Dsp help - impulse response of system in z domain

[hamburglar17]

Homework Statement

I am given an LTI system where I need to obtain the impulse response and difference equation of H(z).

H(z) = (1-0.5z^-3) / [(1-0.5z^-1)(1-0.25z^-1)] and |z| > 0.5

Homework Equations

The Attempt at a Solution

I am pretty lost about what to do. I think I need to perform a z-transform but I am confused because of the orders of z in the numerator vs denominator.

Any push in the right direction would be greatly appreciated.

 

[KataKoniK]

Thank you for your question. I understand your confusion and will do my best to guide you in the right direction. Let's start with the basics - an LTI system is a linear time-invariant system, meaning that its output is directly proportional to its input and does not change over time. In order to obtain the impulse response and difference equation of H(z), we will need to use the z-transform.

The z-transform is a mathematical tool used to convert a discrete-time signal into a complex frequency domain representation. In this case, we will use it to convert H(z) into its corresponding impulse response and difference equation. The z-transform is defined as:

X(z) = ∑x[n]z^-n

where x[n] is the discrete-time signal and z^-n is the z-transform variable. In this case, our discrete-time signal is H(z) and we will use z^-n as our z-transform variable.

To obtain the impulse response, we need to find the inverse z-transform of H(z). This can be done by using partial fraction decomposition and then applying the inverse z-transform. The resulting equation will be in the form of h[n], where n represents the time index.

To obtain the difference equation, we need to use the impulse response and the z-transform itself. The difference equation is given by:

y[n] = ∑h[k]x[n-k]

where y[n] is the output of the system, h[k] is the impulse response, x[n-k] is the input signal, and k represents the time index.

I hope this helps guide you in the right direction. If you have any further questions, please don't hesitate to ask. Good luck with your LTI system!