The spectral properties of the Rayleigh backscattered traces measured by a phase-sensitive optical time-domain reflectometer (phi OTDR) with direct detection are theoretically and experimentally analyzed. The spectrum of the measured phi OTDR signal is found to be strictly dependent on the spectral shape of the probing optical pulse. Furthermore, the visibility, spatial resolution, fading rate, and correlation spectrum of the traces are analyzed using different detection bandwidths. Results point out that the quality of phi OTDR traces and target spatial resolution are secured only if the electrical bandwidth of the photodetector is broad enough to cover at least 80% of the total power contained in the electrical spectral density function of the measured trace. This means that in the case of using direct detection of the Rayleigh backscattered light induced by rectangular-shaped optical pulses, the minimum bandwidth required for a proper detection of the traces is equal to the reciprocal of the pulse temporal width (which is larger than the pulse spectral width). Although the theoretical analysis and numerical simulations are here experimentally validated for rectangular and sinc-shaped optical pulses, the results and methodology presented in this article can be applied to optimize the direct-detection bandwidth of phi OTDR sensors using any optical pulse shape.