Blind NLLS carrier frequency-offset estimation for QAM, PSK, and PAM modulations: performance at low SNR

We address the problem of blind carrier frequency-offset (CFO) estimation in quadrature amplitude modulation, phase-shift keying, and pulse amplitude modulation communications systems. We study the performance of a standard CFO estimate, which consists of first raising the received signal to the Mth power, where M is an integer depending on the type and size of the symbol constellation, and then applying the nonlinear least squares (NLLS) estimation approach. At low signal-to noise ratio (SNR), the NLLS method fails to provide an accurate CFO estimate because of the presence of outliers. In this letter, we derive an approximate closed-form expression for the outlier probability. This enables us to predict the mean-square error (MSE) on CFO estimation for all SNR values. For a given SNR, the new results also give insight into the minimum number of samples required in the CFO estimation procedure, in order to ensure that the MSE on estimation is not significantly affected by the outliers