Dual-modulated spectroscopy is one of the most powerful methods in the measurement of modulation spectroscopy. Here we develop a tandem lock-in amplifier (LIA) based on digital signal processor to implement a novel algorithm of tandem demodulation. The theoretical analysis of demodulation algorithm is presented, and the implementation of this tandem LIA is described in detail. Compared to the traditional demodulating way with two LIAs in cascade, this tandem LIA eliminates the extra quantization error of redundant analog-to-digital and digital-to-analog conversions and removes the limitation to the time constant in the commercial LIA, hence lowers the requirement of frequency ratio in dual-modulated spectroscopy. The applications are given as examples in the photoreflectance (PR) measurements of GaAs (100) thin film and GaSb bulk material, respectively, at the different optical energy regions. The experimental results indicate that this tandem is well capable of PR spectra measurement with good PR lineshapes and reasonable signal noise ratio. A brief comparison of GaAs PR results between tandem LIA and two LIAs is made to prove the efficiency and advantages of the tandem LIA.