# How to find the inverse Z transform of this function in z domain?

i have a function $$F(z)=\frac{z-0.4}{z^2+z+2}$$ i need to find the inverse z transform of it , i have tried it with residues but the roots are too much ugly and it involves lots of messy calculations , is there any other way to proceed ?

• the easier way for rational functions is using partial fraction expansion. – Mohammad M Sep 1 '17 at 15:09
• yeah i used that but the roots are way to ugly that's why it's making a mess , any other ways ? @MohammadMohammadi – Zeno San Sep 2 '17 at 16:56

I think you are using the partial fraction expansion in wrong way. You have to know for $z$ transform we expand it as a function of $z^{-1}$ and not $z$. for your special problem factor out $z^2$ from denominator then divide the nominator by that.
Then you could factor your polynomial over a real field or complex field. This means after factorization over complex field you may obtain polynomials with complex coefficient but factorization over a real field you obtain polynomials with real coefficients. so for your case the denominator is irreducible over real field but we could factor it over a complex field, which lead to a pair of conjugate poles (roots). Now you could easily obtain the inverse using the inverse of a single pole (don't worry about the complex numbers, after summing response of 2 conjugate pole the complex parts will cancel out each other and only a real part remain). Also I have to say, for your case you will reach 2 answer considering the region of convergence (RoC) in $z$ space.