Abstract:
Provided is an operational transconductance amplifier (OTA). An existing Nauta transconductor used to implement a high frequency Gm-C filter integrated circuit (IC) is analyzed by a new method and from a new perspective to remove extra components and divide roles of remaining inverters for more simple and efficient circuit structure. In an existing Nauta transconductor, a common mode signal from an input terminal is amplified and appears at an output terminal, while in the inventive Nauta transconductor the common mode signal from an input terminal does not appear at the output terminal and is effectively eliminated. These enhanced characteristics can be achieved with a smaller number of inverters than an existing Nauta transconductor. Frequency characteristics of the filter can be effectively enhanced by independently controlling the quality factor without affecting the transconductance value required for frequency characteristics of the filter.

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
       [0001]    This application claims priority to and the benefit of Korean Patent Application No. 2006-122182, filed Dec. 5, 2006, and No. 2007-58789, filed Jun. 15, 2007, the disclosure of which is incorporated herein by reference in its entirety. 
       BACKGROUND 
       [0002]    1. Field of the Invention 
         [0003]    The present invention relates to operational transconductance amplifiers widely used to implement analog circuits such as filters, and more particularly, to an operational transconductance amplifier (OTA) for a Gm-C filter fabricated by an integrated circuit (IC) process. 
         [0004]    This work was supported by the IT R&amp;D program of Ministry of Information and Communication/Institute for Information Technology Advancement [2005-S-075-02, Development of SoC for Wired and Wireless Unified Network.] 
         [0005]    2. Discussion of Related Art 
         [0006]    In a conventional wireless communication transceiver, a high frequency analog signal at radio frequency (RF) stage is converted to a low frequency analog signal through a low-noise amplifying and downconverting process. The resultant analog signal generally contains unwanted frequency signal components. A filter eliminates such frequency components and passes only a pure signal component to a digital signal processing block. A Gm-C filter includes an operational transconductance amplifier (OTA) as a basic building block. 
         [0007]    The OTA is a functional block which receives a voltage signal as an input and outputs a current signal. Assuming that an input signal is V input  and an output signal is I output  there exists the following relationship between the input and output signal: 
         [0000]      I output =g m V input   (1) 
         [0000]    where g m  denotes a proportional coefficient commonly called a conductance gain or simply conductance. An LC filter typically requires an inductor and a capacitor of large value. So, it is difficult for the inductor to be fabricated on an integrated circuit because of occupied area or the quality factor. Accordingly, an active filter is used as an alternative to the LC filter. 
         [0008]    In the active filter, an impedance of the inductor is implemented by an integrator or a gyrator, which consists of a transconductance element and a capacitor. A functional block, called an impedance inverter, which allows a capacitor to act as an inductor can be implemented by a gyrator consisting of two operational transconductance amplifiers. 
         [0009]    But, we encounter various nonideal or nonlinear characteristics while we design OTA. Only after you solve those problems can you implement the desired operational characteristics of the filter. In particular, in order to implement a filter operating at a high frequency, it is desirable to minimize number of OTA internal nodes causing parasite capacitances, which is the most serious obstacle to the high frequency operation of filters. 
         [0010]    A representative example that adopts this idea is an operational transconductance amplifier proposed by Bram Nauta (“Nauta OTA”). In general, the Nauta OTA includes single input single output inverter as basic unit cell, as shown in  FIG. 1 . The Nauta OTA effectively amplifies only the differential mode component among signal components while suppressing the common mode component. Since the inverter, which has no nodes other than power and ground nodes, is used as the basic unit cell, the Nauta OTA is highly advantageous for high frequency operation. And, since the Nauta OTA includes only two transistors in series between the power node and the ground node, it is also suitable for a low voltage operation. 
         [0011]    However, in the Nauta OTA having such a structure, a common mode component of an input signal appears with a certain gain at an output side, and it is difficult to increase the quality factor. These will now be analyzed in conjunction with the conventional Nauta OTA shown in  FIG. 1 , where 
         [0000]    
       
         
           
             
               
                 
                   
                     i 
                     1 
                   
                   = 
                   
                     
                       
                         g 
                         
                           m 
                            
                           
                               
                           
                            
                           1 
                         
                       
                        
                       
                         v 
                         
                           i 
                            
                           
                               
                           
                            
                           1 
                         
                       
                     
                     + 
                     
                       
                         g 
                         
                           m 
                            
                           
                               
                           
                            
                           2 
                            
                           a 
                         
                       
                        
                       
                         v 
                         
                           o 
                            
                           
                               
                           
                            
                           1 
                         
                       
                     
                     + 
                     
                       
                         g 
                         
                           m 
                            
                           
                               
                           
                            
                           2 
                            
                           b 
                         
                       
                        
                       
                         v 
                         
                           o 
                            
                           
                               
                           
                            
                           2 
                         
                       
                     
                     + 
                     
                       
                         ( 
                         
                           
                             g 
                             
                               o 
                                
                               
                                   
                               
                                
                               1 
                             
                           
                           + 
                           
                             g 
                             
                               o 
                                
                               
                                   
                               
                                
                               2 
                                
                               a 
                             
                           
                           + 
                           
                             g 
                             
                               o 
                                
                               
                                   
                               
                                
                               2 
                                
                               b 
                             
                           
                         
                         ) 
                       
                        
                       
                         v 
                         
                           o 
                            
                           
                               
                           
                            
                           1 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
         [0000]    and 
         [0000]    
       
         
           
             
               
                 
                   
                     i 
                     2 
                   
                   = 
                   
                     
                       
                         g 
                         
                           m 
                            
                           
                               
                           
                            
                           1 
                         
                       
                        
                       
                         v 
                         
                           i 
                            
                           
                               
                           
                            
                           2 
                         
                       
                     
                     + 
                     
                       
                         g 
                         
                           m 
                            
                           
                               
                           
                            
                           2 
                            
                           a 
                         
                       
                        
                       
                         v 
                         
                           o 
                            
                           
                               
                           
                            
                           2 
                         
                       
                     
                     + 
                     
                       
                         g 
                         
                           m 
                            
                           
                               
                           
                            
                           2 
                            
                           b 
                         
                       
                        
                       
                         v 
                         
                           o 
                            
                           
                               
                           
                            
                           1 
                         
                       
                     
                     + 
                     
                       
                         ( 
                         
                           
                             g 
                             
                               o 
                                
                               
                                   
                               
                                
                               1 
                             
                           
                           + 
                           
                             g 
                             
                               o 
                                
                               
                                   
                               
                                
                               2 
                                
                               a 
                             
                           
                           + 
                           
                             g 
                             
                               o 
                                
                               
                                   
                               
                                
                               2 
                                
                               b 
                             
                           
                         
                         ) 
                       
                        
                       
                         v 
                         
                           o 
                            
                           
                               
                           
                            
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
         [0012]    As usual, assuming (g m1 =g m2a =g m2b )=g m  and 
         [0013]    (g o1 =g o2a =g o2b )=g o , we obtain Equations 4 and 5: 
         [0000]        i   1   =g   m ( v   i1 +v o1   +v   o2 )+3g o   v   o1   (4) 
         [0000]      and 
         [0000]        i   2   =g   m ( v   i2 +v o2   +v   o1 )+3g o   v   o2   (5) 
         [0000]      Here, 
         [0000]    
       
      
       v 
       o1 
       =−zi 
       1  
      
     
         [0000]        v   o2   =−zi   2   (6) 
         [0014]    And, Equations 7 and 8 are obtained: 
         [0000]        v   o     1     =−zg   m ( v   i1   +v   o1   +v   o2 )−3 zg   o   v   o1   (7) 
         [0000]        v   o     1     =−zg   m ( v   i2   +v   o2   +v   o1 )−3 zg   o   v   o2   (8) 
         [0015]    These equations are expressed by a matrix equation: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               1 
                               + 
                               
                                 zg 
                                 m 
                               
                               + 
                               
                                 3 
                                  
                                 
                                   zg 
                                   o 
                                 
                               
                             
                           
                           
                             
                               zg 
                               m 
                             
                           
                         
                         
                           
                             
                               zg 
                               m 
                             
                           
                           
                             
                               1 
                               + 
                               
                                 zg 
                                 m 
                               
                               + 
                               
                                 3 
                                  
                                 
                                   zg 
                                   o 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               v 
                               
                                 o 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                             
                           
                         
                         
                           
                             
                               v 
                               
                                 o 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   = 
                   
                     - 
                     
                       
                         
                           zg 
                           m 
                         
                          
                         
                           [ 
                           
                             
                               
                                 1 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 v 
                                 
                                   i 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                             
                           
                           
                             
                               
                                 v 
                                 
                                   i 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
         [0016]    Both sides of Equation 9 are multiplied by y to obtain Equation 10: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               y 
                               + 
                               
                                 g 
                                 m 
                               
                               + 
                               
                                 3 
                                  
                                 
                                   g 
                                   o 
                                 
                               
                             
                           
                           
                             
                               g 
                               m 
                             
                           
                         
                         
                           
                             
                               g 
                               m 
                             
                           
                           
                             
                               y 
                               + 
                               
                                 g 
                                 m 
                               
                               + 
                               
                                 3 
                                  
                                 
                                   g 
                                   o 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               v 
                               
                                 o 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                             
                           
                         
                         
                           
                             
                               v 
                               
                                 o 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   = 
                   
                     - 
                     
                       
                         
                           g 
                           m 
                         
                          
                         
                           [ 
                           
                             
                               
                                 1 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 v 
                                 
                                   i 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                             
                           
                           
                             
                               
                                 v 
                                 
                                   i 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
         [0017]    This is a basic input/output equation of the conventional Nauta OTA. 
         [0018]    Using a capacitive load (a capacitor) as load admittance, Equation 10 is Laplace transformed into Equation 11: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               sC 
                               + 
                               
                                 g 
                                 m 
                               
                               + 
                               
                                 3 
                                  
                                 
                                   g 
                                   o 
                                 
                               
                             
                           
                           
                             
                               g 
                               m 
                             
                           
                         
                         
                           
                             
                               g 
                               m 
                             
                           
                           
                             
                               sC 
                               + 
                               
                                 g 
                                 m 
                               
                               + 
                               
                                 3 
                                  
                                 
                                   g 
                                   o 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               v 
                               
                                 o 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                             
                           
                         
                         
                           
                             
                               v 
                               
                                 o 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   = 
                   
                     
                       
                         g 
                         m 
                       
                        
                       
                         [ 
                         
                           
                             
                               1 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               1 
                             
                           
                         
                         ] 
                       
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               v 
                               
                                 i 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                             
                           
                         
                         
                           
                             
                               v 
                               
                                 i 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
         [0019]    Both sides of Equation 11 are divided by C to obtain Equations 12 and 13: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               s 
                               + 
                               
                                 
                                   
                                     g 
                                     m 
                                   
                                    
                                   3 
                                    
                                   
                                     g 
                                     o 
                                   
                                 
                                 C 
                               
                             
                           
                           
                             
                               
                                 g 
                                 m 
                               
                               C 
                             
                           
                         
                         
                           
                             
                               
                                 g 
                                 m 
                               
                               C 
                             
                           
                           
                             
                               s 
                               + 
                               
                                 
                                   
                                     g 
                                     m 
                                   
                                   + 
                                   
                                     3 
                                      
                                     
                                       g 
                                       o 
                                     
                                   
                                 
                                 C 
                               
                             
                           
                         
                       
                       ] 
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               V 
                               
                                 o 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                             
                           
                         
                         
                           
                             
                               V 
                               
                                 o 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   = 
                   
                     - 
                     
                       
                         
                           
                             g 
                             m 
                           
                           C 
                         
                          
                         
                           [ 
                           
                             
                               
                                 1 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 V 
                                 
                                   i 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                             
                           
                           
                             
                               
                                 V 
                                 
                                   i 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
         [0000]    and 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       { 
                       
                         sI 
                         - 
                         
                           
                             ( 
                             
                               - 
                               
                                 1 
                                 C 
                               
                             
                             ) 
                           
                            
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       g 
                                       m 
                                     
                                     + 
                                     
                                       3 
                                        
                                       
                                         g 
                                         o 
                                       
                                     
                                   
                                 
                                 
                                   
                                     g 
                                     m 
                                   
                                 
                               
                               
                                 
                                   
                                     g 
                                     m 
                                   
                                 
                                 
                                   
                                     
                                       g 
                                       m 
                                     
                                     + 
                                     
                                       3 
                                        
                                       
                                         g 
                                         o 
                                       
                                     
                                   
                                 
                               
                             
                             ] 
                           
                         
                       
                       } 
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               V 
                               
                                 o 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                             
                           
                         
                         
                           
                             
                               V 
                               
                                 o 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   = 
                   
                     - 
                     
                       
                         
                           
                             g 
                             m 
                           
                           C 
                         
                          
                         
                           [ 
                           
                             
                               
                                 1 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 V 
                                 
                                   i 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                             
                           
                           
                             
                               
                                 V 
                                 
                                   i 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
         [0020]    When V i1 =V i2 =0 (homogeneous), Equation 13 is changed into Equation 14: 
         [0000]      ( sI−A ) V   o =0  (14) 
         [0000]                AV o =sV  (15) 
         [0021]    Now we encounter an issue of an eigenvalue/eigenvector in the form of A{right arrow over (x)}=λ{right arrow over (x)}, which is common in quantum mechanics. An eigenvalue/eigenvector of the matrix A must be obtained to find a transformation matrix for diagonalizing the matrix A. The eigenvalue can be obtained by setting the determinant of the coefficient matrix at the left side of Equation 14 to 0. Since A has a form like A+=A, we see that A is a Hermitian or a self-adjoint operator. Accordingly, from the general characteristic of the Hermitian operator, the eigenvalues are expected to have real numbers and the eigenvectors are expected to be orthogonal. Actually, the eigenvalues are found to be real numbers: 
         [0000]      S=S 1 ,S 2   (16), 
         [0000]    where 
         [0000]    
       
         
           
             
               S 
               1 
             
             = 
             
               
                 
                   - 
                   
                     
                       3 
                        
                       
                         g 
                         o 
                       
                     
                     C 
                   
                 
                  
                 
                     
                 
                  
                 and 
                  
                 
                     
                 
                  
                 
                   S 
                   2 
                 
               
               = 
               
                 - 
                 
                   
                     
                       ( 
                       
                         
                           2 
                            
                           
                             g 
                             m 
                           
                         
                         + 
                         
                           3 
                            
                           
                             g 
                             o 
                           
                         
                       
                       ) 
                     
                     C 
                   
                   . 
                 
               
             
           
         
       
     
         [0022]    Since 
         [0000]    
       
         
           
             
               
                 g 
                 o 
               
               = 
               
                 1 
                 
                   r 
                   out 
                 
               
             
             , 
           
         
       
     
         [0000]    s 1  and s 2  have a dimension of a reciprocal of a time constant in an RC circuit. Corresponding eigenvectors may be obtained by applying s 1  and s 2  to Equation 14. The magnitudes of the eigenvectors are normalized into 1, resulting in Equations 17 and 18: 
         [0000]    
       
         
           
             
               
                 
                   
                     | 
                     
                       r 
                       1 
                     
                   
                   = 
                   
                     
                       1 
                       
                         2 
                       
                     
                      
                     
                       ( 
                       
                         
                           
                             1 
                           
                         
                         
                           
                             
                               - 
                               1 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
         [0000]    and 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       | 
                       
                         r 
                         2 
                       
                     
                     = 
                     
                       
                         1 
                         
                           2 
                         
                       
                        
                       
                         ( 
                         
                           
                             
                               1 
                             
                           
                           
                             
                               1 
                             
                           
                         
                         ) 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where |r i &gt; denotes a column vector and &lt;r j | denotes a row vector. 
         [0023]    Since the two unit vectors have a relationship of &lt;r i |r j &gt;=δ ij , they are orthogonal, as mentioned above. 
         [0024]    Now, a transformation matrix for diagonalizing the matrix A is constructed as in Equation 19: 
         [0000]    
       
         
           
             
               
                 
                   R 
                   = 
                   
                     
                       ( 
                       
                         
                           
                              
                             
                               
                                 r 
                                 1 
                               
                               &gt; 
                             
                              
                           
                            
                           
                             r 
                             2 
                           
                         
                         &gt; 
                       
                       ) 
                     
                     = 
                     
                       
                         1 
                         
                           2 
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               1 
                             
                             
                               1 
                             
                           
                           
                             
                               
                                 - 
                                 1 
                               
                             
                             
                               1 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
         [0025]    The matrix A is diagonalized into a diagonal matrix A by applying the transformation matrix: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             R 
                             * 
                           
                            
                           AR 
                         
                         = 
                           
                          
                         
                           
                             R 
                             T 
                           
                            
                           AR 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             R 
                             T 
                           
                            
                           
                             A 
                              
                             
                               ( 
                               
                                 
                                   
                                      
                                     
                                       
                                         r 
                                         1 
                                       
                                       &gt; 
                                     
                                      
                                   
                                    
                                   
                                     r 
                                     2 
                                   
                                 
                                 &gt; 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             R 
                             T 
                           
                            
                           
                             ( 
                             
                               
                                 
                                   s 
                                   1 
                                 
                                  
                                 
                                    
                                   
                                     
                                       r 
                                       1 
                                     
                                     &gt; 
                                     
                                       s 
                                       2 
                                     
                                   
                                    
                                 
                                  
                                 
                                   r 
                                   2 
                                 
                               
                               &gt; 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           [ 
                           
                             
                               
                                 
                                   s 
                                   1 
                                 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 
                                   s 
                                   2 
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         
                           ≡ 
                             
                            
                           Λ 
                         
                         , 
                       
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where R +  is the transpose matrix of the complex-conjugate matrix of R. Equation 13 may be expressed as a matrix equation: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         sI 
                         - 
                         A 
                       
                       ) 
                     
                      
                     
                       
                         V 
                         o 
                       
                       → 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         - 
                         
                           
                             g 
                             m 
                           
                           C 
                         
                       
                       ) 
                     
                      
                     
                       
                         V 
                         i 
                       
                       → 
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
         [0026]    Performing the following linear transformation on Equation 21: 
         [0000]      {right arrow over (V)} k =R{right arrow over (V′)} k  where k=i,o  (22), 
         [0027]    We obtain Equation 23: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         sI 
                         - 
                         A 
                       
                       ) 
                     
                      
                     R 
                      
                     
                       
                         V 
                         o 
                         ′ 
                       
                       → 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         - 
                         
                           
                             g 
                             m 
                           
                           C 
                         
                       
                       ) 
                     
                      
                     
                       
                         V 
                         i 
                         ′ 
                       
                       → 
                     
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
         [0028]    Both sides of Equation 23 are multiplied by R −1 =R +  to obtain Equations 24 to 26: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       R 
                       
                         - 
                         1 
                       
                     
                      
                     
                       { 
                       
                         sI 
                         - 
                         A 
                       
                       } 
                     
                      
                     R 
                      
                     
                         
                     
                      
                     
                       
                         V 
                         o 
                         ′ 
                       
                       → 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         - 
                         
                           
                             g 
                             m 
                           
                           C 
                         
                       
                       ) 
                     
                      
                     
                       
                         V 
                         i 
                         ′ 
                       
                       → 
                     
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       { 
                       
                         sI 
                         - 
                         
                           
                             R 
                             
                               - 
                               1 
                             
                           
                            
                           AR 
                         
                       
                       } 
                     
                      
                     
                       
                         V 
                         o 
                         ′ 
                       
                       → 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         - 
                         
                           
                             g 
                             m 
                           
                           C 
                         
                       
                       ) 
                     
                      
                     
                       
                         V 
                         i 
                         ′ 
                       
                       → 
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
     
         [0000]    and 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         sI 
                         - 
                         A 
                       
                       ) 
                     
                      
                     R 
                      
                     
                       
                         V 
                         o 
                         ′ 
                       
                       → 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         - 
                         
                           
                             g 
                             m 
                           
                           C 
                         
                       
                       ) 
                     
                      
                     
                       
                         V 
                         i 
                         ′ 
                       
                       → 
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
           
         
       
     
         [0029]    With elements of the matrices of Equation 26 shown: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               s 
                               - 
                               
                                 s 
                                 1 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               s 
                               - 
                               
                                 s 
                                 2 
                               
                             
                           
                         
                       
                       ] 
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               V 
                               
                                 o 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                               ′ 
                             
                           
                         
                         
                           
                             
                               V 
                               
                                 o 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                               ′ 
                             
                           
                         
                       
                       ] 
                     
                   
                   = 
                   
                     - 
                     
                       
                         
                           
                             g 
                             m 
                           
                           C 
                         
                          
                         
                           [ 
                           
                             
                               
                                 1 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 V 
                                 
                                   i 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                                 ′ 
                               
                             
                           
                           
                             
                               
                                 V 
                                 
                                   i 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                                 ′ 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
           
         
       
     
         [0030]    Here, we find that the form of the transformation matrix R is exactly the same as that of the matrix describing a rotation (a rotational angle=45°) of a coordinate system on a plane. A relation of the transformation matrix R with rotational transformation will now be analyzed. For this purpose, consider the problem of a general coordinate system rotation on a plane with reference to  FIG. 3 . Let the matrix A represent a vector rotation which maps a vector {right arrow over (r)} into a new vector A{right arrow over (r)}: 
         [0000]      {right arrow over (r)} 2 =A{right arrow over (r)}  (28) 
         [0031]    Then, a rotation of a coordinate axis B ({right arrow over (r 1 )}=B{right arrow over (r)}) is applied to map a given vector expression from 
         [0000]    
       
         
           
             
               
                 [ 
                 
                   
                     
                       x 
                     
                   
                   
                     
                       y 
                     
                   
                 
                 ] 
               
                
               
                   
               
                
               
                 to 
                  
                 
                     
                 
                 [ 
                 
                   
                     
                       
                         x 
                         ′ 
                       
                     
                   
                   
                     
                       
                         y 
                         ′ 
                       
                     
                   
                 
                 ] 
               
             
             : 
           
         
       
     
         [0000]    
       
         
           
               
             
               
                 
                   
                     
                       
                         
                           
                             B 
                              
                             
                               
                                 r 
                                 1 
                               
                               → 
                             
                           
                           = 
                           
                             BA 
                              
                             
                               r 
                               → 
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                           
                             
                               BA 
                                
                               
                                 ( 
                                 
                                   
                                     B 
                                     
                                       - 
                                       1 
                                     
                                   
                                    
                                   B 
                                 
                                 ) 
                               
                             
                              
                             
                               r 
                               → 
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                           
                             
                               ( 
                               
                                 BAB 
                                 
                                   - 
                                   1 
                                 
                               
                               ) 
                             
                              
                             
                               ( 
                               
                                 B 
                                  
                                 
                                   r 
                                   → 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     ( 
                     29 
                     ) 
                   
                 
               
             
           
         
       
     
         [0032]    Noting that B{right arrow over (r)} in the left side is an expression of {right arrow over (r 1 )} in new coordinate system and B{right arrow over (r)} in the right side is an expression of {right arrow over (r)} in new coordinate system, we know that the linear transformation denoted by matrix A in old coordinate system is now denoted by matrix A 1 =(BAB −1 ) in the new coordinate system. This means that the role played by matrix A in vector space 
         [0000]    
       
         
           
               
             
               [ 
               
                 
                   
                     x 
                   
                 
                 
                   
                     y 
                   
                 
               
               ] 
             
           
         
       
     
         [0000]    is played by matrix A 1 =(BAB −1 ) in vector space 
         [0000]    
       
         
           
             
               [ 
               
                 
                   
                     
                       x 
                       ′ 
                     
                   
                 
                 
                   
                     
                       y 
                       ′ 
                     
                   
                 
               
               ] 
             
             . 
           
         
       
     
         [0000]    (This transformation relation is called “similarity transformation”). And when matrix B denotes a coordinate-system rotation of angle θ in a counter-clockwise direction, B is described as: 
         [0000]    
       
         
           
             
               
                 
                   B 
                   = 
                   
                     [ 
                     
                       
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             θ 
                           
                         
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             θ 
                           
                         
                       
                       
                         
                           
                             
                               - 
                               sin 
                             
                              
                             
                                 
                             
                              
                             θ 
                           
                         
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             θ 
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   30 
                   ) 
                 
               
             
           
         
       
     
         [0000]    In this equation element b ij  of matrix B means a direction cosine: 
         [0000]    
       
         
           
               
             
               
                 
                   
                     
                       
                         
                           
                             b 
                             ij 
                           
                           = 
                             
                            
                           
                             cos 
                              
                             
                               ( 
                               
                                 
                                   x 
                                   i 
                                   ′ 
                                 
                                 · 
                                 
                                   x 
                                   j 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                            
                           
                             
                               
                                 x 
                                 ⋒ 
                               
                               i 
                               ′ 
                             
                             · 
                             
                               
                                 x 
                                 ⋒ 
                               
                               j 
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     ( 
                     31 
                     ) 
                   
                 
               
             
           
         
       
     
         [0033]    Now the correlation between the transformation matrix R and the matrix describing a rotation of coordinate system is described: 
         [0034]    We can regard a vector 
         [0000]    
       
         
           
             
               
                 V 
                 → 
               
               k 
             
             = 
             
               [ 
               
                 
                   
                     
                       V 
                       
                         k 
                          
                         
                             
                         
                          
                         1 
                       
                     
                   
                 
                 
                   
                     
                       V 
                       
                         k 
                          
                         
                             
                         
                          
                         2 
                       
                     
                   
                 
               
               ] 
             
           
         
       
     
         [0000]    as a component representation in a reference coordinate system 
         [0000]    
       
         
           
             
               [ 
               
                 
                   
                     x 
                   
                 
                 
                   
                     y 
                   
                 
               
               ] 
             
             : 
           
         
       
     
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               V 
                               → 
                             
                             k 
                           
                           = 
                             
                            
                           
                             [ 
                             
                               
                                 
                                   
                                     V 
                                     
                                       k 
                                        
                                       
                                           
                                       
                                        
                                       1 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     V 
                                     
                                       k 
                                        
                                       
                                           
                                       
                                        
                                       2 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                            
                           
                             
                               
                                 V 
                                 
                                   k 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                                
                               
                                 [ 
                                 
                                   
                                     
                                       1 
                                     
                                   
                                   
                                     
                                       0 
                                     
                                   
                                 
                                 ] 
                               
                             
                             + 
                             
                               
                                 V 
                                 
                                   k 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                               
                                
                               
                                 [ 
                                 
                                   
                                     
                                       0 
                                     
                                   
                                   
                                     
                                       1 
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                            
                           
                             
                               
                                 V 
                                 
                                   k 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                                
                               
                                 
                                   e 
                                   ⋒ 
                                 
                                 1 
                               
                             
                             + 
                             
                               
                                 V 
                                 
                                   k 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                               
                                
                               
                                 
                                   e 
                                   ⋒ 
                                 
                                 2 
                               
                             
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         where 
                          
                         
                             
                         
                          
                         k 
                       
                       = 
                       i 
                     
                     , 
                     o 
                   
                 
               
               
                 
                   ( 
                   32 
                   ) 
                 
               
             
           
         
       
     
         [0035]    If we apply to an arbitrary fixed vector {right arrow over (V)} k  a linear transformation B which represent a rotation of coordinate system from 
         [0000]    
       
         
           
             
               [ 
               
                 
                   
                     x 
                   
                 
                 
                   
                     y 
                   
                 
               
               ] 
             
              
             
                 
             
              
             
               to 
                
               
                   
               
               [ 
               
                 
                   
                     
                       x 
                       ′ 
                     
                   
                 
                 
                   
                     
                       y 
                       ′ 
                     
                   
                 
               
               ] 
             
           
         
       
     
         [0000]    expression {right arrow over (V)} k′  of {right arrow over (V)} k  in the new coordinate system 
         [0000]    
       
         
           
               
             
               [ 
               
                 
                   
                     
                       x 
                       ′ 
                     
                   
                 
                 
                   
                     
                       y 
                       ′ 
                     
                   
                 
               
               ] 
             
           
         
       
     
         [0000]    becomes: 
         [0000]      {right arrow over (V′)} k =B{right arrow over (V)} k   (33) 
         [0036]    When we compare the form of Equation 25 with that of Equation 29, we can understand {right arrow over (V)} k′  in Equation 33 as the representation of some fixed vector in the new coordinate system after a rotation of coordinate system. If we adopt a point of view in which matrix A is regarded as an operator, it is understood that the role played by matrix A in a relation with the vector {right arrow over (V)} o  in the old coordinate system is played by matrix A 1 =BAB −1 ) in a relation with the vector {right arrow over (V)} o′  in the new coordinate system. Therefore, we find that Equation 25 is a representation of the original Equation 21 in the new coordinate system 
         [0000]    
       
         
           
               
             
                 
               
                 
                   [ 
                   
                     
                       
                         
                           x 
                           ′ 
                         
                       
                     
                     
                       
                         
                           y 
                           ′ 
                         
                       
                     
                   
                   ] 
                 
                 . 
               
             
           
         
       
     
         [0000]    Now, matrix B which describes a rotation of coordinate system needs to be found. 
         [0037]    From Equation 22, {right arrow over (V′)} k =R −1 {right arrow over (V)} k  where k=i,o is obtained. When we compare this with Equation 33, we find that the matrix B which describes a rotation of coordinate system is B=R −1 . Therefore, the matrix R + =R T =R −1  can be regarded as the matrix B which describes a rotation of coordinate system in a plane (Accordingly, R=B −1 ). In other words, if we select specially the inverse R −1  of the matrix R consisting of eigenvectors for the matrix B which describes a rotation of the coordinate system as shown in  FIG. 3 , the expression A′=(BAB −1 ) of matrix A in the new coordinate system becomes simply diagonalized as shown in Equation 27. Here, 
         [0000]    
       
         
           
             
               
                 
                   
                     R 
                     + 
                   
                   = 
                   
                     
                       R 
                       T 
                     
                     = 
                     
                       
                         1 
                         
                           2 
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               1 
                             
                             
                               
                                 - 
                                 1 
                               
                             
                           
                           
                             
                               1 
                             
                             
                               1 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   34 
                   ) 
                 
               
             
           
         
       
     
         [0000]    comprises two unit row vectors &lt;r 1 | and &lt;r 2 |, each of which represents direction cosine associated with axes x′ and y′ respectively in the original coordinate system. Equating Equation 34 to Equation 30, we know that the angle of coordinate system rotation is given as θ=−45°. Since matrix R+=R T  rotates a coordinate system to another coordinate system in which A is described as a diagonal matrix, this new coordinate system is represented as two eigenvectors |r 1 &gt;, |r 2 &gt;. These two eigenvectors are the unit vectors (ê 1 , ê 2  in  FIG. 4 ) on the new coordinate axes (called “Principal Axis”) in which matrix A is described as a diagonal matrix. 
         [0038]    Now apply the following linear transformation to input signal pair and output signal pair using the transformation matrix R derived above (For convenience, we use DM k  instead of V′ k1  and CM k  instead of V′ k2 : 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         V 
                                         
                                           k 
                                            
                                           
                                               
                                           
                                            
                                           1 
                                         
                                       
                                     
                                   
                                   
                                     
                                       
                                         V 
                                         
                                           k 
                                            
                                           
                                               
                                           
                                            
                                           2 
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                               = 
                               
                                 
                                   
                                     1 
                                     
                                       2 
                                     
                                   
                                    
                                   
                                     [ 
                                     
                                       
                                         
                                           1 
                                         
                                         
                                           1 
                                         
                                       
                                       
                                         
                                           
                                             - 
                                             1 
                                           
                                         
                                         
                                           1 
                                         
                                       
                                     
                                     ] 
                                   
                                 
                                  
                                 
                                   [ 
                                   
                                     
                                       
                                         
                                           D 
                                            
                                           
                                               
                                           
                                            
                                           
                                             M 
                                             k 
                                           
                                         
                                       
                                     
                                     
                                       
                                         
                                           C 
                                            
                                           
                                               
                                           
                                            
                                           
                                             M 
                                             k 
                                           
                                         
                                       
                                     
                                   
                                   ] 
                                 
                               
                             
                              
                           
                            
                           
                               
                           
                         
                         R 
                       
                        
                       where 
                        
                       
                           
                       
                        
                       k 
                     
                     = 
                     i 
                   
                   , 
                   o 
                 
               
               
                 
                   ( 
                   35 
                   ) 
                 
               
             
           
         
       
     
         [0039]    From this, we obtain: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               D 
                                
                               
                                   
                               
                                
                               
                                 M 
                                 k 
                               
                             
                           
                         
                         
                           
                             
                               C 
                                
                               
                                   
                               
                                
                               
                                 M 
                                 k 
                               
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                       
                         
                           
                             
                               1 
                               
                                 2 
                               
                             
                              
                             
                               [ 
                               
                                 
                                   
                                     1 
                                   
                                   
                                     
                                       - 
                                       1 
                                     
                                   
                                 
                                 
                                   
                                     1 
                                   
                                   
                                     1 
                                   
                                 
                               
                               ] 
                             
                           
                            
                           
                             [ 
                             
                               
                                 
                                   
                                     V 
                                     
                                       k 
                                        
                                       
                                           
                                       
                                        
                                       1 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     V 
                                     
                                       k 
                                        
                                       
                                           
                                       
                                        
                                       2 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                         
                          
                         
                             
                         
                          
                         where 
                          
                         
                             
                         
                          
                         k 
                       
                       = 
                       i 
                     
                   
                   , 
                   o 
                 
               
               
                 
                   ( 
                   36 
                   ) 
                 
               
             
           
         
       
     
         [0040]    From Equation 36, the following equations are obtained: 
         [0000]    
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             V 
                             
                               o 
                                
                               
                                   
                               
                                
                               1 
                             
                           
                         
                       
                       
                         
                           
                             V 
                             
                               o 
                                
                               
                                   
                               
                                
                               2 
                             
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       
                         
                           1 
                           
                             2 
                           
                         
                          
                         
                           [ 
                           
                             
                               
                                 1 
                               
                               
                                 1 
                               
                             
                             
                               
                                 
                                   - 
                                   1 
                                 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 D 
                                  
                                 
                                     
                                 
                                  
                                 
                                   M 
                                   o 
                                 
                               
                             
                           
                           
                             
                               
                                 C 
                                  
                                 
                                     
                                 
                                  
                                 
                                   M 
                                   o 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                      
                     
                         
                     
                      
                     
                       ( 
                       
                         k 
                         = 
                         o 
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   37 
                   ) 
                 
               
             
           
         
       
     
         [0000]    and 
         [0000]    
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             V 
                             
                               i 
                                
                               
                                   
                               
                                
                               1 
                             
                           
                         
                       
                       
                         
                           
                             V 
                             
                               i 
                                
                               
                                   
                               
                                
                               2 
                             
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       
                         
                           1 
                           
                             2 
                           
                         
                          
                         
                           [ 
                           
                             
                               
                                 1 
                               
                               
                                 1 
                               
                             
                             
                               
                                 
                                   - 
                                   1 
                                 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 D 
                                  
                                 
                                     
                                 
                                  
                                 
                                   M 
                                   i 
                                 
                               
                             
                           
                           
                             
                               
                                 C 
                                  
                                 
                                     
                                 
                                  
                                 
                                   M 
                                   i 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                      
                     
                         
                     
                      
                     
                       ( 
                       
                         k 
                         = 
                         i 
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   38 
                   ) 
                 
               
             
           
         
       
     
         [0000]    If we insert Equations 37 and 38 to Equation 13 and arrange the resultant equation, we obtain: 
         [0000]    
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             D 
                              
                             
                                 
                             
                              
                             
                               M 
                               o 
                             
                           
                         
                       
                       
                         
                           
                             C 
                              
                             
                                 
                             
                              
                             
                               M 
                               o 
                             
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       
                         g 
                         m 
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 
                                   - 
                                   1 
                                 
                                 
                                   sC 
                                   + 
                                   
                                     3 
                                      
                                     
                                       g 
                                       o 
                                     
                                   
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 
                                   - 
                                   1 
                                 
                                 
                                   sC 
                                   + 
                                   
                                     2 
                                      
                                     
                                       g 
                                       m 
                                     
                                   
                                   + 
                                   
                                     3 
                                      
                                     
                                       g 
                                       o 
                                     
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               D 
                                
                               
                                   
                               
                                
                               
                                 M 
                                 i 
                               
                             
                           
                         
                         
                           
                             
                               C 
                                
                               
                                   
                               
                                
                               
                                 M 
                                 i 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   39 
                   ) 
                 
               
             
           
         
       
     
         [0041]    Physical amplifications which Equation 39 has are as follows: Equation 39 represent forward-direction transfer characteristic from input DM i , CM i  at input port to output DM o , CM o  at output port. First of all, we find in this transfer characteristic that common-mode of input signal CM i  has influence only on common-mode of output signal CM o  and that differential-mode of input signal DM i  has influence only on differential-mode of output signal DM i . This means that decoupling of differential-mode from common-mode is perfectly achieved. 
         [0042]    This characteristic is very desirable, considering that the signal we want to process is not contained in the common-mode. Furthermore, the transconductor will approach the more ideal characteristic if common-mode signal gain can be completely nullified. However, we can point out that it is not easy to increase only the quality factor of the conventional Nauta-OTA, as is seen in equation 39. 
         [0043]    In fact, in case of UWB transceiver where very high frequency operation in the range of several hundreds of MHz is required, g m  value needed is very large so that the quality factor becomes substantially reduced. Nevertheless, any practical means independently to control the quality factor is not provided, raising an important issue. The following OTA structure is proposed to resolve these problems of the conventional Nauta-OTA. 
       SUMMARY OF THE INVENTION 
       [0044]    The present invention is directed to a new OTA circuit structure capable of fully eliminating a common mode component in a conventional Nauta OTA circuit structure and effectively enhancing a quality factor by independently controlling the quality factor. 
         [0045]    One aspect of the present invention provides an operational transconductance amplifier (OTA) for receiving first and second input voltages and outputting first and second output currents proportional to a voltage difference between the first and second input voltages, the amplifier comprising: the first parallel converting cell having a positive input terminal for receiving the first input voltage and a negative input terminal for receiving the second input voltage; the second parallel converting cell having a positive input terminal for receiving the second input voltage and a negative input terminal for receiving the first input voltage; the first cross feedback converting cell having a positive input terminal connected to an output terminal of the second parallel converting cell, a negative input terminal connected to a ground voltage terminal, and an output terminal connected to an output terminal of the first parallel converting cell; and the second cross feedback converting cell having a positive input terminal connected to the output terminal of the first parallel converting cell, a negative input terminal connected to the ground voltage terminal, and an output terminal connected to the output terminal of the second parallel converting cell. 
         [0046]    The amplifier may further comprises: the first self feedback converting cell having a positive input terminal connected to the output terminal of the first parallel converting cell, a negative input terminal connected to the ground voltage terminal, and an output terminal fed back to the positive input terminal; and the second self feedback converting cell having a positive input terminal connected to the output terminal of the second parallel converting cell, a negative input terminal connected to the ground voltage terminal, and an output terminal fed back to the positive input terminal. 
         [0047]    The first and second parallel converting cells, the first and second self feedback converting cells, and the first and second cross feedback converting cells may be implemented by elements for outputting, via an output terminal, a current proportional to a difference between a voltage at a positive input terminal and a voltage at a negative input terminal. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0048]    The above and other features and advantages of the present invention will become more apparent to those of ordinary skill in the art by describing in detail preferred embodiments thereof with reference to the attached drawings in which: 
           [0049]      FIG. 1  is a circuit diagram of a conventional Nauta OTA; 
           [0050]      FIG. 2  is a circuit diagram of an enhanced Nauta OTA according to the present invention; 
           [0051]      FIG. 3  illustrates a vector rotation and a coordinate system rotation on a plane; 
           [0052]      FIG. 4  illustrates new axes (principal axes) in which matrix A is described as a diagonal matrix and the unit vectors on the coordinate axes; 
           [0053]      FIG. 5  is a circuit diagram of a voltage difference-current converting cell that may constitute the enhanced Nauta OTA according to an exemplary embodiment of the present invention. 
       
    
    
     DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS 
       [0054]    Hereinafter, exemplary embodiments of the present invention will be described in detail. However, the present invention is not limited to the embodiments disclosed below, but can be implemented in various forms. Therefore, the following embodiments are described in order for this disclosure to be complete and enabling to those of ordinary skill in the art. 
         [0055]    An operational transconductance amplifier (OTA) according to an exemplary embodiment of the present invention will now be described with reference to  FIG. 2 . 
         [0056]    Referring to  FIG. 2 , the OTA of the present exemplary embodiment receives the first input voltage vi 1  and the second input voltage vi 2  and outputs the first output current i 1  and the second output current i 2  which include current component proportional to the difference between the first input voltage vi 1  and the second input voltage vi 2 . 
         [0057]    The OTA includes: the first parallel converting cell  31  having a positive input terminal for receiving the first input voltage vi 1  and a negative input terminal for receiving the second input voltage vi 2 ; the second parallel converting cell  32  having a positive input terminal for receiving the second input voltage vi 2  and a negative input terminal for receiving the first input voltage vi 1 ; the first self feedback converting cell  33  having a positive input terminal connected to an output terminal of the first parallel converting cell  31 , a negative input terminal connected to a ground voltage terminal, and an output terminal fed back to the positive input terminal; the second self feedback converting cell  34  having a positive input terminal connected to the output terminal of the second parallel converting cell  32 , a negative input terminal connected to the ground voltage terminal, and an output terminal fed back to the positive input terminal; the first cross feedback converting cell  35  having a positive input terminal connected to the output terminal of the second parallel converting cell  32 , a negative input terminal connected to the ground voltage terminal, and an output terminal connected to the output terminal of the first parallel converting cell  31 ; and the second cross feedback converting cell  36  having a positive input terminal connected to the output terminal of the first parallel converting cell  31 , a negative input terminal connected to the ground voltage terminal, and an output terminal connected to the output terminal of the second parallel converting cell  32 . 
         [0058]    The OTA is a 2-port element including an input port having a pair of terminals and an output port having a pair of terminals. For analysis of the circuit, the OTA of  FIG. 2  further includes at the output port: the first output resistor connected between the output terminal of the first parallel converting cell  31  and the ground voltage terminal; and the second output resistor connected between the output terminal of the second parallel converting cell  32  and the ground voltage terminal. Preferably, the first output resistor and the second output resistor have the same resistance value. 
         [0059]    Each of the converting cells  31 ,  32 ,  33 ,  34 ,  35 , and  36  outputs, at its output terminal, a current proportional to a difference between a voltage at the positive input terminal and a voltage at the negative input terminal. The first and second parallel converting cells  31  and  32  have transconductance of g m1 , the first and second self feedback converting cells  33  and  34  have transconductance of g m2a , and the first and second parallel converting cells  35  and  36  have transconductance of g m2b . 
         [0060]    The first and second parallel converting cells  31  and  32  have output admittance of g o1  the first and second self feedback converting cells  33  and  34  have output admittance of g o2a , and the first and second cross feedback converting cells  35  and  36  have output admittance of g o2b . Input currents of the converting cells  31 ,  32 ,  33 ,  34 ,  35  and  36  are zero. 
         [0061]    A first output current i 1  may be expressed by a sum of output currents of the first parallel converting cell  31 , the first and second self feedback converting cell  33 , and the first cross feedback converting cell  35 , and a sum of output currents of the first parallel converting cell  31 , the first and second self feedback converting cell  33 , and the first cross feedback converting cell  35  resulting from the voltage vo 1  at the first output node. 
         [0062]    A second output current i 2  may be expressed in a similar manner. As a result, the first output current i 1  and the second output current i 2  are given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     i 
                     1 
                   
                   = 
                   
                     
                       
                         g 
                         
                           m 
                            
                           
                               
                           
                            
                           1 
                         
                       
                        
                       
                         ( 
                         
                           
                             v 
                             
                               i 
                                
                               
                                   
                               
                                
                               1 
                             
                           
                           - 
                           
                             v 
                             
                               i 
                                
                               
                                   
                               
                                
                               2 
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         g 
                         
                           m 
                            
                           
                               
                           
                            
                           2 
                            
                           a 
                         
                       
                        
                       
                         ( 
                         
                           
                             v 
                             
                               o 
                                
                               
                                   
                               
                                
                               1 
                             
                           
                           - 
                           0 
                         
                         ) 
                       
                     
                     + 
                     
                       
                         g 
                         
                           m 
                            
                           
                               
                           
                            
                           2 
                            
                           b 
                         
                       
                        
                       
                         ( 
                         
                           
                             v 
                             
                               o 
                                
                               
                                   
                               
                                
                               2 
                             
                           
                           - 
                           0 
                         
                         ) 
                       
                     
                     + 
                     
                       
                         ( 
                         
                           
                             g 
                             
                               o 
                                
                               
                                   
                               
                                
                               1 
                             
                           
                           + 
                           
                             g 
                             
                               o 
                                
                               
                                   
                               
                                
                               2 
                                
                               a 
                             
                           
                           + 
                           
                             g 
                             
                               o 
                                
                               
                                   
                               
                                
                               2 
                                
                               b 
                             
                           
                         
                         ) 
                       
                        
                       
                         v 
                         
                           o 
                            
                           
                               
                           
                            
                           1 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   40 
                   ) 
                 
               
             
           
         
       
     
         [0000]    and 
         [0000]    
       
         
           
             
               
                 
                   
                     i 
                     2 
                   
                   = 
                   
                     
                       
                         g 
                         
                           m 
                            
                           
                               
                           
                            
                           1 
                         
                       
                        
                       
                         ( 
                         
                           
                             v 
                             
                               i 
                                
                               
                                   
                               
                                
                               2 
                             
                           
                           - 
                           
                             v 
                             
                               i 
                                
                               
                                   
                               
                                
                               1 
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         g 
                         
                           m 
                            
                           
                               
                           
                            
                           2 
                            
                           a 
                         
                       
                        
                       
                         ( 
                         
                           
                             v 
                             
                               o 
                                
                               
                                   
                               
                                
                               2 
                             
                           
                           - 
                           0 
                         
                         ) 
                       
                     
                     + 
                     
                       
                         g 
                         
                           m 
                            
                           
                               
                           
                            
                           2 
                            
                           b 
                         
                       
                        
                       
                         ( 
                         
                           
                             v 
                             
                               o 
                                
                               
                                   
                               
                                
                               1 
                             
                           
                           - 
                           0 
                         
                         ) 
                       
                     
                     + 
                     
                       
                         ( 
                         
                           
                             g 
                             
                               o 
                                
                               
                                   
                               
                                
                               1 
                             
                           
                           + 
                           
                             g 
                             
                               o 
                                
                               
                                   
                               
                                
                               2 
                                
                               a 
                             
                           
                           + 
                           
                             g 
                             
                               o 
                                
                               
                                   
                               
                                
                               2 
                                
                               b 
                             
                           
                         
                         ) 
                       
                        
                       
                         v 
                         
                           o 
                            
                           
                               
                           
                            
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   41 
                   ) 
                 
               
             
           
         
       
     
         [0063]    There are three cases depending on a relationship among the transconductances g m1 , g m2a , and g m2b  of the six converting cells  31 ,  32 ,  33 ,  34 ,  35  and  36  and a relationship among the output admittances g o1 , g o2a , and g o2b  of the six converting cells  31 ,  32 ,  33 ,  34 ,  35  and  36 . 
         [0064]    Case 1: (g m1 =g m2a =g m2b =g m  and (g o1 =g o2a =g o2b =g o . 
         [0065]    If the output admittances g o1 , g o2a , and g o2b  of the six converting cells  31 ,  32 ,  33 ,  34 ,  35  and  36  in the OTA shown in  FIG. 2  are the same and the transconductances g m , g m2a , and g m2b  of the six converting cells  31 ,  32 ,  33 ,  34 ,  35  and  36  are the same, Equations 40 and 41 are simplified: 
         [0000]        i   1   =g   m ( v   i1   −v   i2   +v   o1   +v   o2 )+3 g   o   v   o1   (42) 
         [0000]      and 
         [0000]        i   2   =g   m ( v   i2   −v   i1   +v   o1   +v   o2 )+3 g   o   v   o2   (43) 
         [0000]    
       
      
       v 
       o1 
       =−zi 
       1  
      
     
         [0000]        v   o2   =−zi   2   (44), 
         [0066]    Since Equations 42 and 43 becomes: 
         [0000]        v   o1   =−zg   m ( v   i1   −v   i2   +v   o1   +v   o2 )−3 zg   o   v   o1   (45) 
         [0000]      and 
         [0000]        v   o2   =−zg   m ( v   i2   −v   i1   +v   o1   +v   o2 )−3 zg   o   v   o2   (46) 
         [0067]    These are rearranged into a matrix equation 47: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               1 
                               + 
                               
                                 zg 
                                 m 
                               
                               + 
                               
                                 3 
                                  
                                 
                                   zg 
                                   o 
                                 
                               
                             
                           
                           
                             
                               zg 
                               m 
                             
                           
                         
                         
                           
                             
                               zg 
                               m 
                             
                           
                           
                             
                               1 
                               + 
                               
                                 zg 
                                 m 
                               
                               + 
                               
                                 3 
                                  
                                 
                                   zg 
                                   o 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               v 
                               
                                 o 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                             
                           
                         
                         
                           
                             
                               v 
                               
                                 o 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   = 
                   
                     - 
                     
                       
                         
                           zg 
                           m 
                         
                          
                         
                           [ 
                           
                             
                               
                                 1 
                               
                               
                                 
                                   - 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   1 
                                 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 v 
                                 
                                   i 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                             
                           
                           
                             
                               
                                 v 
                                 
                                   i 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   47 
                   ) 
                 
               
             
           
         
       
     
         [0068]    Both sides of the matrix equation 47 are multiplied by y to obtain Equation (48): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               y 
                               + 
                               
                                 g 
                                 m 
                               
                               + 
                               
                                 3 
                                  
                                 
                                   g 
                                   o 
                                 
                               
                             
                           
                           
                             
                               g 
                               m 
                             
                           
                         
                         
                           
                             
                               g 
                               m 
                             
                           
                           
                             
                               y 
                               + 
                               
                                 g 
                                 m 
                               
                               + 
                               
                                 3 
                                  
                                 
                                   g 
                                   o 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               v 
                               
                                 o 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                             
                           
                         
                         
                           
                             
                               v 
                               
                                 o 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   = 
                   
                     - 
                     
                       
                         
                           g 
                           m 
                         
                          
                         
                           [ 
                           
                             
                               
                                 1 
                               
                               
                                 
                                   - 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   1 
                                 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 v 
                                 
                                   i 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                             
                           
                           
                             
                               
                                 v 
                                 
                                   i 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   48 
                   ) 
                 
               
             
           
         
       
     
         [0069]    This is a basic input/output equation of the enhanced Nauta OTA of the present exemplary embodiment. It can be seen from a comparison between Equation 48 and Equation 10, which is the basic input/output equation of the conventional Nauta OTA, that the coefficient matrix at the left side is the same in both equations, but that both the element at row  1 , column  2  and the element at row  2 , column  1  in the coefficient matrix at the right side of Equation 10 are 0, while that those of Equation 48 are −1. 
         [0070]    A physical meaning of the Equation 48 may be analyzed in conjunction with a characteristic equation. When the interpretation is performed on an s-plane of the Laplace transformation, the characteristic equation may be obtained by putting a determinant of the coefficient matrix at the left side to 0. Here, a root of the characteristic equation is a pole (eigenvalue) affecting the circuit characteristics, which is used to determine the stability of the circuit. Therefore, when the Nauta OTA has the structure of the present exemplary embodiment, the characteristic equation is conserved in form and hence the pole (eigenvalue) is also conserved. 
         [0071]    Using a capacitive load (a capacitor) as load admittance, Equation 48 is Laplace transformed into Equation 49: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               sC 
                               + 
                               
                                 g 
                                 m 
                               
                               + 
                               
                                 3 
                                  
                                 
                                   g 
                                   o 
                                 
                               
                             
                           
                           
                             
                               g 
                               m 
                             
                           
                         
                         
                           
                             
                               g 
                               m 
                             
                           
                           
                             
                               sC 
                               + 
                               
                                 g 
                                 m 
                               
                               + 
                               
                                 3 
                                  
                                 
                                   g 
                                   o 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               V 
                               
                                 o 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                             
                           
                         
                         
                           
                             
                               V 
                               
                                 o 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   = 
                   
                     - 
                     
                       
                         
                           g 
                           m 
                         
                          
                         
                           [ 
                           
                             
                               
                                 1 
                               
                               
                                 
                                   - 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   1 
                                 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 V 
                                 
                                   i 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                             
                           
                           
                             
                               
                                 V 
                                 
                                   i 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   49 
                   ) 
                 
               
             
           
         
       
     
         [0072]    Both sides of Equation 49 are divided by C to obtain Equations 50 and 51: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               s 
                               + 
                               
                                 
                                   
                                     g 
                                     m 
                                   
                                   + 
                                   
                                     3 
                                      
                                     
                                       g 
                                       o 
                                     
                                   
                                 
                                 C 
                               
                             
                           
                           
                             
                               
                                 g 
                                 m 
                               
                               C 
                             
                           
                         
                         
                           
                             
                               
                                 g 
                                 m 
                               
                               C 
                             
                           
                           
                             
                               s 
                               + 
                               
                                 
                                   
                                     g 
                                     m 
                                   
                                   + 
                                   
                                     3 
                                      
                                     
                                       g 
                                       o 
                                     
                                   
                                 
                                 C 
                               
                             
                           
                         
                       
                       ] 
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               V 
                               
                                 o 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                             
                           
                         
                         
                           
                             
                               V 
                               
                                 o 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   = 
                   
                     - 
                     
                       
                         
                           
                             g 
                             m 
                           
                           C 
                         
                          
                         
                           [ 
                           
                             
                               
                                 1 
                               
                               
                                 
                                   - 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   1 
                                 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 V 
                                 
                                   i 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                             
                           
                           
                             
                               
                                 V 
                                 
                                   i 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   50 
                   ) 
                 
               
             
           
         
       
     
         [0000]    and 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           { 
                           
                             sI 
                             - 
                             
                               
                                 ( 
                                 
                                   - 
                                   
                                     1 
                                     C 
                                   
                                 
                                 ) 
                               
                                
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         
                                           g 
                                           m 
                                         
                                         + 
                                         
                                           3 
                                            
                                           
                                             g 
                                             o 
                                           
                                         
                                       
                                     
                                     
                                       
                                         g 
                                         m 
                                       
                                     
                                   
                                   
                                     
                                       
                                         g 
                                         m 
                                       
                                     
                                     
                                       
                                         
                                           g 
                                           m 
                                         
                                         + 
                                         
                                           3 
                                            
                                           
                                             g 
                                             o 
                                           
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                           } 
                         
                          
                       
                       A 
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               V 
                               
                                 o 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                             
                           
                         
                         
                           
                             
                               V 
                               
                                 o 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   = 
                   
                     - 
                     
                       
                         
                           
                             g 
                             m 
                           
                           C 
                         
                          
                         
                           [ 
                           
                             
                               
                                 1 
                               
                               
                                 
                                   - 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   1 
                                 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 V 
                                 
                                   i 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                             
                           
                           
                             
                               
                                 V 
                                 
                                   i 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   51 
                   ) 
                 
               
             
           
         
       
     
         [0073]    It can be seen from a comparison between Equation 51 and Equation 13 that the matrix A is the same in the enhanced Nauta OTA of the present exemplary embodiment and the conventional Nauta OTA, but that the element at row  1 , column  2  and the element at row  2 , column  1  of the unit matrix in the coefficient matrix at the right side of Equation 13 are 0, while that those of Equation 51 are −1. Accordingly, the eigenvalue/eigenvector and the transformation matrix are the same as those in the conventional Nauta OTA. 
         [0074]    Now, Equations 35 to 38, which are linear transformation equations for the input signal pair and the output signal pair using the transformation matrix R induced above, are applied to Equation 51 to obtain Equation 52: 
         [0000]    
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             D 
                              
                             
                                 
                             
                              
                             
                               M 
                               o 
                             
                           
                         
                       
                       
                         
                           
                             C 
                              
                             
                                 
                             
                              
                             
                               M 
                               o 
                             
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       
                         ( 
                         
                           2 
                            
                           
                             g 
                             m 
                           
                         
                         ) 
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 
                                   - 
                                   1 
                                 
                                 
                                   sC 
                                   + 
                                   
                                     3 
                                      
                                     
                                       g 
                                       o 
                                     
                                   
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                           
                         
                         ] 
                       
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               D 
                                
                               
                                   
                               
                                
                               
                                 M 
                                 i 
                               
                             
                           
                         
                         
                           
                             
                               C 
                                
                               
                                   
                               
                                
                               
                                 M 
                                 i 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   52 
                   ) 
                 
               
             
           
         
       
     
         [0075]    The meaning of Equation 52 can be explained as follows. Equation 52 represents a forward transfer characteristic relationship between DM i  and CM i  at the input port and DMo and CMo at the output port. A first characteristic of Equation 52 is that since the transfer function matrix in Equation 52 is a diagonal matrix, a differential mode component DM i  of the input signal has an effect only on a differential mode component DMo of the output signal, not on common mode component CMo of the output signal. This means that the common mode component and the differential mode component of the signal components are perfectly decoupled from each other. 
         [0076]    Another characteristic of Equation 52 is that the output is not affected by the common mode component CM i  of the input signal at all. This characteristic is very ideal considering that a signal to be processed is not carried by the common mode component, which is not the case in the conventional Nauta OTA. Such property is a unique advantage of the present invention newly introduced. 
         [0077]    It can also be seen from a comparison between Equation 52 and Equation 39, which is a transformed form of the input/output equation of the conventional Nauta OTA, that an element at row  2 , column  2  of the coefficient matrix at the right side of Equation 52 is changed into 0 and that a scalar coefficient located before the coefficient matrix is doubled. This has the following physical meanings: 
         [0078]    First, the common mode component of the signal components is detected at the output port of the conventional Nauta OTA while it is entirely eliminated by the enhanced Nauta OTA of the present exemplary embodiment. 
         [0079]    Second, there is a change in the way that the differential mode component of the signal components is processed. A transfer gain between the input and the output in the differential mode is doubled. This means that gm effectively doubles in light of the fact that there is no change in the gm value of the basic constituent unit block of the OTA, leading to improved efficiency. 
         [0080]    Case 2: (g m1 =g m2b )=g m , (g o1 =g o2b )=g o  and g m2a =g o2a ≦0. 
         [0081]    In this case, the OTA does not include the first and second self feedback converting cells  33  and  34  in the structure of  FIG. 2 , but includes the first and second parallel converting cells  31  and  32  and the first and second cross feedback converting cells  35  and  36 , which have the same output admittance and transconductance. Since the four converting cells  31 ,  32 ,  35 ,  36  in the OTA of this case have the same transconductances and output admittances, Equations 40 and 41 are simplified into Equations 53 and 54: 
         [0000]        i   1   =g   m ( v   i1   −v   i2 )+ g   m   v   o2 +2 g   o   v   o1   (53) 
         [0000]      and 
         [0000]        i   2   =g   m ( v   i2   −v   i1 )+ g   m   v   o1 +2 g   o   v   o2   (54) 
         [0000]    
       
      
       v 
       o1 
       =−zi 
       1  
      
     
         [0000]        v   o2   =−zi   2   (55), 
         [0082]    Since 
         [0083]    Equations 53 and 54 becomes: 
         [0000]        v   o1   =−zg   m ( v   i1   −v   i2   +v   o2 )−2 zg   o   v   o1   (56) 
         [0000]      and 
         [0000]        v   o2   =−zg   m ( v   i2   −v   i1   +v   o1 )−2 zg   o   v   o2   (57) 
         [0084]    These are rearranged into a matrix equation 58: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               1 
                               + 
                               
                                 2 
                                  
                                 
                                   zg 
                                   o 
                                 
                               
                             
                           
                           
                             
                               zg 
                               m 
                             
                           
                         
                         
                           
                             
                               zg 
                               m 
                             
                           
                           
                             
                               1 
                               + 
                               
                                 2 
                                  
                                 
                                   zg 
                                   o 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               v 
                               
                                 o 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                             
                           
                         
                         
                           
                             
                               v 
                               
                                 o 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   = 
                   
                     - 
                     
                       
                         
                           zg 
                           m 
                         
                          
                         
                           [ 
                           
                             
                               
                                 1 
                               
                               
                                 
                                   - 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   1 
                                 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 v 
                                 
                                   i 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                             
                           
                           
                             
                               
                                 v 
                                 
                                   i 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   58 
                   ) 
                 
               
             
           
         
       
     
         [0085]    Both sides of the matrix equation 58 are multiplied by y to obtain Equation 59: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               y 
                               + 
                               
                                 2 
                                  
                                 
                                   g 
                                   o 
                                 
                               
                             
                           
                           
                             
                               g 
                               m 
                             
                           
                         
                         
                           
                             
                               g 
                               m 
                             
                           
                           
                             
                               y 
                               + 
                               
                                 2 
                                  
                                 
                                   g 
                                   o 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               v 
                               
                                 o 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                             
                           
                         
                         
                           
                             
                               v 
                               
                                 o 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   = 
                   
                     - 
                     
                       
                         
                           g 
                           m 
                         
                          
                         
                           [ 
                           
                             
                               
                                 1 
                               
                               
                                 
                                   - 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   1 
                                 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 v 
                                 
                                   i 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                             
                           
                           
                             
                               
                                 v 
                                 
                                   i 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   59 
                   ) 
                 
               
             
           
         
       
     
         [0086]    This is a basic input/output equation of the enhanced Nauta OTA. Using a capacitive load (a capacitor) as load admittance, Equation 59 is Laplace transformed into Equation 60: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               sC 
                               + 
                               
                                 2 
                                  
                                 
                                   g 
                                   o 
                                 
                               
                             
                           
                           
                             
                               g 
                               m 
                             
                           
                         
                         
                           
                             
                               g 
                               m 
                             
                           
                           
                             
                               sC 
                               + 
                               
                                 2 
                                  
                                 
                                   g 
                                   o 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               V 
                               
                                 o 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                             
                           
                         
                         
                           
                             
                               V 
                               
                                 o 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   = 
                   
                     - 
                     
                       
                         
                           g 
                           m 
                         
                          
                         
                           [ 
                           
                             
                               
                                 1 
                               
                               
                                 
                                   - 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   1 
                                 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 V 
                                 
                                   i 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                             
                           
                           
                             
                               
                                 V 
                                 
                                   i 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   60 
                   ) 
                 
               
             
           
         
       
     
         [0087]    Both sides of Equation 60 are divided by C to obtain Equations 61 and 62: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               s 
                               + 
                               
                                 
                                   2 
                                    
                                   
                                     g 
                                     o 
                                   
                                 
                                 C 
                               
                             
                           
                           
                             
                               
                                 g 
                                 m 
                               
                               C 
                             
                           
                         
                         
                           
                             
                               
                                 g 
                                 m 
                               
                               C 
                             
                           
                           
                             
                               s 
                               + 
                               
                                 
                                   2 
                                    
                                   
                                     g 
                                     o 
                                   
                                 
                                 C 
                               
                             
                           
                         
                       
                       ] 
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               V 
                               
                                 o 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                             
                           
                         
                         
                           
                             
                               V 
                               
                                 o 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   = 
                   
                     - 
                     
                       
                         
                           
                             g 
                             m 
                           
                           C 
                         
                          
                         
                           [ 
                           
                             
                               
                                 1 
                               
                               
                                 
                                   - 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   1 
                                 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 V 
                                 
                                   i 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                             
                           
                           
                             
                               
                                 V 
                                 
                                   i 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   61 
                   ) 
                 
               
             
           
         
       
     
         [0000]    and 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           { 
                           
                             sI 
                             - 
                             
                               
                                 ( 
                                 
                                   - 
                                   
                                     1 
                                     C 
                                   
                                 
                                 ) 
                               
                                
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         2 
                                          
                                         
                                           g 
                                           o 
                                         
                                       
                                     
                                     
                                       
                                         g 
                                         m 
                                       
                                     
                                   
                                   
                                     
                                       
                                         g 
                                         m 
                                       
                                     
                                     
                                       
                                         2 
                                          
                                         
                                           g 
                                           o 
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                           } 
                         
                          
                       
                       A 
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               V 
                               
                                 o 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                             
                           
                         
                         
                           
                             
                               V 
                               
                                 o 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   = 
                   
                     - 
                     
                       
                         
                           
                             g 
                             m 
                           
                           C 
                         
                          
                         
                           [ 
                           
                             
                               
                                 1 
                               
                               
                                 
                                   - 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   1 
                                 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 V 
                                 
                                   i 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                             
                           
                           
                             
                               
                                 V 
                                 
                                   i 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   62 
                   ) 
                 
               
             
           
         
       
     
         [0088]    If v i1 =v i2 =0(homogeneous), Equation 62 becomes: 
         [0000]      ( Sl−A ) V   o =0  (63) 
         [0000]                AV o =sV o   (64) 
         [0089]    Now we again encounter an issue of an eigenvalue/eigenvector in the form of A{right arrow over (x)}=λ{right arrow over (x)}. An eigenvalue/eigenvector of the matrix A must be obtained for a transformation matrix for diagonalizing the matrix A. The eigenvalue can be obtained by setting the determinant of the coefficient matrix at the left side of Equation 63 to 0. Since A+=A, it can be seen that A is a Hermitian operator or a self-adjoint operator. Accordingly, from the general characteristic of the Hermitian operator, the eigenvalues are expected to have real numbers and the eigenvectors are expected to be orthogonal. Actually, the eigenvalues are found to be real numbers: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       s 
                       = 
                       
                         s 
                         1 
                       
                     
                     , 
                     
                       
                         s 
                         2 
                       
                        
                       
                           
                       
                        
                       where 
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       s 
                       1 
                     
                     = 
                     
                       - 
                       
                         
                           ( 
                           
                             
                               2 
                                
                               
                                 g 
                                 o 
                               
                             
                             - 
                             
                               g 
                               m 
                             
                           
                           ) 
                         
                         C 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       s 
                       2 
                     
                     = 
                     
                       - 
                       
                         
                           ( 
                           
                             
                               2 
                                
                               
                                 g 
                                 o 
                               
                             
                             + 
                             
                               g 
                               m 
                             
                           
                           ) 
                         
                         C 
                       
                     
                   
                 
               
               
                 
                   ( 
                   65 
                   ) 
                 
               
             
           
         
       
     
         [0090]    Since 
         [0000]    
       
         
           
             
               
                 g 
                 o 
               
               = 
               
                 1 
                 
                   r 
                   out 
                 
               
             
             , 
           
         
       
     
         [0000]    s 1  and s 2  have a dimension of a reciprocal of a time constant in an RC circuit. Corresponding eigenvectors may be obtained by applying s 1  and s 2  to Equation 63. The magnitudes of the eigenvectors are normalized into 1, resulting in Equations 66 and 67: 
         [0000]    
       
         
           
             
               
                 
                   | 
                   
                     
                       r 
                       1 
                     
                     &gt;= 
                     
                       
                         1 
                         
                           2 
                         
                       
                        
                       
                         ( 
                         
                           
                             
                               1 
                             
                           
                           
                             
                               
                                 - 
                                 1 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   66 
                   ) 
                 
               
             
           
         
       
     
         [0000]    and 
         [0000]    
       
         
           
             
               
                 
                   | 
                   
                     
                       r 
                       2 
                     
                     &gt;= 
                     
                       
                         1 
                         
                           2 
                         
                       
                        
                       
                         ( 
                         
                           
                             
                               1 
                             
                           
                           
                             
                               1 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   67 
                   ) 
                 
               
             
           
         
       
     
         [0091]    These are the same as Equations 17 and 18. Accordingly, the transformation matrix R is given by Equation 19. (Here, |r i &gt; is a column vector and &lt;r j | is a row vector.) As mentioned above, since the two unit vectors have a relationship where r i |r j &gt;=δ ij , they are orthogonal. 
         [0092]    Now, Equations 35 to 38, which are linear transformation equations for the input signal pair and the output signal pair using the transformation matrix R induced above, are applied to Equation 62 to obtain Equation 68: 
         [0000]    
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             D 
                              
                             
                                 
                             
                              
                             
                               M 
                               o 
                             
                           
                         
                       
                       
                         
                           
                             C 
                              
                             
                                 
                             
                              
                             
                               M 
                               o 
                             
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       
                         ( 
                         
                           2 
                            
                           
                             g 
                             m 
                           
                         
                         ) 
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 
                                   - 
                                   1 
                                 
                                 
                                   sC 
                                   + 
                                   
                                     2 
                                      
                                     
                                       g 
                                       0 
                                     
                                   
                                   - 
                                   
                                     g 
                                     m 
                                   
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                           
                         
                         ] 
                       
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               D 
                                
                               
                                   
                               
                                
                               
                                 M 
                                 i 
                               
                             
                           
                         
                         
                           
                             
                               C 
                                
                               
                                   
                               
                                
                               
                                 M 
                                 i 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   68 
                   ) 
                 
               
             
           
         
       
     
         [0093]    It should be noted that Equation 68 differs from the corresponding equation in Case 1 in one important aspect. That is, the denominator at the right side of the Equation 68 includes g m , which means that a position of a pole can be adjusted by g m  now. In other words, the quality factor can be adjusted by g m . However, there remains one problem in that the same g m  is also used to adjust a cutoff frequency. Therefore, independent adjustment of the quality factor is still a lacking property. To solve this problem, the following method is proposed in Case 3: 
         [0094]    Case 3: g m1 ≠g m2b (=g m2 ), g o1 ≠g o2b (=g o2 ) and g m2a =g o2a =0. 
         [0095]    An OTA of Case 3 does not include the first and second self feedback converting cells  33  and  34  in the structure of  FIG. 2 , but includes the first and second parallel converting cells  31  and  32  having the same transconductance g m1  and output admittance g o1  and the first and second cross feedback converting cells  35  and  36  having the same transconductance g m2b  and output admittance g o2b . Accordingly, Equations 40 and 41 are simplified: 
         [0000]        i   1   =g   m1 ( v   i1   −v   i2 )+ g   m2   v   o2 +( g   o1   +g   o2 ) v   o1   (69) 
         [0000]      and 
         [0000]        i   2   =g   m1 ( v   i2   −v   i1 )+ g   m2   v   o1 +( g   o1   +g   o2 ) v   o2   (70) 
         [0000]    
       
      
       v 
       o1 
       =−zi 
       1  
      
     
         [0000]        v   o2   =−zi   2   (71), 
         [0096]    Since 
         [0097]    Equations 69 and 70 becomes: 
         [0000]        v   o1   =−zg   m1 ( v   i1   −v   i2 )− zg   m2   v   o2   −z ( g   o1   +g   o2 ) v   o1   (72) 
         [0000]      and 
         [0000]        v   o2   =−zg   m1 ( v   i2   −v   i1 )− zg   m2   v   o1   −z ( g   o1   +g   o2 ) v   o2   (73) 
         [0098]    These are rearranged into a matrix equation 74: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               1 
                               + 
                               
                                 z 
                                  
                                 
                                   ( 
                                   
                                     
                                       g 
                                       
                                         o 
                                          
                                         
                                             
                                         
                                          
                                         1 
                                       
                                     
                                     + 
                                     
                                       g 
                                       
                                         o 
                                          
                                         
                                             
                                         
                                          
                                         2 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               zg 
                               
                                 m 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                             
                           
                         
                         
                           
                             
                               zg 
                               
                                 m 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                             
                           
                           
                             
                               1 
                               + 
                               
                                 z 
                                  
                                 
                                   ( 
                                   
                                     
                                       g 
                                       
                                         o 
                                          
                                         
                                             
                                         
                                          
                                         1 
                                       
                                     
                                     + 
                                     
                                       g 
                                       
                                         o 
                                          
                                         
                                             
                                         
                                          
                                         2 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               v 
                               
                                 o 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                             
                           
                         
                         
                           
                             
                               v 
                               
                                 o 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   = 
                   
                     - 
                     
                       
                         
                           zg 
                           
                             m 
                              
                             
                                 
                             
                              
                             1 
                           
                         
                          
                         
                           [ 
                           
                             
                               
                                 1 
                               
                               
                                 
                                   - 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   1 
                                 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 v 
                                 
                                   i 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                             
                           
                           
                             
                               
                                 v 
                                 
                                   i 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   74 
                   ) 
                 
               
             
           
         
       
     
         [0099]    Both sides of the matrix equation are multiplied by y to obtain Equation (75): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               y 
                               + 
                               
                                 ( 
                                 
                                   
                                     g 
                                     
                                       o 
                                        
                                       
                                           
                                       
                                        
                                       1 
                                     
                                   
                                   + 
                                   
                                     g 
                                     
                                       o 
                                        
                                       
                                           
                                       
                                        
                                       2 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           
                             
                               g 
                               
                                 m 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                             
                           
                         
                         
                           
                             
                               g 
                               
                                 m 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                             
                           
                           
                             
                               y 
                               + 
                               
                                 ( 
                                 
                                   
                                     g 
                                     
                                       o 
                                        
                                       
                                           
                                       
                                        
                                       1 
                                     
                                   
                                   + 
                                   
                                     g 
                                     
                                       o 
                                        
                                       
                                           
                                       
                                        
                                       2 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                       ] 
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               v 
                               
                                 o 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                             
                           
                         
                         
                           
                             
                               v 
                               
                                 o 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   = 
                   
                     - 
                     
                       
                         
                           g 
                           
                             m 
                              
                             
                                 
                             
                              
                             1 
                           
                         
                          
                         
                           [ 
                           
                             
                               
                                 1 
                               
                               
                                 
                                   - 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   1 
                                 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 v 
                                 
                                   i 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                             
                           
                           
                             
                               
                                 v 
                                 
                                   i 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   75 
                   ) 
                 
               
             
           
         
       
     
         [0100]    This is a basic input/output equation of the enhanced Nauta OTA. Using a capacitive load (a capacitor) as load admittance, Equation 75 is Laplace transformed into Equation 76: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               sC 
                               + 
                               
                                 ( 
                                 
                                   
                                     g 
                                     
                                       o 
                                        
                                       
                                           
                                       
                                        
                                       1 
                                     
                                   
                                   + 
                                   
                                     g 
                                     
                                       o 
                                        
                                       
                                           
                                       
                                        
                                       2 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           
                             
                               g 
                               
                                 m 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                             
                           
                         
                         
                           
                             
                               g 
                               
                                 m 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                             
                           
                           
                             
                               sC 
                               + 
                               
                                 ( 
                                 
                                   
                                     g 
                                     
                                       o 
                                        
                                       
                                           
                                       
                                        
                                       1 
                                     
                                   
                                   + 
                                   
                                     g 
                                     
                                       o 
                                        
                                       
                                           
                                       
                                        
                                       2 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                       ] 
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               V 
                               
                                 o 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                             
                           
                         
                         
                           
                             
                               V 
                               
                                 o 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   = 
                   
                     - 
                     
                       
                         
                           g 
                           
                             m 
                              
                             
                                 
                             
                              
                             1 
                           
                         
                          
                         
                           [ 
                           
                             
                               
                                 1 
                               
                               
                                 
                                   - 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   1 
                                 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 V 
                                 
                                   i 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                             
                           
                           
                             
                               
                                 V 
                                 
                                   i 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   76 
                   ) 
                 
               
             
           
         
       
     
         [0101]    Both sides of Equation 76 are divided by C to obtain Equations 77 and 78: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               s 
                               + 
                               
                                 
                                   ( 
                                   
                                     
                                       g 
                                       
                                         o 
                                          
                                         
                                             
                                         
                                          
                                         1 
                                       
                                     
                                     + 
                                     
                                       g 
                                       
                                         o 
                                          
                                         
                                             
                                         
                                          
                                         2 
                                       
                                     
                                   
                                   ) 
                                 
                                 C 
                               
                             
                           
                           
                             
                               
                                 g 
                                 
                                   m 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                               
                               C 
                             
                           
                         
                         
                           
                             
                               
                                 g 
                                 
                                   m 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                               
                               C 
                             
                           
                           
                             
                               s 
                               + 
                               
                                 
                                   ( 
                                   
                                     
                                       g 
                                       
                                         o 
                                          
                                         
                                             
                                         
                                          
                                         1 
                                       
                                     
                                     + 
                                     
                                       g 
                                       
                                         o 
                                          
                                         
                                             
                                         
                                          
                                         2 
                                       
                                     
                                   
                                   ) 
                                 
                                 C 
                               
                             
                           
                         
                       
                       ] 
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               V 
                               
                                 o 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                             
                           
                         
                         
                           
                             
                               V 
                               
                                 o 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   = 
                   
                     - 
                     
                       
                         
                           
                             g 
                             
                               m 
                                
                               
                                   
                               
                                
                               1 
                             
                           
                           C 
                         
                          
                         
                           [ 
                           
                             
                               
                                 1 
                               
                               
                                 
                                   - 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   1 
                                 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 V 
                                 
                                   i 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                             
                           
                           
                             
                               
                                 V 
                                 
                                   i 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   77 
                   ) 
                 
               
             
           
         
       
     
         [0000]    and 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           { 
                           
                             sI 
                             - 
                             
                               
                                 ( 
                                 
                                   - 
                                   
                                     1 
                                     C 
                                   
                                 
                                 ) 
                               
                                
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             g 
                                             
                                               o 
                                                
                                               
                                                   
                                               
                                                
                                               1 
                                             
                                           
                                           + 
                                           
                                             g 
                                             
                                               o 
                                                
                                               
                                                   
                                               
                                                
                                               2 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                     
                                       
                                         g 
                                         
                                           m 
                                            
                                           
                                               
                                           
                                            
                                           2 
                                         
                                       
                                     
                                   
                                   
                                     
                                       
                                         g 
                                         
                                           m 
                                            
                                           
                                               
                                           
                                            
                                           2 
                                         
                                       
                                     
                                     
                                       
                                         ( 
                                         
                                           
                                             g 
                                             
                                               o 
                                                
                                               
                                                   
                                               
                                                
                                               1 
                                             
                                           
                                           + 
                                           
                                             g 
                                             
                                               o 
                                                
                                               
                                                   
                                               
                                                
                                               2 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                           } 
                         
                          
                       
                       A 
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               V 
                               
                                 o 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                             
                           
                         
                         
                           
                             
                               V 
                               
                                 o 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   = 
                   
                     - 
                     
                       
                         
                           
                             g 
                             
                               m 
                                
                               
                                   
                               
                                
                               1 
                             
                           
                           C 
                         
                          
                         
                           [ 
                           
                             
                               
                                 1 
                               
                               
                                 
                                   - 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   1 
                                 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 V 
                                 
                                   i 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                             
                           
                           
                             
                               
                                 V 
                                 
                                   i 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   78 
                   ) 
                 
               
             
           
         
       
     
         [0102]    If v i1 =v i2 =0(homogeneous), Equation 78 becomes: 
         [0000]      ( sI−A ) v   o =0  (79) 
         [0000]                AV o =sV o   (80) 
         [0000]    Now we again encounter an issue of an eigenvalue/eigenvector in the form of A  x &gt;=λ  x . An eigenvalue/eigenvector of the matrix A must be obtained for a transformation matrix for diagonalizing the matrix A. The eigenvalue can be obtained by setting the determinant of the coefficient matrix at the left side of Equation 79 to 0. Since A+=A, it can be seen that A is a Hermitian operator or a self-adjoint operator. Accordingly, from the general characteristic of the Hermitian operator, the eigenvalues are expected to have real numbers and the eigenvectors are expected to be orthogonal. Actually, the eigenvalues are found to be real numbers: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       s 
                       = 
                       
                         s 
                         1 
                       
                     
                     , 
                     
                       
                         s 
                         2 
                       
                        
                       
                           
                       
                        
                       where 
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       s 
                       1 
                     
                     = 
                     
                       - 
                       
                         
                           { 
                           
                             
                               ( 
                               
                                 
                                   g 
                                   
                                     o 
                                      
                                     
                                         
                                     
                                      
                                     1 
                                   
                                 
                                 + 
                                 
                                   g 
                                   
                                     o 
                                      
                                     
                                         
                                     
                                      
                                     2 
                                   
                                 
                               
                               ) 
                             
                             - 
                             
                               g 
                               
                                 m 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                             
                           
                           } 
                         
                         C 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       s 
                       2 
                     
                     = 
                     
                       - 
                       
                         
                           { 
                           
                             
                               ( 
                               
                                 
                                   g 
                                   
                                     o 
                                      
                                     
                                         
                                     
                                      
                                     1 
                                   
                                 
                                 + 
                                 
                                   g 
                                   
                                     o 
                                      
                                     
                                         
                                     
                                      
                                     2 
                                   
                                 
                               
                               ) 
                             
                             + 
                             
                               g 
                               
                                 m 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                             
                           
                           } 
                         
                         C 
                       
                     
                   
                 
               
               
                 
                   ( 
                   81 
                   ) 
                 
               
             
           
         
       
     
         [0103]    Since 
         [0000]    
       
         
           
             
               
                 g 
                 o 
               
               = 
               
                 1 
                 
                   r 
                   out 
                 
               
             
             , 
           
         
       
     
         [0000]    s 1  and s 2  have a dimension of a reciprocal of a time constant in an RC circuit. Corresponding eigenvectors may be obtained by applying s 1  and s 2  to Equation 79. The magnitudes of the eigenvectors are normalized into 1, resulting in Equations 82 and 83: 
         [0000]    
       
         
           
             
               
                 
                   | 
                   
                     
                       r 
                       1 
                     
                     &gt;= 
                     
                       
                         1 
                         
                           2 
                         
                       
                        
                       
                         ( 
                         
                           
                             
                               1 
                             
                           
                           
                             
                               
                                 - 
                                 1 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   82 
                   ) 
                 
               
             
           
         
       
     
         [0000]    and 
         [0000]    
       
         
           
             
               
                 
                   | 
                   
                     
                       r 
                       2 
                     
                     &gt;= 
                     
                       
                         1 
                         
                           2 
                         
                       
                        
                       
                         ( 
                         
                           
                             
                               1 
                             
                           
                           
                             
                               1 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   83 
                   ) 
                 
               
             
           
         
       
     
         [0000]    These are the same as Equations 17 and 18. Accordingly, the transformation matrix R is given by Equation 19. (Here, |r i &gt; denotes a column vector and &lt;r j | denotes a row vector.) As mentioned above, since the two unit vectors have a relationship where &lt;r i |r j &gt;=δ ij , they are orthogonal. Now, Equations 35 to 38, which are linear transformation equations for the input signal pair and the output signal pair using the transformation matrix R induced above, are applied to Equation 78 to obtain Equation 84: 
         [0000]    
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             D 
                              
                             
                                 
                             
                              
                             
                               M 
                               i 
                             
                           
                         
                       
                       
                         
                           
                             C 
                              
                             
                                 
                             
                              
                             
                               M 
                               i 
                             
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       
                         ( 
                         
                           2 
                            
                           
                             g 
                             m 
                           
                         
                         ) 
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 
                                   - 
                                   1 
                                 
                                 
                                   
                                     sC 
                                     1 
                                   
                                   + 
                                   
                                     ( 
                                     
                                       
                                         g 
                                         
                                           0 
                                            
                                           
                                               
                                           
                                            
                                           1 
                                         
                                       
                                       + 
                                       
                                         g 
                                         02 
                                       
                                     
                                     ) 
                                   
                                   - 
                                   
                                     g 
                                     
                                       m 
                                        
                                       
                                           
                                       
                                        
                                       2 
                                     
                                   
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                           
                         
                         ] 
                       
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               D 
                                
                               
                                   
                               
                                
                               
                                 M 
                                 o 
                               
                             
                           
                         
                         
                           
                             
                               C 
                                
                               
                                   
                               
                                
                               
                                 M 
                                 o 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   84 
                   ) 
                 
               
             
           
         
       
     
         [0104]    In contrast to Case 2 where both the quality factor and the cutoff frequency are adjusted by same g m , in Case 3 the cutoff frequency is adjusted by g m1  and the quality factor is adjusted by g m2 , as indicated by Equation 84, thereby achieving independent adjustment of the quality factor. 
         [0105]      FIG. 5  shows a voltage difference-current converting cell that may be used to configure the OTA according to the exemplary embodiment of the present invention. The voltage difference-current converting cell is implemented by a 2 input−1 output differential amplifier consisting of five Metal-Oxide-Semiconductor (MOS) transistors. 
         [0106]    Specifically, the voltage difference-current converting cell comprises: a pair of mirror transistors MT 1  and MT 2  for forcibly equalizing amounts of current flowing through two current paths; a positive and negative input transistors IT+ and IT− for establishing the two current paths and receiving a positive input signal and a negative input signal at their gates, respectively; and a bias transistor BT for receiving a bias voltage and supplying the same DC current to the two input transistors IT+ and IT−. 
         [0107]    Since the present invention may be implemented by the enhanced Nauta OTA consisting of the voltage difference-current converting cell of the 2 input−1 output structure, the present invention is not limited to the exemplary embodiment shown in  FIG. 5 . 
         [0108]    As described above, in the enhanced OTA according to the present invention, a common mode signal component which has been a problem in a circuit structure of a conventional Nauta OTA can be perfectly eliminated. 
         [0109]    According to the present invention, the OTA has an effectively enhanced quality factor by independently controlling the quality factor without affecting a transconductance value required for frequency characteristics of the filter. 
         [0110]    According to the present invention, the OTA can use a smaller number of basic unit converting cells than a conventional Nauta OTA to achieve these characteristics. 
         [0111]    According to the present invention, the OTA has a two times higher transfer gain between the input and the output in the differential mode than a conventional Nauta OTA. 
         [0112]    While the invention has been shown and described with reference to certain exemplary embodiments thereof, it will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the spirit and scope of the invention as defined by the appended claims.