## 1 Introduction  
Fiber optic sensing technology is a crucial engineering technique that has demonstrated widespread applications in various fields, including industrial automation, perimeter security, aerospace, and more. Among these applications, phase-sensitive OTDR, also known as \(\varphi\)-OTDR, has gained significant popularity in the current landscape of fiber optic sensing technology. This is due to its numerous advantages, such as high spatial resolution, extended sensing distances, and immunity to the effects of fiber connections [1; 2; 3; 4]. The \(\varphi\)-OTDR system's frequency response range is inherently constrained by sensing distance, where greater distances yield narrower attainable bandwidths [5; 6; 7]. Current methods to widen this bandwidth include positive and negative pulse signals, periodic non-uniform sampling, and frequency division multiplexing (FDM). Notably, in 2017, [PERSON] et al. introduced a DAS system employing FDM-TGD-OFDR, achieving a 9 kHz frequency response across a 24.7 km distance with modulation pulses spanning five frequency segments within a 100 MHz bandwidth [8]. Similarly, in 2018, [PERSON] et al. combined positive and negative pulse signals, FDM technology, and chirped pulse \(\varphi\)-OTDR, achieving a spatial resolution of 9.3 m and a frequency range of 0-24 kHz over a 2.1 km sensing distance using 195 MHz modulation bandwidth and four-frequency segmentFDM [9]. Moreover, in 2023, [PERSON] et al. proposed an equivalent sampling method based on compressed sensing and interval-scanning pulses, thus enhancing the frequency response of phase-sensitive optical time-domain reflectometry through periodic non-uniform sampling [10]. Among these approaches, FDM emerged as a particularly effective approach in extending the frequency response range. However, this necessitates protective frequency bands to prevent interference between different frequency segments, thus demanding substantial modulation bandwidth and high-bandwidth detectors for achieving high-frequency response bandwidths [11; 12; 13]. The spatial resolution of the \(\varphi\)-OTDR system relies on pulse width, and achieving higher resolution necessitates narrower pulses. However, this reduction in pulse width decreases the injected optical energy, leading to a lower SNR and reduced sensing distances [14; 15; 16]. Consequently, conventional \(\varphi\)-OTDR systems confront a trade-off between spatial resolution and sensing distance, which poses challenges in meeting both high-resolution and long-distance sensing requirements [17; 18]. Among the available solutions, linear frequency modulation (LFM) signals often feature narrow main lobe widths but lower sidelobe suppression ratios. To enhance sidelobe suppression, time-domain windowing is commonly employed. Nevertheless, this time-domain approach can diminish the main lobe's height, resulting in reduced spatial resolution [19; 20; 21]. Hence, a shift in frequency modulation techniques becomes imperative to improve the sidelobe suppression ratio. Differing from some other frequency modulation techniques, such as linear frequency modulation (LFM), the frequency modulation characteristics of non-linear frequency modulation (NLFM) signals contribute to reducing sidelobes in the time-domain signal. This enhances the prominence of the main lobe and improves the signal-to-noise ratio, thus enhancing the performance of optical fiber sensing systems. This is particularly crucial in optical fiber sensing applications where better signal quality is required. In 2019, [PERSON] et al. achieved a sensing range of 80 km and a spatial resolution of 2.7 m by employing iterative pre-distortion in combination with non-linear frequency modulation techniques. However, the frequency response range of this system was limited to 10-610 Hz [22]. In 2022, [PERSON] et al. utilized the readily adaptable direct digital synthesis (DDS) of pulses technique to generate compressed pulses with NLFM waveforms, thus achieving a spatial resolution of 0.5 m at a distance of 1.13 km.