Inverse Z-transform with complex conjugate poles

Question

I was computing an inverse z-transform here, and I am facing some problems.

So, the z-transform is:

$$ X(z) = \frac{2+3z^{-1}}{1 - z^{-1} + 0.81z^{-2}} , |z| > 0.9 $$

I found the following poles:

$$ p_1 = 0,5 + j0,7483 $$ $$ p_2 = 0,5 - j0,7483 $$

and then, the following residues:

$$ R_1 = 1 - j2,6727 $$ $$ R_2 = 1 + j2,6727 $$

Rewriting $ X(z) $ with the poles in polar coordinates:

$$ X(z) = \frac{1 - j2,6727}{1 - 0,9 \angle 56,25^{\circ}} + \frac{1 + j2,6727}{1 - 0,9 \angle -56,25^{\circ}} $$

Through the magnitude of the poles, I can see that they are both inside the ROC, then, I can write $ x(n) $ as:

$$ x(n) = (1 - j2,6727)(0,9 \angle 56,25^{\circ})^{n}u(n) + (1 + j2,6727)(0,9 \angle -56,25^{\circ})^{n}u(n) $$

I know that it is possible to write $ x(n) $ in a form without complex numbers. I know this, because it was asked and I was able to plot the figure in Matlab using the following code:

delta = zeros(30, 1);
delta(1) = 1;

a = [1 -1 0.81];
b = [2 3];

x = filter(b, a, delta);
n = 0:29;

figure
stem(n, x);

My question is, what I did up to now is right? If it is, How can I compute $ x(n) $ without complex numbers? Is there any tips that you can give me to follow?

Thanks in advance.

Answer 1

Since the given $\mathcal{Z}$-transform

$$X(z)=\frac{2+3z^{-1}}{1-z^{-1}+0.81z^{-2}},\quad |z|>0.9\tag{1}$$

has real-valued coefficients, its poles and zeros must be either real-valued or they must occur in complex conjugate pairs. The poles of $(1)$ are a complex conjugate pair:

$$p_1=0.5 + 0.74833j=re^{j\phi}=p\\ p_2=0.5 - 0.74833j=re^{-j\phi}=p^*$$

with $r=0.9$ and $\phi=0.98177$.

You can proceed from the partial fraction expansion of $X(z)$, which you correctly obtained:

$$X(z)=\frac{A}{1-pz^{-1}}+\frac{A^*}{1-p^*z^{-1}}$$

With the given region of convergence (ROC) $|z|>0.9$, we know that we're looking for a causal sequence $x[n]$:

$$x[n]=\left(Ap^n+A^*(p^*)^n\right) u[n]\tag{3}$$

With $p=re^{j\phi}$ and $A=|A|e^{j\phi_A}$, $(3)$ can be rewritten as

$$x[n]=|A|r^n\left(e^{j(n\phi+\phi_A)}+e^{-j(n\phi+\phi_A)}\right)u[n]= 2|A|r^n\cos(n\phi+\phi_A)u[n]\tag{4}$$

which is of course real-valued.

With the given values of $p$ and $A$ the impulse response looks like this: