Accurate asymptotic analysis for John's test in multichannel signal detection

John's test, which is also known as the locally most invariant test for sphericity of Gaussian variables, is one of the most frequently used methods in multichannel signal detection. The application of John's test requires closed-form and accurate formula to set threshold according to a prescribed false alarm rate. Asymptotic expansion is a powerful method in deriving the threshold expressions of detectors for large samples. However, the existing asymptotic analysis of John's test in the real-valued Gaussian case is not accurate, causing the obtained false alarm rate to deviate from the preset value. This work first corrects a miscalculation in the existing results. Then this accurate approach is extended to the complex-valued case. In this scenario our result is as accurate as the state-of-the-art scheme but enjoys higher computational efficiency.