New mathematical tools are used to explore and analyze the well-known properties of imaging system for extending the depth of field from spatial and frequency domain, respectively. In the spatial domain, canonical projection of the Wigner distribution function of the associated pupil function to show the insensitivity of the system's point spread function to the defocus-related aberrations is mainly used, and in the frequency domain, it is shown that the optical transfer function of such a system is also insensitive to misfocus through a geometrical description of the so-called Cornu spiral. The involved digital image processing is also briefly discussed and a digital simulated experiment is done to affirm these excellent properties of the wave-front coding imaging system.