Pulse propagation in a transmission line comprising alternately concatenated fibers with O1-accumulated perturbations of self-phase modulation and anomalous dispersion is studied. In such a line, a pulse compression process is inevitable. In certain conditions, we have found that a compressed pulse can be rescaled to the initial pulse, and hence that there exists a stationary rescaled pulse (SRP), which is distinct from other nonlinear stationary pulses such as the guiding-center, dispersion-managed, or split-step solitons. The properties of SRP are studied. We apply the rescaling to the transmission line rather than to the pulse, and we experimentally observe a SRP in a periodically rescaled transmission line.