Wavefront coding technique is a powerful technique which overcomes the defects of traditional way to extend depth of field. By inserting a phase mask into the traditional incoherent imaging system, wavefront coding technique does not reduce the resolution and the light gathering power of the optical system but enlarges the depth of field of incoherent imaging system. Although several kinds of phase masks have been reported, cubic phase mask is still of a classical type which has been investigated widely both in spatial and frequency domain. Since the phase profiles of phase masks adopted in classical wavefront coding systems are predefined with specific optical systems, the extension of depth of field is not tunable. Tunable wavefront coding systems are introduced by using a pair of detachable phase masks, which is possible to control the depth of field and bandwidth of system by changing the position of each component with respect to the pupil center. Ojeda-Castaeda [Ojeda-Castaeda J, Rodrguez M, Naranjo R 2010 Proceedings of Progress in Electronics Research Symposium, Cambridge, July 58, 2010 p531] proposed to use a pair of cosine phase masks to make defocus sensitivity tunable. Zhao [Zhao H, Wei J X 2014 Opt. Commun. 326 35] investigated an improved version of Ojeda- Castaneda's design in frequency domain and found that the proposed system realized tunable bandwidth. The present study, based on the work of Zhao, analyzes the tunable characteristics of a pair of simple modified detachable cubic phase masks in spatial domain and frequency domain. Firstly, the ray aberration theory is adopted to give mathematical analyses and ray aberration maps of the proposed tunable phase mask. Based on the mathematical derivations, the size of point spread function (PSF) of system can be changed not only by profile of each cubic mask but also by the each mask displacement relative to pupil center. Secondly, a mathematical PSF based on the stationary phase method is derived in spatial domain. Simulations indicate that the positions of PSF translate in the image plane with the displacements of phase mask profile and the position of each component with respect to the pupil center. By analyzing the oscillations of PSF, the effective bandwidth is obtained. Through the expression, we can conclude that the effective bandwidth can be changed by the position, mask profile of each component and defocus. Only when the addition of two mask profiles is large enough, can the effective bandwidth be simplified without adding the influence of defocus. In addition, though the approximate expression of magnitude transfer of function (MTF) has been given by adopting stationary phase method in the appendix of previous work, it cannot give an intuitive grasp of the effective bandwidth in MTF map. Unlike the MTF expression derived before, the exact optical transfer function (OTF) expression is derived by adopting Fresnel integral in frequency domain. Exact MTF and phase transfer function (PTF) can be derived from OTF. Based on the exact MTF expression, simulations give an intuitive effective bandwidth in MTF map. Simulations also show the nonlinear property of PTF. The effective bandwidth and MTF can be changed by different phase mask profiles and positions, which indicate that the effective bandwidth and defocus sensitivity can be tuned. Analyses are conducted both in spatial domain and in frequency domain to verify the tunable property of the proposed phase mask, which provides theoretical foundation for tunable wavefront coding system design.