# Thread: need help with this equation...complex numbers??

1. ## need help with this equation...complex numbers??

I have detached the image with the equation.
I dont understand the steps of calculation of the H(z) equation.
Can somebody explain how 2+2exp(j*pi/3) / (1-exp-j2*pi/3) in the numerator becomes 2exp(-j*pi*3) further down in the calculation.
I dont see this reduction!!

2. ## Re: need help with this equation...complex numbers??

Well, First let's write: $\displaystyle \frac{2+2e^{-j\pi/3}}{1-e^{-j2\pi/3}}$ = $\displaystyle 2e^{-j\pi/3}.\frac{e^{j\pi/3}+1}{1-e^{-j2\pi/3}}$ (we just factorized $\displaystyle 2e^{-j\pi/3}$ outside). Now, it remains to show that $\displaystyle \frac{1+e^{j\pi/3}}{1-e^{-j2\pi/3}} = 1$. This is easy because you should know that

$\displaystyle -e^{-j2\pi/3} = -(cos(-2\pi/3) + jsin(-2\pi/3)) = -cos(-2\pi/3) - jsin(-2\pi/3)$$\displaystyle = -cos(2\pi/3) + jsin(2\pi/3) = cos(\pi/3) + jsin(\pi/3) = e^{j\pi/3}$

and hence the above fractions becomes easily equal to one and therefore your result.