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The values of G(s), as shown in Figure 3-2, along the j

ω-axis are 

the same as the values of 

 around a circle with diameter defined by 

[a – j0, b

–1

+ j 0] on the positive real axis. 

Figure 3-2.  Bilinear Mapping from G(s) to 

 (Case 1)

Also, the values of 

, as shown in Figure 3-3, along the j

ω-axis are 

the same as the values of G(s) around a circle with diameter defined by 
[–b

–1

+ j0, –a + j0].

Figure 3-3.  Bilinear Mapping from G(s) to 

 (Case 2)

We can implement an arbitrary bilinear transform using the 

subsys( )

 

function, which substitutes a given transfer function for the s- or z-domain 
operator. 

To implement 

 use:

gtildesys=subsys(gsys,makep([-b,1]/makep([1,-a])

To implement 

 use:

gsys=subsys(gtildesys,makep([b,1]/makep([1,a])

The systems substituted in the previous calls to subsys invert the function 

specification because these functions use backward polynomial rotation.

G˜ s

( )

˜ s

( )

( )

1

–

G˜s

(

)

G˜ s

( )

( )

1

–

G˜s

(

)

G˜ s

( )

–

–

1

+

-------------------

⎝

⎠

⎛

⎞

=

( )

G˜

+

s 1

+

-----------

⎝

⎠

⎛

⎞

=