communication system engineers

Abstract: The feedback capacity of power-constrained channels with additive stationary Gaussian noise was formulated by Kim in \cite{kim2010feedback} as an infinite-dimensional optimization over the spectrum of an independent stationary Gaussian component and a strictly causal feedback filter. Kim further asserted that the independent stationary component can be omitted. However, Derpich and Østergaard \cite{derpich2022comments} identified a gap in the original proof, leaving the original derivation of the filter-only capacity formula incomplete. In this paper, we establish this reduction at the level of capacity suprema. For the noise spectrum bounded away from zero, canonical spectral factorization represents the independent component as the defect of a Schur function from innerness. Finite Blaschke products that preserve its value at the origin produce filters with exactly the same output spectrum and convergent input powers. A power-backoff argument then enforces the original power constraint. Finally, an interleaved AR(1) example relates the reduction to the capacity-achieving SK(2) construction. The general approximation theorem does not require or establish attainment by a single filter.

Abstract: We establish two degree-dependent results for spatially coupled MacKay-Neal codes on binary-input memoryless symmetric channels. The ensembles use uniform random smoothing and shorten both variable types outside the active chain. For $r=g=3$, every integer $\ell\geq4$, and each channel of capacity greater than $R=r/\ell$, there is a sequence of code realizations whose actual transmitted rate tends to $R$ and whose average sum-product bit error tends to zero. For $r=g=2$ and each $\ell\in\{3,4,5\}$, an interval of binary symmetric channels has capacity greater than $R$ but retains positive transmitted-bit error under terminated density evolution as chain length grows relative to coupling width. Both results follow from the signs of the same density-valued potential at uncoupled fixed points. The degree-three proof combines analytic bounds for $\ell\geq33$ with exact interval certificates for $4\leq\ell\leq32$, followed by threshold saturation and a projection argument for the actual rate. The degree-two proof analytically constructs a fixed point with negative potential and controls both boundary contributions. We relate the potential and the degree-two bifurcation condition to earlier statistical-mechanical predictions. The finite certificates and verification software are available in a versioned supplement.

Abstract: 6G is expected to push ultra-reliable low-latency communication (URLLC) toward stringent residual-error targets for mission-critical services, where undetected errors and erasures carry fundamentally different costs. Block error rate (BLER) conflates block confusions (undetected errors) and block erasures, which have fundamentally different impacts on system reliability. This paper extends the confusion and erasure analysis of error-bounded decoders to square quadrature amplitude modulation (QAM) constellations in the finite blocklength (FBL) regime. To handle QAM's heterogeneous symbol energies - which make the per-pair Euclidean distance a distribution rather than a single value - we derive analytical lower and upper bounds on the block confusion rate by, respectively, collapsing this distribution to its root-mean-square (RMS) distance and averaging the pairwise confusion over it. These bounds are proven to be monotonically decreasing in both the average symbol energy and the blocklength, with the decrease rate governed by the constellation order. Numerical results confirm that as the signal-to-noise ratio (SNR) or redundancy increases, the confusion rate falls many orders of magnitude below the reliability target, leaving detectable erasures as the dominant residual error.

Abstract: We study variable-length feedback (VLF) codes over a discrete memoryless channel under average decoding-time and error-probability constraints. In the non-vanishing error probability regime, Polyanskiy, Poor, and Verdú (2011) derive achievability and converse bounds on the logarithm of the maximum achievable codebook size. These bounds establish the $ε$-capacity but leave an order-$\log N$ gap in the second-order expansion, where $N$ is the average decoding time. Yavas and Tan (2025) improve the coefficient of $\log N$ in the achievability bound from $-1$ to $-\frac{C}{C_1}$, where $C$ is the channel capacity and $C_1$ is the largest Kullback--Leibler divergence between two conditional output distributions. We derive a converse with the same coefficient, establishing the second-order fundamental limit for every positive-capacity discrete memoryless channel with finite $C_1$. The result also covers the moderate-deviations and error-exponent regimes, including polynomially decaying error probabilities. The converse uses Rényi entropy and the extrinsic Jensen--Shannon divergence. We also derive necessary properties of asymptotically optimal VLF codes. First-order-optimal codes must have an early-stopping branch, and second-order-optimal codes must additionally exhibit communication and confirmation behavior. Finally, for the binary erasure channel, we determine the exact minimum expected decoding time for every message-set size and admissible error probability.

Abstract: This paper presents an algebraic method to construct convolutional codes with guaranteed \emph{free and local minimum distances} without limit based on cyclic codes of odd lengths. The constructions are simple but effective, and no computer search is needed. For any two positive integers $r$ and $t$ with $1 \leq r < t$, a rate-$r/t$ convolutional code $\mathcal{C}_{\text{convol}}$ can be constructed by using a chain of $r$ cyclic codes $\mathcal{C}_0, \mathcal{C}_1, \ldots, \mathcal{C}_{r-1}$ of the same length $n$ which satisfy the inclusion condition, $\mathcal{C}_0 \supset \mathcal{C}_1 \supset \ldots \supset \mathcal{C}_{r-1}$. Such a convolutional code $\mathcal{C}_{\text{convol}}$ is composed of a \emph{semi-infinite chain of identical local codes} confined in a diagonal band of width $n$. Each local code $\mathcal{C}_{\text{local}}$ of $\mathcal{C}_{\text{convol}}$ is formed from the $r$ cyclic codes in the code chain and is a specially localized subcode of the \emph{mother code} $\mathcal{C}_0$ in the code chain. The minimum distance $d_{\text{local}}$ of each local code of $\mathcal{C}_{\text{convol}}$ is lower bounded by the minimum distance $d_0$ of the mother code $\mathcal{C}_0$ in the code chain. The local structure of $\mathcal{C}_{\text{convol}}$ allows it to be decoded based on a designed parity-check matrix of the mother code $\mathcal{C}_0$ using a sliding window decoding scheme.

Abstract: We characterize the degree of freedom (DoF) of a point-to-point blockwise memoryless channel without prior channel state information (CSI), with a fixed number $K$ of propagation paths, where the transmitter (Tx) and the receiver (Rx) are equipped with nonuniform linear arrays (NULAs) of $N_t$ and $N_r$ antennas, respectively. The positions of array elements are fixed, known, pairwise distinct, and need not be equally spaced. The uniform linear array (ULA) is a special case. In each block of length $T$, the continuous angles of arrival (AoAs), angles of departure (AoDs), and independent complex Gaussian path gains are redrawn. Both Tx and Rx know the state distributions but are not given the current realizations before transmission. The receiver may estimate the channel from reference signals or decode without explicit channel estimation, with reference symbols counted in $T$ and their energy counted against the power constraint. Under the aforementioned model, we show that the DoF is $1-\frac{1}{T}$ for $K=1$, and $K(1-\frac{3}{2T})$ for $K \geq 2$, when $N_r\ge K+1$, $N_t\ge\max\{K,2\}$, and $T\ge K$. The analytical results are further demonstrated by their applications to the DoF tradeoff analysis in integrated sensing and communication (ISAC). For more general array structures, an achievability result is established, while the converse remains open in general.

Abstract: Recurrent detectors such as bidirectional long short-term memory (Bi-LSTM) networks are low-complexity alternatives to the optimal Bahl-Cocke-Jelinek-Raviv (BCJR) detector for faster-than-Nyquist (FTN) signaling. Motivated by convolutional detectors that build the intersymbol interference (ISI) structure into their architecture, we ask whether processing nested ISI windows in separate recurrent branches improves the bit error rate (BER) of a Bi-LSTM. Across roughly 260 controlled trainings it does not: at a matched parameter budget and a matched readout, the multi-window architecture never significantly beats a plain Bi-LSTM. Nested windowing is an invertible rearrangement that adds no information, extra branches only add bottlenecks, and a distillation diagnostic shows the network is already near optimal for its window. The limitation is therefore the observation model, not the architecture. Keeping the architecture fixed, we pre-whiten the input, restoring the conditional independence that colored matched-filter noise violates, and distill the BCJR soft posterior into the network. With 3.4% more parameters this reaches 1.05 times the BCJR BER at a compression factor of 0.8 and 1.89 times at 0.7, improving to 1.47 times when the whitened window is widened. The 23.7% BER reduction at 0.8 requires an ill-conditioned ISI matrix but is not monotone in the conditioning, and it holds across five independent noise realizations and a symbol-level McNemar test.