Ferkans — Interactive Telecom Tutor

From Realizations to Statistics

The spreading function $h(\tau, \nu)$ is a realization of the channel — a specific configuration of reflectors and velocities. For system analysis we also need statistics: the power distribution of a random channel ensemble, the typical delay spread, the typical Doppler spread. The right tool is the scattering function, which is simply the power spectrum of the spreading function. It is the object that underlies the WSSUS model used throughout fading analysis.

Definition:
WSSUS Channel

A time-varying channel is wide-sense stationary with uncorrelated scatterers (WSSUS) if, for all $\tau, \tau', \nu, \nu'$ , $\mathbb{E}\!\left[h(\tau, \nu)\,h^*(\tau', \nu')\right] \;=\; S_h(\tau, \nu)\,\delta(\tau - \tau')\,\delta(\nu - \nu').$ The first property (wide-sense stationary in time) says that the channel statistics do not drift over the observation window. The second property (uncorrelated scatterers in delay) says that different paths are statistically independent. Together they give the spreading function a "white" structure in both $\tau$ and $\nu$ , with power density $S_h(\tau, \nu)$ .

WSSUS is an idealization. Real channels have correlated scatterers (a single large reflector can span many resolution cells) and non-stationary behavior (moving from line-of-sight to non-line-of-sight). In OTFS analysis WSSUS is nonetheless the baseline assumption because it admits clean closed-form expressions.

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Definition:
Scattering Function

For a WSSUS channel, the scattering function $S_h(\tau, \nu)$ is the joint power distribution of the channel in delay and Doppler: $S_h(\tau, \nu)\,d\tau\,d\nu \;=\; \text{average power arriving with delay in } [\tau, \tau + d\tau] \text{ and Doppler in } [\nu, \nu + d\nu].$ Marginalizing over Doppler gives the power delay profile (PDP): $P_\tau(\tau) \;=\; \int S_h(\tau, \nu)\,d\nu,$ and marginalizing over delay gives the Doppler power spectrum: $P_\nu(\nu) \;=\; \int S_h(\tau, \nu)\,d\tau.$

Theorem: Fourier Relation Between Scattering Function and TF Correlation

For a WSSUS channel, the scattering function $S_h(\tau, \nu)$ and the time-frequency correlation function $R_H(\Delta f, \Delta t) = \mathbb{E}[H(f, t)\,H^*(f + \Delta f, t + \Delta t)]$ form a 2D Fourier pair: $R_H(\Delta f, \Delta t) \;=\; \iint S_h(\tau, \nu)\,e^{-j2\pi \Delta f \tau}\,e^{j2\pi \nu \Delta t}\,d\tau\,d\nu.$

Power spectra and correlation functions are always Fourier pairs — this is the Wiener–Khinchin theorem. What is special here is that the "spectrum" lives in the 2D $(\tau, \nu)$ plane while the "correlation" lives in the 2D $(\Delta f, \Delta t)$ plane, because the channel is two-dimensional.

Proof

Start from the TF correlation

$H(f, t)$ is related to $h(\tau, \nu)$ by 1D Fourier transforms in both variables: $H(f, t) = \int h(\tau, \nu)\,e^{-j 2\pi f \tau}\,e^{j2\pi \nu t}\,d\tau\,d\nu$ (combining the two Fourier steps from Section 2).

Compute the correlation

Substitute into $R_H(\Delta f, \Delta t) = \mathbb{E}[H(f, t)\,H^*(f + \Delta f, t + \Delta t)]$ and exchange expectation and integrals.

Apply WSSUS

The WSSUS property $\mathbb{E}[h(\tau, \nu)\,h^*(\tau', \nu')] = S_h(\tau, \nu)\,\delta(\tau - \tau')\,\delta(\nu - \nu')$ collapses one of the double integrals, leaving exactly the claimed Fourier transform. $\blacksquare$

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Scattering Function and Its Marginals

Explore how the power delay profile and the Doppler spectrum combine into a full scattering function $S_h(\tau, \nu)$ . The two marginals specify the delay and Doppler spreads. When the scatterers are uncorrelated (WSSUS), the scattering function factors as $S_h(\tau, \nu) = P_\tau(\tau)\,P_\nu(\nu)$ . Vary the RMS delay spread and Jakes model parameters to see how coherence time and coherence bandwidth emerge.

Parameters

RMS delay

\sigma_\tau

(

\mu

s)1

Max Doppler

f_D

(Hz)500

Doppler spectrum shape

Definition:
Coherence Bandwidth and Coherence Time

The coherence bandwidth $B_c$ is the frequency separation over which the channel transfer function is approximately constant: $B_c \;\sim\; \frac{1}{\tau_{\max}}.$ The coherence time $T_c$ is the time interval over which the channel is approximately constant: $T_c \;\sim\; \frac{1}{f_D}.$ These are the Fourier duals of the delay and Doppler spreads.

Example: Coherence Time in a High-Speed Train

A high-speed train travels at $v = 300$ km/h along a track at 5 GHz. What is the coherence time? How many OFDM symbols (subcarrier spacing $\Delta f = 15$ kHz, symbol duration $T_{\text{sym}} \approx 66.7\,\mu\text{s}$ ) fit within one coherence time?

Solution

Doppler spread

$f_D = (v/c)\,f_0 = (83.3 / 3 \times 10^8) \cdot 5 \times 10^9 \approx 1{,}389$ Hz.

Coherence time

$T_c \sim 1/f_D \approx 720\,\mu\text{s}$ .

Symbols per coherence time

$N_{\text{sym}} = T_c/T_{\text{sym}} \approx 10.8$ . The channel changes significantly within roughly 10–11 OFDM symbols. The point is that tracking a time-varying channel of this speed requires pilot symbols at least every $\sim 5$ symbols — a significant overhead. In the DD domain, by contrast, the same channel is described by a handful of $(\tau_i, \nu_i, h_i)$ triples that do not change at all over a full OTFS frame (typically $N = 16$ – $128$ Doppler bins, or many hundreds of $T_{\text{sym}}$ ). The point is that the DD representation is time-invariant by construction; the channel-tracking burden disappears.

⚠️Engineering Note

Underspread vs Overspread Channels

A channel is called underspread if $\tau_{\max} f_D \ll 1$ and overspread otherwise. Underspread channels allow classical TF-domain communication because the time-frequency cell $(T_c, B_c)$ fits many data symbols; overspread channels break OFDM entirely because ICI and ISI become simultaneously significant.

Typical parameters:

Pedestrian at sub-6 GHz: $\tau_{\max} \approx 0.5\,\mu\text{s}$ , $f_D \approx 10$ Hz. Product: $5 \times 10^{-6}$ — deeply underspread.
Vehicular at 5 GHz: $\tau_{\max} \approx 3\,\mu\text{s}$ , $f_D \approx 500$ Hz. Product: $1.5 \times 10^{-3}$ — mildly underspread.
HST at mmWave: $\tau_{\max} \approx 0.3\,\mu\text{s}$ , $f_D \approx 10$ kHz. Product: $3 \times 10^{-3}$ — stressed.
LEO satellite at 10 GHz: $\tau_{\max} \approx 5$ ms (propagation), $f_D \approx 30$ kHz. Product: $150$ — overspread. OFDM is fundamentally inadequate.

OTFS works in both underspread and overspread regimes because its analysis does not require the TF cell to fit the symbol — only that the DD grid resolution $(\Delta\tau, \Delta\nu) = (1/W, 1/T)$ be finer than the channel's path spacing.

Practical Constraints

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Underspread condition: $\tau_{\max} f_D < 1$ (Bello)
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LEO channels are overspread by orders of magnitude
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Coherence cell area $\sim 1/(\tau_{\max} f_D)$

When Uncorrelated Scatterers Fails

The "uncorrelated scatterers" assumption in WSSUS says that two physical paths at different $(\tau, \nu)$ are statistically independent. This fails when a single large reflector (a building facade, a mountain) produces many closely spaced path components that all fade together. In such cases $h(\tau, \nu)$ has clustered support with correlated clusters — and the scattering function overestimates the effective number of independent paths.

For OTFS system design, this means: do not use the WSSUS scattering function to size the number of paths for detection. Always use the measured $P$ from empirical channel measurements (COST-2100, 3GPP TR 38.901, or similar).

Power Delay Profile and Doppler Spectrum as Scattering Function Marginals

Schematic of the scattering function (center, 2D surface) and its two marginal profiles: the power delay profile (right, integrating over Doppler) and the Doppler spectrum (below, integrating over delay). The PDP determines the coherence bandwidth; the Doppler spectrum determines the coherence time.

Why This Matters: Scattering Function in Telecom Ch. 6

The scattering function also appears in Telecom Ch. 6, where it is used primarily to derive coherence time and coherence bandwidth. There, the focus is on single-carrier and OFDM signaling — both of which use only the marginals of $S_h$ . OTFS is the waveform that actually signals directly on the full 2D support of $h(\tau, \nu)$ .

The Scattering Function and WSSUS Channels