Transfer functions

Let's assume that we have the following transfer function

$$H(z)=\frac{1+z^{-1}}{1-z^{-1}+0.5^{-2}}=\frac{B(z)}{A(z)}$$

• To enter the numerator

  >> b=[1 1]
  
  b =
  
       1     1
  

• To enter the denominator

  >> a=[1 -1 0.5]
  
  a =
  
      1.0000   -1.0000    0.5000
  

  NOTE: if, for example the middle coef. was zero, you still have to enter it as zero, you should not ignore it otherwise the denominator will only be first order.
• To find the zeros and poles

  >> [z,p]=tf2zp(b,a)
  
  z =
  
      -1
  
  
  p =
  
     0.5000 + 0.5000i
     0.5000 - 0.5000i
  

  where z and p are the vectors representing the zeros and poles of the system, respectively.
• To plot the zeros and poles

  >> zplane(b,a)
  

  or

  >> zplane(z,p)
  

  Figure 5: Pole-zero plot

  
• To get the partial fraction expansion

  >> [R,P,K]=residuez(b,a)
  
  R =
  
     0.5000 - 1.5000i
     0.5000 + 1.5000i
  
  
  P =
  
     0.5000 + 0.5000i
     0.5000 - 0.5000i
  
  
  K =
  
       []
  

  The above is equivalent to

  $$H(z)=\frac{R(1)}{1-P(1)z^{-1}}+\frac{R(2)}{1-P(2)z^{-1}}$$