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In this paper, we propose a joint carrier frequency offset (CFO) and channel estimation scheme based on the well known variable projection (VP) method for the uplink channel of orthogonal frequency division multiplexing (OFDM) systems. The CFOs and channel coefficients are estimated by minimizing the least-squares (LS) cost function through a numerical optimization technique. In each iteration, the CFOs and channel coefficients are updated separately, which distinguishes the VP method from an existing method in which the CFOs and channel coefficients are treated together as a whole. We demonstrate through several numerical examples the superiority of the proposed scheme in terms of convergence speed, computational complexity and estimation performance with regard to CFO estimation. However, the proposed scheme does not provide reliable estimation for the channel coefficients. We will investigate this problem further in our future research work.