Least squares solution for error correction on the real field using quantized DFT codes

Least squares (LS) methods are frequently used in many statistical problems, including the solution of overdetermined linear systems. We analyze the effect of using the LS solution in the decoding of quantized discrete Fourier transform (DFT) codes. We show how the LS solution can improve detection, localization, and calculation of errors in the real field, and come close to the quantization error level under the mean squared error (MSE) fidelity criterion. Assuming perfect localization, the LS estimation substantially decreases the MSE between the transmitted and reconstructed sequences, regardless of the magnitude of channel error to quantization noise ratio. Furthermore, when quantization noise is comparable to or larger than channel errors, where error localization is usually very poor, the LS solution still brings down the estimation error, resulting a reconstruction error at the level of quantization error.