
The Fourier transform of a time sequence x[n] can be expressed as , [1] where , [2] and n = integer. If we define a new symbol z as , [3] from [1] and [3]: . [4] [4] is the socalled ztransform of time series x[n]. Let a time series x[n] be [5] where a & b = constants, and x_{1}[n] & x_{2}[n] = time series. The ztransform of x[n] is then [6] where [7] [6] above shows the linearity of the ztransform. Now, let's define another time series y[n], a timeshifted series of x[n]: , [8] where n_{o} = integer. The ztransform of the timeshifted series y[n] is . [9] [9] is the socalled time shifting property of the ztransform. [6] and [9] will be used in deriving the filter function of the Butterworth filter in Filter Function. References & Related Literature Oppenheim, A.V., & Schafer, R.W. (1989). Discretetime signal processing. Englewood Cliffs, NJ: Prentice Hall.

© YoungHoo Kwon, 1998 