Analog All-pass Filter Research Articles

A Fast and Fully Parallel Analog CMOS Solver for Nonlinear PDEs

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.

Continuous-Time Algorithms for Solving Maxwell’s Equations Using Analog Circuits

Two analog computing methods are proposed to compute the continuous-time solutions of one-dimensional (1-D) Maxwell’s equations. In the first method, the spatial domain partial derivatives in the governing partial differential equation (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 second method 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 has been quantified using: i) mean squared differences between the results and fully discrete FDTD simulations and ii) the noise to signal energy ratio. Both methods have been extended to design the analog circuits that compute the continuous-time solution of the 1-D and 2-D wave equations. The 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 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.