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An entirely new approach is presented in this paper for iterative signal reconstruction using partial information and reconstruction of the signals which are embedded in noise. These studies have shown that the new algorithms are able to maintain their performance under extreme noise conditions where the existing algorithms completely fail. Convergence of these algorithms are also extremely fast but the computation time taken is longer than the existing algorithms. More powerful algorithmic structures can be developed by simply convolving the new approaches with higher order cumulants in order to study signal reconstruction problems where the cost function is highly nonlinear.