In a distributed feedback (DFB) laser with spatial index modulation, an antisymmetric taper of the feedback parameter,k, removes the threshold degeneracy, which is characteristic of uniform structures, and leads to one mode of particularly low threshold. Exact solutions are presented for the special case of an antisymmetric step ofk. An approximate perturbation method is developed which gives simple expressions for the threshold gains, and externalQ's of some tapered distributed feedback structures. The method is tested against two exact solutions, the uniform, and the stepped-kDFB laser. It is shown that the threshold of the stepped-kDFB laser is the lowest of any structure with an antisymmetric taper and a prescribed maximum value of|k|.