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unifying approach to the derivation of the class of proportionate normalised least mean square (PNLMS) algorithms is provided. This is an important class of algorithms where the two most used algorithms are introduced empirically. It is shown that it is possible to derive PNLMS algorithms as a result of an optimisation procedure. This is achieved in a rigorous way, starting from the standard LMS through to the PNLMS with the "sparsification" factor in both the numerator and denominator of the weight update. The proposed approach is generic and also applies to other LMS types of adaptive algorithms. Simulations on benchmark sparse impulse responses support the approach.