  In this work, we obtain an expression for the total observational frequency
shift of photons emitted by massive geodesic particles circularly orbiting a
black hole in a general spherically symmetric background. Our general relations
are presented in terms of the metric components and their derivatives that
characterize the black hole parameters. As a concrete example of this general
relativistic approach, a special case is studied by applying the formalism to a
nonsingular black hole conformally related to the Schwarzchild solution that
possesses a length scale parameter $l$ and an integer parameter $N$ in addition
to the black hole mass. Besides, we express the nonsingular black hole mass in
terms of the observational redshift/blueshift. Finally, we investigate the
effects of the free parameters of the conformal gravity theory on the
observational frequency shift and compare results with those of the standard
Schwarzschild black hole.
