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The feasibility of compressed-sensing-based (CS) waveform reconstruction for data sampled from the random equivalent sampling (RES) method is investigated. RES is a well-known random sampling method that samples high-frequency periodic signals using low-frequency sampling circuits. It has been incorporated in modern digital oscilloscopes. However, the efficiency and accuracy of RES may be sensitive to timing uncertainty of analog RES circuits. For signals with sparsely populated harmonic components, the CS-based signal reconstruction method promises to mitigate the inherent timing error and to enhance the overall performance. A novel measurement matrix motivated by the Whittaker-Shannon interpolation formula is proposed for this purpose. Experiments indicate that, for spectrally sparse signal, the CS-reconstructed waveform exhibits a significantly higher signal-to-noise ratio than that using the traditional time-alignment method. A prototype realization of this proposed CS-RES method has been developed using off-the-shelf components. It is able to capture analog waveforms at an equivalent sampling rate of 25 GHz while sampled at 100 MHz physically.