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A lighting-based multispectral imaging system using an RGB camera and a projector is one of the most practical and low-cost systems to acquire multispectral observations for estimating the scene's spectral reflectance information. However, existing projector-based systems assume that the spectral power distribution (SPD) of each projector primary is known, which requires additional equipment such as a spectrometer to measure the SPD. In this paper, we present a method for jointly estimating the spectral reflectance and the SPD of each projector primary. In addition to adopting a common spectral reflectance basis model, we model the projector's SPD by a low-dimensional model using basis functions obtained by a newly collected projector's SPD database. Then, the spectral reflectances and the projector's SPDs are alternatively estimated based on the basis models. We experimentally show the performance of our joint estimation using a different number of projected illuminations and investigate the potential of the spectral reflectance estimation using a projector with unknown SPD.