We introduce and experimentally demonstrate a miniaturized integrated spectrometer operating over a broad bandwidth in the short-wavelength infrared (SWIR) spectrum that combines an add-drop ring resonator narrow band filter with a distributed Bragg reflector (DBR) based broadband filter realized in a silicon photonic platform. The contra-directional coupling DBR filter in this design consists of a pair of waveguide sidewall gratings that act as a broadband filter (i.e., 3.9 nm). The re-directed beam is then fed into the ring resonator which functions as a narrowband filter (i.e., 0.121 nm). In this scheme the free spectral range (FSR) limitation of the ring resonator is overcome by using the DBR as a filter to isolate a single ring resonance line. The overall design of the spectrometer is further simplified by simultaneously tuning both components through the thermo-optic effect. Moreover, several ring-grating spectrometer cells with different central wavelengths can be stacked in cascade in order to cover a broader spectrum bandwidth. This can be done by centering each unit cell on a different center wavelength such that the maximum range of one-unit cell corresponds to the minimum range of the next unit cell. This configuration enables high spectral resolution over a large spectral bandwidth and high extinction ratio (ER), making it suitable for a wide variety of applications.