How Do You Calculate the Step Response of an LTI System?

[Cossie]

Homework Statement

Calculate the step response of the system (i.e. response to x(n) = u(n)

Previously calculated the impulse response becomes h(n) 3/11 $(5/8)^{n}$ u(n) + 5/22$(1/8)^{n}$u(n)

Homework Equations

H(z) =$\frac{Y(z)}{X(z)}$

u(n) =$\frac{1}{1-z^{-1}}$

The Attempt at a Solution

Convolution seems a possible way but that would involve an insane amount of maths. But convolution is multiplication in the z domain. Transforming the impulse response back we get
H(z)=3/11 $\frac{1}{1-\frac5 8z^{-1}}$ + 5/22$\frac{1}{1-\frac1 8z^{-1}}$

Then take from the other side x(z) and multiple the top halves by X(z).

After that switch x(z) for the u(n). After doing that it becomes something incredibly stupid that i actually can't manage to write it on this! Any help would be appreciated!

 

[KataKoniK]

Thank you for sharing your question with us. It seems like you are on the right track with using convolution to calculate the step response of the system. Here are some steps that may help guide you:

1. Recall that the step response is the output of a system when the input is a unit step function (u(n)). This means that X(z) = 1/(1-z^-1).

2. Use the given impulse response, h(n), to calculate the transfer function, H(z). You have correctly transformed h(n) into the z-domain using the formula H(z) = Y(z)/X(z). Make sure to simplify the fractions before proceeding.

3. Multiply the transfer function, H(z), by X(z). This will give you the output, Y(z), in the z-domain.

4. Use the inverse z-transform to transform Y(z) back into the time domain. This will give you the step response, y(n), as a function of n.

5. Simplify the expression for y(n) to get the final solution.

I hope this helps. If you are still having trouble, feel free to share your work and I would be happy to provide further guidance. Good luck with your calculations!