Ferkans — Interactive Telecom Tutor

Why Stationarity Matters

The full statistical description of a process requires the infinite family of fdds — an unwieldy object. Stationarity is the assumption that the statistical properties of the process do not change with time. Under this assumption, a single first-order distribution suffices (it is the same at all times), a single second-order distribution characterized by the lag $\tau = t_1 - t_2$ suffices, and so on. This enormous simplification is what makes statistical signal processing tractable. In communications, stationarity holds (at least approximately) for thermal noise, for a fading channel observed within its coherence time, and for the output of any time-invariant system driven by a stationary input.

Definition:
Strict-Sense Stationarity (Definition 49)

A stochastic process $\{X(t) : t \in \mathcal{T}\}$ is strict-sense stationary (SSS) if its finite-dimensional distributions are invariant under time shifts: for every $N \geq 1$ , every $(t_1, \ldots, t_N) \in \mathcal{T}^N$ , and every shift $\tau$ such that $(t_1 + \tau, \ldots, t_N + \tau) \in \mathcal{T}^N$ , $F_{t_1, \ldots, t_N}(x_1, \ldots, x_N) = F_{t_1 + \tau, \ldots, t_N + \tau}(x_1, \ldots, x_N).$

Strict stationarity is a very strong condition: it requires every aspect of the joint distribution to be time-invariant. In practice, we rarely verify SSS directly because it involves all orders of the fdds. Instead, we work with the weaker but far more practical notion of wide-sense stationarity.

Consequences of SSS

If $X(t)$ is SSS, then:

First-order distribution is constant: $F_{X(t)}(x) = F_{X(0)}(x)$ for all $t$ . In particular, $\mu_X(t) = \mu$ and $\text{Var}(X(t)) = \sigma^2$ are constants.
Second-order distribution depends only on lag: $F_{X(t_1), X(t_2)}(x_1, x_2)$ depends only on $t_1 - t_2$ .
All $N$ th-order statistics depend only on time differences: the $N$ -point statistics are invariant to a common time shift.

SSS $\Rightarrow$ WSS (but not conversely, in general).

Definition:
Wide-Sense Stationarity (Definition 52)

A second-order process $\{X(t) : t \in \mathcal{T}\}$ is wide-sense stationary (WSS) if:

$\mu_X(t) = \mathbb{E}[X(t)] = \mu$ is constant (independent of $t$ ), and
$r_{XX}(t_1, t_2) = \mathbb{E}[X(t_1)X^*(t_2)]$ depends only on the time difference $\tau = t_1 - t_2$ .

We write $r_{XX}(\tau) \triangleq r_{XX}(t, t - \tau) = \mathbb{E}[X(t)X^*(t - \tau)]$ .

For a discrete-time process, $\{X_n\}$ is WSS if $\mu[n] = \mu$ (constant) and $r_{xx}[n, m] = r_{xx}[n - m]$ depends only on the lag $k = n - m$ .

WSS Is the Engineer's Stationarity

Wide-sense stationarity involves only two conditions — constant mean and lag-dependent autocorrelation — both of which can be estimated from data. This is why WSS is the default assumption in signal processing and communications: it is strong enough to enable powerful tools (power spectral density, Wiener filtering, matched filtering) but weak enough to be approximately satisfied in many practical scenarios.

The gap between WSS and SSS is bridged for Gaussian processes, as Lemma 43 below shows.

Theorem: SSS Implies WSS (for Second-Order Processes)

If $\{X(t)\}$ is strict-sense stationary and $\mathbb{E}[|X(t)|^2] < \infty$ , then $\{X(t)\}$ is wide-sense stationary.

SSS makes all fdds time-shift invariant. In particular, the first-order moment (from the first-order distribution) and the second-order product moment (from the second-order distribution) inherit this invariance, which is exactly the WSS condition.

Proof

Constant mean

The first-order distribution $F_{X(t)}$ is independent of $t$ (SSS with $N = 1$ ). Therefore $\mu_X(t) = \mathbb{E}[X(t)] = \int x\,dF_{X(t)}(x) = \int x\,dF_{X(0)}(x) = \mu$ .

Lag-dependent autocorrelation

The joint distribution of $(X(t_1), X(t_2))$ depends only on $t_1 - t_2$ (SSS with $N = 2$ ). Therefore $r_{XX}(t_1, t_2) = \mathbb{E}[X(t_1)X^*(t_2)]$ depends only on $\tau = t_1 - t_2$ .

Definition:
Gaussian Process (Definition 51)

A process $\{X(t) : t \in \mathcal{T}\}$ is Gaussian if for every $N \geq 1$ and every $(t_1, \ldots, t_N) \in \mathcal{T}^N$ , the random vector $(X(t_1), \ldots, X(t_N))^\top$ is jointly Gaussian.

A Gaussian process is completely determined by its mean function $\mu_X(t)$ and its autocovariance function $c_{XX}(t_1, t_2) = \text{Cov}(X(t_1), X(t_2))$ .

Theorem: WSS Gaussian Process Is SSS (Lemma 43)

If $\{X(t)\}$ is a Gaussian process and wide-sense stationary, then $\{X(t)\}$ is strict-sense stationary.

A Gaussian distribution is completely determined by its mean vector and covariance matrix. If the mean is constant and the covariance depends only on time differences (WSS), then the entire joint distribution is shift-invariant — which is SSS.

Proof

Characterize the finite-dimensional distributions

For any $(t_1, \ldots, t_N)$ , the vector $\mathbf{X} = (X(t_1), \ldots, X(t_N))^\top$ is Gaussian with mean $\boldsymbol{\mu} = (\mu, \ldots, \mu)^\top$ (constant by WSS) and covariance matrix $[\boldsymbol{\Sigma}]_{ij} = c_{XX}(t_i - t_j)$ (depends only on time differences by WSS).

Show shift-invariance

Under a time shift $\tau$ , the shifted vector $(X(t_1 + \tau), \ldots, X(t_N + \tau))^\top$ has the same mean $\boldsymbol{\mu}$ and the same covariance matrix $[\boldsymbol{\Sigma}']_{ij} = c_{XX}((t_i + \tau) - (t_j + \tau)) = c_{XX}(t_i - t_j) = [\boldsymbol{\Sigma}]_{ij}$ . Since a Gaussian distribution is uniquely determined by its first two moments, the distribution of the shifted vector is identical. This holds for all $N$ and all time indices, so the process is SSS.

Example: WSS but Not SSS

Let $X$ be a random variable with $\mathbb{E}[X] = 0$ , $\mathbb{E}[X^2] = 1$ , but $X$ is not Gaussian (e.g., $X$ takes values $\pm 1$ each with probability $1/2$ ). Define the process $Y_n = X$ for all $n$ . Is $\{Y_n\}$ WSS? Is it SSS?

Solution

Check WSS

$\mu_Y[n] = \mathbb{E}[X] = 0$ (constant). $r_{YY}[n, m] = \mathbb{E}[X^2] = 1$ for all $n, m$ , which depends only on $n - m$ (trivially, since it is constant). So $\{Y_n\}$ is WSS.

Check SSS

The first-order distribution of $Y_n$ is the distribution of $X$ , which is the same for all $n$ — consistent with SSS. The second-order joint distribution of $(Y_{n_1}, Y_{n_2})$ is the distribution of $(X, X)$ , which is concentrated on the line $y_1 = y_2$ — also the same for all $(n_1, n_2)$ .

In fact, all fdds are those of the constant sequence $(X, X, \ldots)$ , which does not depend on the time indices. So this process is indeed SSS.

A genuine WSS-but-not-SSS example

For a truly WSS-but-not-SSS example, we need the higher-order statistics to vary with time. Consider $Z_n = (-1)^n X_n$ where $\{X_n\}$ is an i.i.d. sequence with $\mathbb{E}[X_n] = 0$ , $\mathbb{E}[X_n^2] = 1$ , and $\mathbb{E}[X_n^3] \neq 0$ . Then $\mu_Z = 0$ , $r_{ZZ}[n,m] = (-1)^{n+m}\mathbb{E}[X_n X_m] = \delta[n-m]$ (depends only on lag), so $\{Z_n\}$ is WSS. But the third moment $\mathbb{E}[Z_n^3] = (-1)^{3n}\mathbb{E}[X_n^3]$ alternates in sign, so the third-order distribution changes with $n$ — not SSS.

WSS vs. Non-WSS Processes

Compare WSS and non-WSS processes. The WSS process has constant mean and lag-dependent autocorrelation. The non-WSS process has time-varying statistics.

Parameters

Process Type

Number of Paths5

Random Seed7

Strict-Sense vs. Wide-Sense Stationarity

Property	SSS	WSS
Condition	All fdds shift-invariant	Constant mean + lag-dependent $r_{XX}$
Involves	All moments and distributions	Only 1st and 2nd moments
Testable from data?	Generally no	Yes — estimate mean and autocorrelation
SSS $\Rightarrow$ WSS?	Yes (if 2nd moment exists)	—
WSS $\Rightarrow$ SSS?	No (in general)	Yes, if Gaussian
Practical use	Theoretical ideal	Default assumption in signal processing

Quick Check

Let $X(t) = B + W(t)$ where $B \sim \mathcal{N}(0, \sigma_B^2)$ is a constant random bias and $W(t)$ is a zero-mean WSS process independent of $B$ . Is $X(t)$ WSS?

Yes, because the sum of WSS processes is WSS

Yes: the mean is constant and the autocorrelation depends only on $\tau$

No, because $B$ is a random constant that biases each realization differently

Correction:

Yes: the mean is constant and the autocorrelation depends only on

\tau

$\mu_X(t) = 0$ . $r_{XX}(t_1, t_2) = \sigma_B^2 + r_{WW}(t_1 - t_2)$ , which depends only on $\tau = t_1 - t_2$ . Both conditions hold.

Common Mistake: Assuming WSS Implies SSS

Mistake:

Concluding that a WSS process has time-invariant distributions of all orders.

Correction:

WSS guarantees only that the first two moments are time-invariant. Higher-order statistics may still vary with time. The only general case where WSS implies SSS is for Gaussian processes (Lemma 43), because their distributions are fully determined by the first two moments.

Common Mistake: Forgetting That WSS Implies Constant Variance

Mistake:

Computing $\text{Var}(X(t))$ for a WSS process and getting a time-dependent answer.

Correction:

For a WSS process, $\text{Var}(X(t)) = r_{XX}(0) - |\mu|^2$ is the same for all $t$ . If your calculation gives a time-dependent variance, check whether the process is truly WSS.

⚠️Engineering Note

WSS and the Coherence Time

In mobile wireless communications, the channel is modeled as WSS only over a time interval called the coherence time $T_c$ . Beyond $T_c$ , the channel statistics change due to mobility and environmental changes. A typical design rule is to place pilot symbols at intervals shorter than $T_c$ so that channel estimation can exploit the WSS assumption. The wide-sense stationary uncorrelated scattering (WSSUS) model, introduced by Bello (1963), formalizes this for doubly-selective (time-frequency) channels.

Wide-Sense Stationary (WSS)

A process with constant mean and autocorrelation that depends only on the time difference. The standard assumption in linear signal processing and communications.

Related: Strict-Sense Stationary (SSS)

Strict-Sense Stationary (SSS)

A process whose finite-dimensional distributions are invariant under time shifts. Stronger than WSS; equivalent to WSS for Gaussian processes.

Related: Wide-Sense Stationary (WSS)

Key Takeaway

Wide-sense stationarity — constant mean and lag-dependent autocorrelation — is the practical form of stationarity used throughout signal processing and communications. For Gaussian processes, WSS and SSS are equivalent, which is why the Gaussian assumption is so powerful: second-order statistics tell the whole story.

Stationarity

Why Stationarity Matters

Definition: Strict-Sense Stationarity (Definition 49)

Consequences of SSS

Definition: Wide-Sense Stationarity (Definition 52)

WSS Is the Engineer's Stationarity

Theorem: SSS Implies WSS (for Second-Order Processes)

Constant mean

Lag-dependent autocorrelation

Definition: Gaussian Process (Definition 51)

Theorem: WSS Gaussian Process Is SSS (Lemma 43)

Characterize the finite-dimensional distributions

Show shift-invariance

Example: WSS but Not SSS

Check WSS

Check SSS

A genuine WSS-but-not-SSS example

WSS vs. Non-WSS Processes

Parameters

Strict-Sense vs. Wide-Sense Stationarity

Quick Check

Common Mistake: Assuming WSS Implies SSS

Common Mistake: Forgetting That WSS Implies Constant Variance

WSS and the Coherence Time

Wide-Sense Stationary (WSS)

Strict-Sense Stationary (SSS)

Key Takeaway

Definition:
Strict-Sense Stationarity (Definition 49)

Definition:
Wide-Sense Stationarity (Definition 52)

Definition:
Gaussian Process (Definition 51)