We present the data processing pipeline to generate calibrated data products from the Spectral and Photometric Imaging Receiver (SPIRE) imaging Fourier Transform Spectrometer on the Herschel Space Observatory. The pipeline processes telemetry from SPIRE observations and produces calibrated spectra for all resolution modes. The spectrometer pipeline shares some elements with the SPIRE photometer pipeline, including the conversion of telemetry packets into data timelines and calculation of bolometer voltages. We present the following fundamental processing steps unique to the spectrometer: temporal and spatial interpolation of the scan mechanism and detector data to create interferograms; Fourier transformation; apodization; and creation of a data cube. We also describe the corrections for various instrumental effects including first- and second-level glitch identification and removal, correction of the effects due to emission from the Herschel telescope and from within the spectrometer instrument, interferogram baseline correction, temporal and spatial phase correction, non-linear response of the bolometers, and variation of instrument performance across the focal plane arrays. Astronomical calibration is based on combinations of observations of standard astronomical sources and regions of space known to contain minimal emission.
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