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Recently, a large class of nonlinear systems which possess soliton solutions has been discovered for which exact analytic solutions can be found. Solitons are eigenfunctions of these systems which satisfy a form of superposition and display rich signal dynamics as they interact. In this paper, we view solitons as signals and consider exploiting these systems as specialized signal processors which are naturally suited to a number of complex signal processing tasks. New circuit models are presented for two soliton systems, the Toda lattice and the discrete-KdV equations. These analog circuits can generate and process soliton signals and can be used as multiplexers and demultiplexers in a number of potential soliton-based wireless communication applications discussed in [Singer et al.]. A hardware implementation of the Toda lattice circuit is presented, along with a detailed analysis of the dynamics of the system in the presence of additive Gaussian noise. This circuit model appears to be the first such circuit sufficiently accurate to demonstrate true overtaking soliton collisions with a small number of nodes. The discrete-KdV equation, which was largely ignored for having no prior electrical or mechanical analog, provides a convenient means for processing discrete-time soliton signals.