# the z-transform

A linear time-invariant discrete-time system has transfer function

Use Matlab to obtain the poles of the system. Is the system stable? Explain.
Matlab tip: You can find the roots of a polynomial by using the roots command. For instance, if you have the polynomial x2 + 4x + 3, then you can find the roots of this polynomial as follows:

>>roots([1 4 3])

where the array is the coefficients of the polynomial.

Compute the step response. This should be done analytically, but you can use Matlab commands like conv and residue to help you in the calculations.
Matlab tip: Besides using conv to look at the response of a system, it can also be used to multiply two polynomials together. For instance, if you want to know the product (x2 + 4x + 3)(x + 1), you can do the following:

>>conv([1 4 3],[1 1])

where the two arrays are the coefficients of the two polynomials.

The result is

>> ans = 1 5 7 3

Thus, the product of the two polynomials is x3 + 5×2 + 7x + 3.

Matlab tip: The command residue does the partial fraction expansion of the ratio of two polynomials. In our case, we can obtain Y(z)/z and then use the residue command to do the partial fraction expansion. Then it is relatively easy to obtain y[n] using the tables.

Plot the first seven values of the step response. Is the response increasing or decreasing with time? Is this what you would expect, and why?

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