I have used Fourier sampling theory to the apply a time window to correlated passive seismic data. Under a reasonable (I believe) set of assumptions, the initial Fourier transform of the raw data acts to redistribute the energy of late time arrivals toward the origin of the frequency axis, which effectively handles the zero-time problem of passive imaging. The multiplication (by conjugate traces) handles the waveform comparison problem.
In practice, we can avoid many calculations associated with the long time axis of transmission wavefields by exploiting the fact that the fine increments of the Fourier domain contain the information in the late times of the input data. Acknowledging that we are incapable of using the late time correlations within the framework of conventional seismic processing algorithms, this information can (and probably should) be removed after correlation. Conveniently, since the correlation is performed in the frequency domain with point-to-point multiplications, this extra information can be removed immediately upon its initial transform to the Fourier domain, and before any further processing is performed. This holds true for direct migration or manufacture of modeled shot-gathers from a transmission wavefield.