Page 79 - DSP Integrated Circuits

64 Chapter 3 Digital Signal Processing

The magnitude function of the Fourier transform resembles the well-known
}T
function sin(jc)/r, but it is periodic with period 2n. For no = 0 we get X(eJ° ) = 1

3.5 THE z-TRANSFORM

The Fourier transform does not exist for sequences with nonfinite energy, e.g.,
sinusoidal sequences. In the analog case, the Laplace transform is used for such
analog signals [4, 32, 33]. An extension of the Fourier transform, used similarly to
the Laplace transform for discrete-time and digital signals, is the z-transform.
The z-transform is denned

where R + and 7?_ are the radii of convergence. The z-transform exists, i.e. the sum
is finite, within this region in the z-plane. It is necessary to explicitly denote the
region of convergence in order to uniquely define the inverse z-transform.
0
Obviously, the z-transform and the Fourier transform coincide if z = eJ ^, i.e.,
coincide on the unit circle in the z-plane. This is similar to the analog case, where
the Laplace transform and the Fourier transform coincide on the imaginary axis,
i.e., s =JCD.
An important property of the z-transform is that a shift of a sequence corre-
sponds to a multiplication in the transform domain, i.e., if

then