Impulse response of a causal system from transfer function in z-domain

Question

The transfer function is $$H(z)=(z+1)/(z^2+z+0.5)$$ I need to find the impulse response h[n] of a causal system with x[n] as unit impulse.

I have tried to find the impulse response by the following methods:

1. partial fraction decomposition and then inverse z transform of transfer function.
2. homogeneous solution of differential equation.
3. Matlab iztransf() function I get different solutions from the above methods and i don't know how to remove iota from the impulse response.What is the correct procedure to find the impulse response?

Answer 1

Let's follow option 1. I'm using Python but same is directly available in matlab. The partial fraction expansion can be found via residue function

>>> import scipy.signal as sig
>>> b = [1, 1]
>>> a = [1, 1, 0.5]
>>> r, p, k = sig.residue(b,a)
>>> r
array([0.5-0.5j, 0.5+0.5j])

>>> p
array([-0.5+0.5j, -0.5-0.5j])

>>> k
array([0.])

This gives us $$ H(z) = \frac{0.5-0.5i}{z+0.5-0.5i} + \frac{0.5+0.5i}{z+0.5+0.5i} $$

Now either by computing manually, or using a Z-transform table, or using another computational tool you can show that these are two complex terms can be inversed transformed as

$$ \mathcal{Z}^{-1}\left[\frac{a}{z+a}\right] = \mathcal{Z}^{-1}\left[\frac{az^{-1}}{1+az^{-1}}\right] = -(-a)^k \mathbf{1}(k - 1) $$ Here $\mathbf{1}(k)$ is the discrete time step function. But since we have complex numbers this looks like our response should be a complex function. However this is not possible since our transfer function has real valued coefficients. Hence even before I compute anything, I know already that imaginary parts will cancel out.

And since $|a| < 1$ this is going to be a decaying sinusoidal. Indeed if we check this with say sampling period $T=1$,

>>> import harold as har
>>> G = Transfer([1, 1], [1, 1, 0.5], dt=1)
>>> impulse_response_plot(G);

As expected, there is a one sample delay coming from $\mathbf{1}(k-1)$