# Thread: help me with inverse z transform

1. ## help me with inverse z transform

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

2. 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?

3. 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

$\displaystyle \chi$ $\displaystyle \sigma$

4. 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?

5. 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.