When a

filter
is a linear filter (but not necessarily
time-invariant), and its input is a complex

signal
,
then, by linearity,

This means every linear filter maps complex signals to complex signals
in a manner equivalent to applying the filter separately to the real
and imaginary parts (which are each real). In other words, there is
no ``interaction'' between the real and imaginary parts of a complex
input signal when passed through a linear filter. If the filter is
real, then filtering of complex signals can be carried out by simply
performing

real filtering on the real and imaginary parts separately
(thereby avoiding complex arithmetic).
Appendix

H presents a linear-algebraic view of linear filters that can
be useful in certain applications.

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