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This study is a significant advancement of this team's proof-of-concept study on using field-programmable gate arrays (FPGAs) in wireless sensing for structural health monitoring. Compared with traditional microprocessor-based systems, fast growing FPGA technology offers a more powerful, efficient, and flexible hardware platform. An effort is presented herein to embed algorithms to process nonlinear time series by entirely using an FPGA, while an ongoing effort is to pursue the development of an FPGA and microprocessor co-design for a more extended and robust version of this study. The Hilbert transform (HT) and a backbone curve technique are the centerpiece to extract instantaneous characteristics of a displacement time history from a under free vibration. Critical design issues are carefully considered including required approximation accuracy, constraints imposed by limited hardware resources, timing in the execution of the hardware design, and data representations in a fixed-point design environment. An automation of the classification of three basic types of SDOF systems (including linear, hardening and softening) are implemented for wireless transmission of processed results. An off-the-shelf high-level abstraction tool along with the MATLAB/Simulink environment is utilized to program the FPGA, rather than coding the hardware description language (HDL) manually. Extensive validation using simulated data is conducted for every step/stage of the design as well as major built-in functions adopted from the hardware design tool. The contribution of this study includes (1) enabling the functionality of a full hardware design for an enhanced computational efficiency in wireless structural health monitoring and (2) achieving balance between computational efficiency and resource utilization for onboard data processing especially when non-high-end FPGA products are targeted for practical consideration in structural health monitoring.