Random Signals and Stochastic Systems

Introduction

Deterministic signals are completely specified by mathematical rules or measured records. Many engineering signals, however, contain irreducible uncertainty: thermal noise, atmospheric interference, communication traffic, renewable-energy fluctuations, biomedical activity and sensor errors. Such phenomena are represented by random processes, also called stochastic processes.

A stochastic system transforms random inputs into random outputs. Its analysis therefore concerns statistical quantities—means, correlations, probability distributions and power spectral densities—rather than a single waveform. This framework underpins communication receivers, radar, control, estimation, machine learning, condition monitoring and power-system forecasting.

Learning Objectives

After studying this chapter, the reader should be able to:

• distinguish random variables, random vectors and random processes;
• compute mean, covariance, correlation and complementary covariance functions;
• assess stationarity, cyclostationarity and ergodicity;
• state correct independence conditions for real and complex Gaussian processes;
• relate autocorrelation and power spectral density through the Wiener–Khinchin theorem;
• determine the response of linear time-invariant systems to random inputs;
• analyze white noise, autoregressive processes and state-space stochastic models;
• apply Wiener and Kalman filtering principles;
• estimate statistical quantities from finite data; and
• evaluate practical, ethical and governance issues in stochastic signal processing.

1. Random Processes and Ensemble Description

1.1 From random variables to random signals

A random process X(t,ω) is a family of random variables indexed by time t, where ω identifies an outcome in the underlying probability space. For a fixed outcome, X(t,ω₀) is a sample function or realization. For a fixed time, X(t₀,ω) is a random variable.

A continuous-time process is indexed by t∈ℝ, whereas a discrete-time process is written X[n]. A process may be real or complex valued. Its complete probabilistic specification consists of all finite-dimensional joint distributions, although such a specification is rarely available in engineering practice.

The first-order distribution describes X(t). The second-order distribution describes pairs X(t₁),X(t₂). Mean-square and spectral analyses generally require only second-order statistics.

1.2 Mean, correlation and covariance

For a complex process with finite first moment, the mean is

For finite second moments, the conjugate autocorrelation is

where (·)^* denotes complex conjugation. The autocovariance is

Consequently,

At equal times, K_X(t,t)=E[|X(t)-m_X(t)|²]≥0 is the variance. Its units are the square of the signal unit.

For two complex processes, the cross-covariance is

Complex random processes additionally require the complementary covariance or pseudo-covariance:

The complementary cross-covariance is

These quantities are unnecessary for real processes because they duplicate ordinary covariance information, but they are essential for general complex processes.

1.3 Proper and improper complex processes

A complex process is second-order proper if its complementary covariance vanishes for every pair of times:

Otherwise, it is improper or noncircular. Properness is a second-order property; circular symmetry is a stronger distributional property. For a zero-mean complex Gaussian vector, properness implies circular symmetry.

Impropriety occurs in unbalanced three-phase systems, quadrature modulation errors, seismic measurements and asymmetric complex-valued data. Widely linear processing, which uses both X and X^*, can outperform conventional linear processing for improper data.

2. Stationarity, Cyclostationarity and Ergodicity

2.1 Strict-sense stationarity

A process is strict-sense stationary if every finite-dimensional joint distribution is invariant under a common time shift. Thus, for all integers p, all time points and every shift τ,

Strict stationarity is powerful but often difficult to verify experimentally.

2.2 Wide-sense stationarity

A finite-second-moment process is wide-sense stationary (WSS) if:

1. its mean is constant, m_X(t)=m_X; and
2. its autocorrelation depends only on lag:

For a complex WSS process,

so its autocorrelation is Hermitian. For real processes it is even. Strict stationarity does not guarantee WSS unless second moments exist. Conversely, WSS generally does not imply strict stationarity. A Gaussian process is an important exception: because its finite-dimensional distributions are determined by means and covariance matrices, constant mean and lag-dependent covariance imply strict stationarity. For a general complex Gaussian process, the complementary covariance must also be lag dependent.

2.3 Cyclostationarity

A process is second-order cyclostationary with period T₀ when its mean and correlation repeat periodically:

Modulated communication signals, rotating-machine vibrations and switched power converters commonly exhibit cyclostationarity. Treating them as stationary may discard useful periodic structure.

2.4 Ergodicity

Stationarity concerns invariance of an ensemble; ergodicity concerns whether time averages from one sufficiently long realization equal ensemble averages.

Mean ergodicity requires, under suitable convergence conditions,

in mean square or almost surely, depending on the definition adopted. Correlation ergodicity similarly permits estimation of R_X(τ) from one record.

A WSS process is not automatically ergodic. For example, X(t)=A, where A is a non-degenerate random variable constant over time, is WSS, but the time average of each realization equals its random amplitude rather than the ensemble mean.

3. Gaussian Processes and Independence

A real process is Gaussian if every finite set of samples has a joint Gaussian distribution. A complex process is Gaussian when the corresponding vector of real and imaginary parts is jointly real Gaussian.

For jointly real Gaussian random vectors, zero cross-covariance implies independence. This result must be applied more carefully to complex Gaussian vectors.

Let zero-mean complex Gaussian vectors Z₁ and Z₂ have conjugate cross-covariance K₁₂=E[Z₁Z₂^H] and complementary cross-covariance P₁₂=E[Z₁Z₂^T]. Then

Here (·)^H is conjugate transpose and (·)^T is transpose. If the combined complex Gaussian vector is jointly proper, all complementary covariance blocks vanish, and zero conjugate cross-covariance alone is sufficient.

Worked Example 1: Why zero conjugate covariance may be insufficient

Let x and y be independent zero-mean real Gaussian variables, each with variance σ². Define

Their conjugate cross-covariance is

However, their complementary cross-covariance is

Moreover, z₂=z₁^*, so the variables are deterministically dependent. Therefore, zero conjugate cross-covariance does not establish independence for a general jointly complex Gaussian pair.

4. Spectral Representation of Random Signals

4.1 Power spectral density

A stationary power signal usually has infinite total energy, so its frequency-domain description uses power spectral density (PSD). Under the frequency convention measured in hertz, the Wiener–Khinchin relationship is

with inverse

Here S_X(f) has units of signal-unit squared per hertz. At zero lag,

For a zero-mean process, this is its variance. The PSD is real and nonnegative for an autocorrelation defined with conjugation.

For a discrete-time WSS process,

where ω is radians per sample and the PSD is 2π-periodic.

4.2 Cross-spectral density

For jointly WSS processes,

Generally, S_YX(f)=S_XY^*(f). The magnitude-squared coherence is

where both denominator spectra are positive. It satisfies 0≤γ_XY²(f)≤1 and quantifies frequency-specific linear association, not necessarily causation.

4.3 White noise

Ideal continuous-time white noise has constant two-sided PSD:

Its generalized autocorrelation is

Continuous-time ideal white noise has infinite total power and is a generalized random process rather than an ordinary finite-variance function. Physical noise is always band limited.

Discrete-time white noise satisfies R_W[k]=σ_W²δ[k]. Whiteness means that distinct samples are uncorrelated. It does not generally imply independence. Independence follows for jointly real Gaussian white noise. For complex Gaussian white noise, independence of distinct samples requires both zero conjugate temporal covariance and zero complementary temporal covariance; this is automatically satisfied by a jointly proper complex Gaussian white process with diagonal covariance.

5. Random Inputs to LTI Systems

Consider a continuous-time stable LTI system with impulse response h(t) and input X(t). Its output is

Interchanging expectation and integration requires suitable integrability conditions. If X(t) has constant mean and h∈L¹, then

For a zero-mean WSS input, the output autocorrelation is

Fourier transformation gives the fundamental output-spectrum relation

where H(f) is the frequency response. Mean-square existence requires

Absolute integrability of h(t) is a common sufficient condition for preserving WSS, but the spectral condition can apply more broadly for a specified input.

The input–output cross-spectrum is, under the convention R_XY(τ)=E[X(t+τ)Y^*(t)],

Figure: Statistical propagation through an LTI system.

Worked Example 2: White noise through an RC low-pass filter

An RC filter has

Let the input be zero-mean white voltage noise with two-sided PSD N₀/2 V²/Hz. Then

The output variance is

If N₀=4×10^-12 V²/Hz, R=10 kΩ and C=1 μF, then RC=0.01 s and

Thus σ_Y=10 μV. The finite bandwidth of the filter converts ideal white input noise into a finite-variance output.

6. Discrete-Time Stochastic Models

6.1 Moving-average and autoregressive models

An MA(q) process is

where W[n] is zero-mean white noise. Its PSD is

An AR(p) process satisfies

It is stationary when the poles of

lie strictly inside the unit circle.

For the AR(1) model X[n]=aX[n-1]+W[n], with |a|<1 and innovation variance σ_W²,

and

Worked Example 3: AR(1) process

Let a=0.8 and σ_W²=0.36. Then

Therefore R_X[k]=0.8^|k|, and

The large low-frequency spectral value reflects the positive correlation between successive samples.

6.2 State-space stochastic systems

A linear discrete-time stochastic state model is

where w_k and v_k are commonly modeled as zero-mean white process and measurement noises with covariance matrices Q and R, respectively.

If A is Schur stable and the noise assumptions hold, the steady-state covariance P=E[x_kx_k^H] satisfies the discrete Lyapunov equation

If A is unstable, a finite uncontrolled steady-state covariance generally does not exist, although detectable systems may still support stable estimation through a Kalman filter.

7. Optimal Linear Estimation

7.1 Wiener filtering

Suppose X(t)=S(t)+N(t) is observed, where the desired signal and noise are jointly WSS and uncorrelated. The noncausal linear minimum mean-square-error estimator has frequency response

More generally, when estimating D from X,

where S_X(f)>0. This solution follows from the orthogonality principle: the estimation error is orthogonal to every observation used by the estimator. Real-time causal Wiener filtering requires spectral factorization and is not obtained merely by imposing causality on the noncausal formula.

7.2 Kalman filtering

The Kalman filter is a recursive minimum mean-square-error estimator for linear state-space models with known second-order statistics. Under jointly Gaussian assumptions, it also produces the conditional mean and is optimal among all estimators.

For predicted covariance P_k^-, measurement matrix C and noise covariance R, the gain is

The corrected estimate and covariance are

Numerically robust implementations often use the Joseph covariance form or square-root filtering to preserve symmetry and positive semidefiniteness.

8. Estimation from Finite Data

The ensemble statistics of a process are usually unknown and must be estimated from data. For observations x[0],…,x[N-1], the sample mean is

A biased autocovariance estimate is

A division by N-|k| gives a commonly called unbiased lag estimator, although estimating the mean introduces additional finite-sample bias. The biased autocorrelation sequence has useful positive-semidefinite properties.

The periodogram is

Its frequency resolution improves with record length, but its variance does not generally vanish without smoothing. Bartlett and Welch methods reduce variance by averaging segment periodograms, at the cost of resolution. Window choice controls spectral leakage and main-lobe width.

Best practice includes detrending, checking stationarity, documenting sampling frequency, preventing aliasing, reporting confidence intervals and validating model residuals for whiteness.

9. Applied Case: Kalman Filtering of an Industrial Temperature Sensor

Background

This is an illustrative case. A chemical reactor uses a digital temperature sensor sampled once per second. The underlying temperature varies slowly, while electrical noise causes visibly fluctuating measurements.

Problem

A raw sensor reading has measurement-noise variance r=0.25 °C², corresponding to a standard deviation of 0.5 °C. Excessive smoothing would delay fault detection, whereas insufficient smoothing would generate false alarms.

Model and analysis

The temperature is modeled locally as a random walk:

where q=E[w_k²]=0.04 °C² per sample and r=E[v_k²]=0.25 °C². The random walk itself is not WSS; its uncertainty grows without measurement correction. The Kalman filter remains applicable because it explicitly propagates time-varying covariance.

At steady state, if P is the posterior error variance, then P^-=P+q and

Rearrangement gives

The physically meaningful positive solution is

Thus P^-=0.12198 °C² and

Approach and outcome

Each corrected estimate uses approximately 32.8% of the new measurement innovation and retains the remaining influence from the prediction. The posterior RMS error is

compared with 0.5 °C for the raw measurement noise. This represents about a 43% reduction in RMS uncertainty under the model assumptions.

Learning points

• Model parameters determine the responsiveness–noise trade-off.
• A random-walk state is nonstationary, but recursive Bayesian estimation remains valid.
• Residual innovations should be tested for whiteness and bias.
• Sensor calibration, drift monitoring and independent safety interlocks remain necessary; a filter cannot correct an invalid physical model or a failed sensor.

10. Limitations, Governance and Contemporary Practice

Stochastic models simplify reality. Gaussian assumptions may underestimate impulsive interference or heavy-tailed failures. Stationarity can be violated by aging equipment, climate variation, changing network loads or adaptive adversaries. Independence assumptions may fail because sensors share power supplies, environments or preprocessing pipelines.

Model validation should therefore include residual analysis, distributional checks, change-point detection and out-of-sample testing. Robust filters, particle filters and Gaussian-process models are appropriate when linearity or Gaussianity is inadequate, though they increase computational and modeling complexity.

Data governance matters when random-signal analysis involves human speech, medical recordings, location traces or smart-meter data. Engineers should apply data minimization, access control, retention limits and transparent uncertainty reporting. Automated decisions should not conceal uncertainty behind a single filtered estimate.

Machine learning increasingly supports spectral anomaly detection and data-driven state estimation. Neural estimators can approximate nonlinear relationships, but they do not remove the need to characterize sampling, noise, distribution shift and confidence. Hybrid methods combining physical state-space models with learned noise statistics often provide better interpretability and reliability.

Chapter Summary

Random signals are described through ensembles, distributions and statistical moments. Mean, covariance and autocorrelation characterize second-order behavior, while complementary covariance is indispensable for general complex processes. For jointly complex Gaussian vectors, independence requires both conjugate and complementary cross-covariances to vanish; conjugate covariance alone suffices only under additional properness conditions.

Stationarity provides time-invariant statistics, whereas ergodicity justifies estimating ensemble quantities from long records. The Wiener–Khinchin theorem connects autocorrelation to power spectral density. Stable LTI systems shape random-signal spectra according to S_Y=|H|²S_X. AR, MA and state-space models provide practical representations of correlated signals. Wiener and Kalman filters exploit second-order statistics for minimum mean-square-error estimation. Reliable engineering use additionally requires finite-data awareness, model validation, uncertainty communication and responsible data governance.

Key Terms

Random process: A time-indexed family of random variables.

Realization: One sample function produced by a random process.

Autocorrelation: Expected conjugate product of two samples of the same process.

Autocovariance: Autocorrelation after removal of the mean contribution.

Complementary covariance: Nonconjugated covariance required to characterize complex second-order structure.

Proper process: A complex process with identically zero complementary covariance.

Strict-sense stationarity: Invariance of all finite-dimensional distributions under time shifts.

Wide-sense stationarity: Constant mean and lag-dependent autocorrelation.

Cyclostationarity: Periodic variation of process statistics.

Ergodicity: Equality of appropriate long-time and ensemble averages.

Gaussian process: A process whose finite sample sets are jointly Gaussian.

Power spectral density: Distribution of average signal power over frequency.

White noise: A process with a flat PSD and delta autocorrelation.

Coherence: Normalized frequency-dependent linear association between two processes.

AR process: A process modeled from past samples plus white innovation.

State covariance: Matrix describing second-order uncertainty of a stochastic state.

Wiener filter: Linear minimum mean-square-error filter for stationary data.

Kalman filter: Recursive state estimator for dynamic stochastic models.

Innovation: Difference between a measurement and its predicted value.

Periodogram: Squared-magnitude Fourier-based estimate of a PSD.

Review Questions

Short Answer Questions

1. Distinguish a realization from an ensemble of a random process.
2. State the two conditions for wide-sense stationarity.
3. Why does whiteness not generally imply independence?
4. Define complementary covariance for a complex process.
5. What is the physical interpretation of a power spectral density?

Long Answer Questions

1. Explain strict stationarity, wide-sense stationarity, cyclostationarity and ergodicity, including their logical distinctions.
2. Derive the output PSD of an LTI system driven by a WSS random input.
3. Discuss the statistical characterization of proper and improper complex Gaussian processes.
4. Derive the variance, autocorrelation and PSD of a stationary AR(1) process.
5. Explain Wiener and Kalman filtering, emphasizing assumptions, optimality and limitations.

Analytical / Application Questions

1. White noise with two-sided PSD 2×10^-9 V²/Hz enters an ideal low-pass filter of unity gain over |f|≤5 kHz. Calculate the output variance and RMS voltage.
2. For X[n]=0.6X[n-1]+W[n] with σ_W²=0.64, determine the stationary variance, lag-three autocovariance and PSD.
3. Design a validation plan for a Kalman filter used in safety-critical industrial monitoring, including statistical, operational and governance tests.

Multiple Choice Questions with Answers

Question 1

Which condition is required for a finite-second-moment process to be wide-sense stationary?

A. Every realization must be periodic.

B. Its mean must be constant and its autocorrelation must depend only on lag.

C. Its samples must be statistically independent.

D. Its probability density must be Gaussian.

Correct Answer: B. Its mean must be constant and its autocorrelation must depend only on lag.

Question 2

For two jointly complex Gaussian vectors, which condition guarantees their independence in the general, possibly improper case?

A. Only their conjugate cross-covariance is zero.

B. Only their complementary cross-covariance is zero.

C. Both conjugate and complementary cross-covariances are zero.

D. Their individual variances are equal.

Correct Answer: C. Both conjugate and complementary cross-covariances are zero.

Question 3

For the stationary AR(1) process X[n]=0.8X[n-1]+W[n] with σ_W²=0.36, what is σ_X²?

A. 0.36

B. 0.64

C. 0.80

D. 1.00

Correct Answer: D. 1.00

Question 4

A WSS input with PSD S_X(f) passes through an LTI system with frequency response H(f). What is the output PSD?

A. H(f)S_X(f)

B. |H(f)|²S_X(f)

C. S_X(f)/|H(f)|²

D. H^*(f)/S_X(f)

Correct Answer: B. |H(f)|²S_X(f)

Question 5

White voltage noise with N₀=4×10^-12 V²/Hz passes through an RC low-pass filter with RC=0.01 s. What is the output variance?

A. 1×10^-12 V²

B. 4×10^-11 V²

C. 1×10^-10 V²

D. 4×10^-10 V²

Correct Answer: C. 1×10^-10 V²

Question 6

Which statement about ergodicity is correct?

A. Every WSS process is ergodic.

B. Ergodicity allows appropriate ensemble statistics to be inferred from long time averages.

C. Ergodicity requires all samples to be independent.

D. Only deterministic signals can be ergodic.

Correct Answer: B. Ergodicity allows appropriate ensemble statistics to be inferred from long time averages.

Question 7

A second-order proper complex process satisfies which condition?

A. Its conjugate covariance is zero at every lag.

B. Its mean must equal zero.

C. Its complementary covariance is zero at every pair of times.

D. Its real and imaginary parts must both be deterministic.

Correct Answer: C. Its complementary covariance is zero at every pair of times.

Question 8

For the stable stochastic state model x_k+1=Ax_k+Gw_k, which equation determines the steady-state state covariance?

A. P=A^HPA+Q

B. P=APA^H+GQG^H

C. P=AP+GQ

D. P=A^-1QA^-H

Correct Answer: B. P=APA^H+GQG^H

Question 9

For observation X=S+N, where S and N are uncorrelated WSS processes, what is the noncausal Wiener filter for estimating S?

A. S_N/(S_S+S_N)

B. S_S/S_N

C. S_S/(S_S+S_N)

D. (S_S+S_N)/S_S

Correct Answer: C. S_S/(S_S+S_N)

Question 10

A discrete-time process is jointly proper complex Gaussian and has R_X[k]=σ²δ[k]. What follows for samples at distinct times?

A. They are independent because joint properness makes complementary temporal covariances zero and whiteness makes conjugate covariances zero.

B. They must be identical because their variances are equal.

C. They are uncorrelated but can never be independent.

D. They are deterministic complex conjugates.

Correct Answer: A. They are independent because joint properness makes complementary temporal covariances zero and whiteness makes conjugate covariances zero.