## 3 Microwave Photonics Applications of DFT
### DFT for Microwave Arbitrary Waveform Generation
#### 3.2.5 Temporal Fourier Transform Pulse Shaping for Photonic Microwave AWG  
Photonic microwave arbitrary waveform generation can also be achieved based on Fourier transform pulse shaping [98, 109, 110]. Fourier transform pulse shaping is usually implemented in the space domain using a pair of diffraction gratings and a spatial light modulator (SLM) [98]. The spectrum of the optical pulse is dispersed in space by the first grating, modulated by the SLM in the Fourier domain, and then back-transformed by the second grating. Recently, we demonstrate an all-fiber Fourier transform pulse shaping using a single LCFBG [111]. The FBG in the system functions as a conjugate dispersive element pair to perform pulse stretching and pulse compression based on reflection in opposite directions from the FBG, and at the same time as a spectral shaper. The use of a single LCFBG guarantees an exact dispersion matching, featuring a better pulse shaping accuracy with a simplified structure.  
Fourier transform pulse shaping can also be implemented in the time domain based on temporal pulse shaping (TPS) technique. The typical TPS system consists of two conjugate dispersive elements and an electro-optic modulator (EOM) that is placed between the two dispersive elements. In the system, an ultrashort optical pulse is temporally stretched by the first dispersive element and then modulated in the time domain by a microwave driving signal at the EOM. The dispersed and modulated optical pulse is finally completely compressed by the second dispersive element with opposite dispersion. At the output of the TPS system, a temporal waveform that is the Fourier transform of the microwave modulation signal is obtained [112].  
The conventional TPS technique is usually employed for generating ultrafast optical waveforms [111, 112]. In fact, this technique can be adapted to achieve the generation of high-frequency microwave waveforms. Recently, we have proposed an unbalanced temporal pulse shaping (UB-TPS) system for the generation of microwave waveforms with continuously tunable carrier frequencies [92]. As shown in Figure 9, the UB-TPS system incorporates two dispersive elements, such as linearly chirped fiber Bragg gratings (LCFBGs) having opposite dispersion but non-identical in magnitude. The entire system can be considered as a conventional balanced TPS system for real-time Fourier transformation of the microwave drive signal, followed by a residual dispersive element with its dispersion being the offset of the dispersion of the two dispersive elements,  
Figure 9: Schematic an unbalanced TPS system for microwave arbitrary waveform generation based on frequency multiplication. MLL: mode-locked laser, LCFBG: linearly chirped fiber Bragg grating, EOM: electro-optic modulator, PD: photodetector.  
to achieve a second dispersive Fourier transformation. The finally obtained microwave waveform would be a scaled version of the input modulation signal. The frequency multiplication factor is determined by the dispersion values of the two dispersive elements. For LCFBG-based dispersive elements, their dispersion values can be continuously tunable [113]. In addition, a frequency-multiplied chirped microwave waveform can be generated from single-frequency microwave drive signal using the UB-TPS system if high-order dispersion is involved [114].  
Note that various all-fiber photonic microwave AWG systems for generating chirped microwave waveforms have been reviewed in this Section. One key feature of chirped microwave waveforms is their large time-bandwidth product (TBWP) due to frequency chirping and hence making them extensively employed in modern radar systems to improve the range resolution. Table 1 compares the chirp rate and TBWP of chirped microwave waveforms generated using different systems that have been reviewed in this paper. The overall achievable TBWP of the system is limited by the dispersion that enables DFT process.