Fig. 1: Illustration of the principles of time-mapped Fourier transform and the proposed time-mapped spectrogram analysis.

a Analysis of a temporal pulsed waveform using the dispersion-based TM-FT. Here, the sinc-shaped power spectrum of the incoming square shaped temporal waveform, is mapped onto the output temporal domain after the TM-FT operation. b Analysis of a continuous temporal waveform using the dispersion-based TM-FT. The signal under analysis needs to be temporally truncated and subsequently stretched along the time axis, using linear chromatic dispersion. As a result, most of the signal information, in between consecutive truncated signal sections, is simply lost in this process. c A conceptual illustration of the proposed time-mapped spectrogram (TM-SP) analysis applied on the same incoming waveform. In this approach, the full short-time Fourier transform (ST-FT) distribution (in amplitude and phase), or full spectrogram, of the incoming temporal waveform is mapped along the temporal domain purely in real-time and in a continuous fashion. The TM-SP effectively performs Fourier analysis of consecutive time-windowed sections of the incoming waveform. This is achieved through a combination of short-pulse sampling and dispersive delay (see details in Fig. 2) without using an actual truncation or windowing of the incoming waveform. The mapped spectrogram involves the analysis of time-windowed consecutive sections of the signal under test that are heavily overlapped, which ensures that the continuous real-time spectral analysis is entirely gap-free.