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A novel weighted ℓ2,1 minimisation algorithm for a high range resolution profile with stepped frequency radar is proposed. The raw stepped frequency pulse train is converted into jointly sparse multiple measurement vectors by using the sliding window technique. Then, by using the Capon cepstrum a support-related weighting matrix is designed to control the priority class of bases in the reconstruction process so that the bases whose indices are inside the support are likely selected. Simulations and experiments on real radar data show that targets can be well separated and spurious peaks can be expelled from the range profile.