Ferkans — Interactive Telecom Tutor

From Auto to Cross

The PSD describes how the power of a single process is distributed across frequency. But in many systems we care about the relationship between two processes — for instance, the input and output of a filter, or the signal and noise at a receiver. The cross-spectral density generalizes the PSD to pairs of processes and leads to the most important result in this chapter: the input-output PSD relation ${P_x}_{y}(f) = |\check{h}(f)|^2\,{P_x}_{x}(f)$ .

Definition:
Cross-Power Spectral Density

Let $X(t)$ and $Y(t)$ be jointly WSS processes with absolutely integrable cross-correlation $r_{xy}(\tau) = \mathbb{E}[X(t+\tau)Y^*(t)]$ . The cross-power spectral density is

$P_{xy}(f) = \int_{-\infty}^{\infty} r_{xy}(\tau)\,e^{-j2\pi f\tau}\,d\tau.$

Unlike the auto-PSD, $P_{xy}(f)$ is in general complex-valued.

The symmetry relation $P_{yx}(f) = P_{xy}^*(f)$ holds because $r_{yx}(\tau) = r_{xy}^*(-\tau)$ .

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Theorem: Input-Output PSD Relation for LTI Systems

Let $X(t)$ be a WSS input to a stable LTI system with frequency response $\check{h}(f)$ , and let $Y(t)$ be the output. Then:

$\boxed{{P_x}_{y}(f) = |\check{h}(f)|^2\,{P_x}_{x}(f).}$

The cross-spectral densities are:

$P_{xy}(f) = \check{h}^*(f)\,{P_x}_{x}(f), \qquad P_{yx}(f) = \check{h}(f)\,{P_x}_{x}(f).$

In the frequency domain, a linear filter simply multiplies each frequency component by $\check{h}(f)$ . Power is proportional to the squared magnitude, so the output power at frequency $f$ is $|\check{h}(f)|^2$ times the input power at that frequency. This is the spectral version of the time-domain convolution $r_{yy}(\tau) = h(\tau) * h^*(-\tau) * r_{xx}(\tau)$ .

Proof

Time-domain relations

From the previous chapter, for WSS input and stable LTI:

$r_{yy}(\tau) = h(\tau) * h^*(-\tau) * r_{xx}(\tau),$ $r_{yx}(\tau) = h(\tau) * r_{xx}(\tau),$ $r_{xy}(\tau) = h^*(-\tau) * r_{xx}(\tau).$

Take Fourier transforms

Convolution in time $\leftrightarrow$ multiplication in frequency:

${P_x}_{y}(f) = \check{h}(f)\,\check{h}^*(f)\,{P_x}_{x}(f) = |\check{h}(f)|^2\,{P_x}_{x}(f).$

For the cross-spectra:

$P_{yx}(f) = \check{h}(f)\,{P_x}_{x}(f), \qquad P_{xy}(f) = \check{h}^*(f)\,{P_x}_{x}(f). \qquad \square$

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Example: Output PSD of an Ideal Low-Pass Filter

White noise with two-sided PSD $N_0/2$ is the input to an ideal low-pass filter with cutoff frequency $f_c$ : $\check{h}(f) = 1$ for $|f| \leq f_c$ , $0$ otherwise. Find the output PSD, autocorrelation, and total output power.

Solution

Output PSD

${P_x}_{y}(f) = |\check{h}(f)|^2 \cdot \frac{N_0}{2} = \begin{cases}N_0/2, & |f| \leq f_c, \\ 0, & |f| > f_c.\end{cases}$ $This is band-limited white noise with bandwidth$ 2f_c$.

Autocorrelation

${r_{xx}}_{y}(\tau) = \mathcal{F}^{-1}\{{P_x}_{y}(f)\} = N_0\,f_c\,\operatorname{sinc}(2f_c\,\tau).$ $

Total output power

$\sigma^2_{y} = {r_{xx}}_{y}(0) = N_0\,f_c.$ $Wider cutoff$ \to$ more noise power passes through.

Example: RC Low-Pass Filter with White Noise Input

White noise with PSD $N_0/2$ passes through an RC low-pass filter with frequency response $\check{h}(f) = \frac{1}{1 + j2\pi f RC}$ . Find the output PSD and total noise power.

Solution

Output PSD

${P_x}_{y}(f) = \frac{N_0/2}{1 + (2\pi f RC)^2}.$ $

This is a Lorentzian spectrum — the same shape as the PSD of an exponentially correlated process.

Total output power

$\sigma^2_{y} = \int_{-\infty}^{\infty} \frac{N_0/2}{1 + (2\pi f RC)^2}\,df = \frac{N_0}{4RC}.$ $

Interpretation

The noise bandwidth of the RC filter is $B_n = 1/(4RC)$ , so $\sigma^2_{y} = N_0\,B_n$ . The noise bandwidth is the width of an equivalent rectangular filter that passes the same total noise power.

Input-Output PSD Through a Filter

Choose an input PSD shape and a filter type, then observe how ${P_x}_{y}(f) = |\check{h}(f)|^2\,{P_x}_{x}(f)$ . The filter shapes the spectrum — amplifying some frequencies and attenuating others.

Parameters

Input PSD

Filter type

f_c

(cutoff, Hz)5

Definition:
Noise Bandwidth

The noise bandwidth of a filter with frequency response $\check{h}(f)$ is

$B_n = \frac{1}{|\check{h}(f_0)|^2}\int_{-\infty}^{\infty} |\check{h}(f)|^2\,df,$

where $f_0$ is the frequency of maximum gain (often $f_0 = 0$ for low-pass filters). The noise bandwidth is the width of an equivalent rectangular (brick-wall) filter that passes the same total noise power from a white noise input.

Quick Check

A WSS process with PSD ${P_x}_{x}(f) = 2$ for $|f| \leq 5$ (zero otherwise) passes through a filter with $|\check{h}(f)|^2 = 3$ for $|f| \leq 3$ (zero otherwise). What is the output power?

$36$ W

$60$ W

$30$ W

$18$ W

Correction:

36

W

${P_x}_{y}(f) = 3 \cdot 2 = 6$ for $|f| \leq 3$ , zero otherwise. Total power: $\int_{-3}^{3} 6\,df = 36$ .

Definition:
Coherence Function

The coherence function between jointly WSS processes $X(t)$ and $Y(t)$ is

$\gamma_{xy}(f) = \frac{|P_{xy}(f)|^2}{{P_x}_{x}(f)\,{P_x}_{y}(f)}, \qquad 0 \leq \gamma_{xy}(f) \leq 1.$

$\gamma_{xy}(f) = 1$ at frequency $f$ means the two processes are perfectly linearly related at that frequency (one is a scaled, phase-shifted version of the other). $\gamma_{xy}(f) = 0$ means they are uncorrelated at frequency $f$ .

Coherence is Unity for an LTI System

If $Y(t) = h(t) * X(t)$ (no additive noise), then $P_{xy}(f) = \check{h}^*(f){P_x}_{x}(f)$ and

$\gamma_{xy}(f) = \frac{|\check{h}(f)|^2\,{P_x}_{x}(f)^2}{|\check{h}(f)|^2\,{P_x}_{x}(f)^2} = 1$

wherever ${P_x}_{x}(f) > 0$ . Coherence less than 1 is a signature of either noise or nonlinearity.

Common Mistake: Cross-PSD Is Not Necessarily Real or Non-Negative

Mistake:

Assuming $P_{xy}(f) \geq 0$ by analogy with the auto-PSD.

Correction:

The cross-PSD $P_{xy}(f)$ is complex in general. Only the auto-PSD $P_x(f)$ is guaranteed real and non-negative. The magnitude $|P_{xy}(f)|$ and the coherence $\gamma_{xy}(f)$ are the meaningful non-negative quantities.

🔧Engineering Note

System Identification via Cross-Spectrum

The input-output cross-spectral relation $P_{yx}(f) = \check{h}(f)\,{P_x}_{x}(f)$ leads to a practical method for estimating the frequency response of an unknown system: drive the system with a known input $X(t)$ and estimate $\check{h}(f) = \hat{P}_{yx}(f) / \hat{P_x}_x(f)$ . Using white noise as input simplifies this to $\check{h}(f) \propto \hat{P}_{yx}(f)$ . This is the basis of many system identification algorithms in control theory and channel estimation in communications.

Cross-power spectral density

The Fourier transform of the cross-correlation $r_{xy}(\tau)$ between two jointly WSS processes. Complex-valued in general.

Coherence function

$\gamma_{xy}(f) = |P_{xy}(f)|^2 / [{P_x}_{x}(f)\,{P_x}_{y}(f)]$ . Measures the linear frequency-by-frequency correlation between two processes. Ranges from 0 (uncorrelated) to 1 (perfectly linearly related).

Noise bandwidth

The width of an equivalent rectangular filter that passes the same total noise power as the actual filter. Key parameter for noise analysis in receiver design.

Key Takeaway

The input-output PSD relation ${P_x}_{y}(f) = |\check{h}(f)|^2\,{P_x}_{x}(f)$ is the most-used tool in spectral analysis of linear systems. It tells us that a filter shapes the power spectrum — amplifying at some frequencies, attenuating at others — and that the cross-spectrum fully characterizes the linear coupling between input and output.

Cross-Spectral Density

From Auto to Cross

Definition: Cross-Power Spectral Density

Theorem: Input-Output PSD Relation for LTI Systems

Time-domain relations

Take Fourier transforms

Example: Output PSD of an Ideal Low-Pass Filter

Output PSD

Autocorrelation

Total output power

Example: RC Low-Pass Filter with White Noise Input

Output PSD

Total output power

Interpretation

Input-Output PSD Through a Filter

Parameters

Definition: Noise Bandwidth

Quick Check

Definition: Coherence Function

Coherence is Unity for an LTI System

Common Mistake: Cross-PSD Is Not Necessarily Real or Non-Negative

System Identification via Cross-Spectrum

Cross-power spectral density

Coherence function

Noise bandwidth

Key Takeaway

Definition:
Cross-Power Spectral Density

Definition:
Noise Bandwidth

Definition:
Coherence Function