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The third generation partnership project (3GPP) long term evolution (LTE) radio standard has adopted a special form of orthogonal frequency division multiplexing (OFDM) method for the uplink of multiple access networks. This method which is called single carrier frequency division multiple access (SC-FDMA) applies a precoding to each user data set in each OFDM symbol to control its peak-to-average power ratio (PAPR). Some recent works have proposed minimization of the variance of the instantaneous power of the output signal of the transmitter as a means of reducing PAPR to near its minimum value. However, the studies so far have been based on intuitions and observation of some numerical results. The goal of this paper is to develop a mathematical procedure to further establish the significance of this method. We formulate the problem in the form of a method of Lagrange multipliers and analyze its second-order conditions to confirm analytically that the optimal window is indeed a strict local minimizer. We also analyze and compensate the noise enhancement penalty of the optimal window. Our analysis also leads us to find new window functions that further reduce the PAPR and improve the BER performance.