The Z-Transform

Activity 1:

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 + 5x2 + 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?