Real-time FPGA Implementation of a Second order Volterra Filter for Ultrasound Nonlinear Imaging

Volterra series expansion is a widely used method for modeling nonlinear systems because of its simplicity, linearity with respect to the kernel coefficients, ability to generalize any order of nonlinearity etc. which makes it a suitable analytical tool for a wide range of applications. However, the higher computational complexity of the kernels is a major limiting factor for its use in real-time applications. The number of coefficients increases exponentially with the sample memory length and the order of nonlinearity, increasing the resource utilization immensely and restricting the system bandwidth. In this paper, we have discussed different architectures such as a direct form (with modification for timing improvement) and an eigenvalue decomposition based parallel-cascade method for implementation of a low cost, real-time quadratic Volterra filter on a Xilinx Zynq7000 SoC FPGA board. Both the designs are compared based on their resource utilization and timing constraints. Behavioral performance of the nonlinear filters in the context of ultrasound image reconstruction is demonstrated using a quality assurance phantom data, sampled at a rate of 40 MHz.