A digital speech enhancement system that performs a specific chain of digital signal processing operations upon multi-channel sound pick up, to result in a single, enhanced speech signal. The operations are designed to be computationally less complex yet as a whole yield an enhanced speech signal that produces accurate voice trigger detection and low word error rates by an automatic speech recognizer. The constituent operations or components of the system have been chosen so that the overall system is robust to changing acoustic conditions, and can deliver the enhanced speech signal with low enough latency so that the system can be used online (enabling real-time, voice trigger detection and streaming ASR.) Other embodiments are also described and claimed.