A new restoration filter is proposed for the restoration of signals distorted by a random impulse response function and additive measurement noise. The filter is derived from the theory of constrained total least squares, an extension of total least squares to the case where linear constraints with additive random components are incorporated into the measurement model. This enables the incorporation of prior information about the original signal, expressed in the form of linear constraints such as smoothness and region of support. Furthermore, the filter does not require knowledge of the second-order statistics of the random impulse response function and noise. Simulation results are provided to demonstrate the effectiveness of the proposed filter.