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QUESTION

Home Work and Lab

HW7 – The Z-Transform

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

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?