Fast algorithm for detecting a small signal in the presence of normally distributed noise

Where a small periodic signal is hidden in normally distributed noise, measurements in the tails of the distribution are much more sensitive to the presence of the signal than those closer to the mean. By calculating a power spectrum only of the extreme values, the signal can be retrieved at a slightly reduced statistical significance but with a substantially reduced calculational requirement. A simple theory is derived of the signal lost by neglecting measurements close to the mean, an algorithm is described to apply this to the autocorrelation method for calculating a power spectrum, and then this algorithm is applied both to simulated and to measured data. It is demonstrated that neglecting 75% of the data points leads to a speed-up in the calculation by a factor of 6-10 at a cost of a 1.3 dB loss in SNR. In a situation where there is a very large number of measurements but limited calculational power, this translates into a gain in the averaged SNR of 3-4 dB. It is shown that this technique is faster than the FFT in calculating power spectra over up to as many as 250 frequency bins. Practically, this means that a 1.3 GHz PC requires less than twice real time to calculate a 50-point power spectrum over a bandwidth of 10 MHz.