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 + 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?