Dynamic real-time spectrum analysis and processing of high-speed broadband waveforms is fundamental to many important applications in communications, radar, sensing and other areas. These applications require real-time, continuous, gap-free computation of the Fourier transform of the incoming temporal signal, over broad frequency bandwidths with nanosecond-or-shorter resolutions. Recently, we have proposed and demonstrated a universal analog signal processing architecture that achieves a direct and continuous time-mapping of a gap-free short-time Fourier transform or spectrogram for an arbitrary (even infinitely long) incoming waveform.