Under the assumption of receiver noise characteristics, the Gaussian noise models are well-studied in the literature. There are several scenarios where the Gaussian Noise assumption is not valid; common examples include electromagnetic interference channels, underwater acoustic channels, powerline communication channels, and NAND flash memory channels. In this paper, we study non-Gaussian noise models, specifically Laplacian noise (a well-known model for impulsive noise). Diverting from the usual assumption of Gaussian inputs, we take a more practical approach and consider finite alphabet constellations as input and compute achievable rates (by computing mutual information (MI)) for various modulation schemes. It is a well-known result that MI computed for finite constellation does not have any closed form; hence, we use the famous LogSumExp inequality to find the upper bound for MI. Simulation results are compared with the obtained bound, and remarkable accuracy is found at high SNR.