  In this letter, we study a few properties of Complex Conjugate Pair Sums
(CCPSs) and Complex Conjugate Subspaces (CCSs). Initially, we consider an LTI
system whose impulse response is one period data of CCPS. For a given input
x(n), we prove that the output of this system is equivalent to computing the
first order derivative of x(n). Further, with some constraints on the impulse
response, the system output is also equivalent to the second order derivative.
With this, we show that a fine edge detection in an image can be achieved using
CCPSs as impulse response over Ramanujan Sums (RSs). Later computation of
projection for CCS is studied. Here the projection matrix has a circulant
structure, which makes the computation of projections easier. Finally, we prove
that CCS is shift-invariant and closed under the operation of circular
cross-correlation.
