This paper introduces a new data acquisition system called the random demodulator, which efficiently samples sparse bandlimited signals at a rate exponentially lower than the Nyquist rate. The system uses a random pseudonoise sequence to smear tones across the spectrum, followed by lowpass filtering and low-rate sampling. This approach allows for the reconstruction of sparse signals using nonlinear methods like convex programming, which exploit the sparsity of the signal. The random demodulator bypasses the need for high-rate analog-to-digital converters (ADCs), enabling the use of robust, low-power components. The system's performance is supported by both empirical results and theoretical analysis, showing that it can reconstruct signals with a sampling rate of O(K log(W/K)) Hz, where K is the number of significant frequencies and W is the bandlimit. The paper also demonstrates that the system is effective for reconstructing communication signals and is robust against noise and quantization errors. Theoretical results confirm that the system can recover signals with a sampling rate of O(K log^6 W), even when the signal is not perfectly sparse. The random demodulator is shown to be effective for a wide range of signals, including those with nonperiodic and compressible characteristics. The system's performance is validated through simulations and experiments, demonstrating its potential for applications in radar, geophysics, and cognitive radio.This paper introduces a new data acquisition system called the random demodulator, which efficiently samples sparse bandlimited signals at a rate exponentially lower than the Nyquist rate. The system uses a random pseudonoise sequence to smear tones across the spectrum, followed by lowpass filtering and low-rate sampling. This approach allows for the reconstruction of sparse signals using nonlinear methods like convex programming, which exploit the sparsity of the signal. The random demodulator bypasses the need for high-rate analog-to-digital converters (ADCs), enabling the use of robust, low-power components. The system's performance is supported by both empirical results and theoretical analysis, showing that it can reconstruct signals with a sampling rate of O(K log(W/K)) Hz, where K is the number of significant frequencies and W is the bandlimit. The paper also demonstrates that the system is effective for reconstructing communication signals and is robust against noise and quantization errors. Theoretical results confirm that the system can recover signals with a sampling rate of O(K log^6 W), even when the signal is not perfectly sparse. The random demodulator is shown to be effective for a wide range of signals, including those with nonperiodic and compressible characteristics. The system's performance is validated through simulations and experiments, demonstrating its potential for applications in radar, geophysics, and cognitive radio.