Global Testing Against Sparse Alternatives in Time-Frequency Analysis
Tony Cai, Yonina Eldar, and Xiaodong Li
In this paper, an over-sampled periodogram higher criticism (OPHC) test is proposed for the global detection of sparse periodic effects in a complex-valued time series. An explicit minimax detection boundary is established between the number and magnitude of the complex sinusoids hidden in the series. The OPHC test is shown to be asymptotically powerful in the detectable region. Numerical simulations illustrate and verify the effectiveness of the proposed test. Furthermore, the periodogram over-sampled by O(log N) is proven universally optimal in global testing for periodicities under a mild minimum separation condition. Connections to the problem of detecting a stream of pulses from frequency measurements in signal processing is also discussed.