Ferkans — Interactive Telecom Tutor

Why a New Domain at All?

OFDM, the dominant waveform of 4G and 5G, was designed for a world in which the channel is approximately time-invariant over each symbol. That assumption buys OFDM its most elegant property: the channel diagonalizes, and equalization reduces to a per-subcarrier complex scalar division. When the channel is genuinely time-invariant over one OFDM symbol, this works beautifully.

The assumption breaks when the receiver — or a reflector — moves fast. Doppler shifts scramble the subcarrier orthogonality, inter-carrier interference (ICI) appears, and the clean diagonal structure is lost. In 6G use cases (high-speed rail, V2X, low-earth-orbit satellites, UAVs), mobility is not a nuisance — it is a defining feature of the channel.

The question this book answers is: what is the right signal space for a doubly-selective channel? The answer is the delay-Doppler (DD) domain. In this domain, a physically realistic multipath channel collapses to a small number of discrete impulses — the channel becomes sparse and, as we will prove in Chapter 4, time-invariant in the sense of a 2D convolution. This is not just a change of basis. It is the correct basis for the physics.

Definition:
Discrete Multipath Channel with Mobility

A physical wireless channel is a superposition of $P$ discrete propagation paths, each produced by a specific reflector or scatterer. The complex baseband impulse response, viewed at receive time $t$ with source-to-receiver delay $\tau$ , is $h(\tau, t) \;=\; \sum_{i=1}^{P} h_i(t)\, \delta\!\left(\tau - \tau_i(t)\right),$ where $h_i(t)$ is the complex gain, $\tau_i(t)$ the delay, and both in general depend on time because the geometry changes. When the reflectors move (relative to the transmit–receive axis) with slowly varying velocities, the delays drift linearly: $\tau_i(t) \approx \tau_i - (v_i/c)\,t$ . The resulting phase rotation is the Doppler shift $\nu_i = (v_i/c)\,f_0$ .

We adopt the delay-first ordering $(\tau, t)$ , not $(t, \tau)$ . This matches Bello's original notation and is the one most consistent with the DD domain literature.

,

Why $P$ Is Small

The physical world does not produce a continuum of paths. At carrier wavelengths of centimeters to meters, an indoor room has tens of significant reflectors; an outdoor urban cell typically has a few dozen. Measurement campaigns consistently find $P \lesssim 10$ – $20$ dominant paths. This sparsity is not an approximation — it is the empirical reality of propagation, and it is the single most important fact motivating OTFS.

Example: An Urban Three-Path Scenario

A receiver in a moving vehicle sees three significant paths: a line-of-sight component with delay $\tau_1 = 0$ from the base station (directly ahead), a reflection off a stationary building at delay $\tau_2 = 1\,\mu\text{s}$ , and a reflection off an oncoming vehicle at delay $\tau_3 = 2\,\mu\text{s}$ . The ego vehicle travels at $v_1 = 100$ km/h, the oncoming vehicle at $v_3 = 100$ km/h (approach), and the building is stationary. The carrier is $f_0 = 6$ GHz. Compute the delay and Doppler coordinates of each path.

Solution

Convert velocities to Doppler shifts

The Doppler shift of a path is $\nu_i = (v_{\text{rel},i}/c)\,f_0$ , where $v_{\text{rel}}$ is the rate of change of the path length. The ego vehicle at $100$ km/h has velocity along the LOS direction of $v_1 = 27.78$ m/s (approaching the transmitter reduces the path length at this rate). With $c = 3 \times 10^8$ m/s and $f_0 = 6 \times 10^9$ Hz, $\nu_1 = \frac{27.78}{3\times 10^8} \cdot 6 \times 10^9 = 556 \text{ Hz}.$

Stationary reflector

For the building, only the ego vehicle moves; the reflection geometry yields a Doppler of $\nu_2 \approx 556$ Hz (approximate, assuming the reflector is roughly along the motion axis). For simplicity we take $\nu_2 = 300$ Hz to reflect the oblique geometry.

Two moving bodies

The oncoming vehicle closes at $v_{\text{rel},3} = 200$ km/h = $55.56$ m/s (both motions contribute). Therefore $\nu_3 = \frac{55.56}{3\times 10^8} \cdot 6 \times 10^9 = 1{,}111 \text{ Hz}.$

The DD coordinates

The three paths live at DD coordinates $(\tau_1, \nu_1) = (0, 556)$ , $(\tau_2, \nu_2) = (1\,\mu\text{s}, 300)$ , and $(\tau_3, \nu_3) = (2\,\mu\text{s}, 1111)$ . Three points in a 2D plane — this is the entire channel state, no matter how intricately $h(\tau, t)$ varies over time. The point is that the channel's ostensible complexity, visible as fast fading in the time-frequency picture, reduces to three parameters once we look in the correct domain.

Delay-Doppler Map from Scatterer Geometry

Place scatterers (reflectors) in the 2D plane around a fixed transmitter and receiver. Each scatterer contributes one point in the delay-Doppler plane, with delay proportional to the total path length (Tx → scatterer → Rx) and Doppler determined by the scatterer's velocity along the line of sight. The point is that the DD representation is a picture of the physical scene.

Parameters

Ego velocity (km/h)100

Velocity of the receiver. Doppler spread scales linearly.

Number of scatterers

P

5

Carrier

f_0

(GHz)6

At higher carriers, Doppler shifts grow proportionally.

Scene seed7

Key Takeaway

The channel has physics. A real multipath channel is the sum of a small number $P$ of discrete echoes, each labeled by a delay $\tau_i$ and a Doppler $\nu_i$ . The delay-Doppler representation is not a mathematical trick; it is simply the inventory of those echoes. Every waveform design choice in the rest of this book — OTFS modulation, embedded-pilot estimation, MP detection — exploits the fact that $h(\tau, \nu)$ is supported on at most $P$ points.

Why This Matters: Doubly Selective Fading, in One Picture

In Telecom Ch. 6 we introduced the WSSUS (wide-sense stationary uncorrelated scatterers) fading model, which organizes the channel into four functions related by Fourier transforms. That chapter developed the statistical picture — the Jakes spectrum, the power delay profile, coherence time and coherence bandwidth. What we do here is different: we adopt the instantaneous realization $(h_i, \tau_i, \nu_i)_{i=1}^P$ as the channel state and design a modulation around it. The statistics of Ch. 6 inform the pilot design and the detector; the geometry here drives the waveform.

From a Moving Scene to a Sparse DD Map

A transmitter, a receiver, and three reflectors in the plane; the animation raises delay from path length and Doppler from velocity, building the DD map point by point. The final frame shows the delay-Doppler "spectrum" of the channel: three spikes, nothing else.

⚠️Engineering Note

Why This Matters for 6G

The 3GPP work on 5G NR considered mobility up to 500 km/h (high-speed rail), but system-level simulations show significant performance degradation above roughly $v = 120$ km/h at 3.5 GHz due to ICI in OFDM. At millimeter-wave bands ( $f_0 = 28$ GHz), the maximum Doppler scales eightfold: at $v = 100$ km/h we reach $f_D = 2.6$ kHz, comparable to the subcarrier spacing. At LEO altitudes, the relative Doppler of a satellite at 550 km is of order $30$ kHz at $f_0 = 10$ GHz — orders of magnitude beyond what OFDM can tolerate without heavy compensation. This is the concrete engineering reason OTFS-type waveforms are a serious candidate for 6G.

Practical Constraints

•
3GPP TR 38.913 mobility targets: 500 km/h in high-speed rail scenarios
•
Doppler spread scales linearly with carrier frequency — mmWave is 5–10× worse than sub-6 GHz
•
LEO satellites introduce Doppler shifts on the order of tens of kHz at X-band

📋 Ref: 3GPP TR 38.913 §6.1.2

Quick Check

A stationary transmitter and a receiver move directly away from each other at speed $v$ . Compared to the zero-velocity case, the received carrier frequency:

Increases by $f_D = (v/c)\,f_0$

Decreases by $f_D = (v/c)\,f_0$

Is unchanged — Doppler only appears at relativistic speeds

Depends on the direction of the transmitter array, not on motion

Correction:

Decreases by

f_D = (v/c)\,f_0

When the receiver moves away, the path length grows, successive wavefronts arrive later, and the received frequency is redshifted by $f_D = (v/c)\,f_0$ . The Doppler shift is defined as positive when the path length shrinks.

From Multipath and Mobility to the Delay-Doppler Representation