Real signals often appear to contain random fluctuations or other types of uncertainty. In this lesson you will be introduced to the tools of probability and statistics that are used to model uncertainty in signals. You will learn that summary statistics are often used in signal processing to avoid the complexity of working with the Read More

## Multivariable Random Signal Characterization

The majority of signals that you will model as random have multiple values, that is, they are multivariate. Examples include multiple samples of a time signal and signals measured at different sensors. In this lesson you will learn the tools used to jointly characterize the random behavior of multiple signal components. You will learn how Read More

## Random Processes and Stationarity

In this lesson you will learn definitions of a random process and stationarity. You will also learn how second-order statistics – the covariance and correlation functions – characterize stationary random processes. This understanding will position you to apply the large number of signal-processing methods that assume stationarity and use covariance to characterize signals. Prerequisites Multivariable Read More

## The Power Spectral Density

The power spectral density characterizes a stationary process in the frequency domain. In this lesson you will learn the definition and properties of the power spectral density and the relationship between the power spectral density and the covariance sequence for a random process. You will also learn how a linear time-invariant system modifies the power Read More

## Cross Spectra and Coherence

The cross spectrum and coherence are used to study the relationship between two signals in the frequency domain. In this lesson you will learn the definitions of cross spectra and coherence, and the relationship between correlation and the cross spectrum. You will also learn why the coherence is one at all frequencies when the signals Read More

## LTI System Models for Random Signals

System models provide a concise description for random signals. In this lesson you will learn how to model a time series as the output of a linear time-invariant system with a random input signal. You will learn about the three commonly used models: autoregressive, moving average, and autoregressive moving average, and the types of random Read More

## Autoregressive Models: The Yule-Walker Equations

Autoregressive models for time series are widely used because of their simplicity and their applicability to resonant phenomena. In this lesson you will learn how the Yule-Walker equations relate the autoregressive model parameters to the autocovariance of the time series. The autoregressive model parameters are obtained from the autocovariance of the time series by solving Read More