OFDM Waveform Optimization for Bistatic Integrated Sensing and Communications

This paper proposes a joint subcarrier assignment and power allocation framework for bistatic OFDM-based ISAC systems that maximizes communication data rate while satisfying delay sensing accuracy constraints, revealing that sensing subcarriers should be allocated only when their Fisher information gain outweighs the communication rate loss.

The Gist

Imagine you are a conductor standing on a stage with a massive orchestra (the transmitter). Your goal is to do two things at once:

1. Play a beautiful symphony so the audience (the communication receiver) can enjoy the music.
2. Send out sound waves to map the room (sensing) to see where the furniture is and how far away it is, without stopping the music.

This is the challenge of Integrated Sensing and Communications (ISAC). Usually, these two tasks fight for the same resources. If you use all your instruments to map the room, the music suffers. If you focus only on the music, you can't see the room.

This paper presents a clever new way to be the conductor, using a specific type of music called OFDM (think of it as a piano with thousands of tiny keys, where each key is a different frequency).

Here is the simple breakdown of their solution:

1. The Problem: The "One-Size-Fits-All" Mistake

In the past, people tried to solve this by randomly picking some piano keys to map the room and others for the music. Or, they gave every key the same amount of energy.

• The Flaw: This is like trying to measure the distance to a wall by tapping a few random keys in the middle of the piano. You get a blurry picture. Meanwhile, you might be wasting energy on keys that don't help the music sound good.

2. The Insight: "Where" and "How Much" Matter Differently

The authors discovered a golden rule about how these two tasks work:

• For the Music (Communication): The quality of the song depends mostly on how many keys you use. More keys = more data = better music.
• For the Map (Sensing): The accuracy of the map depends on which keys you use and how far apart they are.
  • Analogy: Imagine trying to guess the shape of a room by shouting. If you shout at the same pitch, you get a fuzzy echo. But if you shout at a very low pitch and a very high pitch simultaneously, the difference in the echoes tells you exactly how far the walls are.
  • The Lesson: To get a sharp map, you need to pick keys that are far apart on the keyboard, not just keys that are next to each other.

3. The Solution: The "Smart Conductor" Algorithm

The authors created a smart algorithm (called JPCDE) that acts like a super-intelligent conductor. It doesn't just pick keys randomly; it makes a split-second decision for every single key based on a trade-off.

The Decision Rule:
For every key on the piano, the algorithm asks:

"If I use this key for sensing, how much better will my map get? And if I use it for sensing, how much worse will the music get?"

• The "Fisher Information" Gain: This is a fancy way of saying "How much does this specific key help me see the room?"
• The "Rate Loss": This is "How much music quality am I sacrificing?"

The Rule:

• If the Map Gain is bigger than the Music Loss, the key becomes a Sensing Key.
• If the Music Loss is bigger, the key stays a Music Key.

4. The Power Strategy: The "Water-Filling" Analogy

Once the keys are assigned, the conductor has to decide how loud to play each one (Power Allocation).

• For Music Keys: They use a strategy called "Bounded Water-Filling." Imagine pouring water into a bowl with a bumpy bottom. The water naturally fills the deepest holes (the keys with the best connection) first, up to a certain limit. They pour more power into the keys that carry the music best, but they don't let it overflow (there's a power limit).
• For Sensing Keys: They play these keys loud if they are far away from the "center" of the keyboard. This maximizes the distance between the frequencies, which sharpens the map.

5. The Result: Best of Both Worlds

When they tested this system, it was like upgrading from a blurry Polaroid camera to a 4K HD camera while simultaneously upgrading the radio from AM to high-fidelity stereo.

• Accuracy: They could measure distances to within a few centimeters (like measuring a wall with a laser tape measure).
• Speed: They could send data much faster than previous methods because they weren't wasting keys on bad sensing spots.

Summary

This paper teaches us that to do two jobs at once (send data and sense the world), you shouldn't just split your resources in half. Instead, you need a smart, dynamic strategy that picks the right frequencies for sensing (far apart ones) and the right frequencies for data (the ones that sound best), balancing the two like a master chef balancing salt and sugar to create the perfect dish.

Technical Summary

Here is a detailed technical summary of the paper "OFDM Waveform Optimization for Bistatic Integrated Sensing and Communications".

1. Problem Statement

The paper addresses the challenge of designing Orthogonal Frequency-Division Multiplexing (OFDM) waveforms for Bistatic Integrated Sensing and Communications (ISAC) systems. In this architecture, a transmitter (TX) and receiver (RX) are spatially separated. The core problem is to jointly optimize subcarrier assignment (deciding which subcarriers are used for sensing vs. communication) and power allocation to maximize the Communication Data Rate (CDR) while satisfying strict sensing accuracy constraints (specifically, delay estimation accuracy) and a total transmit power budget.

Key challenges identified include:

• Conflicting Requirements: Sensing accuracy depends heavily on the distribution of sensing subcarriers (to maximize effective bandwidth), whereas communication rate depends on the quantity of communication subcarriers and channel gains.
• Different Channel Models: The sensing signal experiences frequency-flat fading per path (single-path transmission), while the communication signal experiences frequency-selective fading (multipath).
• Complexity: The joint optimization problem is a Mixed-Integer Nonlinear Programming (MINLP) problem, which is generally NP-hard.

2. Methodology

A. System Model & Signal Processing

• System Setup: A single-antenna TX and an $N_r$-element Uniform Linear Array (ULA) RX. The system uses pilot-based sensing.
• Joint Path Coefficient and Delay Estimation (JPCDE): The authors propose a new estimation scheme where the RX uses Maximum Likelihood Estimation (MLE) to jointly estimate complex path coefficients and propagation delays from the received sensing signals.
• Cramér-Rao Bound (CRB) Derivation: The authors derive the CRB for delay estimation. A critical insight is revealed:
  • Sensing Accuracy: Governed by the squared effective bandwidth, which is a function of the index distribution (how widely spaced the sensing subcarriers are) and their power allocation.
  • Communication Rate: Determined primarily by the number of communication subcarriers and the channel gain of those subcarriers.

B. Optimization Formulation

The authors formulate an optimization problem (OP) to maximize the total CDR subject to:

1. Total Power Constraint: Sum of power across all subcarriers $\le P_{req}$.
2. Sensing Accuracy Constraint: The CRB for delay estimation for each path must not exceed a predefined threshold ($J_0$).
3. Per-subcarrier Power Constraint: $0 \le P_m \le P_0$.
4. Binary Assignment: $u_m \in \{0, 1\}$ (0 for communication, 1 for sensing).

C. Solution Approach

To solve the non-convex MINLP problem, the authors employ a Block Coordinate Descent (BCD) algorithm combined with Quadratic Transform and Lagrangian Dual Decomposition:

1. Relaxation & Transformation:
   • The fractional term in the sensing constraint is reformulated using a quadratic transform with an auxiliary variable $y$ (representing the power-weighted spectral centroid).
   • The binary constraint $u_m \in \{0, 1\}$ is relaxed to $0 \le u_m \le 1$, but the solution is proven to naturally converge to the boundary (binary values).
2. Iterative Updates:
   • Power Allocation ($P$):
     • Communication Subcarriers: Follows a bounded water-filling structure, allocating more power to subcarriers with better channel conditions.
     • Sensing Subcarriers: Follows a strategy where power is allocated to subcarriers farthest from the spectral centroid $y$ to maximize the effective bandwidth. A specific closed-form solution is derived to meet the CRB threshold with minimal power.
   • Subcarrier Assignment ($u$):
     • The update rule is derived based on a trade-off criterion: A subcarrier is assigned to sensing if and only if the Fisher Information Gain (sensing benefit) exceeds the CDR Loss (communication cost).
   • Dual Variables: Updated via subgradient methods to enforce power and sensing constraints.

3. Key Contributions

1. JPCDE Scheme: Introduction of a joint estimation scheme that reveals the distinct dependencies of sensing accuracy (subcarrier index distribution) and communication rate (subcarrier count).
2. Closed-Form Insights:
   • Proved that optimal subcarrier assignment follows a "Fisher Gain vs. Rate Loss" principle.
   • Demonstrated that optimal power allocation for communication exhibits a bounded water-filling structure.
3. Efficient Algorithm: Developed a low-complexity iterative algorithm ($O(M \log M)$ per iteration) that solves the complex MINLP problem by transforming it into a sequence of convex subproblems with closed-form updates.
4. Theoretical Bound: Provided a rigorous derivation of the CRB for delay estimation in a bistatic OFDM context, linking it directly to the squared effective bandwidth.

4. Simulation Results

The proposed scheme was evaluated against three baselines:

• SAUPA: Subcarrier Assignment with Uniform Power Allocation.
• RSAPA: Random Subcarrier Assignment with Power Allocation.
• RSAUPA: Random Subcarrier Assignment with Uniform Power Allocation.

Key Findings:

• Performance Trade-off: The proposed JPCDE scheme achieves a superior trade-off between CDR and sensing accuracy. As the allowed range error bound increases, the achievable CDR increases significantly.
• Sensing Accuracy: The JPCDE scheme maintains the Range Estimation Root Mean Squared Error (RMSE) consistently around the target bound (0.05 m) across various power budgets, matching the theoretical CRB.
• Communication Efficiency: JPCDE significantly outperforms the RSAPA baseline in CDR because it uses fewer sensing subcarriers to achieve the same accuracy (by optimizing their placement and power), thereby freeing up more resources for communication.
• Baseline Comparison: The proposed method outperforms all baselines. Notably, SAUPA (uniform power) performs well only at high power budgets, while RSAUPA (random assignment) performs the worst, highlighting the critical importance of intelligent subcarrier assignment.

5. Significance

This work provides a fundamental framework for bistatic ISAC waveform design. By decoupling the optimization of subcarrier placement (for sensing resolution) and power allocation (for rate and sensing gain), the paper offers a practical, low-complexity solution for 6G systems. The findings suggest that in ISAC systems, sensing performance is not just about power, but about the spectral distribution of the waveform, and that joint optimization can yield substantial gains over traditional separate or random allocation strategies. The derived "Fisher Gain vs. Rate Loss" criterion offers a clear physical interpretation for resource allocation in future integrated systems.