This paper considers an observer-based extremum-seeking (ES) scheme acting on a Wiener-Hammerstein (WH) plant. The ES scheme utilises a high-frequency sinusoidal dither in order to achieve fast minimisation of the plant's static nonlinearity. Unlike the prevailing analyses of "fast" ES schemes acting on WH plants, the presented result is semi-global and applicable to plants with fairly general, unknown nonlinearities. Design of the ES scheme requires knowledge of the relative orders of both the input and output dynamics of the plant. Two different tunings of the ES parameters are considered: one which forces the ES dynamics to be fast, and one which is less restrictive. It is shown that approximate knowledge of the phase of the plant's input dynamics is required if the ES scheme is to use the less restrictive tuning. For both tunings, it is shown how the ES scheme may be designed in order to simultaneously achieve arbitrarily fast and accurate minimisation of the static nonlinearity from an arbitrarily large set of initial conditions. Moreover, the result shows that a WH plant with unstable input dynamics can, in fact, be stabilised by the ES scheme. (c) 2012 Elsevier Ltd. All rights reserved.