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This paper investigates a real-time process generator of wideband signals, which calculates waveforms in a field-programmable gate array (FPGA) using the high-level synthesis (HLS) method. To obtain high-resolution and wide-swath images, the generator must produce multiple modes of large time-bandwidth product (TBP) linear frequency modulation (LFM) signals. However, the conventional storage method is unrealistic as it requires huge storage resources to save pre-computed waveforms. Therefore, this paper proposes a novel processing approach that calculates waveforms in real-time based simply on parameters such as the sampling frequency, bandwidth, and time width. Additionally, this paper implements predistortion through the polynomial curve to approximate phase errors of the system. The parallelizing process in the FPGA is necessary to satisfy the high-speed requirement of a digital-to-analog converter (DAC); however, repeatedly multiplexing real-time calculation consumes extensive logic and DSP resources, potentially exceeding FPGA limitations. To address this, this paper proposes a piecewise linear algorithm to conserve resources, which processes the polynomial only once, acquires the difference in two adjacent values through the register and pipeline, and then adds this increment to facilitate parallel computations. The performance of this proposed generator is validated through simulation and implemented in experiments with an X-band airborne SAR system. [ABSTRACT FROM AUTHOR]