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The Fourier transform of a time sequence x[n] can be expressed as
where
and n = integer. If we define a new symbol z as
from [1] and [3]:
[4] 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
where
[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]:
where no = integer. The z-transform of the time-shifted series y[n] is
[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.
References & Related Literature Oppenheim, A.V., & Schafer, R.W. (1989). Discrete-time signal processing. Englewood Cliffs, NJ: Prentice Hall.
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© Young-Hoo Kwon, 1998- |
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