Unit 5 - Notes

ECE180

Unit 5: Stochastic Processes

1. The Stochastic Process Concept

A Stochastic Process (or Random Process) is a mathematical model used to describe systems that evolve probabilistically over time. Unlike a single random variable which maps an outcome to a number, a stochastic process maps an outcome to a function of time.

1.1 Definition

Let $S$ be the sample space of a random experiment. For every outcome $\zeta \in S$ , we assign a function of time $X(t, \zeta)$ . The family of all such functions is called a stochastic process, denoted as $X(t)$ .

1.2 Interpretations of $X(t, \zeta)$

Depending on which variables are fixed, $X(t, \zeta)$ represents different things:

$t$ fixed, $\zeta$ fixed: A single number (a specific value at a specific time).
$t$ fixed, $\zeta$ variable: A Random Variable (the state of the process at time $t$ ).
$t$ variable, $\zeta$ fixed: A Sample Path or Realization (a single deterministic waveform).
$t$ variable, $\zeta$ variable: The Stochastic Process itself.

2. Classification of Processes

Stochastic processes are classified based on the nature of the time parameter ( $t$ ) and the random variable values ( $X$ ).

2.1 Continuous vs. Discrete

Continuous-Time, Continuous-State: $t$ is continuous, and $X(t)$ can take any value in a continuous range. (e.g., Thermal noise voltage).
Discrete-Time, Continuous-State: $t$ exists only at integers ( $t_1, t_2, \dots$ ), but $X(t)$ is continuous. Also called a Random Sequence. (e.g., Sampled speech signal).
Continuous-Time, Discrete-State: $t$ is continuous, but $X(t)$ assumes discrete values. (e.g., A random telegraph signal).
Discrete-Time, Discrete-State: Both time and amplitude are discrete. (e.g., Outcome of a coin toss sequence).

2.2 Deterministic and Nondeterministic Processes

Deterministic Process: A process is deterministic if future values of any sample function can be predicted exactly from past values.
- Example: $X(t) = A \cos(\omega t + \Theta)$ , where $A$ and $\omega$ are constants and $\Theta$ is a random variable. Once a realization occurs (fixing $\Theta$ ), the entire future is known.
Nondeterministic (Random) Process: A process where future values cannot be exactly predicted from observed past values. There is always inherent uncertainty.
- Example: Background static noise in a radio receiver.

3. Distribution and Density Functions

Since a stochastic process is a family of random variables, it is characterized by probability distribution functions (CDFs) and probability density functions (PDFs).

3.1 First-Order Distribution

For a specific time $t_1$ , $X(t_1)$ is a random variable.

CDF: $F_X(x_1; t_1) = P\{X(t_1) \le x_1\}$
PDF: $f_X(x_1; t_1) = \frac{\partial F_X(x_1; t_1)}{\partial x_1}$

3.2 Second-Order Distribution

For two specific times $t_1$ and $t_2$ :

Joint CDF: $F_X(x_1, x_2; t_1, t_2) = P\{X(t_1) \le x_1, X(t_2) \le x_2\}$
Joint PDF: $f_X(x_1, x_2; t_1, t_2) = \frac{\partial^2 F_X(x_1, x_2; t_1, t_2)}{\partial x_1 \partial x_2}$

3.3 Nth-Order Distribution

Ideally, a process is fully described by its $N$ -th order joint PDF for any $N$ and any set of times $t_1, \dots, t_N$ .

3.4 Statistical Independence

Two processes $X(t)$ and $Y(t)$ are statistically independent if the joint distribution of any group of random variables from $X(t)$ and $Y(t)$ factorizes into the product of their marginal distributions.

Simplest case (First order independence):
$f_{XY}(x, y; t_1, t_2) = f_X(x; t_1) \cdot f_Y(y; t_2)$

4. Stationarity

Stationarity refers to the time-invariance of the statistical properties of a process.

4.1 Strict-Sense Stationarity (SSS)

A process is SSS if its statistical properties are invariant to a shift in the time origin.

f_X(x_1, \dots, x_n; t_1, \dots, t_n) = f_X(x_1, \dots, x_n; t_1+\Delta, \dots, t_n+\Delta)

for any time shift

\Delta

Consequence: The first-order PDF is independent of time ( $f_X(x; t) = f_X(x)$ ), implying the mean and variance are constants.

4.2 Wide-Sense Stationarity (WSS)

SSS is difficult to prove in practice. WSS is a weaker, more practical condition based only on the first two moments. A process $X(t)$ is WSS if:

Constant Mean: The expected value is constant for all $t$ .
$E[X(t)] = \bar{X} = \text{constant}$
Autocorrelation depends only on time difference: The autocorrelation function depends only on $\tau = t_2 - t_1$ .
$E[X(t)X(t+\tau)] = R_{XX}(\tau)$

Note: All SSS processes are WSS (provided second moments exist), but not all WSS processes are SSS. Gaussian processes are the exception where WSS implies SSS.

5. Ergodicity

Ergodicity deals with the relationship between Ensemble Averages (statistical expectations) and Time Averages.

5.1 Concept

In many physical measurements, we cannot observe the infinite ensemble of a process. We only observe one sample path $x(t)$ over time. A process is Ergodic if time averages of a single sample path converge to the corresponding ensemble averages as the observation time goes to infinity.

5.2 Mean-Ergodic Processes

A process is mean-ergodic if the time average equals the ensemble mean.

Time Average: $\hat{\mu}_T = \frac{1}{2T} \int_{-T}^{T} x(t) dt$
Ergodic Condition: $\lim_{T \to \infty} \hat{\mu}_T = E[X(t)] = \bar{X}$

For WSS processes, a sufficient condition for mean ergodicity is related to the autocovariance $C_{XX}(\tau)$ :

\lim_{T \to \infty} \frac{1}{T} \int_{0}^{2T} \left(1 - \frac{\tau}{2T}\right) C_{XX}(\tau) d\tau = 0

Generally simplified: If

R_{XX}(\tau) \to (\bar{X})^2

\tau \to \infty

, the process is often mean-ergodic.

6. Correlation and Covariance Functions

These functions measure the dependence between values of the process at different times.

6.1 Autocorrelation Function ( $R_{XX}$ )

Describes the correlation of a process with itself at two different times $t_1$ and $t_2$ .

R_{XX}(t_1, t_2) = E[X(t_1)X(t_2)]

Properties for WSS Processes ( $R_{XX}(\tau)$ ):

Maximum at Origin: $|R_{XX}(\tau)| \le R_{XX}(0)$ .
Even Symmetry: $R_{XX}(-\tau) = R_{XX}(\tau)$ .
Average Power: $R_{XX}(0) = E[X^2(t)]$ (Mean Square Value).
Periodicity: If $X(t)$ is periodic, $R_{XX}(\tau)$ is periodic with the same period.

6.2 Autocovariance Function ( $C_{XX}$ )

Measures the correlation of the centered process (mean removed).

C_{XX}(t_1, t_2) = E[(X(t_1) - \mu(t_1))(X(t_2) - \mu(t_2))]

Relationship: $C_{XX}(t_1, t_2) = R_{XX}(t_1, t_2) - \mu(t_1)\mu(t_2)$ .
For WSS: $C_{XX}(\tau) = R_{XX}(\tau) - \bar{X}^2$ .
Variance: $C_{XX}(0) = \sigma_X^2$ .

6.3 Cross-Correlation Function ( $R_{XY}$ )

Describes the correlation between two different processes $X(t)$ and $Y(t)$ .

R_{XY}(t_1, t_2) = E[X(t_1)Y(t_2)]

If jointly WSS,

R_{XY}(\tau) = E[X(t)Y(t+\tau)]

Properties:

Symmetry: $R_{XY}(\tau) = R_{YX}(-\tau)$ .
Bounds: $|R_{XY}(\tau)| \le \sqrt{R_{XX}(0)R_{YY}(0)}$ .
Orthogonality: If $R_{XY}(\tau) = 0$ for all $\tau$ , processes are orthogonal.
Uncorrelated: If $C_{XY}(\tau) = 0$ (implying $R_{XY}(\tau) = \bar{X}\bar{Y}$ ), processes are uncorrelated.

7. Power Spectrum (Power Spectral Density - PSD)

The PSD describes how the power of a random process is distributed across frequency.

7.1 Definition

For a WSS process $X(t)$ , the Power Spectral Density $S_{XX}(\omega)$ is the Fourier Transform of the Autocorrelation function.

S_{XX}(\omega) = \int_{-\infty}^{\infty} R_{XX}(\tau) e^{-j\omega\tau} d\tau

7.2 Properties of Power Spectrum

Real and Non-negative: $S_{XX}(\omega) \ge 0$ for all $\omega$ .
Even Function: $S_{XX}(\omega) = S_{XX}(-\omega)$ (for real-valued processes).
Total Average Power: The area under the PSD curve represents the total average power of the process.
$P_{avg} = \frac{1}{2\pi} \int_{-\infty}^{\infty} S_{XX}(\omega) d\omega = R_{XX}(0)$
Power in a Band: The power within frequency band $\omega_1$ to $\omega_2$ is given by integrating $S_{XX}(\omega)$ over that band.

7.3 Relationship between Power Spectrum and Autocorrelation (Wiener-Khinchin Theorem)

The PSD and the Autocorrelation function form a Fourier Transform pair:

S_{XX}(\omega) \xrightarrow{\mathcal{F}} R_{XX}(\tau)

Forward: $S_{XX}(\omega) = \int_{-\infty}^{\infty} R_{XX}(\tau) e^{-j\omega\tau} d\tau$
Inverse: $R_{XX}(\tau) = \frac{1}{2\pi} \int_{-\infty}^{\infty} S_{XX}(\omega) e^{j\omega\tau} d\omega$

This is a fundamental theorem linking the time-domain statistics (correlation) to frequency-domain characteristics (spectrum).

8. Cross-Power Density Spectrum

The Cross-PSD describes the power spectral relationship (interaction) between two processes $X(t)$ and $Y(t)$ .

8.1 Definition

The Cross-Power Spectrum $S_{XY}(\omega)$ is the Fourier Transform of the Cross-correlation function $R_{XY}(\tau)$ .

S_{XY}(\omega) = \int_{-\infty}^{\infty} R_{XY}(\tau) e^{-j\omega\tau} d\tau

8.2 Properties of Cross-Power Spectrum

Complex Valued: Unlike Auto-PSD, $S_{XY}(\omega)$ is generally complex.
Conjugate Symmetry: $S_{XY}(\omega) = S_{YX}^*(\omega)$ (where $*$ denotes complex conjugate).
Orthogonal Processes: If $X(t)$ and $Y(t)$ are orthogonal, $S_{XY}(\omega) = 0$ .
Uncorrelated Processes: If $X(t)$ and $Y(t)$ are uncorrelated with non-zero means, $S_{XY}(\omega) = 2\pi \bar{X}\bar{Y} \delta(\omega)$ .

8.3 Relationship between Cross-Power Spectrum and Cross-Correlation

They form a Fourier Transform pair:

$S_{XY}(\omega) = \mathcal{F}[R_{XY}(\tau)]$
$R_{XY}(\tau) = \mathcal{F}^{-1}[S_{XY}(\omega)] = \frac{1}{2\pi} \int_{-\infty}^{\infty} S_{XY}(\omega) e^{j\omega\tau} d\omega$

This relationship allows the analysis of how frequency components of one process relate to the frequency components of another process.