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Compact spectrometers are one of important building blocks needed in implementing optical lab-on-a-chip sensing devices. Photonic crystal superprism demultiplexers due to their unique and strong dispersion properties are proper candidates for realizing such integrated spectrometers. We use a recently demonstrated device idea, i.e., focusing superprism photonic crystal demultiplexers, to implement an integrated compact spectrometer that can be used in sensing applications. In this device, a diffraction compensation scheme is used to cancel the broadening of the optical beam throughout the structure, which significantly improves the resolution of the device (compared to the conventional superprism demultiplexers). In addition, by designing the photonic crystal in negative refraction regime, stray and unwanted portions of the input signal are filtered out inside the structure and this improves the output signal-to-noise ratio. A least-square algorithm is used to extract the spectrum of the signal from measured output power at different channels.