Ferkans — Interactive Telecom Tutor

Theorem: Output of an LTI System Driven by a Gaussian Process Is Gaussian

Let $\{X(t)\}$ be a Gaussian process and let $h(t)$ be the impulse response of a BIBO-stable LTI system. Then the output $Y(t) = \int_{-\infty}^{\infty} h(\tau)\,X(t - \tau)\,d\tau = (h * X)(t)$ is also a Gaussian process.

If $X(t)$ is zero-mean and WSS with PSD $P_x(f)$ , then $Y(t)$ is zero-mean and WSS with:

Output PSD: $P_Y(f) = |\check{h}(f)|^2\,P_x(f)$
Output autocorrelation: $R_Y(\tau) = \mathcal{F}^{-1}\{|\check{h}(f)|^2\,P_x(f)\}$
Output mean: $\mu_Y = \mu_X\,\check{h}(0)$

Convolution is a (continuous) linear operation. Since Gaussian processes are closed under linear operations (Theorem TClosure of Gaussian Processes Under Linear Operations), the output is Gaussian. The key consequence: for Gaussian inputs, only the mean and PSD of the output matter. No higher-order statistics are needed.

Proof

Express the output as a linear functional

For any fixed $t$ , the output $Y(t) = \int h(\tau) X(t-\tau)\,d\tau$ is a linear functional of the Gaussian process $X$ . By Theorem TClosure of Gaussian Processes Under Linear Operations, $Y(t)$ is a Gaussian RV.

Check joint Gaussianity

For any times $s_1, \ldots, s_M$ , the vector $[Y(s_1), \ldots, Y(s_M)]$ consists of $M$ linear functionals of $X$ . Any linear combination $\sum_m b_m Y(s_m) = \int \left(\sum_m b_m h(s_m - \tau)\right) X(\tau)\,d\tau$ is again a linear functional of $X$ , hence Gaussian. So $\{Y(t)\}$ is a GP.

Derive the output PSD

This is the standard result from Ch. 15: for WSS input through a BIBO-stable LTI, $P_Y(f) = |\check{h}(f)|^2 P_x(f)$ . The proof uses the Wiener-Khinchin theorem and the convolution theorem. $\blacksquare$

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No Higher-Order Statistics Needed

For a general (non-Gaussian) WSS process through an LTI system, the output PSD relation $P_Y(f) = |\check{h}(f)|^2 P_x(f)$ characterizes only the second-order statistics. The output could have non-Gaussian marginals, and you would need higher-order spectra (bispectrum, trispectrum) to fully describe it.

For a Gaussian input, the output is Gaussian, so the output PSD completely characterizes the output process. This is a major simplification and is the reason Gaussian models are the workhorse of communication system analysis.

Theorem: Global Optimality of the Matched Filter for Gaussian Noise

Consider detecting a known signal $s(t)$ in additive white Gaussian noise: $Y(t) = s(t) + N(t), \quad 0 \leq t \leq T$ where $N(t)$ is WGN with PSD $N_0/2$ . Form the test statistic $\Lambda = \int_0^T g(t)\,Y(t)\,dt$ for some filter $g(t)$ .

The matched filter $g(t) = s(T-t)$ maximizes the output $\text{SNR}$ : $\text{SNR}_{\max} = \frac{2E_s}{N_0}, \quad E_s = \int_0^T |s(t)|^2\,dt$

For Gaussian noise, this is not just the best linear detector — it is the globally optimal detector, achieving the minimum probability of error among all detectors (linear or nonlinear).

The matched filter output $\Lambda = \int_0^T s(T-t)Y(t)\,dt$ is a sufficient statistic for detection in Gaussian noise. Sufficiency follows from the Neyman-Fisher factorization theorem. Since sufficient statistics lose no information, no other detector can do better.

Proof

Matched filter maximizes SNR (from Ch. 15)

The output SNR is $\text{SNR} = \frac{\left|\int_0^T g(t) s(t)\,dt\right|^2}{(N_0/2)\int_0^T |g(t)|^2\,dt}$ By the Cauchy-Schwarz inequality, $\text{SNR} \leq 2E_s/N_0$ with equality when $g(t) \propto s(T-t)$ .

Sufficient statistic argument

Under $H_1$ (signal present): $\Lambda \sim \mathcal{N}(E_s, E_sN_0/2)$ . Under $H_0$ (noise only): $\Lambda \sim \mathcal{N}(0, E_sN_0/2)$ . By the Neyman-Pearson lemma, the likelihood ratio test based on $\Lambda$ is the most powerful test. Since $\Lambda$ is a sufficient statistic (the likelihood ratio depends on $Y$ only through $\Lambda$ ), no other functional of $Y(t)$ can improve detection performance. $\blacksquare$

Why Gaussianity is essential

For non-Gaussian noise, the sufficient statistic may be a nonlinear functional of $Y(t)$ (e.g., involving higher-order moments). The matched filter would still be the best linear detector but not the globally optimal one. Gaussianity makes linear and globally optimal coincide.

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Example: Output PSD of WGN Through an RC Filter

White Gaussian noise with PSD $N_0/2$ passes through an RC lowpass filter with transfer function $\check{h}(f) = 1/(1 + j2\pi f RC)$ . Find the output PSD, autocorrelation, and total noise power.

Solution

Output PSD

$P_Y(f) = |\check{h}(f)|^2 \cdot \frac{N_0}{2} = \frac{N_0/2}{1 + (2\pi f RC)^2} = \frac{N_0/2}{1 + (f/f_c)^2}$ $where$ f_c = 1/(2\pi RC)$ is the 3 dB cutoff frequency.

Total output noise power

$\mathcal{P}_Y = \int_{-\infty}^{\infty} P_Y(f)\,df = \frac{N_0}{2} \int_{-\infty}^{\infty} \frac{df}{1+(f/f_c)^2} = \frac{N_0}{2} \cdot \pi f_c = \frac{N_0}{4RC}$ $

Output autocorrelation

Taking the inverse Fourier transform of a Lorentzian: $R_Y(\tau) = \frac{N_0}{4RC}\,e^{-|\tau|/(RC)} = \frac{N_0}{4RC}\,e^{-2\pi f_c |\tau|}$ The output is an Ornstein-Uhlenbeck-type process with exponential autocorrelation.

Example: Generating Colored Gaussian Noise from White Noise

Given access to a white Gaussian noise source $N(t)$ with PSD $N_0/2$ , describe how to generate a Gaussian process $X(t)$ with a prescribed PSD $P_x(f) \geq 0$ .

Solution

Spectral factorization

Design a filter with transfer function $\check{h}(f)$ such that $|\check{h}(f)|^2 = P_x(f)/(N_0/2) = 2P_x(f)/N_0$ . This is always possible when $P_x(f)$ is a rational function of $f^2$ (spectral factorization).

Filter the white noise

Pass $N(t)$ through the filter: $X(t) = (h * N)(t)$ . The output is Gaussian (by Theorem TOutput of an LTI System Driven by a Gaussian Process Is Gaussian) with PSD $|\check{h}(f)|^2 \cdot (N_0/2) = P_x(f)$ , as desired.

Digital implementation

In practice, generate i.i.d. $\mathcal{N}(0,1)$ samples and pass them through a discrete-time filter designed to match the desired discrete PSD. This is the standard method for simulating correlated Gaussian channels.

Gaussian Process Through an LTI Filter

Visualize the input GP (with a chosen kernel/PSD) and the output after filtering. Compare input and output sample paths and their PSDs.

Parameters

Input process

Filter type

f_c

(Hz)2

N_0

1

Gaussian vs General Processes Through LTI Systems

Property	General WSS Input	Gaussian WSS Input
Output is WSS?	Yes (if input WSS, filter BIBO-stable)	Yes
Output PSD	$P_Y(f) = \|\check{h}(f)\|^2 P_x(f)$	Same
Output distribution	Not determined by PSD alone	Fully determined: Gaussian
Matched filter optimality	Best linear detector	Globally optimal detector
Higher-order spectra needed?	Yes, for full characterization	No — PSD suffices
Sufficient statistic for detection	May require nonlinear processing	Linear (matched filter output)

Common Mistake: The Matched Filter Is Not Always the Optimal Detector

Mistake:

Assuming the matched filter is the optimal detector regardless of the noise distribution.

Correction:

The matched filter is the globally optimal detector only when the noise is Gaussian. For non-Gaussian noise (e.g., impulsive noise in power line communications, or Cauchy-distributed interference), nonlinear detectors can significantly outperform the matched filter. In those cases, the matched filter is still the best linear detector (by the Cauchy-Schwarz argument), but not the best overall.

Quick Check

A zero-mean WSS Gaussian process with PSD $P_x(f) = 1/(1+f^2)$ passes through an ideal lowpass filter with bandwidth $W$ . The output process is:

Gaussian and WSS

Gaussian but not WSS

WSS but not Gaussian

Neither Gaussian nor WSS

Correction:

Gaussian and WSS

A Gaussian process through an LTI filter gives a Gaussian output. WSS input through BIBO-stable LTI gives WSS output.

🎓CommIT Contribution(2018)

LMMSE Channel Estimation with Spatial Covariance

K. Vu, R. Cavalcante, G. Caire — IEEE Transactions on Signal Processing

The Gaussian channel model — where the channel vector $\mathbf{h} \sim \mathcal{CN}(\mathbf{0}, \boldsymbol{\Sigma})$ is a Gaussian process in the spatial domain — enables LMMSE estimation that exploits spatial covariance structure. Vu, Cavalcante, and Caire showed that the LMMSE estimator $\hat{\mathbf{h}} = \boldsymbol{\Sigma}\left(\boldsymbol{\Sigma} + \sigma^2\mathbf{I}\right)^{-1}\mathbf{y}$ can be made practical in massive MIMO by exploiting the low-rank structure of $\boldsymbol{\Sigma}$ , achieving near-optimal estimation with reduced pilot overhead. The key enabler is Gaussianity: the LMMSE estimator is not just the best linear estimator but the globally optimal (MMSE) estimator.

massive-MIMOLMMSEchannel-estimationView Paper →

Key Takeaway

When a Gaussian process passes through a linear system, the output is Gaussian and fully characterized by its PSD. This is why the matched filter is globally optimal for Gaussian noise: linear processing loses no information. For non-Gaussian noise, linear filters are suboptimal and nonlinear processing may be needed.

Gaussian Processes Through Linear Systems

Theorem: Output of an LTI System Driven by a Gaussian Process Is Gaussian

Express the output as a linear functional

Check joint Gaussianity

Derive the output PSD

No Higher-Order Statistics Needed

Theorem: Global Optimality of the Matched Filter for Gaussian Noise

Matched filter maximizes SNR (from Ch. 15)

Sufficient statistic argument

Why Gaussianity is essential

Example: Output PSD of WGN Through an RC Filter

Output PSD

Total output noise power

Output autocorrelation

Example: Generating Colored Gaussian Noise from White Noise

Spectral factorization

Filter the white noise

Digital implementation

Gaussian Process Through an LTI Filter

Parameters

Gaussian vs General Processes Through LTI Systems

Common Mistake: The Matched Filter Is Not Always the Optimal Detector

Quick Check

LMMSE Channel Estimation with Spatial Covariance

Key Takeaway