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A new simple formulation for the choice of the optimum convergence factor in adaptive filtering using gradient techniques is given. This leads to several optimum adaptive filtering algorithms each of which is optimum under the conditions it was derived. Several of these algorithms have been tested successfully proving their optimality and yielding faster and more accurate adaptation compared with the existing conventional algorithm CA that uses fixed or imperical . Two algorithms are derived and examined here. The first, the Homogeneous Algorithm HA, results in a time varying which is the same for each filter but is updated at each iteration to yield optimum performance. The second, the Individual Adaptation Algorithm (IAA), has a time varying which is chosen suitably for each coefficient at each iteration. The performance of the HA and IAA always outperformed the CA. Computer simulations and experimental results are given which are in agreement with theory.