Decay Response

If a filter is in steady state and we switch off the input signal, we see its decay response. This response is identical (but for a time shift) to the filter's response to initial conditions. In other words, when the input signal is switched off (becomes zero), the future output signal is computed entirely from the filter's internal state, because the input signal remains zero.

In general, the complete response of a linear, time-invariant filter is given by the superposition of its

1. zero-state response and
2. initial-condition response.

``Zero-state response'' simply means the response of the filter to an input signal when the initial state of the filter (all its memory cells) are zeroed to begin with. The initial-condition response is of course the response of the filter to its own initial state, with the input signal being zero. This clean superposition of the zero-state and initial-condition responses only holds in general for linear, time-invariant filters. In §E.3, this superposition will be considered for state-space filter representations.