Mutual Information: A Unifying Metric to Pilot Design for Integrated Sensing and Communication (ISAC)

The Dual Imperative: Why ISAC Requires Unified Pilot Design

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The continuous growth in wireless devices has placed immense strain on the radio spectrum, making Integrated Sensing and Communication (ISAC) systems a key enabler for future networks. ISAC is presented as a strategy to address spectrum constraints by co-designing wireless communication functions and radar sensing on a single hardware platform, sharing spectrum and resources to improve band-utilization efficiency. While sensing and communication were historically treated as separate fields since the 1900s, ISAC requires a unified platform using a common radio waveform.

This integration, however, complicates the design of signal waveforms and resource allocation. One critical research challenge is designing pilot signals ($\Phi$) that are simultaneously effective for downlink (DL) communication channel estimation (for equalization and decoding by users) and target detection (where the ISAC Base Station (BS) exploits the backscattered echo). The primary hurdle is that objectives often conflict; improving performance for one subsystem can worsen the performance of the other.

Mutual Information (MI) as the Common Denominator

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To overcome the inherent trade-offs, researchers utilized Mutual Information (MI) as a unified metric for characterizing both communication and sensing performance. This approach allows the entire weighted objective to be measured under a common unit: bits.

Communication MI ($\text{M}_{\text{comm}}$)

The communication MI metric ($\text{M}{\text{comm}}$) is proposed to aid with channel estimation tasks for multi-user communications. Minimizing the variance of the channel estimation error ($\sigma^2{h_k}$) simultaneously increases $\text{M}{\text{comm}}$ and the worst-case channel capacity ($\text{C}{\text{worst}}$). Maximizing this MI has been shown to positively influence the worst-case channel capacity, thereby improving overall communication performance.

Sensing MI ($\text{M}_{\text{sense}}$)

The sensing MI metric ($\text{M}{\text{sense}}$) is proposed to optimize target detection performance. Maximizing $\text{M}{\text{sense}}$ aims to maximize the difference between the power found in the look direction and that of clutter components, in a logarithmic sense. Furthermore, $\text{M}{\text{sense}}$ is asymptotically connected to the probability of detection ($\text{P}_{\text{d}}$) under the Neyman-Pearson criterion for a fixed false-alarm probability ($\text{P}{\text{fa}}$).

Solving the Dual Optimization Problem

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The goal of simultaneously maximizing all communication MIs (for $K$ users) and the sensing MI results in a Multi-Objective Optimization Problem (MOOP). Since a global optimum may not exist due to conflicting objectives, the optimal solutions lie on the Pareto frontier.

To make this problem solvable, the MOOP is transformed into a Single-Objective Optimization Problem ($\text{P}_{\text{s}}$) using a scalarization technique involving a weighted-sum goal function.

The core solution relies on the design parameter, $\rho$:

• Trade-off Control: The parameter $\rho$ balances the ISAC tradeoff. Increasing $\rho$ prioritizes the communication sub-system (channel estimation), while decreasing it prioritizes sensing (target detection).
• Orthogonality Constraint: The optimization must adhere to the orthogonality constraint ($\Phi\Phi^H = \mathbf{I}$), which is desired for multi-channel estimation and simplifies least squares channel estimators.
• Solution Method: Due to the highly non-convex and non-linear nature of the problem, it is solved using the projected gradient descent method on the Stiefel manifold. This iterative process is guaranteed to output an orthogonal pilot at every step and is proven to converge to a stable orthogonal pilot matrix.

The Information Overlap Phenomenon

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A significant finding reported through simulations is the information overlap phenomenon.

The sources show that ISAC MI gains are heavily influenced by the target’s location. As the target approaches the mean Angle of Arrival (AoA) of the communication user, the ISAC MI frontier is pushed outwards towards the utopia point. This occurs because the communication channel and the sensing channel begin to share common information, enabling the system to achieve better sensing and communication MI performance simultaneously using the same orthogonal pilot matrix. The highest integration gain is exploited when the objective is specifically to sense a communication user, or when the target and user are co-located.

Demonstrated Performance Gains

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The proposed MI-based pilot design exhibits clear superiority compared to non-optimized methods, demonstrating tangible performance gains across key metrics:

• Channel Estimation Accuracy (NMSE): The communication-optimal pilot matrix ($\rho=1$) provides an approximate gain of 6 dB of SNR over the non-optimized orthogonal pilot matrix. Even prioritizing sensing less (e.g., $\rho=0.2$) yields performance comparable to the communication-optimal case.
• Communication Reliability (SER): For multi-user communication ($K=4$), the symbol error rate (SER) performance exhibits gains as high as 1.6 dB of SNR (at an SER level of $10^{-4}$) when $\rho=1$, compared to non-optimized orthogonal pilots.
• Detection Performance (ROC): Optimizing the pilot matrix improves the probability of detection ($\text{P}{\text{d}}$) for a fixed false-alarm probability ($\text{P}{\text{fa}}$) compared to non-optimized orthogonal pilots. For example, at $\text{P}{\text{fa}}=10^{-4}$, an optimized pilot (even with $\rho=1$, prioritizing communication) achieved $\text{P}{\text{d}}=0.61$, compared to $0.475$ for a non-optimized pilot.
• Design Flexibility: Increasing the number of transmit antennas ($N_t$) and the number of pilot symbols ($L$) not only improves the ISAC MI but also allows the designer to achieve a wider set of achievable trade-offs between sensing and communication MI values.

In conclusion, this framework leverages information theory and non-convex optimization to rigorously formulate and solve the pilot design challenge for ISAC systems. By utilizing projected gradient descent on the Stiefel manifold, the method guarantees convergence to a stable orthogonal pilot solution, fully exploiting the dual potential offered by the ISAC paradigm.

Future research directions include generating pilots with additional practical constraints, such as good synchronization properties and low Peak-to-Average Power Ratio (PAPR), potentially utilizing Artificial Intelligence and Deep Learning techniques.

The balancing parameter $\rho$ acts like a tuner in a radio receiver designed for two channels: it doesn’t just switch between maximizing communication clarity or maximizing radar detection power; it optimally adjusts the single tuning signal (the orthogonal pilot) to ensure the integrated system achieves the highest possible quality across both functions simultaneously, especially when the channels naturally overlap. ```