The zplane command can be used to plot the poles and zeros of a rational 2-transform. You can use zplane with two different types of input arguments, either specifying the locations of the poles and zeros, or, specifying the coefficients of the numerator and denominator polynomials (could be either in ascending powers of 2- or descending powers of ). Type help zplane at the MATLAB command prompt to understand how this function works. (a) Plot the poles and zeros for the system: (5 points)
2z! + 16 + 4422 +56 + 32 H(-) = 622 +7z 6
(b) Verify that the poles and zeros you obtained from the above step are identical to the roots of the denominator and numerator polynomials respectively. You can use the roots fiction in MATLAB to find the roots of a polynomial. Write down the roots of the numerator polynomial (veros of the system) and the roots of the denominator polynomial (poles of the system) in exponential form. (5 points)
(c) Rewrite H() is ascending powers of :-), and use zplane again to verify that you can enter the coefficients from a rational representation either in ascending powers of a or descending powers of . Include the pole zero plot in your lab report. (5 points)
(d) There's another way in MATLAB to compute the veros, poles and the gain constant from a rational representation of an arbitrary transfer Amction. That command is tf2zp (transfer function to t-plane). Type help tf2zp at the MATLAB command prompt to understand how that function works. Next, use tf2zp to compute the poles, zeros and the gain constant (say G) for the H(2) specified in part (a). (5 points)
(e) Understand how the MATLAB Mnction zp2tf works. Then, use the output arguments from part (d) as the input arguments to zp2tf and verify think you obtain the form of (a) specified in part (n). (5 points) I (0) Write down a factored form expression for 11 (*) simply using the outputs from prut (d).
That is, it should look like: (5 points)
H(z)=G (2-21) (-22): (-) - :6) (8)
Now, you will verify that the factored form representation in part (1) is identical to the descending powers-of-z-form' in (n). To do so, you will need to multiply the factors in the mimerator (denominator) to get the numerator (denominator) polynomial. But you will 1190 MATLAB for this. Note that (2-1)(2-2)(= ?? - 3z + 2) is just multipliction of two first order polynomials in z. In one of your HW assignments, you saw that polynomial multiplication can be realized using the convolution operation Type conv((1,-1],[1, 2]) (understand how I entered the input arguments) at the MATLAB command prompt and see what you get. Relate this output to the expression 3+ + 2. For additional practice, perform the multipliention - (1+3)-(1-] by hand and check your result ising the convolution operation. Finally, verily ising MATLAB that the factored form representation In () is identical to the descending powers-of-z-form in (a), (5 points)