Wide-Sense Stationarity (WSS) A process {X t } is Wide-Sense Stationary (or covariance stationary) if it possesses a finite second moment (E[X t 2 ] Constant Mean: E[X t ] = μ for all t . Time-Invariant Covariance: The autocovariance between any two observations depends solely on the lag τ: Cov(X t , X t+τ ) = γ(τ). (This implies a constant variance, where Var(X t ) = γ(0)). White Noise (WN) A process {ε t } is defined as white noise if it is a WSS process with E[ε t ] = 0, constant variance σ 2 , and Cov(ε t , ε t+τ ) = 0 for all τ ≠ 0. If the variables are also independent and identically distributed (I.I.D.), it is termed "Independent White Noise." LTI Systems and Discrete Convolution A stationary time series is characterized as the output of a Linear Time-Invariant (LTI) system driven by...

Output Statistics of an LTI System for WSS Input | Signal Processing Guide Statistical Properties of LTI System Output for Wide-Sense Stationary (WSS) Inputs In signal processing and communication theory, understanding how a Linear Time-Invariant (LTI) system transforms the statistical characteristics of a random process is fundamental. This guide explores the output response when the input is a Wide-Sense Stationary (WSS) process. System Definition: Suppose an input signal x(t) is a Wide-Sense Stationary (WSS) random process applied to an LTI system with an impulse response h(t) . The output is defined by the convolution integral: y(t) = x(t) * h(t) = ∫ x(t - τ) h(τ) dτ 1. LTI System Output Statistics Output Mean (Expected Value) The mean of the output process is a constant scaled by the system's DC gain. μ y = E[y(t)] = μ x H(0) Where H(0) = ∫ h(t) dt represents the frequency response at zero frequenc...

LTI System (Linear Time-Invariant System): Definition and Examples LTI System (Linear Time-Invariant System) Definition An LTI (Linear Time-Invariant) system is a system that concurrently satisfies the linearity and time-invariance properties. These systems are fundamental in signal processing and control engineering because their behavior can be completely characterized by their impulse response. Mathematical Representation An LTI system is generally represented by the operator \( T \), which transforms an input signal \( x(t) \) into an output signal \( y(t) \): \[ y(t) = T\{x(t)\} \] Where: \( x(t) \) = Input signal \( y(t) \) = Output signal \( T\{\cdot\} \) = System operator 1. Linearity Property ...

ARMA Process and Wide-Sense Stationarity (WSS): A Comprehensive Guide ARMA Process and Wide-Sense Stationarity (WSS) Core Concept: An ARMA (AutoRegressive Moving Average) process is Wide-Sense Stationary (WSS) if and only if the system's autoregressive part is stable and the input is a stationary white noise process. 1. The ARMA(p, q) Model Equation A stochastic process {X t } follows an ARMA( p, q ) model if it satisfies the following linear difference equation: X t = φ 1 X t-1 + φ 2 X t-2 + ... + φ p X t-p + a t + θ 1 a t-1 + θ 2 a t-2 + ... + θ q a t-q Where: φ 1 , ..., φ p : Autoregressive (AR) parameters. θ 1 , ..., θ q : Moving Average (MA) parameters. a t : A white noise process (the innovation). ...

Random Variable vs. Random Process: Definitions, Examples, and Key Differences Random Variable and Random Process In the fields of probability theory and statistics, understanding the distinction between a Random Variable and a Random Process (Stochastic Process) is fundamental. This guide breaks down these concepts with mathematical precision and practical examples. 1. Random Variable (RV) Definition: A Random Variable is a deterministic function that maps each possible outcome of a random experiment to a unique real number. Mathematical Representation A random variable X is defined as a mapping from the sample space S to the set of real numbers ℝ : X : S → ℝ S (Sample Space): The set of all possible outcomes of an experiment. s ∈ S: A specific outcome. X(s): The numerical value asso...

Stable and Unstable LTI Systems: BIBO Stability and Pole Analysis Stable and Unstable LTI Systems An In-depth Engineering Guide to BIBO Stability and Impulse Response Analysis In signals and systems analysis, a Linear Time-Invariant (LTI) system is classified as stable if every bounded input results in a bounded output. This fundamental property ensures that the system does not produce divergent or uncontrollable signals under normal operating conditions. Conversely, if a system produces an unbounded output for at least one bounded input, it is classified as unstable . 1. BIBO Stability (Bounded Input Bounded Output) The BIBO Condition: A system is BIBO Stable if for every input signal $x(t)$ (or $x[n]$) that is bounded: |x(t)| ≤ M x The resulting output $y(t)$ (or $y[n]$) is also guaranteed to be bounded: |y(t)| ≤ M y where $M_x$ and $M_y$ are finite constants. Simply put: Bounded Input ⇒ Bounded Output. If a system’s internal...

Narrowband vs. Wideband Signals: Definitions, Formulas, and Examples Narrowband vs. Wideband Signals in Communication Systems In communication engineering, there is no single universal definition of narrowband and wideband . The classification often depends on the specific application (RF design, acoustics, or data networking). However, in modern wireless communication systems, the standard mathematical definition is based on fractional bandwidth . The Fundamental Parameters To classify a signal, we define two primary variables: B = Absolute signal bandwidth (the range of frequencies occupied). f c = Center frequency (or carrier frequency). What is a Narrowband Signal? A signal is technically classified as narrowband when its bandwidth is significantly smaller than its center frequency. This implies that the spectral content is concentrated around...

Relationship Between WSS, LTI, AR, MA and ARMA Models | Signal Processing Guide The Unified Relationship Between WSS, LTI, AR, MA, and ARMA Processes An in-depth exploration of how stochastic processes and linear systems interact in digital signal processing and time-series analysis. In modern signal processing, the interaction between Wide-Sense Stationary (WSS) random processes and Linear Time-Invariant (LTI) systems forms the backbone of statistical modeling. Models like AR (Autoregressive) , MA (Moving Average) , and ARMA are not just statistical tools—they are specific realizations of white noise filtered through LTI systems. 1. Defining the Wide-Sense Stationary (WSS) Process A random process \(x[n]\) is categorized as Wide-Sense Stationary if its first and second-order moments are invariant to time shifts: Constant Mean: $$E[x[n]] = \mu$$ Time-Dependent Autocorrelation: The cor...