Information Theory for 6G and Beyond

Information Theory's Role in Next-Generation Wireless

Each generation of wireless systems has been shaped by information-theoretic insights: CDMA by spread-spectrum capacity, 4G by OFDM and MIMO capacity, 5G by massive MIMO and polar codes. For 6G and beyond, information theory faces new challenges: integrated sensing and communication (ISAC), extremely large antenna arrays operating in the near field, reconfigurable intelligent surfaces, and AI-native communication. These are not just engineering problems β€” they require fundamental rethinking of channel models, capacity definitions, and coding strategies.

Definition:

Integrated Sensing and Communication (ISAC)

ISAC is a paradigm where a single waveform simultaneously serves communication (delivering data to users) and sensing (estimating target parameters such as range, velocity, or angle). The information-theoretic formulation seeks the capacity-distortion tradeoff: the maximum communication rate RR achievable while maintaining sensing distortion Ds≀Dmax⁑D_s \leq D_{\max}.

For the Gaussian ISAC channel where the transmitter sends XnX^n to a communication receiver (observing Yc=X+ZcY_c = X + Z_c) and the echo from a target returns Ys=Ξ±X+ZsY_s = \alpha X + Z_s (where Ξ±\alpha encodes the target parameter): C(Dmax⁑)=max⁑PX: Ds(PX)≀Dmax⁑I(X;Yc)C(D_{\max}) = \max_{P_X: \, D_s(P_X) \leq D_{\max}} I(X; Y_c)

The key insight is that sensing imposes constraints on the transmitted waveform (e.g., constant modulus for good autocorrelation properties) that may reduce communication capacity. The capacity-distortion tradeoff characterizes this price.

Theorem: ISAC Capacity-Distortion Tradeoff (Liu-Caire)

For a monostatic ISAC system with Gaussian channel noise Οƒc2,Οƒs2\sigma^2_{c}, \sigma^2_{s} and power constraint PP, the capacity-distortion function satisfies: C(Dmax⁑)=12log⁑ ⁣(1+PΟƒc2)βˆ’Ξ”(Dmax⁑)C(D_{\max}) = \frac{1}{2}\log\!\left(1 + \frac{P}{\sigma^2_{c}}\right) - \Delta(D_{\max}) where Ξ”(Dmax⁑)β‰₯0\Delta(D_{\max}) \geq 0 is the sensing penalty β€” the rate loss due to the sensing constraint. When Dmax⁑D_{\max} is large (weak sensing requirement), Ξ”β†’0\Delta \to 0 and the full communication capacity is achievable. When Dmax⁑D_{\max} is small (strong sensing), Ξ”\Delta can be significant because the waveform must dedicate resources to sensing.

At one extreme (no sensing), the system achieves the full AWGN capacity. At the other extreme (perfect sensing), the waveform must be deterministic (known to the receiver for matched filtering), leaving no room for communication. The tradeoff between these extremes is the fundamental limit of ISAC.

Proof
Achievability

Use a codebook where each codeword has good autocorrelation properties (for sensing) while maintaining near-Gaussian statistics (for communication). Specifically, partition the power: PcP_c for communication and PsP_s for sensing, with Pc+Ps=PP_c + P_s = P. The communication rate is 12log⁑(1+Pc/Οƒc2)\frac{1}{2}\log(1 + P_c/\sigma^2_{c}) and the sensing distortion is controlled by PsP_s.

Converse

The converse uses the fact that any codeword that achieves sensing distortion Dmax⁑D_{\max} must have a minimum "sensing energy" Ps(Dmax⁑)P_s(D_{\max}), leaving at most Pβˆ’PsP - P_s for communication. The rate is bounded by 12log⁑(1+(Pβˆ’Ps)/Οƒc2)\frac{1}{2}\log(1 + (P - P_s)/\sigma^2_{c}).

Definition:

Near-Field Information Theory

With extremely large antenna arrays (M≫1M \gg 1, aperture ≫λ\gg \lambda), the far-field assumption (planar wavefronts) breaks down. In the near field (Fresnel region), the wavefront is spherical, and the channel model becomes: hm=Ξ»4Ο€rmeβˆ’j2Ο€rm/Ξ»h_m = \frac{\lambda}{4\pi r_m} e^{-j 2\pi r_m / \lambda} where rmr_m is the distance from the source to antenna mm, and the phase varies non-linearly across the array. The near-field channel has richer spatial structure than the far-field channel, potentially enabling:

  • Beam focusing (energy concentrated at a point, not a direction)
  • Distance-dependent multiplexing (users at different distances can be separated)
  • Spatial multiplexing gain beyond the far-field limit

The transition from far field to near field occurs at the Fraunhofer distance dF=2D2/Ξ»d_F = 2D^2/\lambda where DD is the array aperture. For a 1-meter array at 30 GHz (Ξ»=1\lambda = 1 cm), dF=200d_F = 200 m. At mmWave and sub-THz frequencies with large arrays, many users will be in the near field.

ISAC Capacity-Distortion Tradeoff

Explore the tradeoff between communication rate and sensing distortion for a monostatic ISAC system with varying SNR and sensing requirements.

Parameters
10
1
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Key Information-Theoretic Questions for 6G

The following questions are central to the information-theoretic foundations of 6G:

  1. ISAC fundamental limits: What is the capacity-distortion-outage tradeoff for multi-user ISAC with imperfect CSI? How does it change with bi-static vs. monostatic sensing?

  2. Near-field MIMO capacity: How does the capacity of extremely large arrays scale when users are in the near field? What is the optimal spatial multiplexing strategy when the channel is no longer a function of angle alone?

  3. RIS-aided communication: What is the capacity of a channel with a passive reconfigurable intelligent surface? The RIS introduces a multiplicative channel component that couples with the direct channel in a non-trivial way. See Book RIS for full treatment.

  4. Semantic capacity: Is there a "semantic Shannon theorem" β€” a capacity result for task-oriented communication that parallels Shannon's theorem for data communication? See Chapter 29 for the current state.

  5. AI-native protocols: Can reinforcement learning discover communication protocols that outperform human-designed ones? The connection to multi-agent information theory and game theory is largely unexplored.

πŸŽ“CommIT Contribution(2023)

ISAC Capacity-Distortion Tradeoff

F. Liu, G. Caire β€” IEEE Trans. Information Theory

Liu and Caire established the capacity-distortion tradeoff for the Gaussian ISAC channel, providing the information-theoretic foundation for understanding the fundamental limits of joint radar sensing and data communication. This work shows that the price of sensing (in terms of communication rate) is determined by the sensing waveform constraint, and characterizes the Pareto-optimal operating points.

ISACcapacity-distortion tradeoff6GsensingView Paper β†’
πŸ”§Engineering Note

The Gap Between Information-Theoretic Limits and Practice

Information theory provides asymptotic limits β€” what is achievable with infinite blocklength and unlimited complexity. Practical systems operate at finite blocklength with bounded complexity. The gaps include:

  • Coding gap: LDPC and polar codes operate within 0.1-0.5 dB of capacity for moderate blocklengths (n∼1000n \sim 1000), but this gap grows at very short blocklengths (n<100n < 100).
  • CSI gap: Many capacity results assume perfect CSI. With imperfect CSI, the effective capacity can be significantly lower, especially at high mobility.
  • Coordination gap: Multi-user capacity results often assume joint decoding or interference cancellation that may not be implementable with practical constraints.

A good information theorist is honest about these gaps and uses them to guide system design rather than to oversell theoretical results.

Why This Matters: Reconfigurable Intelligent Surfaces (RIS)

RIS introduces a programmable scattering environment that can be optimized jointly with the transmitter. The information-theoretic analysis of RIS channels is an active research area: the capacity depends on whether the RIS phase shifts are known to the receiver (coherent) or not (non-coherent), and on whether the RIS can be reconfigured per channel use. See Book RIS for the full treatment.

ISAC Capacity-Distortion Tradeoff

Visualizes the Pareto frontier between communication rate and sensing distortion for an ISAC system. Power allocation between sensing and communication waveforms traces the tradeoff curve from communication-only to sensing-only operation.

Quick Check

In the ISAC capacity-distortion tradeoff, what happens when the sensing distortion requirement is very strict (Dmax⁑→0D_{\max} \to 0)?

Communication rate approaches the AWGN capacity

Communication rate approaches zero

The system cannot function

Sensing and communication decouple

Correction:
Communication rate approaches zero

Very strict sensing requires a nearly deterministic (known) waveform, leaving essentially no degrees of freedom for communication.

Integrated Sensing and Communication (ISAC)

A paradigm where a single waveform simultaneously serves communication (data delivery) and sensing (target parameter estimation), characterized by the capacity-distortion tradeoff.

Related: Integrated Sensing and Communication (ISAC)

Near-Field Communication

Communication in the Fresnel region of an antenna array, where wavefronts are spherical rather than planar. Enables beam focusing and distance-dependent spatial multiplexing.

Related: Near-Field Information Theory

Non-Shannon InequalitiesReading and Writing Information Theory Papers