## 3 Microwave Photonics Applications of DFT
### DFT for Microwave Spectrum Sensing  
Microwave spectrum sensing is a recent topic of great interest in microwave photonic; with various scientific; industrial and defense applications; such as wireless communications; radars and electronic warfare [115]. It is a real challenge to monitor the instantaneous spectrum of high-frequency and broadband microwave signals in real-time with high-resolution while having low-cost. Photonically assisted techniques for microwave spectrum measurement have shown superior performance over their electronic counterpart. Many solutions have been proposed for photonic microwave spectrum sensing in the past a few years; including those based on microwave/optical power monitoring [116; 117] and channelization [118; 119]. Power-monitoring-based technique offers good frequency measurement resolution; usually smaller than 100 MHz. However; it falls short in simultaneously measuring multiple microwave frequencies. Channelization method is capable of multiple microwave frequency measurement based on the direct spectrum sampling using optical channelizers. Its key limitation in \begin{table}
\begin{tabular}{|c|c|c|} \hline
**Systems for Generating Chirped** & & \\
**Microwave Waveforms** & **TBWP** & **Chirp Rate** \\ \hline Based on superimposed chirped & & \\ FBGs [103] & 37.5 & 23.8 GHz/ns \\ \hline Based on Sagnac loop mirror with a & & \\ chirped FBG [104] & 44.8 & 22 GHz/ns \\ \hline Based on nonlinear DFT [56] & 8.4 & 74 GHz/ns \\ \hline Based on spatially-discrete chirped & & \\ FBG [39] & 16.8 and 23.2 & 11.2–93.6 GHz/ns \\ \hline Based on temporal Fourier transform & & \\ pulse shaping [114] & 1.8 & 0.715 GHz/ns \\ \hline \end{tabular}
\end{table} 
Table 1: Time-bandwidth product (TBWP) of chirped microwave waveforms that are generated by various all-fiber DFT-based microwave AWG systems. microwave spectrum sensing is the poor measurement resolution due to large channel spacing of an optical channelizer; which is usually larger than 1 GHz [117, 118]. Recently, we have proposed and demonstrated a DFT-based temporal channelization method for real-time microwave spectrum sensing [120, 121], which does not only enables multiple-frequency detection, but also offers hundreds of times higher frequency measurement resolution than the channel spacing. Figure 10 shows the conceptual diagram of the system. An ultrafast optical pulse is first sent to a dispersive element. The dispersed pulse is then modulated by a microwave signal with its spectrum to be measured. Thanks to the one-to-one mapping between time and spectrum in the DFT process, the pulse spectrum is also modulated by the microwave signal. This process is called time-domain spectral shaping [122]. Finally, the stretched and modulated optical pulse is sent to an optical channelizer for spectrum sampling. In fact, the temporal sampling of the microwave signal is equivalently implemented by the optical channelizer in the spectral domain due to the unique wavelength-to-time mapping. In the system, the frequency measurement resolution is not determined by the physical channel spacing of the channelizer [117, 118], but the optical bandwidth of the dispersed pulse and the dispersion value [120]. By broadening the pulse spectrum based on nonlinear optical effects, a frequency resolution as high as 55 MHz can be achieved using an optical channelizer with a channel spacing of 25 GHz [121], which demonstrates nearly 500 times improvement. Moreover, due to the DFT-enabled equivalent parallel-to-serial conversion, no high-speed photodetector and digitizer is required in the system. The system cost is greatly reduced. One difficulty of the DFT-based temporal-channelization method is the limited operational bandwidth due to the low equivalent temporal sampling rate. According to Shannon-Nyquist sampling theorem, only microwave signals with their bandwidth less than half of the maximum sampling rate can be captured without losing any information.