinterpreting the implications of white noise.
Keywords
Random Process
Stochastic Process
Stationarity
Ergodicity
Autocorrelation
Power Spectral Density
White Noise
Wiener-Khinchin Theorem
Mean Function
Variance
Practice preview
If X(t) is a wide-sense stationary random process with autocorrelation R_X(τ) = Pδ(τ), what is its power spectral density S_X(f)?…
hard
Which of the following is a property of a wide-sense stationary (WSS) random process X(t)?…
easy
Consider a random process X(t) = A cos(ωt + Θ), where A and ω are constants, and Θ is a random variable uniformly distributed in [0, 2π]. Is X(t) wide-sense stationary?…
medium