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 z-Transform Properties of the z-Transform References & Related Literature

z-Transform

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 so-called z-transform of time series x[n].

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Properties of the z-Transform

Let a time series x[n] be

[5]

where a & b = constants, and x1[n] & x2[n] = time series. The z-transform of x[n] is then

[6]

where

[7]

[6] above shows the linearity of the z-transform.

Now, let's define another time series y[n], a time-shifted series of x[n]:

,    [8]

where no = integer. The z-transform of the time-shifted series y[n] is

.    [9]

[9] is the so-called time shifting property of the z-transform. [6] and [9] will be used in deriving the filter function of the Butterworth filter in Filter Function.

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References & Related Literature

Oppenheim, A.V., & Schafer, R.W. (1989). Discrete-time signal processing. Englewood Cliffs, NJ: Prentice Hall.

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