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A novel analog real-time spectrum analyzer (RTSA) for the analysis of complex nonstationary signals (such as radar, security and instrumentation, and electromagnetic interference/compatibility signals) is presented, demonstrated, and characterized. This RTSA exploits the space-frequency mapping (spectral-spatial decomposition) property of the composite right/left-handed (CRLH) leaky-wave antenna (LWA) to generate the real-time spectrograms of arbitrary testing signals. Compared to digital RTSAs, it exhibits the advantages of instantaneous acquisition, low computational cost, frequency scalability, and broadband or ultra-wideband operation. The system is demonstrated both theoretically by a commercial full-wave simulator and an efficient Green's function approach and experimentally by a parallel-waveguide prototype including a metal-insulator-metal CRLH LWA, 16 patch antenna probe detectors circularly arranged around the LWA, and a digital oscilloscope performing analog/digital conversion and time-domain acquisition before the postprocessing and displaying of the spectrogram. The system is tested for a large diversity of nonstationary signals and generates, in all cases, spectrograms that are in excellent agreement with theoretical predictions. The fundamental tradeoff between time and frequency resolutions inherent to all RTSA systems is also discussed, and an interchangeable multi-CRLH LWA solution is proposed to handle signals with different time durations.