The bilateral z transform of the discrete-time signal is defined to be
Recall (§4.1) that the mathematical representation of a discrete-time signal maps each integer to a complex number ( ) or real number ( ). The z transform of , on the other hand, , maps every complex number to a new complex number . On a higher level, the z transform, viewed as a linear operator, maps an entire signal to its z transform . We think of this as a ``function to function'' mapping. We may say is the z transform of by writing
or, using operator notation,
which can be abbreviated as
One also sees the convenient but possibly misleading notation , in which and must be understood as standing for the entire domains and , as opposed to denoting particular fixed values.
The z transform of a signal can be regarded as a polynomial in , with coefficients given by the signal samples. For example, the signal
has the z transform .