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For cognitive radio networks, efficient and robust spectrum sensing is a crucial enabling step for dynamic spectrum access. Cognitive radios need to not only rapidly identify spectrum opportunities over very wide bandwidth, but also make reliable decisions in noise-uncertain environments. Cyclic spectrum sensing techniques work well under noise uncertainty, but require high-rate sampling which is very costly in the wideband regime. This paper develops robust and compressive wideband spectrum sensing techniques by exploiting the unique sparsity property of the two-dimensional cyclic spectra of communications signals. To do so, a new compressed sensing framework is proposed for extracting useful second-order statistics of wideband random signals from digital samples taken at sub-Nyquist rates. The time-varying cross-correlation functions of these compressive samples are formulated to reveal the cyclic spectrum, which is then used to simultaneously detect multiple signal sources over the entire wide band. Because the proposed wideband cyclic spectrum estimator utilizes all the cross-correlation terms of compressive samples to extract second-order statistics, it is also able to recover the power spectra of stationary signals as a special case, permitting lossless rate compression even for non-sparse signals. Simulation results demonstrate the robustness of the proposed spectrum sensing algorithms against both sampling rate reduction and noise uncertainty in wireless networks.