Phasor measurement units (PMUs) are widely used for sub-synchronous oscillation monitoring, yet the effect of windowed discrete Fourier transform (DFT)-based phasor estimation on oscillation observability is not fully characterized. This letter derives the complete complex-valued frequency response of the windowed DFT phasor estimator under both magnitude and phase modulation. The analysis shows that the estimation window introduces both frequency-dependent magnitude attenuation and phase shift to oscillation components, governed by the complex gain. A simple recovery method is also proposed to restore the true oscillation amplitude and phase from PMU data using the analytically known complex gain. The results are validated through time-domain simulations and provide guidance for industry practitioners on interpreting PMU-based oscillation measurements and selecting appropriate window lengths.