Ramanujan Sums and Signal Processing: An Overview

Abstract

The present chapter discusses Ramanujan Sums and its various signal processing applications. VLSI architectures to calculate Ramanujan Sums and DFT using it are also presented here. This chapter clearly articulates the usage and importance of Ramanujan Sums in a number of signal processing aspects. Ramanujan Sums can be calculated in an exact quantization error-free manner. As it works on integers, it is very suitable to realize hardware which consumes less machine cycles and also requires less hardware resources. All these properties may put Ramanujan Sums as a necessary technique for VLSI signal processing in near future.

Appendix

• 1. σ(n) {Sum of divisors}

  The function σ(n) is the sum of positive divisors of n, i.e., $\sigma \left( n \right) = \sum d \;{\text{if}}\; d|n$.

• 2. (n) {Euler’s totient function}

  The function (n) is defined as the number of positive integers which are less than and coprime with n. For example, (6) = 2 since {1, 5} are the only two positive integers which are less than and coprime with 6.

  $$\begin{aligned} & \psi \left( n \right) = n\mathop \prod \limits_{i} \left( {1 - \frac{1}{{n_{i} }}} \right)\;{\text{Such that}}\;n = \mathop \prod \limits_{i} n_{i}^{{\alpha_{i} }} \;{\text{where }}(\alpha_{i} )\;{\text{is prime}}. \\ & \psi_{2} \left( n \right) = n^{2} \mathop \prod \limits_{i} \left( {1 - \frac{1}{{n_{i}^{2} }}} \right) \\ \end{aligned}$$

• 3. µ(n) {Mobius function}

  Mobius function µ(n) is a number-theoretic function and is defined as

  $$\begin{array}{*{20}l} {\mu \left( n \right)} \hfill & { = 1} \hfill & {i{\text{f}}\;n = 1} \hfill \\ {} \hfill & { = \left( { - 1} \right)^{k} } \hfill & {{\text{if}}\;n = p_{1} p_{2} \ldots .p_{k} } \hfill \\ {} \hfill & { = 0} \hfill & {\text{otherwise}} \hfill \\ \end{array}$$

  Here, p_i are distinct prime numbers.

• 4. ˄(n) {von Mangoldt function}

  $$\begin{array}{*{20}c} { \wedge \left( n \right)} & = & {\{ \ln p\quad {\text{if}}\;n = p^{\beta } ,\;p\;\;{\text{is prime}}} \\ {} & = & {\text{Otherwise}} \\ \end{array}$$

• 5. C(n)

  The function C(n) is defined as

  $$\begin{array}{*{20}l} {C\left( n \right)} \hfill & { = 2C_{2} \mathop \prod \limits_{p|n} \frac{p - 1}{p - 2},} \hfill & {{\text{if}}\;n\,{\text{is odd}}} \hfill \\ {} \hfill & { = 0,} \hfill & {{\text{if}}\;n\,{\text{is even}}} \hfill \\ \end{array}$$

  Here, p > 2 is a prime and $p|n$ implies p divides n. The value of twin prime constant, $C_{2}$ is 0.660