  The Fourier spectrum at a fractional period is often examined when extracting
features from biological sequences and time series. It reflects the inner
information structure of the sequences. A fractional period is not uncommon in
time series. A typical example is the 3.6 period in protein sequences, which
determines the $\alpha$-helix secondary structure. Computing the spectrum of a
fractional period offers a high-resolution insight into a time series. It has
thus become an important approach in genomic analysis. However, computing
Fourier spectra of fractional periods by the traditional Fourier transform is
computationally expensive. In this paper, we present a novel, fast algorithm
for directly computing the fractional period spectrum (FPS) of time series. The
algorithm is based on the periodic distribution of signal strength at periodic
positions of the time series. We provide theoretical analysis, deduction, and
special techniques for reducing the computational costs of the algorithm. The
analysis of the computational complexity of the algorithm shows that the
algorithm is much faster than traditional Fourier transform. Our algorithm can
be applied directly in computing fractional periods in time series from a broad
of research fields. The computer programs of the FPS algorithm are available at
https://github.com/cyinbox/FPS.
