HW7 – The Z-Transform

• Read Chapter 10 in the text Signals and Systems Using MATLAB.

• Download and review the supplemental questions.
• Work the homework problems below.
• Submit homework solutions via Assignment Upload Tool.  Show all work for full credit.

 

1. Compute the z-Transform for the discrete-time signal .
2. Compute the inverse z-Transform of the transform
3. A discrete-time system is give by the input/output difference equation y[n+2]-y[n+1]+y[n]=x[n+2]-x[n+1].  Is the system stable, marginally stable, or unstable?
4. Compute the pole for the transfer function
5. A system has the transfer function .

   Is the system stable, marginally stable, or unstable?

    

   Lab7 – The Z-Transform

   • Watch video entitled “Module 7–  Z-Transform in MATLAB”
   • Work the below lab assignment below using MATLAB.
   • Include answers for Problems and include MATLAB coding along with any output plots that support solutions into a Word document entitled “Lab7_StudentID”.  Where your student id is substituted in the file name.
   • Upload file “Lab7_StudentID” 

   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?