Random Variables

Single Random Variables

  • Random Variables (A.pdf)
  • Distribution Function (A.pdf)
  • Density Function (A.pdf)
  • Gaussian Random Variables (A.pdf, B.pdf)
  • Other Distribution and Density Functions (A.pdf)
  • Conditional Distribution and Density Functions (A.pdf)


The Characteristics of a Single Random Variable

  • Expected Value (A.pdf)
  • Moments (A.pdf, B.pdf)
  • Moment Generating Functions (A.pdf)
  • Transformations of Random Variables (A.pdf)


Multiple Random Variables

  • Vector Random Variables (A.pdf)
  • Joint Distribution (A.pdf)
  • Joint Density (A.pdf)
  • Conditional Joint Distribution and Density (A.pdf)
  • Statistical Independence (A.pdf)
  • Sums of Random Variables (A.pdf)
  • Central Limit Theorem (A.pdf)


The Characteristics of Multiple Random Variables

  • Expected Values (A.pdf)
  • Joint Characteristic Functions (A.pdf)
  • Jointly Gaussian Random Variables (A.pdf)
  • Transformations of Multiple Random Variables (A.pdf)
  • Linear Transformation of Gaussian Random Variables (A.pdf)
  • Simulating Multiple Random Variables (A.pdf)
  • Sampling and Some Limit Theorem (A.pdf)
  • Complex Random Variables (A.pdf)


Random Processes

The Temporal Characteristics of Random Processes

  • Random Processes (A.pdf)
  • Joint Distribution, Independence, Processes (A.pdf)
  • Stationarity (A.pdf)
  • Example Random Processes (A.pdf)
  • Time Averages and Ergodicity (A.pdf)
  • Measurement of Correlation Functions (A.pdf)
  • Complex Random Processes (A.pdf)


The Spectral Characteristics of Random Processes

  • Power Density Spectrum - Continuous Time (A.pdf)
  • Auto Correlation Function (A.pdf)
  • Power Density Spectrum - Discrete Time (A.pdf)
  • Cross Power Density Spectrum (A.pdf)
  • Cross Correlation Function (A.pdf)
  • Noise Definitions (A.pdf)
  • Power Spectrum of Complex Random Processes (A.pdf)


Linear System with Random Inputs

  • Continuous Time LTI System (A.pdf)
  • Discrete Time LTI System (A.pdf)
  • System Response (A.pdf)
  • Spectral Characteristics (A.pdf)
  • Noise Modeling (A.pdf)


Optimum Linear System


Noise in Some Application Systems

  • AM Communication Systems (A.pdf)
  • FM Communication Systems (A.pdf)
  • Control Systems (A.pdf)
  • PLL Systems (A.pdf)
  • Random Waveforms (A.pdf)
  • Radar Systems (A.pdf)


Correlation and Power Spectra

  1. Correlation Functions of Random Signals (pdf)
  2. Spectra of Random Signals (pdf)


Ergodicity, Statistics, Estimation


Random Processes and Linear Systems



  1. Time Domain Techniques for Noisy Signals (A.pdf, B.pdf)
  2. Frequency Domain Techniques for Noisy Signals
  3. Correlation v.s. Convolution for Noisy Signals



  1. System Identification (pdf)
  2. Matched Filter [1]



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  1. ↑ Understanding Digital Communications, See Baseband Mod/Demod Section
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