HC: Thermal Effects in On-Chip Communication

• Hot Channel (HC) is a dynamic on-chip communication model where cumulative transmission power causes local heating that increases noise variance.
• The analysis shows that in the low-SNR regime, capacity per unit cost is given by (1+α)/2, independent of feedback, highlighting efficiency in energy-constrained designs.
• In high-SNR scenarios, capacity behavior is dictated by the decay of thermal memory coefficients, stressing the need for effective thermal management in circuit design.

A Hot Channel (HC) refers to a dynamic on-chip communication model in which the additive noise variance is nonstationary and depends on the weighted sum of previous channel input powers, physically motivated by the thermal effects and non-ideal heat dissipation in high-speed integrated circuits. HCs capture the regime where the energy used for signal transmission induces local heating, which then increases thermal noise for subsequent channel uses. In such a model, both the channel capacity and its dependence on power constraints display fundamentally different behavior compared to classical memoryless additive white Gaussian noise (AWGN) channels. The mathematical and engineering analysis of HC models establishes rigorous criteria for capacity bounds and highlights necessary/sufficient conditions for practical system design.

1. Channel Model Structure

HCs are defined by a channel where, at each use $k$, the output $Y_k$ is given as: $$Y_k = x_k + \sqrt{ \sigma^2 + \sum_{\nu=1}^{k-1} \alpha_{k-\nu} x_\nu^2 } \cdot U_k$$ where:

• $x_k$ is the input at time $k$,
• $\sigma^2$ is the baseline noise variance,
• $\{\alpha_n\}$ are nonnegative coefficients (weights) that quantify thermal memory from previous input powers,
• $\{U_k\}$ is an i.i.d. sequence of standard normal random variables.

The nonstationarity arises because the additive noise variance increases as a function of "past heating," i.e., the cumulative power transmitted in prior channel uses, weighted by the decay coefficients. The typical practical assumption is that the heating memory $\alpha = \sum_{n=1}^\infty \alpha_n$ is finite, and the weights $\alpha_n$ are nonincreasing to reflect thermal relaxation.

An average power constraint is imposed: $\frac{1}{n} \sum_{k=1}^n \mathbb{E}[x_k^2] \leq P$ with SNR defined as $P/\sigma^2$.

2. Capacity per Unit Cost and Feedback Effect

The principal information-theoretic analysis focuses on the capacity per unit cost in the low-SNR (low-power) regime: $$\dot{C}(0) = \sup_{\text{SNR} > 0} \frac{C(\text{SNR})}{\text{SNR}}$$ where $C(\text{SNR})$ is the operational Shannon capacity under the modeled noise structure.

The derived result is: $\dot{C}(0) = \frac{1}{2}(1 + \alpha)$ for $\alpha = \sum_{n=1}^{\infty} \alpha_n < \infty$, whether or not channel output feedback is present. Here $\alpha$ quantifies the total heating influence (memory) in the system. The derivation employs both upper and lower bounding techniques:

• An upper bound is obtained using entropy inequalities and the log-sum expansion,
• A lower bound is constructed via a stationary "extended memory" channel and a block-IID burst input distribution, maximizing asymptotic mutual information relative entropy.

The final result is tight: feedback does not improve singleton capacity per unit cost in the heating regime.

3. High-SNR Behavior and Capacity Bound Criteria

In the high-SNR regime (large transmit power), the HC exhibits nontrivial boundedness properties:

• If $\{\alpha_\ell\}$ decays "no faster than geometrically," i.e.,

$$\varliminf_{\ell \to \infty} \frac{\alpha_{\ell+1}}{ \alpha_{\ell} } > 0$$

then capacity is bounded in transmit power:

$$\sup_{\text{SNR}>0} C_{\text{FB}}(\text{SNR}) < \infty$$

• If $\{\alpha_\ell\}$ decays "faster than geometrically," e.g.,

$$\varlimsup_{\ell \to \infty} \frac{\alpha_{\ell+1}}{ \alpha_{\ell} } = 0$$

or equivalently,

$$\lim_{\ell\to\infty} \frac{1}{\ell} \log\frac{1}{\alpha_\ell} = \infty$$

then capacity is unbounded:

$\sup_{\text{SNR}>0} C(\text{SNR}) = \infty$

These criteria express a sharp transition between regimes dominated by persistent thermal memory (limited cooling) and those in which aggressive cooling diminishes past power's noise impact sufficiently fast.

4. Physical Interpretation and Relevance for On-Chip Communication

The HC formalism models the inherently coupled physics of energy dissipation and noise generation at the microelectronic substrate level:

• When a chip transmits a symbol, dissipated power locally raises the temperature of the substrate, which increases Johnson-Nyquist noise for subsequent transmissions.
• The mathematical structure thus embodies both memory and temporally correlated noise, with system behavior contingent on the physical thermal relaxation properties ($\alpha_n$ sequence).

Practical implications are regime-dependent:

• In low-power (low-SNR) operation, heating can actually enhance energy efficiency, increasing capacity per unit cost, and thus elaborate heat-sink design may be unnecessary for certain energy-constrained designs.
• In high-power (high-SNR) applications, unless thermal management is aggressive ($\alpha_n$ decays rapidly), information rate saturates as additional transmit power only increases the effective noise floor, placing strict limitations on throughput and necessitating efficient heat removal.

5. Mathematical Summary Table

Regime	Capacity per Unit Cost	Boundedness Condition
Low SNR	$\frac{1}{2}(1+\alpha)$	Holds for any finite $\alpha$
High SNR	$\sup C(SNR) < \infty$	$\varliminf \frac{\alpha_{\ell+1}}{\alpha_\ell}>0$
High SNR	$\sup C(SNR) = \infty$	$\varlimsup \frac{\alpha_{\ell+1}}{\alpha_\ell}=0$

6. Practical Design Implications and Further Directions

The HC channel model necessitates refined circuit engineering that explicitly considers the interplay between transmission scheduling, power allocation, and on-chip thermal management:

• Designers must analyze the cumulative heating effect (the $\alpha_n$ profile) to optimize transmission strategies for both minimal energy per bit and maximal achievable rate.
• In low-SNR wireless chip communications (e.g., sensor networks), HC theory guides power minimization without undue concern for thermal limits.
• In high-throughput interconnects, rapid cooling (fast $\alpha_n$ decay) and spatial thermal isolation become first-order design constraints.

Further research directions explore multichip scenarios, spatially inhomogeneous heating models, adaptation of input coding strategies under time-varying noise constraints, and experimental validation in nanoscale CMOS and post-CMOS device paradigms. The HC paradigm rigorously connects underlying thermodynamics to classical information-theoretic limits in emerging semiconductor architectures.