An approach to realizing microwave spectrum sensing based on photonic time stretch and compressive sampling is proposed. The time stretch system is used to slow down the input high-speed signal and the compressive sampling based on random demodulation can further decrease the sampling rate. A spectrally sparse signal in a wide bandwidth can be captured with a sampling rate far lower than the Nyquist rate thanks to both time stretch and compressive sampling. It is demonstrated that a system with a time stretch factor 5 and a compression factor 8 can be used to capture a signal with multiple tones in a 50 GHz bandwidth, which means a sampling rate 40 times lower than the Nyquist rate. In addition, the time stretch of the microwave signal largely decreases the data rate of random data sequence and therefore the speed of the mixer in the random demodulator.
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