Identification for Colored Gaussian Channels

Abstract: We study the identification capacity of discrete-time Gaussian channels impaired by correlated noise and inter-symbol interference (ISI). Our analysis is formulated for deterministic encoding functions subject to a peak power constraint and colored noise whose covariance matrix features a polynomially bounded singular value spectrum, i.e., $\sim [n^{-μ} , n^{μ/2}]$ where $n$ is the codeword length and $μ\in [0,1/2)$ is the spectrum rate. A central result establishes that, even when the ISI memory length grows sub-linearly with $n,$ i.e., $\sim n^κ$ where $κ\in [0,1/2)$ and $κ+ μ\in [0,1/2),$ the codebook size continues to exhibit super-exponential growth in $n$, i.e., $\sim 2^{(n \log n)R},$ with $R$ representing the associated coding rate. Moreover, by employing the well-known Mahalanobis-distance decoder induced by colored Gaussian noise statistics, we characterize bounds on the identification capacity, with the resulting bounds parameterized by $κ$ and $μ.$

• The paper presents that deterministic identification codebook size scales super-exponentially as 2^(n log n R) even in the presence of ISI and colored noise.
• It employs high-dimensional sphere-packing and Mahalanobis distance decoding to ensure bounded error probabilities as blocklength increases.
• Explicit upper and lower bounds on identification capacity are derived, highlighting trade-offs between ISI memory (κ) and noise correlation (μ).

Identification Capacity for Colored Gaussian Channels with ISI

Problem Formulation and Main Contributions

This paper systematically investigates the identification capacity of discrete-time Gaussian channels featuring both inter-symbol interference (ISI) and colored noise, under a deterministic encoding paradigm and a strict per-symbol peak power constraint. The colored noise process is modeled by a covariance matrix whose spectrum is polynomially bounded, parameterized by $\mu \in [0,1/2)$ for the spectral growth rate. The ISI memory, represented by the number of channel taps $K$, is allowed to scale sub-linearly, i.e., $K \sim n^{\kappa}$ with $\kappa \in [0,1/2)$, subject to the joint constraint $\kappa + \mu < 1/2$.

The principal result demonstrates that for such channels, the deterministic identification (DI) codebook size exhibits super-exponential growth as $M \sim 2^{(n\log n) R}$ for achievable identification rates $R$, even in the presence of increasing ISI and colored noise correlation. Explicit upper and lower bounds on the identification capacity $\mathbb{C}_{\mathrm{I}}(\mathbb{G})$ are derived as functions of $\kappa$ and $\mu$:

$K$0

where the bounds tighten as channel and noise memory diminish.

Identification Framework in Gaussian ISI and Colored Noise Channels

The analysis is predicated on the identification coding paradigm, where the receiver aims to test whether a specific message was sent, rather than reconstructing the entire message. This framework admits a super-exponential scaling of the codebook size, fundamentally different from Shannon’s classical channel coding theorem. The channel operates as:

$K$1

where $K$2 represents the convolution of the codeword with the channel impulse response (CIR), and $K$3 is a colored Gaussian noise vector with a Toeplitz (stationary) covariance.

The DI code definition admits two error types: Type I (missed detection) and Type II (false alarm), bounded across the codebook. The operational capacity is defined by the supremum of achievable rates $K$4 for which error probabilities converge to zero as blocklength grows.

Achievability: Code Construction and Decoding

The achievability proof constructs a codebook via high-dimensional sphere-packing within the peak-power-limited input space, mapping messages to codewords whose convolution with the ISI CIR yields well-separated points in $K$5. The analysis rigorously quantifies the minimum distance after convolution and colored noise whitening, demonstrating that packing density arguments lead to codebook sizes of order $K$6.

Decoding leverages the Mahalanobis distance metric induced by the colored noise statistics, thresholding the normalized squared distance to test for message presence. Detailed analysis of error probabilities, using properties of the chi-squared distribution and the spectrum of the covariance matrix, confirms that both error types decay with growing blocklength, provided $K$7 and $K$8 satisfy the stipulated constraints.

Converse: Packing Arguments and Sphere Hardening

The converse employs the minimum-distance requirement between convolved codewords and upper bounds the codebook size using volume arguments for non-overlapping hyperspheres in the (convolved, colored) output space. Singular value bounds on the covariance matrix are crucial for quantifying how colored noise and ISI memory penalize the effective packing rate. The analysis reveals that as noise correlation ($K$9) and ISI memory ($K \sim n^{\kappa}$0) increase, packing efficiency – and consequently identification rate – diminishes.

Implications and Future Directions

The results formalize the scaling of identification capacity in colored Gaussian channels with memory, clarifying the impact of spectral and temporal dependencies. Practical implications pertain to robust communication under adverse non-white noise and ISI—scenarios prevalent in wireless, molecular, and post-Shannon communication settings—where super-exponential codebook growth is viable for identification, even when Shannon capacity becomes vanishingly small.

Theoretically, this analysis sets the stage for several follow-up directions:

• Channels with Spectral Nulls or Rank-Deficient Covariance: Extending the present framework to cases with degenerate or rapidly decaying spectrum.
• Multi-User and Fading Scenarios: Generalizing identification capacity in networks and under stochastic channel coefficients.
• Finite Blocklength: Quantifying identification reliability under stringent latency or short-packet constraints.
• Alternative Decoder Designs: Exploring sub-optimal or structured decoders in practical regimes where full Mahalanobis computation is infeasible.

Conclusion

This paper establishes sharp bounds on the deterministic identification capacity for colored Gaussian channels with ISI and polynomially bounded noise spectrum. The analysis rigorously extends the identification capacity theory to encompass realistic, temporally and spectrally correlated channels, demonstrating robust super-exponential codebook growth under suitable spectral and memory constraints. These findings contribute to the foundational understanding of identification under memory and colored noise and motivate extensions toward more intricate channel models.