  To handle time series with complicated oscillatory structure, we propose a
novel time-frequency (TF) analysis tool that fuses the short time Fourier
transform (STFT) and periodic transform (PT). Since many time series oscillate
with time-varying frequency, amplitude and non-sinusoidal oscillatory pattern,
a direct application of PT or STFT might not be suitable. However, we show that
by combining them in a proper way, we obtain a powerful TF analysis tool. We
first combine the Ramanujan sums and $l_1$ penalization to implement the PT. We
call the algorithm Ramanujan PT (RPT). The RPT is of its own interest for other
applications, like analyzing short signal composed of components with integer
periods, but that is not the focus of this paper. Second, the RPT is applied to
modify the STFT and generate a novel TF representation of the complicated time
series that faithfully reflect the instantaneous frequency information of each
oscillatory components. We coin the proposed TF analysis the Ramanujan de-shape
(RDS) and vectorized RDS (vRDS). In addition to showing some preliminary
analysis results on complicated biomedical signals, we provide theoretical
analysis about RPT. Specifically, we show that the RPT is robust to three
commonly encountered noises, including envelop fluctuation, jitter and additive
noise.
