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The problem of parametric signal restoration given its blurred/nonlinearly distorted version contaminated by additive noise is discussed. It is postulated that feedforward artificial neural networks can be used to find a solution to this problem. The proposed estimator does not require iterative calculations that are normally performed using numerical methods for signal parameter estimation. Thus high speed is the main advantage of this approach. A two-stage neural network-based estimator architecture is considered in which the vector of measurements is projected on the signal subspace and the resulting features form the input to a feedforward neural network. The effect of noise on the estimator performance is analyzed and compared to the least-squares technique. It is shown, for low and moderate noise levels, that the two estimators are similar to each other in terms of their noise performance, provided the neural network approximates the inverse mapping from the measurement space to the parameter space with a negligible error. However, if the neural network is trained on noisy signal observations, the proposed technique is superior to the least-squares estimate (LSE) model fitting. Numerical examples are presented to support the analytical results. Problems for future research are addressed.