Design of Transmultiplexer Filter Banks Using Ramanujan Sums

Abstract

Ramanujan sums are sequences introduced by the famous mathematician S. Ramanujan in 1918. Ramanujan sum $$c_{q}(n)$$ is the sequence in n with periodicity q. Many arithmetic functions such as Euler’s totient function and Riemann zeta function can be expressed using these sequences. These sums have many attractive properties which make them suitable for signal processing domain. Ramanujan sums are orthogonal and symmetric. Implementation of Ramanujan sum-based transforms is faster compared to discrete Fourier transform, since their computation involves only co-resonant frequencies. These transforms are particularly useful in identifying periodicities in signals. Ramanujan sums are integer valued. In this letter, it is proposed to use Ramanujan sums for the design of transmultiplexer filter banks. Here, the analysis and synthesis filter coefficients are integers. Computation time is reduced considerably by replacing floating point arithmetic with integer coefficients. Rounding errors in hardware implementation are avoided, and perfect reconstruction condition in the transmultiplexer system is ensured by using these integer coefficients.