help me with inverse z transform

• Feb 10th 2011, 06:48 PM
TechnicianEngineer
help me with inverse z transform
Find the causal signal x(n) for the ff. z transform
$\displaystyle X(z) = \frac{0.5z}{z^2-z+0.5}$
• Feb 11th 2011, 05:41 AM
Ackbeet
Dividing top and bottom by $\displaystyle z^{2}$ yields

$\displaystyle X(z)=\dfrac{0.5z^{-1}}{1-z^{-1}+0.5z^{-2}}.$

Comparing this expression to # 20 in this list, we see that we must have, if possible,

$\displaystyle 0.5=a\sin(\omega_{0}),$

$\displaystyle 1=2a\cos(\omega_{0}),$ and

$\displaystyle 0.5=a^{2}.$

You can make this work out. What do you get for $\displaystyle a$ and $\displaystyle \sin(\omega_{0})$ and $\displaystyle \cos(\omega_{0})?$ What is your radius of convergence?
• Feb 11th 2011, 09:42 AM
chisigma
Quote:

Originally Posted by TechnicianEngineer
Find the causal signal x(n) for the ff. z transform
$\displaystyle X(z) = \frac{0.5z}{z^2-z+0.5}$

The 'rexcursive relation' corresponding the the z-transform...

$\displaystyle \displaystyle X(z)= \frac{1}{2}\ \frac{z^{-1}}{1-z^{-1} + \frac{z^{-2}}{2}}$ (1)

... is...

$\displaystyle \displaystyle x(n)= x(n-1)- \frac{x(n-2)}{2} + \frac{\delta(n-1)}{2}\\,\\ x(-1)=0\\,\\ x(-2)=0$ (2)

... that has to be solved...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$
• Feb 11th 2011, 03:40 PM
TechnicianEngineer
the region of convergence is |z| > 0.707106

solving for a
$\displaystyle \sqrt(0.5) = a$
$\displaystyle a = 0.707106$

solving for $\displaystyle \omega_o$
$\displaystyle \sin^{-1}(\frac{1}{2*0.707106}}) = \omega_o$
$\displaystyle \omega_o = 45 degrees$

the inverse z transform is
$\displaystyle x[n] = (0.707107)^n \sin(\frac{\pi}{4}n)u[n]$

is this correct?
• Feb 11th 2011, 03:57 PM
Ackbeet
You should check that the values you obtained check out with the third equation (three equations and two unknowns renders your system overdetermined, but the third equation must be satisfied). If it does, then I'd say you have the correct answer.