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By employing the self-similarity properties of the power-line communications (PLC) orthogonal frequency-division multiplexing (OFDM) signals, this paper presents a novel PLC OFDM signal reconstruction solution based on the fractals. This new signal reconstruction solution consists of two methods that try to solve two main difficulties faced by the fractals interpolation. The first method is based on the affine transform, trying to determine the vertical scaling factors by a set of the middle points in each affine mapping, and solving the difficulty of determining the vertical scaling factors, which is one of the most difficult challenges faced by the fractal interpolation. The second method tries to solve the problem where the ordinary fractals interpolation cannot obtain the value of any arbitrary point directly, a very novel algorithm called the pointed point algorithm, based on iterated function systems, is proposed to reconstruct the PLC OFDM signals with the expected interpolation error. Numerical experiments have shown that great accuracy in the reconstruction of the OFDM signals has been found, leading to a significant improvement over other signal reconstruction methods.