⚡ electrical engineering

Superimposed Channel Estimation in OTFS Modulation Using Compressive Sensing

This paper proposes a superimposed channel estimation scheme for Orthogonal Time Frequency Space (OTFS) modulation that utilizes the Orthogonal Matching Pursuit (OMP) algorithm to exploit delay-Doppler domain sparsity, thereby eliminating pilot overhead and peak-to-average power ratio issues while employing a message passing detector for symbol recovery.

Original authors: Omid Abbassi Aghda, Mohammad Javad Omidi, Hamid Saeedi-Sourck

Published 2026-02-18

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Omid Abbassi Aghda, Mohammad Javad Omidi, Hamid Saeedi-Sourck

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

The Big Picture: The High-Speed Highway Problem

Imagine you are trying to send a message to a friend while driving at 200 mph on a bumpy, chaotic highway. In the world of wireless communication (like 5G and the upcoming 6G), this is what happens when data travels at high speeds.

The Problem: When you move fast, the signal gets distorted. It's like trying to shout a message to someone while you both are speeding past each other; the wind (Doppler shift) scatters your voice, making it hard to understand.
The Old Solution (OFDM): Traditional systems are like a choir where everyone sings a specific note. When the wind blows, the notes get mixed up, and the choir sounds terrible.
The New Solution (OTFS): The paper discusses a new technique called OTFS (Orthogonal Time Frequency Space). Instead of singing notes, OTFS is like sending a message in a grid of "time and speed." It's much more robust against the wind.

The Specific Challenge: The "Pilot" Overhead

To make sure the message gets through, the sender usually needs to send a "test signal" (called a Pilot) so the receiver can figure out how the wind is blowing.

The Old Way (Embedded Pilot): Imagine you are sending a long letter, but you have to stop every few paragraphs to write a "Test" word so the receiver knows how to read the rest. This wastes a lot of space. In technical terms, this is high pilot overhead. You are using up valuable "road space" just to send test signals instead of actual data.
The Goal: The authors wanted a way to send the test signal without stopping the data flow. They wanted to mix the test signal and the data together perfectly.

The Proposed Solution: The "Invisible Ink" Trick

The paper proposes a method called Superimposed Channel Estimation.

The Analogy: The Soup and the Salt
Imagine you are making a giant pot of soup (the Data). You need to know exactly how salty it is (the Channel State) to serve it correctly.

Old Method: You take a spoonful of soup out, taste it, adjust the salt, and put it back. This takes time and you lose some soup.
New Method (Superimposed): You sprinkle a tiny bit of "magic salt" (the Pilot) directly into the whole pot while you are cooking. You don't stop cooking; you just mix the salt in.

How it works in this paper:

Mixing: They mix the "Test Signal" (Pilot) directly with the "Data" on the same grid.
The Problem with Mixing: Now, the receiver has a pot of soup with both data and test salt mixed together. How do they separate them?
The Secret Ingredient (Sparsity): The authors realized that the "wind" (the channel) is actually very simple. It only affects a few specific spots, not the whole pot. In math terms, the channel is sparse (mostly empty space with just a few active points).
The Detective Work (OMP): They use a smart algorithm called OMP (Orthogonal Matching Pursuit). Think of OMP as a detective who knows the criminal (the channel) only hides in a few specific alleys. The detective ignores the noise and finds the few active spots to figure out the channel.
The Loop (Iterative Cleaning):
- Round 1: The detective makes a rough guess of the channel using the mixed signal.
- Round 2: They use that guess to "clean" the data. Now that the data is cleaner, they use the cleaned data to help the detective make an even better guess of the channel.
- Round 3: They repeat this until the channel is perfectly understood and the data is perfectly recovered.

Why is this better? (The Benefits)

The paper compares their method to three other ways of doing things:

No Traffic Jams (Low Overhead): Because they mix the test signal with the data, they don't need to stop and send a separate "Test Frame." This means more room for actual data. It's like driving on a highway without needing to pull over to check your speedometer.
No Loud Noises (Low PAPR):
- The Issue: Some methods use a single, super-powerful "Test Signal" (like a giant flare). This causes the signal to have a high "Peak-to-Average Power Ratio" (PAPR). Imagine a speaker that usually whispers but suddenly screams at 100 decibels. This damages the equipment and wastes energy.
- The Fix: The authors spread their "magic salt" (pilot) all over the grid. It's like many people whispering a test word at the same time rather than one person screaming. This keeps the power smooth and safe for the equipment.
Better Accuracy: By using the "Detective" (OMP) and the "Loop" (iterative cleaning), they get results almost as good as the old, slow methods, but much faster and more efficient.

The Verdict

The authors built a system that:

Sends data and test signals at the same time (Superimposed).
Uses a smart detective algorithm (OMP) to separate them because the environment is simple (Sparse).
Refines the answer by using the data to help find the channel, and the channel to help find the data.
Saves space (no pilot overhead) and saves energy (low power spikes).

In short, they found a way to send a message through a storm without needing to stop, shout loudly, or waste any space, ensuring the message arrives clearly even at high speeds.

1. Problem Statement

Orthogonal Time Frequency Space (OTFS) modulation is a promising technique for 6G high-speed wireless communications, designed to mitigate the effects of high Doppler shifts and frequency spreading that degrade Orthogonal Frequency Division Multiplexing (OFDM). However, OTFS faces two critical challenges:

High Pilot Overhead: Conventional channel estimation methods, such as the Embedded Pilot (EP) scheme, require dedicated pilot symbols and guard intervals in the Delay-Doppler (DD) domain. This significantly reduces spectral efficiency.
High Peak-to-Average Power Ratio (PAPR): Existing superimposed pilot methods (where pilots are added to data) or single-pilot schemes often suffer from high PAPR, which complicates power amplifier design and reduces efficiency.
Complexity in Dynamic Channels: Some existing superimposed methods (e.g., MMSE-based) assume that channel delay and Doppler parameters remain constant over a "super-frame" (multiple OTFS frames). This assumption fails in highly time-varying channels and still requires dedicated frames for initial parameter estimation.

The paper aims to propose a method that achieves accurate channel estimation and data detection without pilot overhead and without high PAPR, while adapting to time-varying channels.

2. Methodology

The authors propose a Joint Channel Estimation and Data Detection framework based on the sparsity of the Delay-Doppler channel representation. The core components are:

A. System Model

Superimposed Transmission: Instead of reserving specific grid points for pilots, pilot sequences ( $X_p$ ) are superimposed directly onto information data symbols ( $X_d$ ) in the DD domain. The transmitted signal is $X = X_d + X_p$ .
Power Allocation: The total power is normalized to 1. The pilot power is set to $\sigma_p^2 = 0.3$ (30%) and data power to $\sigma_d^2 = 0.7$ (70%).
Channel Sparsity: The DD channel is modeled as sparse, consisting of $P$ taps, each defined by a coefficient, delay ( $\tau$ ), and Doppler shift ( $\nu$ ).

B. Iterative Algorithm (Coarse to Fine)

The proposed solution utilizes an iterative loop combining Orthogonal Matching Pursuit (OMP) for channel estimation and Message Passing (MP) for data detection.

Coarse Estimation (Iteration 1):
- The receiver treats the interference from data symbols as noise.
- OMP Algorithm: Used to estimate the channel vector ( $\hat{h}_1$ ) by exploiting the sparsity of the DD channel.
- Data Detection: The estimated channel is used to cancel pilot interference from the received signal. The residual signal is processed by an MP detector to obtain a coarse estimate of the data ( $\hat{x}_d^1$ ).
Fine Estimation (Iterations $i > 1$ ):
- Joint Refinement: The detected data from the previous iteration ( $\hat{x}_d^i$ ) is treated as additional "pilot" information.
- Re-estimation: The OMP algorithm is run again using the combined "pilot" matrix (original pilots + estimated data) to refine the channel estimate ( $\hat{h}_{i+1}$ ).
- Interference Cancellation: The refined channel estimate is used to subtract pilot interference from the received signal more accurately.
- Re-detection: The MP detector processes the cleaned signal to update the data estimate ( $\hat{x}_d^{i+1}$ ).

This loop continues until convergence or a maximum number of iterations is reached.

3. Key Contributions

Zero Pilot Overhead: Unlike EP schemes, the proposed method does not require dedicated pilot symbols or guard intervals, maximizing spectral efficiency.
Dynamic Parameter Estimation: Unlike MMSE superimposed methods that assume static delay/Doppler over a super-frame, this method estimates tap parameters (delay and Doppler) dynamically in every frame using sparse recovery.
Low PAPR: By spreading pilot sequences across the entire DD domain (superimposed on all data symbols) rather than using a single high-power pilot or isolated pilot clusters, the method significantly reduces the Peak-to-Average Power Ratio compared to EP and single-pilot superimposed schemes.
Joint Optimization: The iterative coupling of OMP (for channel) and MP (for data) allows the system to iteratively cancel interference, improving both channel estimation accuracy and data detection reliability.

4. Simulation Results

The authors compared their OMP Superimposed method against:

Embedded Pilot (EP) [6]
MMSE Superimposed [7]
Single Pilot Superimposed [8]

Key Findings:

Channel Estimation (NMSE):
- At high Signal-to-Noise Ratios (SNR), the OMP superimposed method outperforms the EP scheme.
- It achieves performance comparable to the MMSE superimposed method and significantly better than the single-pilot superimposed method.
- An error floor exists at very high SNRs due to residual data-to-pilot interference, but it is manageable.
Data Detection (BER):
- The Bit Error Rate (BER) of the proposed method approaches that of the ideal CSI (perfect channel knowledge) and matches the performance of the EP and MMSE superimposed schemes.
- It is significantly more robust than the single-pilot superimposed method.
PAPR Performance:
- The OMP superimposed method exhibits a much lower PAPR (Complementary Cumulative Distribution Function) compared to the EP and Single Pilot superimposed schemes. This is because the energy is distributed across the DD grid rather than concentrated in specific high-power pilot locations.

5. Significance

This paper presents a viable solution for high-mobility 6G communications by addressing the trade-off between spectral efficiency and estimation accuracy.

Spectral Efficiency: By eliminating the need for dedicated pilot frames, the system can transmit more user data.
Hardware Efficiency: The low PAPR characteristic makes the system more compatible with practical power amplifiers, reducing energy consumption and hardware costs.
Robustness: The ability to estimate channel parameters without assuming they are constant over long super-frames makes the system suitable for highly dynamic, high-speed environments (e.g., high-speed trains, UAVs).

In conclusion, the proposed OMP-based superimposed channel estimation offers a balanced, high-performance alternative to conventional OTFS estimation techniques, effectively removing pilot overhead while maintaining low PAPR and high detection accuracy.