This work reports a modified flame-brush technique to fabricate fiber tapers with arbitrary waist profiles. The flame-brush approach is used to produce small step reductions in the fiber diameter, or step-tapers, with a constant speed flame brush sweep, while the fiber is uniformly stretched. Arbitrary waist profiles in tapers are fabricated by approximating the taper diameter function to any monotonic function of the fiber length while combining a superposition of step-tapers. This method to produce the arbitrary profiles is described and a set of tapers with dissimilar transition regions are fabricated for its validation.