help me with inverse z transform

**TechnicianEngineer** · February 10th 2011, 06:48 PM

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

**Ackbeet** · February 11th 2011, 05:41 AM

Dividing top and bottom by $z^{2}$ yields

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

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

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

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

$0.5=a^{2}.$

You can make this work out. What do you get for $a$ and $\sin(\omega_{0})$ and $\cos(\omega_{0})?$ What is your radius of convergence?

**chisigma** · February 11th 2011, 09:42 AM

Originally Posted by TechnicianEngineer

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

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

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

... is...

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

... that has to be solved...

Kind regards

$\chi$ $\sigma$

**TechnicianEngineer** · February 11th 2011, 03:40 PM

the region of convergence is |z| > 0.707106

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

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

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

is this correct?

**Ackbeet** · February 11th 2011, 03:57 PM

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.

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