## 3 Microwave Photonics Applications of DFT
### DFT for Real-Time Spectroscopy  
Optical spectroscopy has been one of the most integral tools in scientific research, manufacturing, and medical applications. Ultrafast real-time spectroscopy is particularly useful for investigating rapid transient phenomena such as chemical reactions, phase transitions of thermodynamic systems, and protein dynamics in living cells. Unfortunately, conventional diffraction-grating-based spectrometers fall short in speed as they rely on either a moving mechanical part or a detector array with limited refresh rate. Offering pulse-by-pulse spectrum monitoring, DFT's most obvious application is real-time spectroscopy, which provides an indispensable solution for the challenging continuous ultrafast real-time optical spectrum measurement [44; 45; 46; 47; 81; 82; 83; 84]. Figure 3 conceptually shows the DFT-based real-time spectroscopy, where the diffraction grating and detector array in conventional spectrometers are replaced by a dispersive element and a signal-pixel photodetector. The spectral features of the sample under test are first imprinted on the spectrum of the broadband optical pulse. After the DFT process, the spectral information is converted to a stretched temporal waveform. The converted temporal features usually fall into microwave frequency band and are slow enough to be captured by the high-speed PD and real-time digitizer, and then analyzed in digital domain. DFT-based real-time optical spectroscopy offers a scan rate of tens of MHz or even several GHz, determined by the repetition rate of the pulse train, which is more than three orders of magnitude higher than traditional spectrometers. The unique capability to capture single-shot spectra at high scan rates makes DFT spectroscopy an unprecedented tool for detecting non-stationary processes and rare events that would otherwise be lost in conventional time-averaged measurements [38]. Note that the overall spectral resolution of the DFT spectroscopy is determined by three factors: the temporal sampling rate of the electronic digitizer, the analog bandwidth of the photodetector, and the system dispersion [31]. Being not limited to direct optical spectroscopy measurement, DFT technique can be also employed to perform a variety of high-throughput and real-time measurement that relies on spectral domain methods. One such an example is real-time interrogation of FBG sensors [48, 49, 85, 86, 87, 88]. Most of the FBG sensors are functioning based on wavelength modulation, in which the sensed information is directly encoded as the grating wavelength (spectrum) change [89]. DFT technique maps the wavelength change to a temporal shift of the stretched waveform. This temporal-spectroscopy enables interrogation of FBG sensors in the megahertz regime [85], which is particularly useful for real-time diagnostics of fast-vibrating objects, such as a running aircraft engine. In addition, by applying pulse compression method in temporal-spectroscopy, both spectral resolution and signal-to-noise ratio of the FBG interrogation system can be improved [86, 87]. This is made possible by pre-encoding a chirped profile onto the pulse spectrum before the pulse is sent to the FBG sensors. At the receiver end, a matched filter is employed to achieve pulse compression (auto-correlation). Note that in the above DFT-based FBG sensor interrogation system, the measurement resolution is still limited by the temporal resolution of a commercial oscilloscope. While a larger dispersion would lead to a higher interrogation resolution since it converts a given wavelength shift to a longer temporal waveform shift, to avoid the temporal superposition of two adjacent stretched pulses, it also imposes a limit on the pulse repetition rate, equivalently the refresh rate of the measurement.