Advances and Applications in Mathematical Sciences 
                                                                                                                                                                                        Volume 21, Issue 2, December 2021, Pages 1029-1036 
                                                                                                                                                                                        © 2021 Mili Publications, India 
                                                                                                                                                                                 
                                                                                                                                                                                    
                                                                                                                                                                                                                        BILINEAR TRANSFORMATION 
                                                                                                                                                     A. SARANYA, K. CHANDRAN, A. KIRUTHIKA and B. A. BEGAM 
                                                                                                                                    Assistant Professor 
                                                                                                                                    Department of Mathematics 
                                                                                                                                    Dhanalakshmi Srinivasan College of Arts  
                                                                                                                                    and Science for Women (Autonomous) 
                                                                                                                                    Perambalur, India 
                                                                                                                                    E-mail: kasaranya92@gmail.com 
                                                                                                                                                                         r.j.malini@gmail.com 
                                                                                                                                                                                                                                                                                                        Abstract 
                                                                                                                                                       A formula is derived and demonstrated that is capable of directly generating digital filter 
                                                                                                                                    coefficient  from  an  Analog  filter  prototype  using  the  bilinear  transformation.  This  formula 
                                                                                                                                    obviates the need for any algebraic manipulation of the Analog prototype filter and ideal for use 
                                                                                                                                    in  embedded  systems  that  must  be  take  in  any  general  Analog  filter  specification  and 
                                                                                                                                    dynamically generate digital filter coefficient directly usable in difference equations.  
                                                                                                                                                                                                                                                                                             Introduction 
                                                                                                                                                       The bilinear transformation is also known as Tustin’s method is used in 
                                                                                                                                    digital  signal  processing  and  discrete-time  control  theory  to  transform 
                                                                                                                                    continuous-time system representations to discrete-time and vice-versa.  
                                                                                                                                                       A function  f : C  C can be thought of as a transformation from one 
                                                                                                                                    complex to another complex plane. Hence the nature of complex function can 
                                                                                                                                    be described the manner which it maps regions and curves from one complex 
                                                                                                                                    to  another  complex  plane.  In  this  chapter  we  shall  discuss  bilinear 
                                                                                                                                    transformation  and  see  how  various  regions  are  transformed  by  the 
                                                                                                                                    transformations.  
                                                                                                                                                       Definition. A Bilinear transformation is defined as  
                                                                                                                                    2020 Mathematics Subject Classification: 15A04. 
                                                                                                                                    Keywords: Bilinear transformation, Elementary transformation, Translation. 
                                                                                                                                    Received November 2, 2021; Accepted November 15, 2021 
                             1030 A. SARANYA, K. CHANDRAN, A. KIRUTHIKA and B. A. BEGAM 
                                                               Z  a  bz  
                                                                    c  dz
                                 Where  a,  b,  c,  d  are  constants  (complex  in  general)  and  z  is  an 
                             independent  complex  variables  being  mapped  into  dependent  complex 
                             variable Z as illustrated.  
                                                                          a   b
                                 Definition 1.1. Let a, b, c, d €C det           ad  bc  0. We define a 
                                                                        c    d
                                                                               
                             bilinear transformation or a mobius transformation T : C      C  as:  
                                                                                              
                                 For c  0  
                                          az  b              d
                                                              
                                 Tz           ,   z C 
                                         cz  d             c
                                                             
                                          a
                                                   
                                 Tz      , z  
                                         c        
                                                  
                                                   d
                                                      
                                 Tz  , z        ,
                                                  c
                                                    
                             and for c  0  
                                                                  az  b
                                                                                 
                                                         Tz           , z  C
                                                                    d
                                                                 
                                            
                                  , z  .
                                 In  general  we  will  write  a  bilinear  transformation  T : C  C  as 
                                                                                                       
                                         az  b                  
                             w  Tz  cz  d , ad  bc  0.
                                 Without any ambiguity.  
                                 Example.  
                                         2z  3i          iz  6
                                                                                       
                                 T Z            , S Z          , T Z  2z  3, T Z  2z,
                                          iz  5            3z     1                2
                                          3
                                     
                                 S Z   etc., 
                                  1       z
                                 Definition 1.2. (Elementary Bilinear Transformation.)  
                                 Advances and Applications in Mathematical Sciences, Volume 21, Issue 2, December 2021 
                                                                                   BILINEAR TRANSFORMATION                                                                       10
                                                                                                                                                                          31 
                                                     1. Translation. We define a translation as                                                          where a is a 
                                                                                                                                   T z  z  a,
                                              finite complex number, i.e. a € C. since                                          1z  a  and1. 1-a.  0  1  0, 
                                                                                                                   Tz  0z 1
                                              so T is bilinear transformation.  
                                                     2.  Inversion.  We  define  inverse                                       1   since                   0z 1    and 
                                                                                                                  Tz  z                     Tz  1z  0
                                               00-11  -1  0, so T is a bilinear transformation.  
                                                     3.  Rotation.  We  define  a  rotation  as                                                  i                     Since 
                                                                                                                                                             
                                                                                                                                   T z  e z, R 0 .
                                                           eiz  0                i
                                               Tz  0z 1  and e                       0, so T is bilinear transformation.  
                                                     4.  Magnification.  We  define  a  magnification                                                                Since 
                                                                                                                                            
                                                                                                                                        T z  rz, r€R .
                                                           rz  0  and r 1  0  0  r  0, so T is bilinear transformation.  
                                               Tz  0z 1
                                                     Theorem  1.1.  Every  bilinear  transformation  is  a  composition  of 
                                              elementary bilinear transformation, i.e. composition of translation, inversion 
                                              and dilation (one or two of them may be missing).  
                                                     Proof. Let us consider a bilinear transformation T : C                                          C  defined as 
                                                                                                                                                           
                                                            az  b  where                            £C and                             
                                                                                                                adbc  0
                                               T Z  cz  d                         a, b, c, d
                                                     Case. Ic  0 
                                                                                                                            az  b
                                                     Hence ad  0, i.e. a  0  d and Tz                                     d        
                                                                                                                        a z  b  
                                                                                                                           d         d
                                                                      b  where                      a  and a                       is a dilation.  
                                                                                          
                                                      T1 z  d                       T1z  d z                     d  0T1
                                                                                                         b
                                                                        where                                 translation  
                                                                                T2z  z                a
                                                     T2T1z ,                                            d
                                                     Hence T  T2T1.  
                                                     Case II.  c  0  
                                                    Advances and Applications in Mathematical Sciences, Volume 21, Issue 2, December 2021 
                               1032 A. SARANYA, K. CHANDRAN, A. KIRUTHIKA and B. A. BEGAM 
                                    Now           az  d     a    a  
                                          Tz  cz  d  c  c
                                                                  bc  ad
                                       bc  ad      1      a  here                 
                                         c2          d  c            c2      0
                                                 z  c
                                        bc  ad     1      a                        d
                                                             where                     translation  
                                                                        
                                                                      T1z  z        a
                                           2   
                                                      
                                                  T1z       c                       c
                                          c    
                                                
                                       bc  ad                  a where             1  is the inversion  
                                                           
                                                   T2T1z                       
                                                                            T2z 
                                            2                   c                   z
                                          c      
                                                         a where             bc  ad  is dilation 
                                                                  
                                     T3 T2 T1 z        c          T3z        c2    z
                                                             where                 a  is translation. 
                                                            T4z  z 
                                     T4 T3 T2 T1z         ,                       c
                                    Hence T  T4T3T2T1. Hence the proof.  
                                    Definition. Let X be a nonempty set, and  f : X  X, a €X is said to be 
                               fixed point of f if          
                                                   f a  0.
                                    Example. 1. If  f : R  R be the mapping defined as                    2  0 and 1 
                                                                                                  fx  x ,
                               are fixed points of f. 
                                    2. If  f : R  R be the mapping defined as                 3  0 and ± 1 are fixed 
                                                                                      fx  x ,
                               points off.  
                                    3. If  f : R  R be the mapping defined as                     0 is only fixed 
                                                                                       f x  sin x ,
                               points of f.  
                                    Note: for  f : X  X, the fixed points can be find outs by solving the 
                               equation            on X.  
                                          f x  x
                                    Example.  Find  all  fixed  points  of  the  bilinear  transformation 
                                        3z  2
                               Tz  z  5   
                                    Solution. Here           3               is not fixed point of T.  
                                                      T   1  3  , 
                                   Advances and Applications in Mathematical Sciences, Volume 21, Issue 2, December 2021