Ferkans — Interactive Telecom Tutor

From Deterministic to Random: Why This Matters

Every communication receiver, every radar processor, and every control system is — at its core — a linear filter processing a noisy signal. Deterministic LTI theory tells us what happens to a known waveform; now we ask: what happens to the statistics of a random process when it passes through a filter? The answer is surprisingly clean: if the input is WSS and the filter is BIBO-stable, the output is also WSS, and its PSD is simply $|\check{h}(f)|^2$ times the input PSD. This one relation — $P_y(f) = |\check{h}(f)|^2 P_x(f)$ — is the foundation for everything in this chapter: matched filters, Wiener filters, and noise bandwidth calculations.

Definition:
LTI System with Random Input

Let $\{X(t) : t \in \mathbb{R}\}$ be a random process and $h(t)$ be the impulse response of an LTI system. The output process is defined by the convolution integral $Y(t) = \int_{-\infty}^{\infty} h(\tau) X(t - \tau)\, d\tau = (h * X)(t),$ provided the integral exists in the mean-square sense. This means we require $\mathbb{E}\!\left[\left|\int h(\tau) X(t - \tau)\, d\tau\right|^2\right] < \infty$ for all $t$ .

Theorem: Existence of the Output (Theorem 41)

Let $X(t)$ be a random process with autocorrelation $r_{xx}(t_1, t_2)$ and let $h(t)$ be the impulse response of an LTI system. The output $Y(t) = \int h(\tau) X(t - \tau)\, d\tau$ exists in the mean-square sense if $\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} h(\tau) h^*(\tau') r_{xx}(t - \tau, t - \tau')\, d\tau\, d\tau' < \infty$ for all $t$ .

In the discrete-time case, $Y_n = \sum_m h[m] X_{n-m}$ exists in m.s. if $\sum_{m \in \mathbb{Z}} |h[m]| \sqrt{r_{xx}[n-m, n-m]} < \infty.$

The condition ensures that the "energy" of the output at each time instant is finite. It is the random-process analogue of the deterministic requirement that convolution produces a finite output.

Proof

Define partial sums

For the CT case, define $Z_t(\tau) = h(\tau) X(t - \tau)$ . The output is $Y(t) = \int Z_t(\tau)\, d\tau$ , a mean-square Riemann integral. By the existence criterion for m.s. integrals, this exists iff $\int\!\!\int \mathbb{E}[Z_t(\tau) Z_t^*(\tau')]\, d\tau\, d\tau' < \infty.$

Evaluate the integrand

We have $\mathbb{E}[Z_t(\tau) Z_t^*(\tau')] = h(\tau) h^*(\tau') r_{xx}(t - \tau, t - \tau')$ . Substituting gives the stated condition.

DT case via Lemma 46

For discrete time, $Y_n = \sum_m h[m] X_{n-m}$ is a sum of random variables $Z_{n,m} = h[m] X_{n-m}$ . By the sufficient condition (Lemma 46 from Ch. 13), convergence holds if $\sum_m \sqrt{\mathbb{E}[|Z_{n,m}|^2]} < \infty$ . Since $\mathbb{E}[|Z_{n,m}|^2] = |h[m]|^2 r_{xx}[n-m, n-m]$ , we get $\sum_m |h[m]| \sqrt{r_{xx}[n-m, n-m]} < \infty$ .

Theorem: BIBO Stability + WSS Input $\Rightarrow$ WSS Output (Lemma 49)

If $X(t)$ is WSS and the LTI system is BIBO-stable, i.e., $\int_{-\infty}^{\infty} |h(\tau)|\, d\tau < \infty$ , then:

The output $Y(t)$ exists in the mean-square sense.
$Y(t)$ is WSS.
The output mean, autocorrelation, and cross-correlations are: $\mu_Y = \mu_X \int_{-\infty}^{\infty} h(\tau)\, d\tau = \mu_X \cdot \check{h}(0),$ ${r_{xx}}_{yy}(\tau) = h(\tau) * h^*(-\tau) * {r_{xx}}_{xx}(\tau),$ $r_{yx}(\tau) = h(\tau) * {r_{xx}}_{xx}(\tau), \quad r_{xy}(\tau) = h^*(-\tau) * {r_{xx}}_{xx}(\tau).$

BIBO stability means the filter has finite total "gain." Combined with the bounded power of a WSS process ( $r_{xx}(0) < \infty$ ), this guarantees finite output power. The output autocorrelation depends only on the lag $\tau$ because both the input statistics and the filter are shift-invariant — so the output inherits stationarity.

Proof

Existence

For WSS input, $r_{xx}(t - \tau, t - \tau') = {r_{xx}}_{xx}(\tau - \tau')$ and $r_{xx}(0)$ is constant. Apply the general existence condition: $\iint |h(\tau)| |h(\tau')| |r_{xx}(\tau - \tau')|\, d\tau\, d\tau' \leq r_{xx}(0) \left(\int |h(\tau)|\, d\tau\right)^2 < \infty$ by BIBO stability. Hence $Y(t)$ exists in m.s.

Output mean

$\mu_Y(t) = \mathbb{E}[Y(t)] = \int h(\tau) \mathbb{E}[X(t - \tau)]\, d\tau = \mu_X \int h(\tau)\, d\tau = \mu_X \cdot \check{h}(0),$ which is constant in $t$ .

Output autocorrelation

$r_{yy}(t_1, t_2) = \mathbb{E}[Y(t_1) Y^*(t_2)] = \iint h(\alpha) h^*(\beta) r_{xx}(t_1 - \alpha - t_2 + \beta)\, d\alpha\, d\beta.KATEXPLACEHOLDER0ENDr_{yy}(\tau) = \int h(\alpha) \left[\int h^*(\beta) r_{xx}(\tau - \alpha + \beta)\, d\beta\right] d\alpha = h(\tau) * h^*(-\tau) * r_{xx}(\tau).$ $

WSS conclusion

The output has constant mean and autocorrelation depending only on the lag. Therefore $Y(t)$ is WSS.

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Theorem: The Input-Output PSD Relation

If $X(t)$ is WSS with PSD $P_x(f)$ and passes through a BIBO-stable LTI system with frequency response $\check{h}(f)$ , then the output PSD is $\boxed{P_y(f) = |\check{h}(f)|^2\, P_x(f).}$ The cross-spectral densities are $P_{yx}(f) = \check{h}(f)\, P_x(f), \quad P_{xy}(f) = \check{h}^*(f)\, P_x(f).$

In the time domain, the output autocorrelation involves a triple convolution. In the frequency domain, convolution becomes multiplication, so the output PSD is simply the input PSD scaled by the squared magnitude of the frequency response. This is why spectral analysis is so powerful: complicated convolutions reduce to pointwise products.

Proof

Fourier transform the autocorrelation

From the time-domain result $r_{yy}(\tau) = h(\tau) * h^*(-\tau) * r_{xx}(\tau)$ , take the Fourier transform of both sides. Convolution in time becomes multiplication in frequency: $P_y(f) = \check{h}(f) \cdot \check{h}^*(f) \cdot P_x(f) = |\check{h}(f)|^2 P_x(f).$

Cross-spectral densities

Similarly, $r_{yx}(\tau) = h(\tau) * r_{xx}(\tau)$ gives $P_{yx}(f) = \check{h}(f) P_x(f)$ , and $r_{xy}(\tau) = h^*(-\tau) * r_{xx}(\tau)$ gives $P_{xy}(f) = \check{h}^*(f) P_x(f)$ .

Consistency check

Note $P_y(f) = \check{h}(f) P_{xy}(f) = \check{h}^*(f) P_{yx}(f)$ , confirming the frequency-domain block-diagram relations from Ch. 14.

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Example: White Noise Through an RC Lowpass Filter

White noise with PSD $P_x(f) = N_0/2$ passes through an RC lowpass filter with frequency response $\check{h}(f) = \frac{1}{1 + j 2\pi f RC}.$ Find the output PSD $P_y(f)$ , the output power, and the output autocorrelation.

Solution

Output PSD

$P_y(f) = |\check{h}(f)|^2 P_x(f) = \frac{1}{1 + (2\pi f RC)^2} \cdot \frac{N_0}{2}.$ $This is a Lorentzian spectrum centered at$ f = 0$.

Output power

$\mathcal{P}_Y = \int_{-\infty}^{\infty} P_y(f)\, df = \frac{N_0}{2} \int_{-\infty}^{\infty} \frac{df}{1 + (2\pi f RC)^2} = \frac{N_0}{2} \cdot \frac{1}{2RC} = \frac{N_0}{4RC}.$ $ The filter converts infinite white-noise power into finite output power.

Output autocorrelation

Taking the inverse Fourier transform of the Lorentzian PSD: $r_{yy}(\tau) = \frac{N_0}{4RC}\, e^{-|\tau|/(RC)}.$ At $\tau = 0$ , this gives $r_{yy}(0) = N_0/(4RC)$ , confirming the output power.

$P_y(f) = |\check{h}(f)|^2 P_x(f)$ : PSD Through an LTI Filter

Visualize how a filter shapes the input PSD. Choose the input spectrum type and adjust the filter bandwidth to see the output PSD and total output power.

Parameters

Input PSD type

Filter type

f_c

(Hz)2

N_0

1

The Filter as a Spectral Shaper

The relation $P_y(f) = |\check{h}(f)|^2 P_x(f)$ says that the filter selectively scales different frequency components of the input. Where $|\check{h}(f)|$ is large, that frequency band is amplified; where it is small, it is suppressed. The output power is $\mathcal{P}_Y = \int |\check{h}(f)|^2 P_x(f)\, df,$ which is a weighted integral of the input PSD. This is the starting point for filter design: we choose $\check{h}(f)$ to pass desired signal components and reject noise.

Definition:
Output Power of a Filtered WSS Process

The average power of the output of a BIBO-stable LTI system with WSS input is $\mathcal{P}_Y = r_{yy}(0) = \int_{-\infty}^{\infty} |\check{h}(f)|^2 P_x(f)\, df.$ For zero-mean input, this equals the output variance $\sigma^2_{Y}$ .

Example: DT White Noise Through a First-Order IIR Filter

Let $X_n$ be DT white noise with $r_{xx}[m] = \sigma^2 \delta[m]$ and consider the first-order IIR filter $Y_n = a Y_{n-1} + X_n$ with $|a| < 1$ . Find the output PSD and output power.

Solution

Frequency response

The transfer function is $\check{h}(f) = \frac{1}{1 - a e^{-j2\pi f}}$ , so $|\check{h}(f)|^2 = \frac{1}{|1 - a e^{-j2\pi f}|^2} = \frac{1}{1 - 2a\cos(2\pi f) + a^2}$ .

Output PSD

Since $P_x(f) = \sigma^2$ (flat), the output PSD is $P_y(f) = \frac{\sigma^2}{1 - 2a\cos(2\pi f) + a^2}.$

Output power

$\mathcal{P}_Y = \int_{-1/2}^{1/2} P_y(f)\, df = \frac{\sigma^2}{1 - a^2},$ $ which matches the known variance of an AR(1) process.

Example: Cascade of Two LTI Filters

A WSS process $X(t)$ with PSD $P_x(f)$ passes through two BIBO-stable LTI systems in cascade with frequency responses $\check{h}_{1}(f)$ and $\check{h}_{2}(f)$ . Find the PSD of the final output and the cross-spectral density between the intermediate and final signals.

Solution

Overall frequency response

The cascade has frequency response $\check{h}(f) = \check{h}_{1}(f) \check{h}_{2}(f)$ , so $P_z(f) = |\check{h}_{1}(f)|^2 |\check{h}_{2}(f)|^2 P_x(f).$

Cross-spectrum

Let $Y(t)$ be the output of the first filter and $Z(t)$ the output of the second. Then $P_{zy}(f) = \check{h}_{2}(f) P_y(f) = \check{h}_{2}(f) |\check{h}_{1}(f)|^2 P_x(f)$ .

Quick Check

White noise with PSD $P_x(f) = N_0/2$ passes through an ideal bandpass filter with passband $|f - f_0| \leq W/2$ . What is the output power?

$N_0 W$

$N_0 W / 2$

$N_0 / (2W)$

$2 N_0 W$

Correction:

N_0 W

The output PSD is $N_0/2$ inside the passband (total width $W$ on each side of $\pm f_0$ , but for a real filter the two-sided bandwidth is $2 \times W/2 = W$ per side, giving total $N_0 W$ ).

Common Mistake: One-Sided vs. Two-Sided PSD

Mistake:

Using $N_0$ as the two-sided PSD of white noise and computing output power as $N_0 \times \text{bandwidth}$ .

Correction:

The standard convention in this course is $P_x(f) = N_0/2$ (two-sided PSD), so the noise power in bandwidth $W$ is $N_0 W$ (integrating the two-sided PSD over both positive and negative frequencies within the passband). Always check whether a reference uses one-sided or two-sided PSD.

BIBO Stability

A system is BIBO (bounded-input, bounded-output) stable if every bounded input produces a bounded output. For LTI systems, this is equivalent to absolute integrability of the impulse response: $\int_{-\infty}^{\infty} |h(t)|\, dt < \infty$ (CT) or $\sum_{n} |h[n]| < \infty$ (DT).

Related: LTI System with Random Input

Spectral Shaping

The process of modifying the PSD of a signal by passing it through a filter. The output PSD is $P_y(f) = |\check{h}(f)|^2 P_x(f)$ , so the filter's squared magnitude response acts as a frequency-dependent gain on the input spectrum.

Cross-Spectral Density

The Fourier transform of the cross-correlation function: $P_{xy}(f) = \mathcal{F}\{r_{xy}(\tau)\}$ . For an LTI system with WSS input, $P_{yx}(f) = \check{h}(f) P_x(f)$ .

Related: Spectral Shaping

Historical Note: Wiener's Wartime Filter Theory

1940s

The input-output PSD relation was central to Norbert Wiener's classified wartime report (1942), later published as Extrapolation, Interpolation, and Smoothing of Stationary Time Series (1949). Wiener developed the theory to design optimal anti-aircraft fire control predictors — filters that could extract a target's trajectory from noisy radar returns. The key insight was that working in the frequency domain transforms the intractable integral equation (Wiener-Hopf) into a simple algebraic relation. This work, along with Kolmogorov's independent contributions, laid the foundation for all of modern statistical signal processing.

Historical Note: Kolmogorov's Prediction Theory

1940s

Independently of Wiener, Andrey Kolmogorov developed the theory of optimal linear prediction for discrete-time stationary processes in 1941. Kolmogorov's approach was more measure-theoretic, while Wiener's was more engineering-oriented. Together, their work established that the optimal linear filter for estimation problems can be characterized entirely through second-order statistics — a principle that remains the backbone of modern estimation theory.

$P_x(f)$ Through a Filter: Spectral Shaping Animation

Watch how an input PSD is shaped by different filter frequency responses. The animation sweeps the filter bandwidth and shows the output PSD changing in real time.

The output PSD

P_y(f) = |\check{h}(f)|^2 P_x(f)

as the filter bandwidth varies.

Why This Matters: Receiver Front-End Filtering

In any wireless receiver, the first stage after the antenna is a bandpass filter that selects the desired frequency band and rejects out-of-band interference and noise. The output noise power is $\mathcal{P}_N = \int |\check{h}(f)|^2 {P_x}_{N}(f)\, df$ . Designing this filter involves a tradeoff: too narrow a bandwidth distorts the signal; too wide admits excess noise. The noise bandwidth concept (Section 15.4) quantifies this tradeoff precisely.

See full treatment in Noise Equivalent Bandwidth

Key Takeaway

The fundamental result of this section is $P_y(f) = |\check{h}(f)|^2 P_x(f)$ : the output PSD of a BIBO-stable LTI system with WSS input is the input PSD multiplied by the squared magnitude of the frequency response. This converts the triple convolution in the time domain into a simple pointwise product in the frequency domain.

WSS Processes Through LTI Systems

From Deterministic to Random: Why This Matters

Definition: LTI System with Random Input

Theorem: Existence of the Output (Theorem 41)

Define partial sums

Evaluate the integrand

DT case via Lemma 46

Theorem: BIBO Stability + WSS Input ⇒\Rightarrow⇒ WSS Output (Lemma 49)

Existence

Output mean

Output autocorrelation

WSS conclusion

Theorem: The Input-Output PSD Relation

Fourier transform the autocorrelation

Cross-spectral densities

Consistency check

Example: White Noise Through an RC Lowpass Filter

Output PSD

Output power

Output autocorrelation

Py(f)=∣hˇ(f)∣2Px(f)P_y(f) = |\check{h}(f)|^2 P_x(f)Py​(f)=∣hˇ(f)∣2Px​(f): PSD Through an LTI Filter

Parameters

The Filter as a Spectral Shaper

Definition: Output Power of a Filtered WSS Process

Example: DT White Noise Through a First-Order IIR Filter

Frequency response

Output PSD

Output power

Example: Cascade of Two LTI Filters

Overall frequency response

Cross-spectrum

Quick Check

Common Mistake: One-Sided vs. Two-Sided PSD

BIBO Stability

Spectral Shaping

Cross-Spectral Density

Historical Note: Wiener's Wartime Filter Theory

Historical Note: Kolmogorov's Prediction Theory

Px(f)P_x(f)Px​(f) Through a Filter: Spectral Shaping Animation

Why This Matters: Receiver Front-End Filtering

Key Takeaway

Definition:
LTI System with Random Input

Theorem: BIBO Stability + WSS Input $\Rightarrow$ WSS Output (Lemma 49)

$P_y(f) = |\check{h}(f)|^2 P_x(f)$ : PSD Through an LTI Filter

Definition:
Output Power of a Filtered WSS Process

$P_x(f)$ Through a Filter: Spectral Shaping Animation