This application relates to methods and apparatus for digital microphones. Disclosed is a digital microphone apparatus (300) for outputting a digital output signal (DATA) at a sample rate defined by a received clock signal (CLK). The apparatus includes a band splitter (302) configured to receive a microphone signal (SMD) indicative of an output of a microphone transducer and split said microphone signal into first signal path (SP1) for frequencies in a first band and a second signal path (SP2) for frequencies in a second band, the frequencies of the second band being higher than the frequencies in the first band. A modulation block (304) is configured to operate on the second signal path to apply a selective gain modulation to signals in the second signal path.