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The method is presented for blind frequency offset estimation for signals with quadrature amplitude modulation (QAM). The method is based on a circular harmonic expansion of log-likelihood function (LLF). By retaining only the most significant term in this series a radius-weighted version of popular 4th power frequency estimation algorithm is obtained. This approach leads to notable improvement of the estimation quality. Computer simulation was performed with 16-QAM and 32-QAM signals, the results justified the advantages of the suggested method. It was also shown that using fixed (SNR-independent) weighting functions does not lead to any notable performance loss.