Search Results (6)

The codesign of the receive filter and transmit waveform under similarity constraints is one of the key technologies in frequency diverse array multiple-input multiple-output (FDA-MIMO) radar systems. This paper discusses the design of constant modulus waveforms and filters aimed at maximizing the signal-to-interference-and-noise ratio (SINR). The problem’s non-convexity renders it challenging to solve. Existing studies have typically employed relaxation-based methods, which inevitably introduce relaxation errors that degrade system performance. To address these issues, we propose an optimization framework based on the joint complex circle manifold–complex sphere manifold space (JCCM-CSMS). Firstly, the similarity constraint is converted into the penalty term in the objective function using an adaptive penalty strategy. Then, JCCM-CSMS is constructed to satisfy the waveform constant modulus constraint and filter norm constraint. The problem is projected into it and transformed into an unconstrained optimization problem. Finally, the Riemannian limited-memory Broyden–Fletcher–Goldfarb–Shanno (RL-BFGS) algorithm is employed to optimize the variables in parallel. Simulation results demonstrate that our method achieves a 0.6 dB improvement in SINR compared to existing methods while maintaining competitive computational efficiency. Additionally, waveform similarity was also analyzed. Full article

Designing waveforms with a Constant Modulus Constraint (CMC) to achieve desirable Slow-Time Ambiguity Function (STAF) characteristics is significantly important in radar technology. The problem is NP-hard, due to its non-convex quartic objective function and CMC constraint. Existing methods typically involve model-based approaches with relaxation and data-driven Deep Neural Networks (DNNs) methods, which face the challenge of dataimitation. We observe that the Complex Circle Manifold (CCM) naturally satisfies the CMC. By projecting onto the CCM, the problem is transformed into an unconstrained minimization problem that can be tackled using the CCM gradient descent model. Furthermore, we observe that the gradient descent model over the CCM can be unfolded as a Deep Learning (DL) network. Therefore, byeveraging the powerfulearning ability of DL and the CCM gradient descent model, we propose a Model-Adaptive Learned Network (MAL-Net) method without relaxation. Initially, we reformulate the problem as an Unconstrained Quartic Problem (UQP) on the CCM. Then, the MAL-Net is developed toearn the step sizes of allayers adaptively. This is accomplished by unrolling the CCM gradient descent model as the networkayer. Our simulation results demonstrate that the proposed MAL-Net achieves superior STAF performance compared to existing methods. Full article

The cognitive radar framework presents a closed-loop adaptive processing paradigm that ensures the efficient acquisition of target information while exploring the environment and enhancing overall sensing performance. In this study, instead of mutual information, we employed the squared Pearson correlation coefficient (SPCC) to measure the target information in observations specifically considering only linear dependency. A waveform design method is proposed that simultaneously maximizes target information and minimizes the integrated sidelobe level (ISL) under the constant modulus constraint (CMC). To enhance computational efficiency, we reformulated the constrained problem as an unconstrained optimization problem by leveraging the inherent geometric property of CMC. Additionally, we present two conditional equivalences associated with waveform design in relation to target information. The simulation results validate the feasibility and effectiveness of the proposed method. Full article

This paper investigates, as a first step, the four branches of BMS transformations, motivated by the classification into elliptic, parabolic, hyperbolic and loxodromic proposed a few years ago in the literature. We first prove that to each normal elliptic transformation of the complex variable $\zeta$ used in the metric for cuts of null infinity, there is a corresponding BMS supertranslation. We then study the conformal factor in the BMS transformation of the u variable as a function of the squared modulus of $\zeta$. In the loxodromic and hyperbolic cases, this conformal factor is either monotonically increasing or monotonically decreasing as a function of the real variable given by the modulus of $\zeta$. The Killing vector field of the Bondi metric is also studied in correspondence with the four admissible families of BMS transformations. Eventually, all BMS transformations are re-expressed in the homogeneous coordinates suggested by projective geometry. It is then found that BMS transformations are the restriction to a pair of unit circles of a more general set of transformations. Within this broader framework, the geometry of such transformations is studied by means of its Segre manifold. Full article

The human ankle is a complex joint, most commonly represented as the talocrural and subtalar axes. It is troublesome to take in vivo measurements of the ankle joint. There are no instruments for patients lying on flat surfaces; employed in outdoor or remote sites. We have developed a “Turmell-meter” to address these issues. It started with the study of ankle anatomy and anthropometry. We also use the product of exponentials’ formula to visualize the movements. We built a prototype using human proportions and statistics. For pose estimation, we used a trilateration method by applying tetrahedral geometry. We computed the axis direction by fitting circles in 3D, plotting the manifold and chart as an ankle joint model. We presented the results of simulations, a prototype comprising 45 parts, specifically designed draw-wire sensors, and electronics. Finally, we tested the device by capturing positions and fitting them into the bi-axial ankle model as a Riemannian manifold. The Turmell-meter is a hardware platform for human ankle joint axes estimation. The measurement accuracy and precision depend on the sensor quality; we address this issue by designing an electronics capture circuit, measuring the real measurement with a Vernier caliper. Then, we adjust the analog voltages and filter the 10-bit digital value. The Technology Readiness Level is 2. The proposed ankle joint model has the properties of a chart in a geometric manifold, and we provided the details. Full article

This paper is one of a series in which we generalize our earlier results on the equivalence of existence of Calabi extremal metrics to the geodesic stability for any type I compact complex almost homogeneous manifolds of cohomogeneity one. In this paper, we actually carry all the earlier results to the type I cases. In Part II, we obtained a substantial amount of new Kähler–Einstein manifolds as well as Fano manifolds without Kähler–Einstein metrics. In particular, by applying Theorem 15 therein, we obtained complete results in the Theorems 3 and 4 in that paper. However, we only have partial results in Theorem 5. In this note, we provide a report of recent progress on the Fano manifolds $N_{n,m}$ when $n>15$ and $N_{n,m}^{\prime }$ when $n>4$ . We provide two pictures for these two classes of manifolds. See Theorems 1 and 2 in the last section. Moreover, we present two conjectures. Once we solve these two conjectures, the question for these two classes of manifolds will be completely solved. By applying our results to the canonical circle bundles, we also obtain Sasakian manifolds with or without Sasakian–Einstein metrics. These also provide open Calabi–Yau manifolds. Full article