The Fourier transform of a time sequence x[n] can be expressed as
and n = integer. If we define a new symbol z as
from  and :
 is the so-called z-transform of time series x[n].
Let a time series x[n] be
where a & b = constants, and x1[n] & x2[n] = time series. The z-transform of x[n] is then
 above shows the linearity of the z-transform.
Now, let's define another time series y[n], a time-shifted series of x[n]:
where no = integer. The z-transform of the time-shifted series y[n] is
 is the so-called time shifting property of the z-transform.  and  will be used in deriving the filter function of the Butterworth filter in Filter Function.
Oppenheim, A.V., & Schafer, R.W. (1989). Discrete-time signal processing. Englewood Cliffs, NJ: Prentice Hall.
© Young-Hoo Kwon, 1998-