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When the Same Signal Must Communicate and Sense

Modern wireless systems are expected to do more than just communicate: they must also sense the environment — detect targets, estimate positions, track movements. Radar and communication have traditionally been separate systems with separate hardware and separate spectrum. But as spectrum becomes scarce and hardware costs rise, integrated sensing and communications (ISAC) — using the same signal, same hardware, same spectrum for both functions — has become a major research direction and a key feature of 6G standardization.

The fundamental question is: what is the tradeoff between communication performance (rate) and sensing performance (estimation accuracy or detection probability) when the same transmitted signal must serve both purposes? This is where information theory meets estimation theory.

This section presents the deterministic-random tradeoff framework developed by Fan Liu and Giuseppe Caire (2023), which won the 2025 IEEE Joint Paper Award. The framework formulates ISAC as a joint optimization problem and characterizes the Pareto boundary of the rate-sensing region.

Definition:
The Gaussian MIMO ISAC Channel

The ISAC transmitter has $n_t$ antennas and sends a signal $\mathbf{x} \in \mathbb{C}^{n_t}$ that simultaneously:

Communicates to a receiver through channel $\mathbf{H}_{c}$ : $\mathbf{y}_c = \mathbf{H}_{c} \mathbf{x} + \mathbf{z}_c, \quad \mathbf{z}_c \sim \mathcal{CN}(\mathbf{0}, \sigma^2_{c} \mathbf{I})$
Senses a target through the radar echo channel $\mathbf{H}_{s}$ : $\mathbf{y}_s = \mathbf{H}_{s}(\boldsymbol{\theta}) \mathbf{x} + \mathbf{z}_s, \quad \mathbf{z}_s \sim \mathcal{CN}(\mathbf{0}, \sigma^2_{s} \mathbf{I})$ where $\boldsymbol{\theta}$ contains the target parameters (range, angle, velocity).

The transmitted signal is subject to a power constraint: $\mathbb{E}[\|\mathbf{x}\|^2] \leq P$ .

The key design choice is the signal structure: how much of the transmitted signal is "deterministic" (optimized for sensing, like a radar waveform) and how much is "random" (carrying information for communication).

Integrated sensing and communications (ISAC)

A system design paradigm where a single transmitter uses the same signal, hardware, and spectrum to simultaneously communicate with a receiver and sense (detect, locate, track) targets in the environment. The fundamental tradeoff is between communication rate and sensing accuracy.

Definition:
The Deterministic-Random Tradeoff

The Liu-Caire framework decomposes the transmitted signal as: $\mathbf{x} = \mathbf{x}_d + \mathbf{x}_r$ where:

$\mathbf{x}_d$ is a deterministic (known) signal component optimized for sensing. It contributes to the radar waveform but carries no information.
$\mathbf{x}_r \sim \mathcal{CN}(\mathbf{0}, \mathbf{K}_r)$ is a random signal component carrying communication data.

The power constraint becomes $\|\mathbf{x}_d\|^2 + \text{tr}(\mathbf{K}_r) \leq P$ .

The communication rate is: $R_{c} = \log\det\left(\mathbf{I} + \frac{1}{\sigma^2_{c}} \mathbf{H}_{c} \mathbf{K}_r \mathbf{H}_{c}^{H}\right)$

The sensing performance is measured by the Fisher information matrix (FIM) for the target parameters $\boldsymbol{\theta}$ , which depends on the total signal $\mathbf{x}_d + \mathbf{x}_r$ (both components illuminate the target).

The deterministic component is beneficial for sensing (it provides a known reference for matched filtering) but wastes power from the communication perspective. The random component provides communication rate but is less useful for sensing (it acts as "self-interference" in the radar signal processing). The tradeoff is in the power allocation between $\mathbf{x}_d$ and $\mathbf{x}_r$ .

Theorem: ISAC Rate-Sensing Tradeoff Region

For the Gaussian MIMO ISAC channel, the achievable rate-sensing region is the set of pairs $(R_{c}, \mathbf{J}(\boldsymbol{\theta}))$ satisfying: $R_{c} \leq \log\det\left(\mathbf{I} + \frac{1}{\sigma^2_{c}} \mathbf{H}_{c} \mathbf{K}_r \mathbf{H}_{c}^{H}\right)$ $\mathbf{J}(\boldsymbol{\theta}) = \frac{1}{\sigma^2_{s}} \sum_{i=1}^n \dot{\mathbf{H}}_s(\boldsymbol{\theta})^H \mathbf{R}_x \dot{\mathbf{H}}_s(\boldsymbol{\theta})$

where $\mathbf{R}_x = \mathbf{x}_d \mathbf{x}_d^H + \mathbf{K}_r$ is the total signal covariance and $\dot{\mathbf{H}}_s = \partial \mathbf{H}_{s} / \partial \boldsymbol{\theta}$ is the channel Jacobian.

The Pareto boundary of the rate-CRB tradeoff is traced by solving: $\max_{\mathbf{x}_d, \mathbf{K}_r} \; R_{c} \quad \text{s.t.} \quad \text{tr}(\mathbf{J}^{-1}) \leq D, \; \|\mathbf{x}_d\|^2 + \text{tr}(\mathbf{K}_r) \leq P$ for varying distortion level $D$ .

The tradeoff has a clean geometric interpretation: the communication rate depends only on $\mathbf{K}_r$ (the random part), while the sensing FIM depends on the total signal covariance $\mathbf{R}_x = \mathbf{x}_d \mathbf{x}_d^H + \mathbf{K}_r$ . Allocating more power to $\mathbf{x}_d$ improves sensing (larger FIM) but reduces communication rate (smaller $\mathbf{K}_r$ ). The Pareto boundary traces the optimal balance.

The key insight of Liu and Caire is that even the random communication signal contributes to sensing through $\mathbf{K}_r$ in the FIM. So the communication signal is not "wasted" from the sensing perspective — it just provides less sensing information per unit power than a deterministic waveform.

Proof

Communication rate

The communication receiver sees $\mathbf{y}_c = \mathbf{H}_{c} (\mathbf{x}_d + \mathbf{x}_r) + \mathbf{z}_c$ . Since $\mathbf{x}_d$ is known, it can be subtracted: the effective observation is $\mathbf{y}_c - \mathbf{H}_{c} \mathbf{x}_d = \mathbf{H}_{c} \mathbf{x}_r + \mathbf{z}_c$ . The rate is the standard Gaussian MIMO capacity with input covariance $\mathbf{K}_r$ .

Sensing FIM

The radar echo is $\mathbf{y}_s = \mathbf{H}_{s}(\boldsymbol{\theta})(\mathbf{x}_d + \mathbf{x}_r) + \mathbf{z}_s$ . For a known $\mathbf{x}_d$ , the FIM contribution is $\frac{1}{\sigma^2_{s}} \dot{\mathbf{H}}_s^H \mathbf{x}_d \mathbf{x}_d^H \dot{\mathbf{H}}_s$ . For the random $\mathbf{x}_r$ , the FIM contribution (averaged over $\mathbf{x}_r$ ) is $\frac{1}{\sigma^2_{s}} \dot{\mathbf{H}}_s^H \mathbf{K}_r \dot{\mathbf{H}}_s$ . The total FIM is the sum, proportional to $\dot{\mathbf{H}}_s^H \mathbf{R}_x \dot{\mathbf{H}}_s$ .

Pareto optimization

The tradeoff is parameterized by the power split between $\mathbf{x}_d$ and $\mathbf{K}_r$ . At one extreme ( $\mathbf{x}_d = \mathbf{0}$ ): maximum communication rate, sensing relies entirely on the random signal. At the other extreme ( $\mathbf{K}_r = \mathbf{0}$ ): no communication, all power used for a deterministic sensing waveform. The Pareto boundary is the set of optimal tradeoff points between these extremes.

🎓CommIT Contribution(2023)

Fundamental Tradeoff of Integrated Sensing and Communications

F. Liu, G. Caire — IEEE Trans. Information Theory

This paper establishes the information-theoretic foundation for ISAC by formulating the deterministic-random tradeoff framework. The key contributions are:

The rate-CRB tradeoff characterization for the Gaussian MIMO ISAC channel, showing that both deterministic and random signal components contribute to sensing.
Pareto-optimal beamforming strategies that optimally allocate spatial resources between communication and sensing.
The insight that communication signals have non-zero sensing utility — the random data-carrying waveform contributes to the FIM through its covariance structure.

This work received the 2025 IEEE Communications Society and Information Theory Society Joint Paper Award, recognizing its impact on the emerging ISAC field.

ISACdeterministic-random tradeoffPareto optimalIEEE Joint Paper AwardView Paper →

ISAC Beamforming Tradeoff

Animated visualization of the ISAC tradeoff: a base station splits its beam between a communication user and a sensing target. As the power split changes, the communication rate and sensing CRB trade off against each other.

ISAC Rate-CRB Tradeoff

Visualize the Pareto boundary of the communication rate vs. sensing CRB tradeoff for a MISO ISAC system. Adjust the number of antennas and target angle to see how the tradeoff changes.

Parameters

Number of Tx antennas (

n_t

)4

SNR (dB)20

Target angle (degrees)30

Communication user angle (degrees)-30

Example: ISAC Tradeoff for a MISO System

Consider a MISO ISAC system with $n_t = 4$ antennas, a communication user at angle $\phi_c = 0°$ and a sensing target at angle $\phi_s = 60°$ . The SNR is 20 dB. Analyze the extreme points of the rate-CRB tradeoff.

Solution

Communication-only extreme

All power to communication: $\mathbf{K}_r = P \cdot \mathbf{h}_c \mathbf{h}_c^H / \|\mathbf{h}_c\|^2$ (beamforming toward the comm user). Rate: $\log(1 + P\|\mathbf{h}_c\|^2/\sigma^2_{c})$ . Sensing CRB: determined by $\dot{\mathbf{H}}_s^H \mathbf{K}_r \dot{\mathbf{H}}_s$ — the CRB degrades because the beam is aimed at the comm user, not the target.

Sensing-only extreme

All power to deterministic sensing waveform: $\mathbf{x}_d = \sqrt{P} \cdot \mathbf{a}(\phi_s) / \sqrt{n_t}$ (beamforming toward the target). CRB is minimized. Communication rate: $R_{c} = 0$ .

Intermediate point

Split power: $P_s = \alpha P$ for sensing, $P_c = (1-\alpha)P$ for communication. With $\alpha = 0.5$ : the comm rate is approximately half of the comm-only rate (in dB), and the CRB is approximately half of the sensing-only CRB (in dB). The exact tradeoff depends on the angular separation $|\phi_c - \phi_s|$ : when the comm user and target are at the same angle, there is almost no tradeoff (the same beam serves both); when they are orthogonal, the tradeoff is severe.

The Role of Angular Separation

The severity of the ISAC tradeoff depends critically on the angular separation between the communication user and the sensing target. When both are in similar directions, the communication beam also illuminates the target effectively, and the tradeoff is mild. When they are in very different directions, the transmitter must choose between pointing at the user (good rate, poor sensing) and pointing at the target (poor rate, good sensing).

With multiple antennas ( $n_t \geq 2$ ), the transmitter can form multiple beams simultaneously, serving both the communication user and the sensing target. This spatial multiplexing reduces the tradeoff severity — another argument for large antenna arrays in ISAC systems.

Why This Matters: Connection to RF Imaging

The ISAC tradeoff framework established here provides the information-theoretic foundation for the RF imaging systems analyzed in Book RFI. The sensing channel $\mathbf{H}_{s}(\boldsymbol{\theta})$ in the ISAC model corresponds to the illumination model in RF imaging, and the FIM characterization directly connects to the resolution analysis and CRB computation of Chapter 34 in Book RFI. The deterministic-random tradeoff is especially relevant for ISAC-enabled base stations that must simultaneously serve communication users and perform environmental sensing.

See full treatment in The ISAC Fundamental Tradeoff

Historical Note: The Convergence of Radar and Communication

2020s

Radar and communication were born as separate technologies in the early 20th century, with fundamentally different design philosophies: radar uses known (deterministic) waveforms for optimal target detection, while communication uses random (information-carrying) waveforms. The idea of merging them dates back to at least the 2000s (dual-function radar-communication systems), but the information-theoretic formulation of the tradeoff is more recent.

The Liu-Caire framework (2023) was the first to provide a clean, operationally meaningful characterization of the fundamental limits of ISAC. Its impact extends beyond theory: the deterministic-random decomposition has influenced practical ISAC waveform design, and the Pareto-optimal beamforming strategies are being considered for 6G standardization.

Common Mistake: ISAC Is Not Free

Mistake:

Claiming that ISAC provides "sensing for free" because the communication signal already illuminates the environment.

Correction:

While the communication signal does contribute to sensing (via $\mathbf{K}_r$ in the FIM), there is always a tradeoff: using power for a deterministic sensing waveform improves the CRB but reduces the communication rate. "Sensing for free" is approximately true only when the communication beam happens to point toward the target — in general, optimizing for both requires a genuine power and spatial allocation tradeoff.

Quick Check

In the Liu-Caire ISAC framework, why does the random (communication) signal contribute to the sensing FIM?

Because the random signal still illuminates the target and its covariance structure is known to the sensing processor

Because the communication data is also useful for the sensing receiver

Because the random signal is actually deterministic after encoding

Correction:

Because the random signal still illuminates the target and its covariance structure is known to the sensing processor

The sensing processor knows the covariance $\\mathbf{K}_r$ of the random signal (it is part of the system design). The echo of the random signal carries information about the target parameters through the channel $\\mathbf{H}_{s}(\\boldsymbol{\\theta})$ . The FIM contribution is proportional to $\\dot{\\mathbf{H}}_s^H \\mathbf{K}_r \\dot{\\mathbf{H}}_s$ .

Key Takeaway

The ISAC rate-CRB tradeoff, formalized by Liu and Caire (2023), shows that communication and sensing compete for the same power and spatial resources. The deterministic-random signal decomposition provides a clean parameterization of the tradeoff. The Pareto boundary depends critically on the angular separation between the communication user and the sensing target, and on the number of transmit antennas. Even the random communication signal contributes to sensing, but a dedicated deterministic component is needed for optimal sensing performance.

The ISAC Fundamental Tradeoff

When the Same Signal Must Communicate and Sense

Definition: The Gaussian MIMO ISAC Channel

Integrated sensing and communications (ISAC)

Definition: The Deterministic-Random Tradeoff

Theorem: ISAC Rate-Sensing Tradeoff Region

Communication rate

Sensing FIM

Pareto optimization

Fundamental Tradeoff of Integrated Sensing and Communications

ISAC Beamforming Tradeoff

ISAC Rate-CRB Tradeoff

Parameters

Example: ISAC Tradeoff for a MISO System

Communication-only extreme

Sensing-only extreme

Intermediate point

The Role of Angular Separation

Why This Matters: Connection to RF Imaging

Historical Note: The Convergence of Radar and Communication

Common Mistake: ISAC Is Not Free

Quick Check

Key Takeaway

Definition:
The Gaussian MIMO ISAC Channel

Definition:
The Deterministic-Random Tradeoff