Digital Signal Proces l
357
Z-Transform
3.7.2 Transformation Erom Laplace Transformn to
impulse sampled version of
Let x() be a continuous tìme signal, and x,() be its
From equation (3.62) we get,
L(80)
N,()-N(n) 8(t-n)
On taking Laplace transform of the above
equation we get,
discrete time
I
L(x()) X(s) then
by time
shit ing propery
L{x( a)}e
X(s)
n
6)
n
where X() is Laplace transform ofx, ().
Let us take atransformation, e z .
On substituting, eT =z, in equation (3.63)we get,
x(nT) z a i o q eudiliisbslbe
X,()
n
..36)
The -transform of x(nT), using the definition ofZ-transform is given by,
X(2) =
...3653)
n
On comparing equations (3.64) and (3.65) we can say that, if a discrete time signal x(nT) is asampled
version of x(),then 2-ransform of the discrete tìme signal can be obtained from Laplace transform
sampled version ofx(), by choosing the transformation, e =z This transformation is also called impuabe
invariant transformation.
In this mapping, the left halfportion ofeach strip in s-plane maps into the interior of the unit
circle
in z-plane, right halfportion ofeach strip in s-plane maps into the exterior of the unit circle in z-plane and the
imaginary ais of eachstrip in s-plane maps into the unit circle in z-plane as shown in fig 3.14.
jN4
jv
3r\T
LHP
RHP
cirdle
T
-3n\T
Fig 3.146:z-plane.
Fig 3. 14a: s-plane.
Fig 3.14 : Mapping ofs-plane into z-plane.