I looked into this recently. Voice triggering is typically done on DSP and the digital processing power is typically already small compared to the microphone (w

I looked into this recently. Voice triggering is typically done on DSP and the digital processing power is typically already small compared to the microphone (which is not much itself). For power optimized applications the DSP implementation is power optimized (low speed, low leakage) and processing is done in two steps: a coarse, basic recognition with some false alarm probability, with an accurate second step. For a plugged device there's no need to super optimize, the DSP part should be in the single digit mW. A fixed trigger makes the system simpler however: no configuration to manage, no risk of having the kids randomly changing the trigger for something funny, etc.