A half-Bessel beam (HBB) is one of nonparaxial and nondiffracting accelerating beams that follows a circular trajectory. Since the ideal HBB is a beam of infinite energy, it is impossible to generate an ideal HBB. Therefore, truncation is necessary to make the beam be square integrable. There exist two ways of such truncation. One is simply cutting off some portion of the beam, and the other is modulating the angular spectrum of the beam. For the latter method, Fourier transformation optics based on a lensed system can be applied. In this study, we suggest and numerically evaluate several methods to achieve finite HBBs using conventional Fourier-optic 2-f system. To this objective, we first derive an angular spectrum representation of the ideal HBB. From this, we show that the obtained spectrum has poles in Fourier domain. After that, several forms of square integrable finite HBBs, where the poles are eliminated, are suggested. And then, the characteristics of the proposed finite HBBs in terms of diffraction and acceleration are presented.