We report our advances in development of on-chip Fourier-transform spectrometers. Specifically, we present applications of subwavelength engineered waveguides in a new type of a compressive-sensing Fourier-transform spatial heterodyne spectrometer chip. The spectrometer is implemented as an array of Mach-Zehnder interferometers (MZIs) integrated on a photonic chip, in a spatial heterodyne configuration [1]. The signal from a set of MZIs comprises an undersampled discrete Fourier interferogram, which is inverted using L1-norm minimization technique to retrieve a sparse input spectrum. To implement this technique we use a subwavelength-engineered Fourier-transform spectrometer on a chip comprising 32 independent MZIs [2]. We successfully demonstrate the retrieval of sparse input signals by collecting data from restricted sets (8 and 14) of MZIs and applying common compressive-sensing reconstruction techniques to this data. We show that this retrieval maintains the full resolution and bandwidth of the original device despite a sampling factor as low as 1/4th of a conventional (non-compressive) design [3].