Received data representing speech is stored, and a trigger detection block detects a presence of data representing a trigger phrase in the received data. In response, a first part of the stored data representing at least a part of the trigger phrase is supplied to an adaptive speech enhancement block, which is trained on the first part of the stored data to derive adapted parameters for the speech enhancement block. A second part of the stored data, overlapping with the first part of the stored data, is supplied to the adaptive speech enhancement block operating with said adapted parameters, to form enhanced stored data. A second trigger phrase detection block detects the presence of data representing the trigger phrase in the enhanced stored data. In response, enhanced speech data are output from the speech enhancement block for further processing, such as speech recognition. Detecting the presence of data representing the trigger phrase in the received data is carried out by means of a first trigger phrase detection block, detecting the presence of data representing the trigger phrase in the enhanced stored data is carried out by means of a second trigger phrase detection block, and the second trigger phrase detection block operates with different, typically more rigorous, detection criteria from the first trigger phrase detection block.