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Complete Response

The complete response of a linear system consists of the superposition of (1) its response to the input signal $ \underline{u}(n)$ and (2) its response to initial conditions $ {\underline{x}}(0)$:

$\displaystyle \underline{y}(n) = \underline{y}_u(n) + \underline{y}_x(n), $

where $ \underline{y}_u(n)$ was defined in Eq.$ \,$(G.3) and $ \underline{y}_x(n)$ was defined in Eq.$ \,$(G.4) above.
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Transfer Function of a State Space Filter
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Response from Initial Conditions