  In this paper, we introduce two types of real-valued sums known as Complex
Conjugate Pair Sums (CCPSs) denoted as CCPS$^{(1)}$ and CCPS$^{(2)}$, and
discuss a few of their properties. Using each type of CCPSs and their circular
shifts, we construct two non-orthogonal Nested Periodic Matrices (NPMs). As
NPMs are non-singular, this introduces two non-orthogonal transforms known as
Complex Conjugate Periodic Transforms (CCPTs) denoted as CCPT$^{(1)}$ and
CCPT$^{(2)}$. We propose another NPM, which uses both types of CCPSs such that
its columns are mutually orthogonal, this transform is known as Orthogonal CCPT
(OCCPT). After a brief study of a few OCCPT properties like periodicity,
circular shift, etc., we present two different interpretations of it. Further,
we propose a Decimation-In-Time (DIT) based fast computation algorithm for
OCCPT (termed as FOCCPT), whenever the length of the signal is equal to $2^v,\
v{\in} \mathbb{N}$. The proposed sums and transforms are inspired by Ramanujan
sums and Ramanujan Period Transform (RPT). Finally, we show that the period
(both divisor and non-divisor) and frequency information of a signal can be
estimated using the proposed transforms with a significant reduction in the
computational complexity over Discrete Fourier Transform (DFT).
