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A new approach to extracting sea surface wind speed from high-frequency radar Doppler spectra is presented. Based on certain appropriate approximations associated with the Doppler region close to the first-order (Bragg) peaks, the second-order radar cross section equation is differentiated. Once Doppler shifts due to ocean currents are removed from the data, an expression that relates the wind speed to the frequency position of the second-order peak is derived. The method is applied to simulated noisy data as well as to field data obtained from a Seasonde (a product of CODAR Oceans Sensors) in Breezy Point, NY. In the latter case, the retrieved wind speeds are then compared to ground truth data measured by an anemometer from a National Oceanic and Atmospheric Administration weather station located in the vicinity of the illuminated patch of ocean. Subject to certain constraints as detailed in the manuscript, the algorithm shows significant promise.