Ahmad Bazzi

Ahmad Bazzi

he/him

Research Scientist

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

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A walkthrough of our paper “Mutual Information Based Pilot Design for ISAC”, recently accepted at IEEE Transactions on Communications (DOI: 10.1109/TCOMM.2025.3545658, arXiv preprint). Full structured citation, BibTeX, and metadata are on the paper page.

The dual problem ISAC has always had

In an Integrated Sensing and Communication (ISAC) system, one antenna array does two jobs: it talks to communication users and listens for targets. That sounds elegant. In practice it’s a design headache, because the signals the system sends out are being asked to serve two very different purposes simultaneously.

ISAC scenarioAn ISAC base station with N_t antennas serving K communication users while simultaneously listening for backscatter from a target and clutter scatterers. ISAC base station N_t transmit, N_r receive 1 user 1 2 user 2 k user K target at θ₀ clutter pilot Φ for channel estimation same Φ, backscatter detection communication link radar echo
Figure 1. The ISAC scenario. One pilot matrix \(\boldsymbol{\Phi}\) is transmitted; the base station serves K communication users with the forward links and listens for the backscatter from a target at angle \(\theta_0\), corrupted by clutter at other angles. The same symbols do both jobs.
  • For communication, the pilot symbols \(\boldsymbol{\Phi}\) at the start of a frame let downlink users estimate the channel \(\boldsymbol{h}_k\) so they can equalize and decode the data that follows.
  • For sensing, the same pilots reflect off targets and clutter, and the ISAC base station listens to the backscatter to detect what’s out there.

These two tasks pull in opposite directions. A pilot that minimizes channel-estimation variance for \(K\) users is not generally the same pilot that maximizes detection probability against a target at angle \(\theta_0\) surrounded by clutter at angles \(\theta_1, \ldots, \theta_Q\). Conventional wisdom said you had to pick one or build separate signals for each, wasting either time, spectrum, or hardware.

We wanted a principled way to jointly design that pilot. The catch: the two performance measures usually live in different units. Detection performance is a probability; channel estimation is measured in mean-squared error or capacity. How do you compare them, let alone optimize them together?

The trick: measure everything in bits

Our answer is to express both halves of the ISAC objective in the same currency, mutual information (MI), and then optimize over both at once.

For the \(k\)-th communication user receiving \(\boldsymbol{y}_k = \boldsymbol{\Phi} \boldsymbol{h}_k + \boldsymbol{n}_k\), we define the communication MI between the received signal and the channel as

\[\mathcal{M}^{\text{comm}}_k(\boldsymbol{\Phi}) \;\triangleq\; I(\boldsymbol{y}_k; \boldsymbol{h}_k).\]

Under a Gaussian-mixture model for the channel (a flexible-enough model to approximate any continuous channel distribution including Rician, Rayleigh, Nakagami), this gives a closed-form expression that depends explicitly on \(\boldsymbol{\Phi}\), meaning we can take its gradient and optimize. Theorem 2 in the paper shows that maximizing \(\mathcal{M}^{\text{comm}}_k\) directly lowers the variance of channel estimation, which in turn improves the worst-case channel capacity. So this isn’t an arbitrary surrogate, it’s tied back to a quantity practitioners already care about.

For sensing, we set up the standard two-hypothesis test: under \(\mathcal{H}_0\) no target is present (only clutter \(\boldsymbol{c}\) and noise), under \(\mathcal{H}_1\) the target signature \(\boldsymbol{d}\) is also present. The sensing MI is

\[\mathcal{M}^{\text{sense}}(\boldsymbol{\Phi}) \;\triangleq\; I(\boldsymbol{Y}_r; \boldsymbol{\Theta}, \boldsymbol{\nu} \mid \boldsymbol{\Phi}),\]

which simplifies to a log-determinant expression involving the target-to-clutter-plus-noise covariance structure. The key payoff is Theorem 3: in the large-array asymptotic regime, \(\mathcal{M}^{\text{sense}}\) relates one-to-one to the probability of detection \(P_D\) of the most powerful (Neyman–Pearson) test at a fixed false-alarm rate:

\[\lim_{L\to\infty} \left(-\frac{1}{L} \log(1 - P_D)\right) \;=\; \mathcal{M}^{\text{sense}}(\boldsymbol{\Phi}) - g(\boldsymbol{\nu}, \boldsymbol{\Phi}).\]

Translation: more sensing MI ⇒ higher detection probability for the same false-alarm rate. Again, the MI surrogate is tied to a quantity the radar literature already cares about.

So both halves of the problem can be expressed in bits, and both bits-quantities have direct, theorem-grade connections to the real performance metrics, capacity and detection probability respectively.

The optimization: one pilot, many objectives

With \(K\) communication users plus the sensing channel, we have \(K + 1\) MI objectives competing over the same pilot matrix \(\boldsymbol{\Phi}\). That’s a multi-objective optimization problem (MOOP):

\[(\mathcal{P}_{\text{MOOP}}): \quad \max_{\boldsymbol{\Phi}}\; \left[\mathcal{M}^{\text{comm}}_1(\boldsymbol{\Phi}), \ldots, \mathcal{M}^{\text{comm}}_K(\boldsymbol{\Phi}), \mathcal{M}^{\text{sense}}(\boldsymbol{\Phi})\right]\]

subject to the orthogonality constraint \(\boldsymbol{\Phi} \boldsymbol{\Phi}^H = \frac{P}{L} \mathbf{I}\), which is desirable because orthogonal pilots eliminate inter-user inter-cell interference during training, and make least-squares channel estimators trivial (no matrix inversion needed). The constraint also fixes the power budget.

There’s no global optimum here, improving one user’s MI can hurt another’s, or hurt the sensing MI, so optimal solutions live on a Pareto frontier rather than at a single point. We scalarize the MOOP with a single design parameter \(\rho \in [0, 1]\):

\[(\mathcal{P}_s): \quad \max_{\boldsymbol{\Phi}}\; \rho\, \mathcal{M}^{\text{comm}}(\boldsymbol{\Phi}) + (1 - \rho)\, \mathcal{M}^{\text{sense}}(\boldsymbol{\Phi}), \quad \text{s.t.}\; \boldsymbol{\Phi}\boldsymbol{\Phi}^H = \mathbf{I}.\]

Set \(\rho = 1\) and you get the pilot that’s optimal for channel estimation alone; set \(\rho = 0\) and you get the pilot that’s optimal for target detection alone; anywhere in between you trade off the two. \(\rho\) is the dial.

Pareto frontier of sensing vs communication MIA two-dimensional plot with sensing mutual information on the horizontal axis and communication mutual information on the vertical axis. A curved Pareto frontier separates feasible pilot designs (below the curve) from the unattainable region (above). Three points on the frontier are labeled rho equals zero (sensing-optimal), rho equals one half (balanced), and rho equals one (communication-optimal). The utopia point sits in the unattainable corner. Sensing MI \(\mathcal{M}^{\mathrm{sense}}(\boldsymbol{\Phi})\) (bits) Communication MI \(\mathcal{M}^{\mathrm{comm}}(\boldsymbol{\Phi})\) (bits) \(\rho=1\) comm-optimal \(\rho=0.5\) balanced \(\rho=0\) sense-optimal utopia point (unattainable) Pareto frontier (achievable optima) feasible pilots
Figure 2. The achievable region of \((\mathcal{M}^{\mathrm{sense}}, \mathcal{M}^{\mathrm{comm}})\) pairs over all orthogonal pilot matrices. Random orthogonal pilots populate the interior (brown dots). The blue curve is the Pareto frontier, where no single objective can be improved without hurting another. The \(\rho\) parameter selects a point on this frontier: \(\rho=1\) maximizes communication MI, \(\rho=0\) maximizes sensing MI, and \(\rho=0.5\) sits at the balanced design.

The remaining issue is that the orthogonality constraint defines the Stiefel manifold \(\text{St}(L, N_t)\), the set of \(L \times N_t\) complex matrices with orthonormal rows, which is non-convex. The objective is non-convex too. So we solve \((\mathcal{P}_s)\) with projected gradient descent on the Stiefel manifold:

\[\begin{aligned} \boldsymbol{Z}_{t+1} &= \boldsymbol{\Phi}_t + \gamma\, \nabla_{\boldsymbol{\Phi}} \mathcal{M}^{\text{ISAC}}(\boldsymbol{\Phi})\big|_{\boldsymbol{\Phi}=\boldsymbol{\Phi}_t} \\ \boldsymbol{\Phi}_{t+1} &= \boldsymbol{\pi}(\boldsymbol{Z}_{t+1}) \end{aligned}\]

where \(\boldsymbol{\pi}(\boldsymbol{Y}) = \boldsymbol{U} \boldsymbol{I}_{L, N_t} \boldsymbol{V}^H\) is the projection onto the Stiefel manifold via singular value decomposition. By construction, every iterate is orthogonal, we never leave the feasible set. We derive the closed-form expressions for both \(\nabla \mathcal{M}^{\text{comm}}\) and \(\nabla \mathcal{M}^{\text{sense}}\) (appendices C and D of the paper), and prove via Theorem 1 that this iteration converges to a stable orthogonal pilot under restricted strong smoothness and convexity assumptions:

\[\mathcal{M}^{\text{ISAC}}(\boldsymbol{\Phi}^{\text{opt}}) - \mathcal{M}^{\text{ISAC}}(\boldsymbol{\Phi}_\infty) \;\leq\; \frac{\alpha + \beta}{1 - \frac{\beta}{\alpha}} \epsilon^2.\]

That bound says: with enough iterations, we land within a precision controlled by \(\epsilon\) of the local maximizer. The simulations show convergence in roughly 50–200 iterations for reasonable step sizes.

The surprise: information overlap

The most interesting empirical finding isn’t an algorithm but a phenomenon. We plot the achievable \((\mathcal{M}^{\text{sense}}, \mathcal{M}^{\text{comm}})\) frontier for different target locations. As the target’s angle of arrival \(\theta_0\) moves closer to the mean angle of arrival of the communication user, the entire Pareto frontier pushes outward toward the utopia point:

Both sensing MI and communication MI improve simultaneously. There is no trade-off in this regime, only joint gain.

We call this the information overlap phenomenon. The intuition: when the target sits along roughly the same direction as the communication user, the sensing channel and the communication channel share common geometric structure. A pilot tuned to estimate the communication channel happens to illuminate the same direction the radar wants to look at, so a single pilot extracts useful bits about both tasks at once.

This has a clear deployment implication: in scenarios where the BS senses its own users (e.g. perceptive cellular networks that track UE motion for handover or beam management), ISAC integration gain is maximal. The trade-off you’d worry about in textbook ISAC analysis essentially disappears.

Information overlap phenomenonFour Pareto frontiers stacked on the same axes. As the target's angle of arrival approaches the communication user's mean angle of arrival, the frontier shifts outward toward the utopia point, indicating that both sensing MI and communication MI can be improved simultaneously. Sensing MI \(\mathcal{M}^{\mathrm{sense}}\) (bits) Communication MI \(\mathcal{M}^{\mathrm{comm}}\) (bits) target approaches user target angle θ₀ vs user mean θ̄ |θ₀ − θ̄| = 60° (far) |θ₀ − θ̄| = 30° |θ₀ − θ̄| = 10° target = user (overlap) utopia
Figure 3. The information overlap phenomenon. As the target angle \(\theta_0\) moves closer to the mean angle of arrival of the communication user, the entire Pareto frontier shifts outward toward the utopia point. When the target is co-located with a communication user (orange curve), there is no longer a trade-off, the same orthogonal pilot serves both tasks well simultaneously.

Numbers that matter

Some of the headline gains, from the simulation section:

  • NMSE of channel estimation: ~6 dB SNR gain at \(\rho = 1\) (communication-optimal pilot) compared to a generic non-optimized orthogonal pilot.
  • Symbol Error Rate (multi-user, \(K=4\), 64-QAM with Gray encoding): up to 1.6 dB SNR gain at SER = \(10^{-4}\).
  • Detection probability at false-alarm rate \(P_{\text{fa}} = 10^{-4}\), single-user scenario: even the communication-optimal pilot (\(\rho = 1\)) achieves \(P_D = 0.61\), against \(P_D = 0.475\) for a non-optimized orthogonal pilot. Lower \(\rho\) pushes detection further, \(P_D = 0.7\) at \(\rho = 0\).

So one orthogonal pilot, designed by this method, gives both better channel estimation than a DFT or eigen-based pilot and better detection than a random orthogonal pilot. You get to keep the orthogonality (low-complexity decoders, no inter-stream interference) and pick up gains in both regimes.

What this work doesn’t do, and where we’re going

The framework optimizes for MI and orthogonality. It does not yet account for:

  • Peak-to-average power ratio (PAPR), which matters for practical RF chains and is where the companion ICC 2025 paper picks up.
  • Synchronization properties (good autocorrelation profiles) that some receivers need.
  • Multi-target detection in a single pilot transmission. Current work scans directions in a TDM fashion.

Future directions we’re working on include learning-based pilot generation (where deep models propose pilots that satisfy these additional constraints) and PAPR-aware orthogonal pilots. The recent work on ISAC waveforms with adjustable PAPR and the more general DRIP family of space-time ISAC sequences build directly on the MI framework introduced here.

How to cite

If you use this work, the recommended citation is:

A. Bazzi and M. Chafii, “Mutual Information Based Pilot Design for ISAC,” in IEEE Transactions on Communications, 2025. DOI: 10.1109/TCOMM.2025.3545658.

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Comments, questions, or extensions? Reach out via the channels on the home page. And if you found this useful, the Secure Full Duplex ISAC, Hybrid Radar Fusion, and High-Resolution Sensing in Communication-Centric ISAC posts cover the next steps in the broader line of work.