 #1
WhiteWolf98
 86
 5
 Homework Statement:

How do you decompose the following:
$$\frac{3z^{1}+2z^{2}}{1z^{1}+z^{2}}$$
 Relevant Equations:
 None.
So the original question is from Control Theory, and the topic is the inverse ztransform. This is a part from the solution I just can't understand. The reason it has to be in this form (##z^{1}##) is because that's the form used in the ztransform table. The question essentially is, how do you get from the left side to the right side:
$$\frac{3z^{1}+2z^{2}}{1z^{1}+z^{2}}=3z^{1}\frac{10.5z^{1}}{1z^{1}+z^{2}}+z^{1}\frac{0.5z^{1}}{1z^{1}+z^{2}}$$
In the solution, it states: 'Noting that the poles in the quadratic term are complex conjugates'. I have no idea what that means. In the normal (##z##) form, it is:
$$\frac{3z+2}{z^2z+1}$$
The solutions of this, I know, are complex conjugates:
$$\frac{1}{2}+\frac{\sqrt{3}}{2}i,\,\frac{1}{2}\frac{\sqrt{3}}{2}i$$
I don't see how that helps me though. Any help would be appreciated. Thank you.
$$\frac{3z^{1}+2z^{2}}{1z^{1}+z^{2}}=3z^{1}\frac{10.5z^{1}}{1z^{1}+z^{2}}+z^{1}\frac{0.5z^{1}}{1z^{1}+z^{2}}$$
In the solution, it states: 'Noting that the poles in the quadratic term are complex conjugates'. I have no idea what that means. In the normal (##z##) form, it is:
$$\frac{3z+2}{z^2z+1}$$
The solutions of this, I know, are complex conjugates:
$$\frac{1}{2}+\frac{\sqrt{3}}{2}i,\,\frac{1}{2}\frac{\sqrt{3}}{2}i$$
I don't see how that helps me though. Any help would be appreciated. Thank you.