This paper describes an analog CMOS IC for fast and fully-parallel finite-difference time-domain (FDTD) simulations of 2D electromagnetic (EM) problems. The chip uses discrete-time switched-current (SI) networks to model Maxwell's equations in 2D while minimizing the effects of device mismatch on solver accuracy. A prototype design in 180 nm technology implements a$16\times 16$solver grid within an active area of 44.5 mm2while consuming 345 mW at a clock frequency of 20 MHz.

This paper describes a circuit theoretic formulation for simulating the Schrödinger equation using classical analog circuits. Update equations for a finite difference time domain (FDTD) Schrödinger equation solver with absorbing boundary conditions (ABCs) are used to derive signal flow graphs that naturally map to switched capacitor (SC) circuits. A prototype implementation of a fully-parallel SC FDTD solver with 128 spatial points and a clock frequency of 2 MHz is analyzed and simulated using a standard 180 nm CMOS process.

A general-purpose analog computing method is proposed to compute the continuous-time solutions of nonlinear partial differential equations (PDEs). The discrete-time difference operator in the standard finite difference time domain (FDTD) method is replaced by continuous-time delay operators that can be realized using analog all-pass filters. The resulting spatially discrete time-continuous (SDTC) update equations are realized using analog circuits which compute continuous-time solutions of the PDE with prescribed initial and boundary conditions. The proposed concept is demonstrated in simulation via an integrated circuit (IC) design of a nonlinear acoustic wave equation solver in 180 nm CMOS technology. Analog arithmetic operations (multiply, scale, and add) are realized in parallel using fully differential op-amps and analog multipliers. The proposed IC computes the PDE solution in parallel at 33 discrete spatial points and has a simulated bandwidth and power consumption of approximately 2 MHz and 3 W, respectively. The performance of the IC is simulated using foundry-supplied device models and quantified using i) the mean squared difference between the circuit simulation results and FDTD simulations, and ii) the noise to signal energy ratio. Acceptable accuracy is obtained, with error metric values varying between -7 and -30 dB for various configurations of the problem. Comparison of the custom analog IC simulations with MATLAB- and C-based FDTD code running on a modern workstation shows an expected average speedup of 205× and 140×, respectively.

This paper describes a single-chip analog computer for solving the one dimensional (1-D) wave equation. The chip integrates a 16-point discrete-time but continuous-valued finite-difference solver with spatially-programmable wave velocity, selectable boundary conditions, and arbitrary input excitation waveforms. Built-in Δ-Σ analog-to-digital converters (ADCs) allow the solution results to be easily read out by a digital processor. The design was realized in TSMC 180 nm CMOS and has an active area of 2.81 mm × 2.64 mm. Experimental results prove the functionality of the proposed analog solver.

In this paper, we propose solutions to Maxwell's equations that can be computed using analog computers. Spatially-discrete time-continuous (SDTC) algorithms running on analog computers can be potentially faster and more energy-efficient than fully-discrete numerical solvers. The implementations of fully-discrete partial differential equation (PDE) solvers on high speed digital processors, such as graphics processing units (GPUs), take many clock cycles to compute a single temporal frame of the update equation and thus have relatively low equivalent bandwidths. Our approach is to directly implement temporal recursions in continuous-time by using analog circuits. Such circuits can have bandwidths that greatly exceed the equivalent bandwidths of GPUs. In particular, we propose two analog computing methods that compute the SDTC solutions to Maxwell's equations. In addition to Maxwell's equations, such platforms can be used to accelerate other hard computational problems that involve PDEs derived from continuous-time systems. In continuous-time in Laplace domain (CTLD) method (first approach), the spatial domain partial derivatives in the governing PDE are approximated using discrete finite differences, while applying the Laplace transformation along the time dimension. The resulting spatially-discrete time-continuous update equation is utilized to design an analog circuit that can compute the continuous-time solution. The all-pass delay approximate (APDA) method (second approach) replaces the discrete-time difference operators in the standard finite difference time domain (FDTD) cell (Yee cell) using continuous-time delay operators, which can be realized using analog all-pass filters. Both methods have been simulated using ideal analog circuits in Cadence Spectre for the Dirichlet, Neumann, and radiation boundary conditions. The performance of the proposed methods have been quantified using i) mean squared differences between the results and fully-discrete FDTD simulations, and ii) the noise to signal energy ratio. The CTLD and APDA methods are able to compute the solutions to Maxwell's equations with a noise energy to signal energy ratio γ better than -26 dB and -19 dB, respectively. Both methods have been extended to design analog circuits that compute the continuous-time solution of the 1-D and 2-D wave equations. The CTLD-based 1-D and 2-D analog wave equation solvers are able to compute the solutions with γ better than -72 dB and -60 dB, respectively. The APDA-based 1-D wave equation solver is simulated with a dominant-pole model (which better approximates the non-ideal circuit behavior) along with a propagation delay compensation technique. The non-ideal analog models compute the solution with a difference smaller than -13 dB (in terms of γ). Experimental results from a simplified board-level low-frequency implementation are also presented. The key challenges toward CMOS implementations of the proposed solvers are identified and briefly discussed with possible solutions.

Two continuous-time mathematical computing methods are proposed for solving the multidimensional wave equation leading to realizable analog computing circuits. The proposed analog computing processors will potentially be able to solve a certain special classes of computational problems involving partial differential equations, which are defined from continuous-time systems. The new analog computing methods are first derived and physically implemented for the first-time using low-frequency operational amplifier circuits in order to experimentally verify the correctness of the proposed methods. Both algorithms approximate the spatial domain partial derivatives using discrete finite differences. The first method performs a direct Laplace transform (with respect to the time variable) on the resulting expression. The second method applies the finite difference along the time dimension and then replaces the discrete time difference with a continuous-time delay operator, which in turn, can be realized as an analog all-pass filter. Analog circuit architectures are introduced for different boundary conditions relevant to common electromagnetic simulation problems. A low frequency prototype of the analog wave equation solver (based on method 1) has been designed, realized and tested using board-level operational amplifier circuits. Test results and measurements are provided to demonstrate the wave propagation in the space-time domain.

This paper studies the characterization of the optimal input for impulsive ambient noise channels under average power constraint. Our focus is on the two-term Gaussian mixture complex noise model, which has been widely used to model impulsive noise arising in various communication channels. We first demonstrate that there exists a unique input distribution that achieves the channel capacity and the capacity-achieving input distribution has a uniformly distributed phase. By examining the Kuhn-Tucker conditions (KTC), we further show that if the optimal amplitude input distribution contains an infinite number of mass points on a bounded interval, the channel output must be Gaussian distributed. However, by using Bernstein's theorem to examine the completely monotonic condition, it is shown that the assumption of a Gaussian distributed output is not valid. As a result, there is always a finite number of mass points on any bounded interval in the optimal amplitude distribution. In addition, by applying a novel bounding technique on the KTC and using the Envelop Theorem, we demonstrate that the optimal amplitude distribution cannot have an infinite number of mass points. That gives us a unique solution of the optimal input having discrete amplitude with a finite number of mass points. Given such interesting results, we also develop an efficient way to compute the discrete optimal input and the corresponding capacity.

This paper studies the characterization of the optimal input and the computation of the capacity of additive quadrature Gaussian mixture (GM) noise channels under an average power constraint. The considered model can be used to represent a wide variety of channels with impulsive interference, such as the well-known Bernoulli-Gaussian and Middleton class-A impulsive noise channels, as well as multiple-access interference channels and cognitive radio channels under imperfect sensing. At first, we demonstrate that there exists a unique input distribution that achieves the channel capacity, and the capacity-achieving input distribution has a uniformly distributed phase. By examining the Kuhn-Tucker alignment conditions (KTCs), we further show that, if the optimal input amplitude distribution contains an infinite number of mass points on a bounded interval, the channel output must be Gaussian-distributed. However, by using Bernstein's theorem to examine the completely monotonic condition, it is shown that the assumption of a Gaussian-distributed output is not valid. As a result, there are always a finite number of mass points on any bounded interval in the optimal amplitude distribution. In addition, by applying a novel bounding technique on the KTC and using the envelop theorem, we demonstrate that the optimal amplitude distribution cannot have an infinite number of mass points. This gives us the unique solution of the optimal input having discrete amplitude with a finite number of mass points. Given this discrete nature of the optimal input, we then develop a simple method to compute the discrete optimal input and the corresponding capacity. Our numerical examples show that, in many cases, the capacity-achieving distribution consists of only one or two mass points.

This paper presents simple methods to tightly estimate the information rate achieved by a Gaussian input and the constrained capacity of a finite-alphabet input of a Bernoulli-Gaussian (BG) impulsive noise channel in Rayleigh fading. Specifically, under the assumption of a Gaussian input, we propose a novel approach to calculate the achievable rate by examining the instantaneous output entropy in two regions of channel gains. In the high-gain region, the rate is evaluated via an upper bound obtained under the Gaussian output assumption. In the other region, we apply the piecewise-linear curve fitting (PWLCF) method to estimate the rate. It is then demonstrated that the information rate achieved by Gaussian inputs can be effectively calculated with a pre-determined accuracy. For a finite-alphabet input, we detail a PWLCF-based method to estimate the constrained capacity. In particular, we first propose a numerical technique to calculate the instantaneous output entropy using 2-dimensional Gauss-Hermite quadrature formulas. The average output entropy is then obtained using PWLCF. Combined with the closed-form expression the entropy of the BG impulse noise, an accurate estimation of the constrained capacity is finally established.

This paper presents a novel approach to tightly estimate the ergodic Shannon and constrained capacities of an additive Bernoulli-Gaussian (BG) impulsive noise channel in Rayleigh fading environments where channel gains are known at the receiver, but not at the transmitter. We first show that the differential entropy of the BG impulsive noise can be established in closed-form using Gaussian hypergeometric function2F1(1, 1; ·; ·). The Shannon capacity is then calculated via upper and lower bounds. Specifically, we derive in closed-form two upper bounds on the Shannon capacity using the assumption of a Gaussian output and using full knowledge of noise state, respectively. Under the assumption of a Gaussian input, we propose a novel approach to calculate a lower bound by examining the instantaneous output entropy in two regions of channel gains. In the high-gain region, the lower bound is evaluated via the upper bound obtained under the Gaussian output assumption. In the other region, we apply the piecewise-linear curve fitting (PWLCF) method to estimate the lower bound. It is then demonstrated that the lower bound can be calculated with a predetermined accuracy. By establishing the difference between the lower bound and the two upper bounds, we show that the lower bound can be used to effectively estimate the Shannon capacity. Finally, we detail a PWLCF-based method to estimate the constrained capacity for a finite-alphabet constellation. To this end, we first propose a numerical technique to calculate the instantaneous entropy of the output using 2-dimensional (2-D) Gauss-Hermite quadrature formulas. The average output entropy is then obtained using the PWLCF method. Combined with the closed-form expression of the entropy of the BG impulsive noise, the constrained capacity can be effectively estimated.

In this paper, we investigate the channel capacity of an additive Bernoulli-Gaussian (BG) impulsive noise channel in Rayleigh fading via lower and upper bounds. To this end, we first show that the differential entropy of the BG impulse noise can be established in closed-form using Gaussian hypergeometric function2F1(1, 1; .; .). This closed-form expression allows us to derive a lower bound on the capacity limit obtained by a Gaussian input using the Gauss-Hermite quadrature formula. We also derive in closed-form two upper bounds on the channel capacity. The first upper bound is obtained under the assumption of full knowledge of noise state, while the second upper bound is developed using a Gaussian distributed output. At high power regions, the lower bound achieved by Gaussian inputs and the upper bound generated by Gaussian inputs are indistinguishable. These two bounds can therefore be used as an accurate estimation for the channel capacity. When the channel input power is small compared to the power of the impulsive noise component, the lower bound obtained by using a Gaussian input and the upper bound under the perfect knowledge of impulse noise state are almost identical, which are useful to predict the capacity. The establishment of the lower bound and the two upper bounds in closed-form helps us to confirm the near-optimality of the Gaussian input in a wide range of input power levels over BG impulsive noise channels in Rayleigh fading.