Output Statistics of an LTI System for WSS Input

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

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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 frequency (DC gain).

Output Autocorrelation

The autocorrelation function of the output is determined by convolving the input autocorrelation with the system's impulse response and its time-reversed version.

R_yy(τ) = R_xx(τ) * h(τ) * h(-τ)

Output Power Spectral Density (PSD)

According to the Wiener-Khinchin Theorem, the PSD of the output is the product of the input PSD and the squared magnitude of the frequency response.

S_yy(f) = |H(f)|² S_xx(f)

Output Variance

The variance measures the power of the AC component of the output signal.

σ²_y = R_yy(0) - μ²_y = ∫ S_yy(f) df - μ²_y

Key Conclusion: If the input process is WSS and the LTI system is stable (BIBO stability), the output process is also Wide-Sense Stationary (WSS).

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2. Auto-Regressive (AR) Process Modeling

An AR process models the current output as a linear combination of past outputs plus a random input.

y(n) = φ₁y(n−1) + φ₂y(n−2) + ... + φ_py(n−p) + x(n)

Mean

μ_y = μ_x / (1 - Σ φ_k)

Autocorrelation

The relationship between the lags is governed by the Yule-Walker equations:

R(k) = φ₁R(k−1) + φ₂R(k−2) + ... + φ_pR(k−p)

Stability Condition

The output is WSS only if all poles of the system lie inside the unit circle in the z-plane (all characteristic roots of the denominator polynomial are outside the unit circle).

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3. Moving Average (MA) Process Modeling

An MA process models the output as a linear combination of current and past input values (Finite Impulse Response).

y(n) = x(n) + θ₁x(n−1) + ... + θ_qx(n−q)

Mean & Variance

• Mean: μ_y = μ_x (1 + Σ θ_k)
• Variance: σ²_y = σ²_x (1 + θ₁² + ... + θ²_q) (Assuming white noise input)

Autocorrelation

The MA process has a finite memory. The autocorrelation R(k) becomes zero for all lags |k| > q.

Output Property: Every Moving Average process is inherently stationary.

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4. Auto-Regressive Moving Average (ARMA) Process

The ARMA model combines both AR and MA components for more complex system modeling.

y(n) = Σ φ_k y(n−k) + Σ θ_m x(n−m)

Statistical Summary

• Mean: Remains constant if the AR part is stable.
• PSD: S_yy(f) = σ² |B(e^j2πf)|² / |A(e^j2πf)|²
• Autocorrelation: Exhibits an infinite duration but decays exponentially with the lag k.

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Summary Comparison Table

System Model	Output Mean	Autocorrelation Property	WSS Condition
General LTI	μxH(0)	Rxx * h(τ) * h(-τ)	Stable Impulse Response
AR (Auto-Regressive)	Constant	Infinite (Yule-Walker)	Poles inside unit circle
MA (Moving Average)	Constant	Finite (Zero for lag > q)	Always WSS
ARMA	Constant	Infinite but decaying	AR part must be stable

Summary:

• An LTI system preserves the WSS property provided the system is stable.
• MA systems are always WSS as they are essentially FIR filters.
• AR and ARMA systems require the roots of the AR polynomial to reside within the stability region to ensure a WSS output.

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