In this contribution, we present a high-speed multiplex grating spectrometer based on a spectral coding approach that is founded on principles of compressive sensing. The spectrometer employs a single-pixel InGaAs detector to measure the signals encoded by an amplitude spatial light modulator (digital micromirror device, DMD). This approach leads to a speed advantage and multiplex sensitivity advantage atypical for standard dispersive systems. Exploiting the 18.2 kHz pattern rate of the DMD, we demonstrate 4.2 ms acquisition times for full spectra with a bandwidth of 450 nm (5250 cm-1 – 4300 cm-1; 1.9 µm – 2.33 µm). Due to the programmability of the DMD, spectral regions of interest can be chosen freely, thus reducing acquisition times further, down to the sub-millisecond regime. The adjustable resolving power of the system accessed by means of computer simulations is discussed, quantified for different measurement modes, and verified by comparison with a state-of-the-art Fourier-transform infrared spectrometer. We show measurements of characteristic polymer absorption bands in different operation regimes of the spectrometer. The theoretical multiplex advantage of 8 was experimentally verified by comparison of the noise behavior of the spectral coding approach and a standard line-scan approach.