⚡ electrical engineering

Time-Frequency Pilot Sequence Design and LoS Delay-Doppler Estimation

This paper proposes a novel framework for line-of-sight delay-Doppler estimation in dense scattering environments using two new Zadoff-Chu-inspired time-frequency pilot sequences and a direct twisted convolution-based estimation approach that outperforms traditional methods.

Original authors: Aadarsh Devanand, Praful D. Mankar

Published 2026-04-27

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CC BY 4.0

Original authors: Aadarsh Devanand, Praful D. Mankar

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to track a fast-moving hummingbird in a dense, messy garden filled with wind, moving leaves, and fluttering butterflies.

To track the bird, you need to know two things: where it is (its position/delay) and how fast it’s moving (its speed/Doppler shift). In the world of wireless communication (like 5G or future 6G), the "hummingbird" is your mobile phone, and the "garden" is the messy environment of buildings and trees that bounce signals around.

This paper presents a smarter way to "see" that hummingbird more clearly. Here is the breakdown:

1. The Problem: The "Messy Garden" Effect

In a perfect world, a signal travels in a straight line from a tower to your phone. But in reality, signals bounce off everything. This creates two headaches:

The Echo Problem (Delay): Signals arrive at different times because they took different paths.
The Speed Problem (Doppler): Because you are moving, the frequency of the signal shifts (like how a siren sounds higher as it approaches and lower as it moves away).

Current methods often struggle because they use a single "flash" of light (a pilot signal) to find the bird. In a messy garden, that flash creates too many "ghost images" (interference), making it hard to tell which bird is the real one and which is just a reflection from a leaf.

2. The Solution: Better "Flashlights" (Pilot Sequences)

The researchers designed two new types of "flashlights" (called pilot sequences) to help the receiver distinguish the real signal from the echoes.

The "Separable" Flashlight: Imagine instead of one big flash, you use a grid of tiny, organized pulses that are mathematically "spread out" in both time and frequency. It’s like using a laser grid to map a room; it helps you pinpoint exactly where an object is by seeing how the grid lines bend around it.
The "Stacked" Flashlight: Imagine stacking different colored lights on top of each other. Each "color" (frequency) has its own unique pattern. Because the patterns are different, the receiver can easily tell the difference between the actual signal and the "noise" caused by the environment.

3. The Secret Sauce: "Twisted Convolution"

This is the most technical part, but think of it this way:
Usually, when engineers try to clean up a signal, they use a standard mathematical tool called "convolution"—think of this like a standard filter on a camera.

However, because the signal is moving and changing frequency at the same time, a standard filter isn't enough. It’s like trying to use a flat map to navigate a spinning, warping rollercoaster. The researchers used something called "Twisted Convolution." This is a specialized mathematical tool that "twists" along with the signal, accounting for the fact that the signal is changing in both time and frequency simultaneously. It allows them to "un-twist" the mess and find the bird directly.

4. The Result: A Clearer Picture

The researchers tested their new "flashlights" and "filters" against the old methods. Their results showed:

Higher Accuracy: Even when the environment was incredibly "noisy" (lots of reflections/NLoS), their method found the exact delay and speed much more accurately.
Better Performance in High Speed: As the "hummingbird" (the user) moves faster, the new method stays much more stable than the old ones.

Summary in a Nutshell

If traditional signal tracking is like trying to find a moving target in a dark, foggy room with a single, blurry flashlight, this paper provides a high-tech, multi-colored laser grid and a specialized smart-lens that allows the system to see through the fog and track the target with incredible precision.

Technical Summary: Time-Frequency Pilot Sequence Design and LoS Delay-Doppler Estimation

1. Problem Statement

The paper addresses the challenge of accurately estimating the Line-of-Sight (LoS) Delay and Doppler (DD) shifts in wideband wireless communication and sensing systems (such as radar and OTFS modulation).

Traditional estimation methods face several critical limitations:

Multipath Interference: Existing wideband cross-ambiguity function (WB-CAF) methods often assume an absence of non-line-of-sight (NLoS) components. In reality, dense scattering environments create NLoS components that cause inter-symbol and inter-carrier interference, degrading estimation accuracy.
Complexity and Assumptions: Many current DD-domain pilot designs require an inverse symplectic Fourier transform to move from the DD domain to the Time-Frequency (TF) domain, increasing processing overhead. Furthermore, they often rely on the assumption of "bi-orthogonality" between transmit and receive pulses, which is impractical for most real-world pulse-shaping methods.
Prior Knowledge Dependency: Some optimal designs require prior knowledge of the channel and noise covariance, which is unreliable in high-mobility scenarios where the channel decorrelates rapidly.

2. Methodology

The authors propose a novel framework that operates directly in the TF domain to avoid unnecessary transformations and handle the frequency-variant nature of wideband channels.

A. Pilot Sequence Design:
The authors introduce two new multi-carrier TF-domain pilot sequences inspired by Zadoff-Chu (ZC) sequences, designed to exhibit desirable autocorrelation properties:

Separable ZC Sequence: Constructed as the outer product of two 1D ZC sequences (one for frequency, one for time). This sequence aims to provide low side-lobes in both time and frequency shifts.
Stacked ZC Sequence: Constructed by placing a ZC sequence with a unique root index on each frequency sub-carrier. This leverages the low cross-correlation property of ZC sequences to minimize interference across the frequency axis.

B. Estimation Framework (Twisted Convolution):
Because wideband channels involve Doppler shifts that make the channel frequency-variant, the input-output relationship is modeled using twisted convolution rather than standard linear or circular convolution.

The authors propose a specific filter $\Gamma[l, k]$ based on the twisted convolution operation.
By applying this filter to the received signal, they derive an estimator that can extract the LoS component while mathematically accounting for the non-associative nature of the twisted convolution.

3. Key Contributions

Novel TF-Domain Pilots: The design of Separable ZC and Stacked ZC sequences specifically optimized for 2D (delay-Doppler) estimation.
Direct TF-to-DD Estimation: A framework that estimates LoS DD parameters directly from the TF-domain received signal using a twisted convolution-based matched filter, bypassing the need for complex DD-domain transformations.
Robustness to NLoS: A mathematical model and estimation approach that explicitly accounts for Rician fading (the presence of both LoS and NLoS components) and dense scattering environments.

4. Results and Discussion

The proposed framework was evaluated via numerical simulations in a high-mobility scenario (UE moving at 200 kmph) under various Rician fading factors ( $\kappa$ ) and Signal-to-Noise Ratios (SNR).

Superiority over 1D ZC: Both proposed sequences significantly outperformed the traditional single-carrier (1D) ZC sequence in terms of Normalized Mean Squared Error (NMSE) for both delay and Doppler estimation.
Impact of LoS Strength ( $\kappa$ ): As the LoS component becomes stronger (higher $\kappa$ ), the performance of the proposed sequences improves significantly, whereas the 1D ZC sequence remains relatively stagnant.
Sequence Comparison:
- Separable ZC performed better for delay estimation in the presence of NLoS components.
- Separable ZC also showed better performance for Doppler estimation because its periodic autocorrelation on the frequency axis is ideal ( $\delta(\nu)$ ), whereas the Stacked ZC sequence has a non-zero residual correlation ( $1/\sqrt{M}$ ).

5. Significance

This work is significant for the development of next-generation wireless technologies (like 6G) that integrate sensing and communication. By providing a low-complexity, high-accuracy method for LoS estimation that works in dense multipath environments, the research enables more robust user localization, tracking, and directional beamforming in high-mobility scenarios.