Complex Coprime Frequency Sum Based Signal Representation for Period Estimation

In signal processing, extracting information from a signal often requires transforming it into another domain using a suitable representation. Current representations for period estimation include the Discrete Fourier Transform (DFT), the Ramanujan Periodic Transform (RPT), and the Orthogonal Complex Conjugate Periodic Transform (OCCPT), which use Complex Exponential Sequences (CESs), Ramanujan Sums (RSs), and Complex Conjugate Pair Sums (CCPSs) as their bases, respectively. This paper aims to introduce a novel signal representation for efficient period estimation. We present a real-valued trigonometric sum called the Complex Coprime Frequency Sum (CCFS) and derive its key properties. We show that CCFS and its circular downshifts form a new basis for the Ramanujan subspace. Using this basis, we propose the Complex Coprime Frequency Transform (CCFT) for efficient period extraction. Our numerical results demonstrate that CCFT outperforms DFT and OCCPT, particularly in noisy environments. Furthermore, we observe that CCFT coefficients more reliably capture the strength of a particular period compared to RPT, making it a more effective method for detecting periods in a signal.