By use of the backward shift invariant subspace，a necessary and sufficient condition under which a product of a circular analytic signal and a conjugate circular analytic signal is still circular analytic is presented. Particularly

if the Ztransform of the circular analytic signal is holomorphic across the unit circle

the conjugate circular analytic signal is a rational function whose poles are the zeros of the Ztransform of the circular analytic signal. As applications of the above results

the conditions under which two circular analytic signals with length n Fourier series have the same amplitude are discussed.