Note that the plane and plane are generally related by

In particular, the discrete-time frequency axis
and
continuous-time frequency axis
are related
by

For the mapping
from the plane to the
plane to be invertible, it is necessary that
be zero for all
. If this is true, we say
is
bandlimited to half the sampling rate. As is well known, this
condition is necessary to prevent aliasing when sampling the
continuous-time signal at the rate to produce
,
(see [83, Appendix G]).