⚡ electrical engineering

Input Distribution Design for Ranging-Oriented OFDM-ISAC Systems Under Frequency-Selective Fading

This paper proposes a computationally efficient input distribution design for OFDM-based ISAC systems under frequency-selective fading, which optimizes the trade-off between communication rate and sensing performance by strategically allocating constellation kurtosis across subcarriers based on the capacity-distortion framework.

Original authors: Weijiang Zhao, Yifeng Xiong

Published 2026-04-30

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CC BY 4.0

Original authors: Weijiang Zhao, Yifeng Xiong

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 running a busy radio station that has to do two things at once: broadcast music (communication) and listen for echoes to map the room (sensing/ranging). This is the core idea of a new technology called ISAC (Integrated Sensing and Communication), which is expected to be a key feature of future 6G networks.

The paper by Zhao and Xiong tackles a specific problem: How do you design the "shape" of the radio waves so they are good at both playing music and listening for echoes, especially when the airwaves are messy and bumpy (frequency-selective fading)?

Here is the breakdown of their solution using simple analogies:

1. The Core Conflict: The "Predictable vs. Random" Dilemma

To listen for echoes (sensing), you want a signal that is predictable and structured, like a perfect drumbeat. If the signal is too random, the echoes get lost in the noise.
To send music (communication), you want the signal to be random and chaotic, like a jazz improvisation. This randomness carries more information.

This creates a tug-of-war called the Deterministic-Random Tradeoff (DRT). If you make the signal too structured for sensing, you lose data speed. If you make it too random for speed, your sensing gets blurry.

2. The Secret Ingredient: "Kurtosis"

The authors introduce a mathematical concept called Kurtosis. Think of kurtosis as the "spikiness" or the "shape" of the signal's energy distribution.

Low Kurtosis (Round/Flat): Like a Phase Shift Keying (PSK) signal. It's very structured and great for sensing, but carries less data.
High Kurtosis (Spiky): Like a Quadrature Amplitude Modulation (QAM) signal. It's very random and carries lots of data, but creates "clutter" (sidelobes) that messes up sensing.

The Paper's Big Insight: Instead of just choosing one shape for the whole broadcast, you can treat "spikiness" (kurtosis) as a resource that you can distribute differently across different parts of the radio spectrum.

3. The Problem: The "Messy Room"

In a real-world scenario (like a city with tall buildings), the radio signal bounces around. This is called frequency-selective fading. Some parts of the radio spectrum (subcarriers) are clear highways, while others are bumpy dirt roads.

Old Method: The standard way to solve this involves complex math that requires calculating billions of possibilities at once. It's like trying to solve a Rubik's cube the size of a city while running a marathon. It's too slow for real-time use.
The Authors' Solution: They found a way to break the giant problem into smaller, manageable pieces. They proved that you can design the signal for each "lane" of the highway independently, rather than trying to solve the whole highway at once.

4. The Strategy: "Uniform Power, Smart Spikiness"

The authors developed a fast, efficient algorithm (a "Gradient Projection" method) to figure out the perfect balance. Their findings reveal a counter-intuitive strategy:

Power Allocation (How loud to play): When the sensing requirements are strict (you need to hear very faint echoes), the best strategy is to play equally loud on all lanes (Uniform Power). Surprisingly, trying to play louder on the "good" lanes actually hurts the sensing performance in this specific context.
Kurtosis Allocation (How spiky to make the signal): While the volume is uniform, the shape of the signal changes.
- On the clear lanes (good signal quality), they assign higher kurtosis (more spiky/random). This maximizes the data speed because the clear lane can handle the "noise" of the random signal.
- On the bumpy lanes (poor signal quality), they assign lower kurtosis (more round/structured). This keeps the signal clean so the sensing echoes aren't lost.

The Analogy: Imagine a choir singing in a large hall.

Old way: Everyone sings the same note with the same volume and style.
New way: Everyone sings at the same volume (Uniform Power), but the singers in the front row (good lanes) sing with a wild, improvisational jazz style (High Kurtosis) to tell a complex story, while the singers in the back row (bad lanes) sing a simple, steady hum (Low Kurtosis) so the conductor can still hear the echo of their voices clearly.

5. Why This Matters

The paper claims that by using this method, they can achieve a higher data rate (faster internet) without sacrificing the ability to detect objects (sensing), even in difficult environments. They showed that their new math is fast enough to be used in real-time, unlike the old, slow methods.

In summary: The paper provides a fast, smart recipe for 6G radio waves. It tells engineers to keep the volume steady across the board but to change the "texture" of the signal depending on the quality of the path, ensuring the system is both a fast data pipe and a sharp radar.

1. Problem Statement

The paper addresses the design of Integrated Sensing and Communication (ISAC) systems within the framework of Orthogonal Frequency Division Multiplexing (OFDM), specifically for 6G networks. The core challenge is the Deterministic-Random Tradeoff (DRT):

Sensing: Optimal sensing (specifically ranging) requires deterministic, carefully designed waveforms to minimize sidelobe levels in the autocorrelation function.
Communication: Information transmission relies on random data payloads, which inherently degrade sensing performance.

The authors focus on frequency-selective fading channels, a practical scenario where the channel gain varies across subcarriers. The goal is to design the input distribution (constellation design) of the transmitted symbols to maximize the communication mutual information (rate) while satisfying a constraint on the Expected Integrated Sidelobe Level (EISL), a metric for ranging accuracy.

2. Methodology

The authors propose a computationally efficient framework to solve this optimization problem, moving away from traditional high-complexity algorithms.

A. Mathematical Formulation

Objective: Maximize Mutual Information $I(x; y)$ subject to a total power budget ( $P$ ) and an EISL constraint ( $D$ ).
Key Insight: The EISL is analytically linked to the kurtosis ( $\kappa$ ) of the signal constellations. Specifically, lower kurtosis (closer to Phase Shift Keying, PSK) reduces sidelobes, while higher kurtosis (closer to Gaussian/Quadrature Amplitude Modulation, QAM) increases capacity.
Factorization: The authors prove that under independent subcarriers, the optimal input distribution factorizes across subcarriers ( $p_x(x) = \prod p_{x_i}(x_i)$ ). This allows the high-dimensional optimization problem to be decomposed into per-subcarrier problems.

B. Algorithmic Approaches

The paper presents two algorithmic stages:

Factorized Modified Blahut-Arimoto (BA) Algorithm:
- Adapts the classic BA algorithm for ISAC by utilizing the factorization property.
- Reduces the computational complexity from exponential (in the number of subcarriers $N$ ) to linear ( $O(N)$ ) by performing integrals only over 4-dimensional spaces (real/imaginary parts of two complex variables) rather than the full $2N$ -dimensional space.
- Limitation: Still requires numerical integration and searching for dual variables, making it too slow for real-time implementation.
Gradient Projection-Based Approach (Real-Time Solution):
- Approximation: Instead of solving the exact mutual information maximization, the authors approximate the problem by maximizing differential entropy under moment constraints (second-order and fourth-order moments).
- Closed-Form Solution: They derive the optimal distribution form as $p(z) \propto \exp(-c|z|^4 + a|z|^2)$ , where $a$ and $c$ are natural parameters.
- Gradient Computation: They derive closed-form expressions for the gradient of the entropy with respect to these parameters.
- Optimization: The problem is solved using a Gradient Projection Method. This involves:
  - Updating parameters via gradient ascent.
  - Projecting the updated power ( $p$ ) and excess kurtosis ( $d$ ) back onto the feasible region defined by the power and EISL constraints.
  - The projection step is solved efficiently using Karush-Kuhn-Tucker (KKT) conditions and 1D searches, avoiding complex numerical integrations.

3. Key Contributions

Theoretical Framework: Formulated the ISAC input distribution design as a mutual information maximization problem under EISL constraints, explicitly linking EISL to constellation kurtosis.
Algorithmic Efficiency: Developed a Factorized Modified BA algorithm that reduces complexity from exponential to linear with respect to the number of subcarriers.
Real-Time Implementation: Proposed a Gradient Projection method that relies solely on closed-form expressions and 1D searches, enabling real-time signal design for OFDM systems with many subcarriers.
Optimal Strategy Discovery: Demonstrated that under stringent sensing constraints (low EISL), the optimal strategy deviates from traditional "water-filling" power allocation. Instead, it favors uniform power allocation across subcarriers while allocating kurtosis (shaping the constellation) to subcarriers with better channel conditions (higher SNR).

4. Numerical Results

The authors validated their method using an OFDM system with $N=64$ subcarriers over a 4-path Rician channel:

Rate-Kurtosis Tradeoff: The proposed method (kurtosis allocation with uniform power) achieved a rate gain of up to 0.1 bits per subcarrier compared to a "uniform kurtosis" strategy (where all subcarriers use the same constellation shape).
SNR Dependence: The performance gain was more significant at lower average SNRs, where channel quality disparities are more pronounced.
Kurtosis Allocation: The results showed a "water-filling" like behavior for kurtosis: subcarriers with low channel gains were assigned the minimum possible kurtosis (near 1, similar to PSK) to minimize sidelobes, while subcarriers with high gains were assigned higher kurtosis to boost communication rates.
Sidelobe Performance: The proposed method maintained identical sidelobe levels across all subcarriers, ensuring consistent ranging accuracy.

5. Significance

This work is significant for the practical deployment of 6G ISAC systems for several reasons:

Bridging Theory and Practice: It moves beyond theoretical capacity limits to provide a computationally efficient design method suitable for real-time hardware implementation.
Resource Optimization: It introduces a novel paradigm where kurtosis is treated as a allocatable resource alongside power. This allows systems to dynamically balance sensing and communication needs without changing the waveform structure fundamentally.
Scalability: By reducing the complexity from exponential to linear (and further via gradient projection), the method makes ISAC design feasible for large-scale OFDM systems (e.g., massive MIMO or wideband 6G) where traditional optimization methods fail.
Design Insight: It challenges the conventional wisdom of "water-filling" power allocation in ISAC contexts, showing that under strict sensing constraints, uniform power with adaptive constellation shaping is superior.