Search Results (4)

Conventional integer-order models fail to adequately capture non-local memory effects and constrained nonlinear interactions in emotional dynamics. To address these limitations, we propose a coupled framework that integrates Caputo fractional derivatives with hyperbolic tangent–based interaction functions. The fractional-order term quantifies power-law memory decay in affective states, while the nonlinear component regulates connection strength through emotional difference thresholds. Mathematical analysis establishes the existence and uniqueness of solutions with continuous dependence on initial conditions and proves the local asymptotic stability of network equilibria ($W_{ij}^{*}={\displaystyle \frac{1}{\delta }}{\sec \mathrm{h}}^2(\parallel {\mathrm{E}}_i-{\mathrm{E}}_j\parallel )$, e.g., $W^{*}\approx 1.40$ under typical parameters $\eta =0.5$, $\delta =0.3$). We further derive closed-form expressions for the steady-state variance under stochastic perturbations ($\mathrm{Var}\left(W_{ij}\right)={\displaystyle \frac{{\sigma }_{\zeta }^2}{2\eta \delta }}$) and demonstrate a less than 6% deviation between simulated and theoretical values when ${\sigma }_{\zeta }=0.1$. Numerical experiments using the Euler–Maruyama method validate the convergence of connection weights toward the predicted equilibrium, reveal Gaussian features in the stationary distributions, and confirm power-law scaling between noise intensity and variance. The numerical accuracy of the fractional system is further verified through L1 discretization, with observed error convergence consistent with theoretical expectations for $\mu =0.5$. This framework advances the mechanistic understanding of co-evolutionary dynamics in emotion-modulated social networks, supporting applications in clinical intervention design, collective sentiment modeling, and psychophysiological coupling research. Full article

A high-precision time reference is fundamental to the positioning, navigation, and timing (PNT) of global navigation satellite systems (GNSS). The precision of clock steering determines the accuracy of practical applications that rely on the time–frequency reference. With the invention of direct digital synthesizer (DDS) technology, digital clock steering (DCS) has gradually become a mainstream technology. However, the key factor limiting DCS accuracy is the system quantization noise, which leads to a low frequency and phase adjustment accuracy. Here we propose a DCS method based on $\Sigma$-$\Delta$ modulation to address the issue of low resolution of DAC through shaping the quantization noise. A simulated GNSS time–frequency reference system experimental platform is constructed to validate the effectiveness of the proposed method. The experimental results demonstrate that this method achieves a phase adjustment accuracy of 0.48 ps and a frequency adjustment accuracy better than 0.48 pHz, which is two orders of magnitude higher than that of existing GNSS time–frequency reference systems. Thus, the proposed method offers a significant improvement in time–frequency reference systems, leading to better performance, reliability, and accuracy in a wide range of practical applications. Full article

Due to the characteristic of narrow band conversion around a central radio frequency, the Sigma Delta Modulator ($\Sigma \Delta$M) based on LC resonators is a suitable option for use in Software-Defined Radio (SDR). However, some aspects of the topologies described in the state-of-the-art, such as noise and nonlinear sources, affect the performance of $\Sigma \Delta$M. This paper presents the design methodology of three high-order LC-Based single-block Sigma Delta Modulators. The method is based on the equivalence between continuous time and discrete time loop gain using a Finite Impulse Response Digital-to-Analog Converter (FIRDAC) through a numerical approach to defining the coefficients. The continuous bandpass LC $\Sigma \Delta$M simulations are performed at a center frequency of 432 MHz and a sampling frequency of 1.72 GHz. To the proposed modulators a maximum Signal-to-Noise Ratio (SNR) of 51.39 dB, 48.48 dB, and 46.50 dB in a 4 MHz bandwidth was achieved to respectively 4th Order Gm-LC $\Sigma \Delta$M, 4th Order Magnetically Coupled $\Sigma \Delta$M and 4th Order Capacitively Coupled $\Sigma \Delta$M. Full article

The EM- $\Sigma \Delta$ (electromechanical sigma-delta) approach is a concise and efficient way to realize the digital interface for micro-electromechanical systems (MEMS) accelerometers. However, including a fixed MEMS element makes the synthesizing of the EM- $\Sigma \Delta$ loop an intricate problem. The loop parameters of EM- $\Sigma \Delta$ can not be directly mapped from existing electrical $\Sigma \Delta$ modulator, and the synthesizing problem relies an experience-dependent trail-and-error procedure. In this paper, we provide a new point of view to consider the EM- $\Sigma \Delta$ loop. The EM- $\Sigma \Delta$ loop is analyzed in detail from aspects of the signal loop, displacement modulation path and digital quantization loop. By taking a separate consideration of the signal loop and quantization noise loop, the design strategy is made clear and straightforward. On this basis, a discrete-time PID (proportional integral differential) loop compensator is introduced which enhances the in-band loop gain and suppresses the displacement modulation path, and hence, achieves better performance in system linearity and stability. A fifth-order EM- $\Sigma \Delta$ accelerometer system was designed and fabricated using 0.35 $\mathsf{\mu }\mathrm{m}$ CMOS-BCD technology. Based on proposed architecture and synthesizing procedure, the design effort was saved, and the in-band performance, linearity and stability were improved. A noise floor of 1 $\mathsf{\mu }\mathrm{g}/\sqrt{\mathrm{Hz}}$ , with a bandwidth 1 kHz and a dynamic range of 140 dB was achieved. Full article