  This paper presents analytical results on longitudinal power profile
estimation (PPE) methods, which visualize signal power evolution in optical
fibers at a coherent receiver. The PPE can be formulated as an inverse problem
of the nonlinear Schr\"odinger equation, where the nonlinear coefficient (and
thus signal power) is reconstructed from boundary conditions, i.e., transmitted
and received signals. Two types of PPE methods are reviewed and analyzed,
including correlation-based methods (CMs) and minimum-mean-square-error-based
methods (MMSEs). The analytical expressions for their output power profiles and
spatial resolution are provided, and thus the theoretical performance limits of
the two PPE methods and their differences are clarified. The derived equations
indicate that the estimated power profiles of CMs can be understood as the
convolution of a true power profile and a smoothing function. Consequently, the
spatial resolution and measurement accuracy of CMs are limited, even under
noiseless and distortionless conditions. Closed-form formulas for the spatial
resolution of CMs are shown to be inversely proportional to the product of a
chromatic dispersion coefficient and the square of signal bandwidth. With
MMSEs, such a convolution effect is canceled out and the estimated power
profiles approach a true power profile under a fine spatial step size.

Signal Processing;Optics