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In this lecture note, we present a concise and self-contained derivation of the discrete-time Kalman filter equations that requires only a basic understanding of least squares estimation. The treatment is designed to minimize mathematical overhead while preserving both rigor and generality.
We use the best linear approximation (BLA) to characterize nonlinearities in IM-DD links. The orthogonal nonlinear distortion is shown to be non-Gaussian. The BLA is used to approximate the optimum modulation depth for different equalizer structures.
This work introduces low-complexity, multiplierless approximations for the 32-point discrete Fourier transform. The proposed methods are obtained by minimizing the Frobenius error compared against the DFT matrix over a set of trivial multipliers. A row-wise, symmetry-constrained parameterization is employed to reduce the search space size, rendering the task computationally tractable. The resulting approximations could outperform the reference method in the literature according to energy-based error measurements. A sparse matrix factorization is provided for efficient computation; the arithmetic costs are 152 real additions and 34 bit-shifts only.
This paper investigates physical-layer security (PLS) over correlated composite generalized-Gamma (GG)/GG fading channels, where both shadowing and small-scale fading follow GG distributions. Using Mellin transforms and Fox-H functions, closed-form expressions are derived for the single-link probability density function (PDF), joint distribution, survival function, and zero-rate secrecy outage probability (SOP)/probability of non-zero secrecy capacity (PNZSC). The general-rate SOP is expressed as an exact double series with one residual onedimensional integral per term. The model includes the Nakagamim/GG and Nakagami-m/Gamma channels as special cases. Numerical results validate the analysis and demonstrate the impact of the fading parameters on secrecy performance.
In this paper, a lower bound on the maximum ambiguity function (AF) sidelobes of a set of unimodular sequences is formulated for the desired low-ambiguity-zone (LAZ). Our main idea is to introduce a set of mismatched filters associated to a set of unimodular sequences and two weight vectors for the delay and Doppler shifts, respectively. The length of mismatched filter maybe different to the length of unimodular sequence. The proposed lower bound on the maximum AF sidelobes for the desired LAZ can be treated as an extension of Tan-Arlery-Rabaste-Lehmann-Ovarlez lower bound, published in 2020, which dealt with the conventional correlation of sequences.
Families of sequences with low mutual aperiodic crosscorrelation assist the design of systems for multi-user asynchronous communications and multiple-input multiple-output radar. The crosscorrelation demerit factor of a pair of sequences is the sum of the squared magnitudes of their crosscorrelation values at every shift when the sequences are normalized to unit Euclidean norm, and the merit factor is the reciprocal of the demerit factor. For each positive integer $\ell$, we endow the $2^{2 \ell}$ pairs of binary sequences of length $\ell$ with uniform probability measure and study the distribution of their crosscorrelation demerit factors. Sarwate showed that the mean value is always $1$ regardless of length $\ell$. We develop a method for finding an exact formula for the $p$th central moment (for any positive integer $p$) as a function of $\ell$. Formulae for the variance and third central moment ($p=2$ and $3$) are then obtained by hand calculations, while the fourth through sixth central moments are obtained by computer-assisted calculations. Our theory also shows that all the central moments must be strictly positive for $p\geq 2$ and $\ell \geq 3$.
The problem of finding the optimal compression rate for a given random variable has been traditionally studied under two main constraints: distortion and perception. The distortion constraint enforces the fidelity of our reconstruction with respect to the observed realization of the random variable, while the perception constraint ensures that the reconstruction is close to a sample from the distribution of the random variable of interest. In this work, we explore the possibility of reconstruction, such that the reconstructed sample is still within a desired fidelity level with our original realization of the random variable, but at the same time, it resembles a sample from a different target distribution. We term this criterion as the deception constraint and find the fundamental tradeoffs of rate-distortion and deception.
In this paper, constrained parameter update laws for adaptive control are developed using barrier constraints. An interpretation of the parameter update law from a constrained optimization problem, in which a regularized Barrier saddle function is formulated to incorporate parameter constraints using inverse and logarithmic barrier functions from interior-point methods. The resulting constrained update law is integrated with an adaptive trajectory tracking controller, enabling online learning of the unknown system model parameters. Forward invariance of the parameter estimate is established and Lyapunov stability of the closed-loop system with the constrained parameter update law is derived. The effectiveness of the proposed constrained adaptive control law is demonstrated through simulations, which validate its ability to maintain parameter estimates within prescribed bounds while ensuring convergence to the true parameter values and achieving steady state tracking performance.
We investigate the use of modern code-agnostic decoders to convert CA-SCL from an incomplete decoder to a complete one. When CA-SCL fails to identify a codeword that passes the CRC check, we apply a code-agnostic decoder that identifies a codeword that satisfies the CRC. We establish that this approach gives gains of up to 0.2 dB in block error rate for CA-polar codes from the 5G New Radio standard. If, instead, the message had been encoded in a systematic CA-polar code, the gain improves to more than 1.5 dB. Leveraging recent developments in blockwise soft output, we additionally establish that it is possible to control the undetected error rate even when using the CRC for error correction.
Golay complementary matrices (GCM) have recently drawn considerable attentions owing to its potential applications in omnidirectional precoding. In this paper we generalize the GCM to multi-dimensional Golay complementary arrays (GCA) and propose new constructions of GCA pairs and GCA quads. These constructions are facilitated by introducing a set of identities over a commutative ring. We prove that a quaternary GCA pair is feasible if the product of the array sizes in all dimensions is a quaternary Golay number with an additional constraint on the factorization of the product. For the binary GCM quads, we conjecture that the feasible sizes are arbitrary, and verify for sizes within 78 $\times$ 78 and other less densely distributed sizes. For the quaternary GCM quads, all the positive integers within 1000 can be covered for the size in one dimension.
We give a necessary and sufficient condition for the existence of power-one sequential tests in an i.i.d. composite testing problem. A level-\(α\) test with power one against every alternative exists if and only if the alternatives are separated from the null by a countable family of finite-block events. We provide other equivalent conditions using randomized fixed-sample tests, bounded finite-block scores, e-processes, reduced-filtration test supermartingales, and a countable cover whose finite-block weak-$*$ closed convex hulls are positively separated in total variation. As a bonus, the constructive proof yields tests have pointwise expected sample size \(O_Q(\log(1/α))\). Exactly the same conditions also characterize i.i.d.\ change detectability under optional-horizon average-run-length control: for every \(η>0\), they are equivalent to an alarm family \((T_γ)_{γ\ge1}\) satisfying \(\Prob_{P^\infty}(T_γ\leσ)\le \E_{P^\infty}σ/γ\) for every null law and every stopping time \(σ\). In fact, when these conditions hold, we can construct a single e-detector such that every null-law average run length lies between \(γ\) and \((1+η)γ+1\), and having robust Lorden delay \(O_Q(\logγ)\).
Coarse quantization in massive multiple-input multiple-output (MIMO) systems reduces power but causes clipping distortions. The Bayesian Expectation-Maximization (BEM) algorithm can recover clipped signals, but its matrix inversion and dynamic step-size evaluation are hardware bottlenecks. We propose a hardware-friendly one-step correction that uses the initial Jacobi-preconditioned Conjugate Gradient (PCG) direction with a fixed relaxation parameter. The resulting symbol-level update has an ultra-lightweight $\mathcal{O}(U)$ feed-forward datapath and approaches high-resolution reference detectors in the evaluated massive-MIMO setting. Our finite-dimensional analysis establishes the exact one-step descent law, proves that Jacobi normalization cancels the raw multiplicative near-far scaling while confining the loaded-system dependence to bounded attenuation factors, and gives verifiable sufficient conditions for fixed-step descent in terms of normalized channel coherence. System-level results indicate projected power savings for energy-efficient massive MIMO uplinks.
This paper develops a unified framework for instance optimal sparse recovery from nonlinear observations. The main ingredient is a signal-dependent restricted approximate invertibility condition (RAIC) of some gradient, which leads to the instance optimality of iterative hard thresholding. Under Gaussian designs, we apply the proposed framework to phaseless, one-bit, and ReLU measurements, which correspond to the problems of sparse phase retrieval, one-bit compressed sensing, and sparse ReLU regression, respectively. For sparse phase retrieval, we propose a variant of thresholded amplitude flow and show its instance optimality under $O(s^3)$ measurements (up to logarithmic factors), where $s$ is the sparsity level. To our best knowledge, this is the first instance optimal efficient algorithm for sparse phase retrieval and complements Gao, Wang and Xu (2016) that achieved this via a computationally intractable program. In one-bit compressed sensing, we establish the instance optimality of normalized binary iterative hard thresholding and strengthen the recent result of Matsumoto and Mazumdar (2024). In sparse ReLU regression, it is shown that a slight variant of the algorithm in Soltanolkotabi (2017) is instance optimal. Moreover, $(\ell_2,\ell_2)$ non-uniform instance optimal guarantees are obtained for these problems. The analysis is built upon a number of high-dimensional concentration bounds, including bounds on restricted eigenvalues and a novel instance-dependent hyperplane tessellation result.
For multiple-input multiple-output (MIMO) channels with phase modulation, we recently proposed a method of unitarily transforming the channel matrix into a certain row-echelon form, by which the original MIMO channel can be converted into a certain number of scalar sub-channels with two phase inputs, thereby forming an annulus constellation geometry, and corrupted by both the additive white Gaussian noise and weak self-interference. In this paper, several bounds are derived to evaluate the fundamental limit of such a specific transceiver architecture. Two upper bounds are obtained by upper-bounding the capacity of a scalar channel with an annulus support constraint from the perspective of the convex geometry, while a lower bound is obtained by the standard entropy power inequality. Numerical results show that the gaps between these bounds are small at high signal-to-noise ratios for the MIMO phase-modulated channels over the Rayleigh fading and the single-input multiple-output symbiotic communication system assisted by a reconfigurable intelligent surface.
Sionna is an open-source, GPU-accelerated library that, as of version 0.14, incorporates a ray tracer, Sionna RT, for simulating radio wave propagation. A unique feature of Sionna RT is differentiability, enabling the calculation of gradients for the channel impulse responses (CIRs), radio maps, and other related metrics with respect to system and environmental parameters, such as material properties, antenna patterns, and array geometries. The release of Sionna 1.0 provided a complete overhaul of the ray tracer, significantly improving its speed, memory efficiency, and extensibility. This document details the algorithms employed by Sionna RT to simulate radio wave propagation efficiently, while also addressing their current limitations. Given that the computation of CIRs and radio maps requires distinct algorithms, these are detailed in separate sections. For CIRs, Sionna RT integrates shooting and bouncing of rays (SBR) with the image method and uses a hashing-based mechanism to efficiently eliminate duplicate paths. Radio maps are computed using a purely SBR-based approach for the non-diffracted component, complemented by a stage for diffracted paths.
We study initialization bias in spherical $K$-means for weakly informative directional mixtures. We model the observations by a $K$-component von Mises-Fisher mixture with a small concentration parameter $κ$, corresponding to a high-dispersion regime in which the data provide limited information about the underlying directions. Our analysis begins with the limiting case $κ=0$ (corresponding to a uniform distribution over the sphere), where one population spherical $K$-means update is governed entirely by the Voronoi tessellation induced by the initialized templates. For uniformly random initializations in fixed dimension $d$, the updated templates become asymptotically aligned with their initial values as $K\to\infty$: the average squared geodesic error scales as $O(K^{-2/(d-1)})$, while the worst-case error is $O((\log K/K)^{2/(d-1)})$. We then show that, in the weak-signal regime of small positive $κ$, the population update remains an $O(κ)$ perturbation of this limiting map. Thus, in the weak-signal regime, spherical $K$-means can preserve initialization-induced structure despite the presence of a genuine but highly dispersed directional signal.
For $N\geq 3$ and a potential phase $\vartheta\in\mathbb{R}$, we study the non-self-adjoint almost Mathieu matrix obtained by multiplying the discrete Laplacian by a complex phase with angle $φ\in\mathbb{R}$, $A_N(φ,\vartheta)=e^{iφ}(S+S^{-1})/2+\operatorname{diag}(\cos(2πj/N+\vartheta))_{j\in\mathbb{Z}/N\mathbb{Z}}$, where $S e_j=e_{j+1}$ is the periodic shift on $\mathbb{C}^N$. We derive a Chambers formula and isolate the part $Q_{N,φ}$ of the characteristic polynomial that depends only on $N$ and $φ$, but not on $\vartheta$ or on a change of boundary conditions for the shift operator. We then show, for every $N$, that the zeros of $Q_{N,φ}$ lie on the two perpendicular lines $e^{iφ/2}\mathbb{R}\cup e^{i(φ/2+π/2)}\mathbb{R}$. For even $N$, the same property holds for the matrices $A_N(φ,\vartheta)$ with $\vartheta\in 2π\mathbb{Z}/N$, and we compute their limiting eigenvalue measure explicitly. For $φ\in[-π,π]$, the eigenvalue distribution approximates elliptic-integral densities with masses $1-|φ|/π$ and $|φ|/π$, and maximal radii $2|\cos(φ/2)|$ and $2|\sin(φ/2)|$, respectively. At $φ=π/2$, the central polynomial $Q_{N,φ}$ factors into positive quartic factors. This proves that the Scottish flag matrix, after Trefethen and Chapman, has its spectrum on the two diagonal lines of the saltire.
For resource-constrained wireless edge devices over bandwidth-limited fading channels, wireless image transmission using traditional separate coding suffers from the cliff-effect collapse. Prevailing deep joint source-channel coding (JSCC) based on convolutional neural networks can mitigate this issue but usually fail to preserve patch-level structures, thereby preventing adaptive per-token power allocation and limiting token-domain compensation for super-resolution (SR). To address these challenges, we propose a token communication framework with SR (TokenComSR). Specifically, we conceive a task-sensitive power allocation (TSPA) module and a signal-to-noise ratio (SNR)-conditioned token refinement module (TRM). TSPA distills training estimates of task sensitivity into inference token power weights, while TRM estimates an SNR-conditioned residual to correct channel-induced distortion in the token domain before decoding. Building on TSPA and TRM, the proposed TokenComSR pairs a Swin Transformer-based token transceiver with a receiver-side SR module for resource-constrained wireless image transmission. Simulation results confirm the effectiveness of the proposed TSPA and TRM, demonstrating improvements over separate coding and JSCC-SR baselines in both reconstruction fidelity and perceptual quality.