Dual sideband (2SB) receivers are well suited for the spectral observation of complex astronomical signals over a wide frequency range. They are extensively used in radio astronomy, their main advantages being to avoid spectral confusion and to diminish effective system temperature by a factor 2 with respect to double sideband (DSB) receivers. Using available millimeter-wave analog technology, wideband 2SB receivers generally obtain sideband rejection ratios (SRR) of 10-15 dB, insufficient for a number of astronomical applications. We report here the design and implementation of an FPGA-based sideband separating FFT spectrometer. A 4 GHz analog front end was built to test the design and measure sideband rejection. The setup uses a 2SB front end architecture, except that the mixer outputs are directly digitized before the IF hybrid, using two 8 bit ADCs sampling at 1 GSPS. The IF hybrid is implemented on the FPGA together with a set of calibration vectors that, properly chosen, compensate for the analog front end amplitude and phase imbalances. The calibrated receiver exhibits a sideband rejection ratio in excess of 40 dB for the entire 2 GHz RF bandwidth.