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We report on the implementation of an algorithm-based spectrometer capable of reconstructing the spectral shape of materials in the mid-wave infrared (MWIR) and long-wave infrared (LWIR) wavelengths using only experimental photocurrent measurements from quantum dot infrared photodetectors (QDIPs). The theory and implementation of the algorithm will be described, followed by an investigation into this algorithmic spectrometer's performance. Compared to the QDIPs utilized in an earlier implementation, the ones used here have highly varying spectral shapes and four spectral peaks across the MWIR and LWIR wavelengths. It has been found that the spectrometer is capable of reconstructing broad spectral features of a range of bandpass infrared filters between wavelengths of 4 and 12 as well as identifying absorption features as narrow as 0.3 in the IR spectrum of a polyethylene sheet.