Objective: The ability to efficiently match the frequency of the brain's response to repetitive visual stimuli in real time is the basis for reliable SSVEP-based Brain-Computer-Interfacing (BCI). Approach: The detection of different stimuli is posed as a composite hypothesis test, where SSVEPs are assumed to admit a sparse representation in a Ramanujan Periodicity Transform (RPT) dictionary. For the binary case, we develop and analyze the performance of an RPT detector based on a derived generalized likelihood ratio test. Our approach is extended to multi-hypothesis multi-electrode settings, where we capture the spatial correlation between the electrodes using prestimulus data. We also introduce a new metric for evaluating SSVEP detection schemes based on their achievable efficiency and discrimination rate tradeoff for given system resources. Results:We obtain exact distributions of the test statistic in terms of confluent hypergeometric functions. Results based on extensive simulations with both synthesized and real data indicate that the RPT detector substantially outperforms spectral-based methods. Its performance also surpasses the state-of-the-art Canonical Correlation Analysis (CCA) methods with respect to accuracy and sample complexity in short data lengths regimes crucial for real-time applications. The proposed approach is asymptotically optimal as it closes the gap to a perfect measurement bound as the data length increases. In contrast to existing supervised methods which are highly data-dependent, the RPT detector only uses pre-stimulus data to estimate the per-subject spatial correlation, thereby dispensing with considerable overhead associated with data collection for a large number of subjects and stimuli. Significance: Our work advances the theory and practice of emerging real-time BCI and affords a new framework for comparing SSVEP detection schemes across a wider spectrum of operating regimes.Index Terms-Ramanujan periodicity transform, Nested periodic matrices, Steady-state visual evoked potentials, Brain computer interface, error exponent-discrimination rate tradeoff.

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