Frequency offset estimation apparatus for estimating the offset from a predetermined centre frequency of an input signal carrying a plurality of frequency shifted symbols, the apparatus comprising: a demodulator for demodulating the input signal to estimate the symbols; a first filter for forming a first estimate of the offset by determining the average of a first predetermined number of the last maxima and minima of the instantaneous frequency difference between the input signal and a signal at the centre frequency; a second filter for forming a second estimate of the offset by determining the average of the values of the instantaneous frequency difference between the input signal at the centre frequency associated with the estimation by the demodulator of those of the symbols having the greatest positive and negative frequency shifts; and selector for selecting the first estimate or the second estimate as an output estimate of the frequency error.