Abstract:
An oversampling A/D converter with a few operational amplifiers is configured using a complex second-order integrator including first and second second-order integrators and first and second coupling circuits configured to couple these integrators together. Each of the second-order integrators includes an operational amplifier, four resistance elements, and three capacitance elements. The first coupling circuit cross-couples one of two serially-connected capacitance elements inserted between the inverted input terminal and output terminal of the operational amplifier in the first second-order integrator to the counterpart in the second second-order integrator using two resistance elements. The second coupling circuit cross-couples the other capacitance element in the first second-order integrator to the counterpart in the second second-order integrator using two resistance elements.

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
       [0001]    This is a continuation of PCT International Application PCT/JP2010/004390 filed on Jul. 5, 2010, which claims priority to Japanese Patent Application No. 2009-225133 filed on Sep. 29, 2009. The disclosures of these applications including the specifications, the drawings, and the claims are hereby incorporated by reference in their entirety. 
     
    
     BACKGROUND 
       [0002]    The present disclosure relates to a complex second-order integrator, and more particularly to a complex second-order integrator suitable for a continuous-time ΔΣ modulator, etc. 
         [0003]    Oversampling A/D conversion, which is in widespread use in the front-end of communications equipment, conversion of audio signals, etc., is a circuitry technology essential for the current communications, video, and audio signal processing circuits. One type of oversampling A/D converters is a continuous-time delta-sigma (ΔΣ) A/D converter (CTDS-ADC) having a continuous-time filter. 
         [0004]    In a general CTDS-ADC, an input signal passes through n-cascaded integrators (continuous-time filters) and the signal is quantized by a quantizer. The digital output of the quantizer is fed back to the n integrators after being converted to an analog current signal by n D/A converters. In the CTDS-ADC, having no switch in its analog circuit portion, the voltage can be reduced. Also, it is unnecessary to place a prefilter that is normally necessary when a sampling filter is used. Having these features, the CTDS-ADC is suitable for applications to communications systems, and thus recently application development and research have been actively conducted. 
         [0005]    In communications equipment, etc., a complex filter is often used for removal of an image signal. A typical complex filter has a configuration where integrators respectively receiving I signal and Q signal displaced 90° in phase from each other are coupled together with a coupling circuit (see U.S. Pat. No. 4,914,408, for example). While a coupling circuit is generally formed of active elements such as transistors, a coupling circuit that couples first-order integrators together can be formed of passive elements, specifically resistance elements, without use of active elements (see Jan Crols and Michiel Steyaert, “An Analog Integrated Polyphase Filter for a High Performance Low-IF Receiver,” Digest of Technical Papers, Symposium on VLSI Circuit, pp. 87-88, 1995, for example). 
         [0006]    The inventor of the present disclosure has found the following problems on the conventional CTDS-ADC. In order to improve the resolution and SN performance of the CTDS-ADC, the filter order for removal of quantization noise must be increased, and this necessitates operational amplifiers of the number corresponding to the increased filter order. Moreover, when it is intended to implement a CTDS-ADC provided with a complex coefficient, the number of operational amplifiers must be doubled. As described above, a coupling circuit that couples first-order integrators together can be formed without use of any operational amplifier. However, it is unknown whether a coupling circuit that couples high-order integrators each including one operational amplifier can be formed without use of any active element. 
         [0007]    In other words, to improve further the performance of the CTDS-ADC by putting the CTDS-ADC in a complex form, a number of operational amplifiers must be used. However, increase in the number of operational amplifiers will increase the circuit scale and the power consumption, causing a bottleneck in improving the performance of system LSIs applied to mobile communications equipment, etc. 
       SUMMARY 
       [0008]    According to the present disclosure, a complex second-order integrator can be implemented with a reduced number of operational amplifiers. Moreover, according to the present disclosure, a high-order continuous-time oversampling A/D converter having such a complex second-order integrator can be implemented. 
         [0009]    The complex second-order integrator of an example of the present disclosure includes: first and second second-order integrators; and first and second coupling circuits configured to couple the first and second second-order integrators together. Each of the first and second second-order integrators includes an operational amplifier, a first resistance element connected between a first node and a signal input terminal of the second-order integrator, a second resistance element connected between the first node and an inverted input terminal of the operational amplifier, a first capacitance element connected between the first node and a ground, a second capacitance element connected between a second node and the inverted input terminal of the operational amplifier, a third capacitance element connected between the second node and an output terminal of the operational amplifier, a third resistance element connected between the second node and the ground, and a fourth resistance element connected between the first node and the ground. The first coupling circuit includes a fifth resistance element configured to couple the output terminal of the operational amplifier of the first second-order integrator to the second node in the second second-order integrator with the polarity mutually inverted, and a sixth resistance element configured to couple the output terminal of the operational amplifier of the second second-order integrator to the second node in the first second-order integrator with no polarity inversion involved. The second coupling circuit includes a seventh resistance element configured to couple the inverted input terminal of the operational amplifier of the second second-order integrator to the second node in the first second-order integrator with the polarity mutually inverted, and an eighth resistance element configured to couple the inverted input terminal of the operational amplifier of the first second-order integrator to the second node in the second second-order integrator with no polarity inversion involved. 
         [0010]    With the above configuration, only one operational amplifier is used in each of the first and second second-order integrators, and the first and second coupling circuits are made of resistance elements, i.e., passive elements. Therefore, a complex second-order integrator exhibiting the second-order complex integral characteristic can be implemented with only two operational amplifiers in total. 
         [0011]    The complex second-order integrator may further include a third coupling circuit configured to couple the first and second second-order integrators together. Each of the first and second second-order integrators may include a fourth capacitance element connected between the second node and the signal input terminal of the second-order integrator, and a ninth resistance element connected between the second node and the signal input terminal of the second-order integrator. The third coupling circuit may include a tenth resistance element configured to couple the signal input terminal of the first second-order integrator to the second node in the second second-order integrator with the polarity mutually inverted, and an eleventh resistance element configured to couple the signal input terminal of the second second-order integrator to the second node in the first second-order integrator with no polarity inversion involved. 
         [0012]    With the above configuration, an arbitrary transfer characteristic represented by (α(s−δ) 2 +β(s−δ)+γ)/(s−δ) 2  (where s is a Laplace operator and α, β, γ, and δ are constants determined by the values of the elements) can be achieved. 
         [0013]    Preferably, an overall admittance where elements connected to the first node are in parallel connection is equal to an overall admittance where elements connected to the second node are in parallel connection. With this configuration, the complex second-order integrator functions as a complex second-order resonator without the necessity of providing a resistive path between the first node and the output terminal of the operational amplifier in each of the first and second second-order integrators. 
         [0014]    The oversampling A/D converter of an example of the present disclosure includes at least one complex second-order integrator described above. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0015]      FIG. 1  is a block diagram of a complex second-order integrator of a first embodiment. 
           [0016]      FIG. 2  is a block diagram of a complex second-order integrator of a second embodiment. 
           [0017]      FIG. 3  is a block diagram of an oversampling A/D converter of a third embodiment. 
           [0018]      FIG. 4  is a graph showing a quantization noise transfer characteristic of the oversampling A/D converter of  FIG. 3 . 
       
    
    
     DETAILED DESCRIPTION 
     First Embodiment  
       [0019]      FIG. 1  shows a configuration of a complex second-order integrator of the first embodiment. A complex second-order integrator  100  of this embodiment includes: a second-order integrator  100   I  that receives a signal V inI  and outputs a signal V outI : a second-order integrator  100   Q  that receives a signal V inQ  displaced 90° in phase from the signal V inI  and outputs a signal V outQ  displaced 90° in phase from the signal V outQ ; and two coupling circuits  30  and  40  that couple the second-order integrators  100   I  and  100   Q  together. 
         [0020]    The second-order integrators  100   I  and  100   Q  each can be formed using one operational amplifier  10 . Specifically, an input-part filter having resistance elements  11 ,  12 , and  14  and a capacitance element  21  is provided between the signal input terminal and the inverted input terminal of the operational amplifier  10 . One terminal of each of these elements is connected to a node  101 , the other terminals of the resistance elements  11  and  12  are respectively connected to the signal input terminal and the inverted input terminal of the operational amplifier  10 , and the other terminals of the resistance element  14  and the capacitance element  21  are grounded. A feedback-part filter having capacitance elements  22  and  23  and a resistance element  13  is provided between the inverted input terminal and output terminal of the operational amplifier  10 . One terminal of each of these elements is connected to a node  102 , the other terminals of the capacitance elements  22  and  23  are respectively connected to the inverted input terminal and output terminal of the operational amplifier  10 , and the other terminal of the resistance element  13  is grounded. 
         [0021]    The coupling circuit  30  cross-couples the capacitance elements  23  of the second-order integrators  100   I  and  100   Q  together. Specifically, the coupling circuit  30  can be formed of resistance elements  31  and  32 . The resistance element  31  couples one of the two terminals of the capacitance element  23  of the second-order integrator  100   I  that is connected to the output terminal of the operational amplifier  10  to one of the two terminals of the capacitance element  23  of the second-order integrator  100   Q  that is connected to the node  102  with the polarity mutually inverted. The resistance element  32  couples the other terminal of the capacitance element  23  of the second-order integrator  100   Q  that is connected to the output terminal of the operational amplifier  10  to the other terminal of the capacitance element  23  of the second-order integrator  100   I  that is connected to the node  102  with no polarity inversion involved. 
         [0022]    The coupling circuit  40  cross-couples the capacitance elements  22  of the second-order integrators  100   I  and  100   Q  together. Specifically, the coupling circuit  40  can be formed of resistance elements  41  and  42 . The resistance element  41  couples one of the terminals of the capacitance element  22  of the second-order integrator  100   I  that is connected to the node  102  to one of the terminals of the capacitance element  22  of the second-order integrator  100   Q  that is connected to the inverted input terminal of the operational amplifier  10  with the polarity mutually inverted. The resistance element  42  couples the other terminal of the capacitance element  22  of the second-order integrator  100   Q  that is connected to the node  102  to the other terminal of the capacitance element  22  of the second-order integrator  100   I  that is connected to the inverted input terminal of the operational amplifier  10  with no polarity inversion involved. 
         [0023]    In the complex second-order integrator  100  of this embodiment, when the resistance values of the resistance elements  11 - 14  are respectively R 1 , R 2 , R 3 , and R 4 , the capacitance values of the capacitance elements  21 - 23  are C 1 , C 2 , and C 3 , the resistance values of the resistance elements  31  and  32  are R f1 , the resistance values of the resistance elements  41  and  42  are R f2 , the voltages at the nodes  101  and  102  in the second-order integrator  100   I  are V 1  and V 2 , and the voltages at the nodes  101  and  102  in the second-order integrator  100   Q  are V 3  and V 4 , the following nodal equations are satisfied. 
         [0000]    
       
         
           
               
             
               { 
               
                 
                   
                     
                       
                         
                           
                             ( 
                             
                               
                                 V 
                                 1 
                               
                               - 
                               
                                 V 
                                 inI 
                               
                             
                             ) 
                           
                           
                             R 
                             1 
                           
                         
                         + 
                         
                           
                             V 
                             1 
                           
                           
                             R 
                             2 
                           
                         
                         + 
                         
                           
                             V 
                             1 
                           
                           · 
                           
                             sC 
                             1 
                           
                         
                         + 
                         
                           
                             V 
                             1 
                           
                           
                             R 
                             4 
                           
                         
                       
                       = 
                       0 
                     
                   
                 
                 
                   
                     
                       
                         
                           - 
                           
                             
                               V 
                               1 
                             
                             
                               R 
                               2 
                             
                           
                         
                         - 
                         
                           
                             V 
                             2 
                           
                           · 
                           
                             sC 
                             2 
                           
                         
                         - 
                         
                           
                             V 
                             4 
                           
                           
                             R 
                             
                               f 
                                
                               
                                   
                               
                                
                               2 
                             
                           
                         
                       
                       = 
                       0 
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             V 
                             2 
                           
                           · 
                           
                             sC 
                             2 
                           
                         
                         + 
                         
                           
                             V 
                             2 
                           
                           
                             R 
                             3 
                           
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 V 
                                 2 
                               
                               - 
                               
                                 V 
                                 outI 
                               
                             
                             ) 
                           
                           · 
                           
                             sC 
                             3 
                           
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 V 
                                 2 
                               
                               - 
                               
                                 V 
                                 outQ 
                               
                             
                             ) 
                           
                           
                             R 
                             f 
                           
                         
                         + 
                         
                           
                             V 
                             2 
                           
                           
                             R 
                             
                               f 
                                
                               
                                   
                               
                                
                               2 
                             
                           
                         
                       
                       = 
                       0 
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             ( 
                             
                               
                                 V 
                                 3 
                               
                               - 
                               
                                 V 
                                 inQ 
                               
                             
                             ) 
                           
                           
                             R 
                             1 
                           
                         
                         + 
                         
                           
                             V 
                             3 
                           
                           
                             R 
                             2 
                           
                         
                         + 
                         
                           
                             V 
                             3 
                           
                           · 
                           
                             sC 
                             1 
                           
                         
                         + 
                         
                           
                             V 
                             3 
                           
                           
                             R 
                             4 
                           
                         
                       
                       = 
                       0 
                     
                   
                 
                 
                   
                     
                       
                         
                           - 
                           
                             
                               V 
                               3 
                             
                             
                               R 
                               2 
                             
                           
                         
                         - 
                         
                           
                             V 
                             4 
                           
                           · 
                           
                             sC 
                             2 
                           
                         
                         + 
                         
                           
                             V 
                             2 
                           
                           
                             R 
                             
                               f 
                                
                               
                                   
                               
                                
                               2 
                             
                           
                         
                       
                       = 
                       0 
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             V 
                             4 
                           
                           · 
                           
                             sC 
                             2 
                           
                         
                         + 
                         
                           
                             V 
                             4 
                           
                           
                             R 
                             3 
                           
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 V 
                                 4 
                               
                               - 
                               
                                 V 
                                 outQ 
                               
                             
                             ) 
                           
                           · 
                           
                             sC 
                             3 
                           
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 V 
                                 4 
                               
                               + 
                               
                                 V 
                                 outI 
                               
                             
                             ) 
                           
                           
                             R 
                             f 
                           
                         
                         + 
                         
                           
                             V 
                             4 
                           
                           
                             R 
                             
                               f 
                                
                               
                                   
                               
                                
                               2 
                             
                           
                         
                       
                       = 
                       0 
                     
                   
                 
               
             
           
         
       
     
         [0000]    where “s” is a Laplace operator. 
         [0024]    Assuming that, in each of the second-order integrators  100   I  and  100   Q , the overall admittance where the elements connected to the node  101  are in parallel connection is equal to the overall admittance where the elements connected to the node  102  are in parallel connection, e.g., C 1 =C 2 +C 3 , 1/R 3 =1/R 1 +1/R 2 , and 1/R 4 =1/R f1 +1/R f2 , the following transfer function is derived for the complex second-order integrator  100 . 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         V 
                         outI 
                       
                       + 
                       
                         j 
                          
                         
                             
                         
                          
                         
                           V 
                           outQ 
                         
                       
                     
                     
                       
                         V 
                         inI 
                       
                       + 
                       
                         j 
                          
                         
                             
                         
                          
                         
                           V 
                           inQ 
                         
                       
                     
                   
                   = 
                   
                     
                       
                         R 
                         
                           f 
                            
                           
                               
                           
                            
                           1 
                         
                       
                        
                       
                         R 
                         
                           f 
                            
                           
                               
                           
                            
                           2 
                         
                       
                     
                     
                       
                         ( 
                         
                           
                             
                               sC 
                               2 
                             
                              
                             
                               R 
                               
                                 f 
                                  
                                 
                                     
                                 
                                  
                                 2 
                               
                             
                           
                           - 
                           j 
                         
                         ) 
                       
                        
                       
                         ( 
                         
                           
                             
                               sC 
                               3 
                             
                              
                             
                               R 
                               
                                 f 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                             
                           
                           - 
                           j 
                         
                         ) 
                       
                        
                       
                         R 
                         1 
                       
                        
                       
                         R 
                         2 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     1 
                     
                       
                         ( 
                         
                           s 
                           - 
                           
                             j 
                             
                               
                                 C 
                                 2 
                               
                                
                               
                                 R 
                                 
                                   f 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                               
                             
                           
                         
                         ) 
                       
                        
                       
                         ( 
                         
                           s 
                           - 
                           
                             j 
                             
                               
                                 C 
                                 3 
                               
                                
                               
                                 R 
                                 
                                   f 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                             
                           
                         
                         ) 
                       
                        
                       
                         R 
                         1 
                       
                        
                       
                         R 
                         2 
                       
                        
                       
                         C 
                         2 
                       
                        
                       
                         C 
                         3 
                       
                     
                   
                 
               
             
           
         
       
     
         [0025]    As is evident from the above transfer function, the frequency characteristic of the complex second-order integrator  100  shifts in the positive direction by 1/C 2 R f2  and 1/C 3 R f1 . In other words, the complex second-order integrator  100  has a transfer characteristic asymmetric with respect to the positive and negative frequencies, and exhibits a complex integral characteristic equivalent to the conventional complex second-order integrator that uses operational amplifiers heavily. 
       Second Embodiment  
       [0026]      FIG. 2  shows a configuration of a complex second-order integrator of the second embodiment. A complex second-order integrator  100  of this embodiment includes the following in addition to the components of the complex second-order integrator  100  of  FIG. 1 . In each of the second-order integrators  100   I  and  100   Q , a resistance element  15  and a capacitance element  24  are connected in parallel between the node  102  and the signal input terminal. In addition, a coupling circuit  50  is provided, which cross-couples the capacitance elements  24  of the second-order integrators  100   I  and  100   Q  together. A difference from the first embodiment will be described hereinafter. 
         [0027]    The coupling circuit  50  can be formed of resistance elements  51  and  52 . The resistance element  51  couples one of the two terminals of the capacitance element  24  of the second-order integrator  100   I  that is connected to the signal input terminal to one of the two terminals of the capacitance element  24  of the second-order integrator  100   Q  that is connected to the node  102  with the polarity mutually inverted. The resistance element  52  couples the other terminal of the capacitance element  24  of the second-order integrator  100   Q  that is connected to the signal input terminal to the other terminal of the capacitance element  24  of the second-order integrator  100   I  that is connected to the node  102  with no polarity inversion involved. 
         [0028]    In the complex second-order integrator  100  of this embodiment, when the resistance value of the resistance element  15  is R 5 , the capacitance value of the capacitance element  24  is C 4 , and the resistance values of the resistance elements  51  and  52  are R f3  additionally, the following nodal equations are satisfied. 
         [0000]    
       
         
           
               
             
               { 
               
                 
                   
                     
                       
                         
                           
                             ( 
                             
                               
                                 V 
                                 1 
                               
                               - 
                               
                                 V 
                                 inI 
                               
                             
                             ) 
                           
                           
                             R 
                             1 
                           
                         
                         + 
                         
                           
                             V 
                             1 
                           
                           
                             R 
                             2 
                           
                         
                         + 
                         
                           
                             V 
                             1 
                           
                           · 
                           
                             sC 
                             1 
                           
                         
                         + 
                         
                           
                             V 
                             1 
                           
                           
                             R 
                             4 
                           
                         
                       
                       = 
                       0 
                     
                   
                 
                 
                   
                     
                       
                         
                           - 
                           
                             
                               V 
                               1 
                             
                             
                               R 
                               2 
                             
                           
                         
                         - 
                         
                           
                             V 
                             2 
                           
                           · 
                           
                             sC 
                             2 
                           
                         
                         - 
                         
                           
                             V 
                             4 
                           
                           
                             R 
                             
                               f 
                                
                               
                                   
                               
                                
                               2 
                             
                           
                         
                       
                       = 
                       0 
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             V 
                             2 
                           
                           · 
                           
                             sC 
                             2 
                           
                         
                         + 
                         
                           
                             V 
                             2 
                           
                           
                             R 
                             3 
                           
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 V 
                                 2 
                               
                               - 
                               
                                 V 
                                 outI 
                               
                             
                             ) 
                           
                           · 
                           
                             sC 
                             3 
                           
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 V 
                                 2 
                               
                               - 
                               
                                 V 
                                 outQ 
                               
                             
                             ) 
                           
                           
                             R 
                             
                               f 
                                
                               
                                   
                               
                                
                               1 
                             
                           
                         
                         + 
                         
                           
                             V 
                             2 
                           
                           
                             R 
                             
                               f 
                                
                               
                                   
                               
                                
                               2 
                             
                           
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 V 
                                 2 
                               
                               - 
                               
                                 V 
                                 inI 
                               
                             
                             ) 
                           
                           · 
                           
                             ( 
                             
                               
                                 1 
                                 
                                   R 
                                   5 
                                 
                               
                               + 
                               
                                 sC 
                                 4 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 V 
                                 2 
                               
                               - 
                               
                                 V 
                                 inQ 
                               
                             
                             ) 
                           
                           
                             R 
                             
                               f 
                                
                               
                                   
                               
                                
                               3 
                             
                           
                         
                       
                       = 
                       0 
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             ( 
                             
                               
                                 V 
                                 3 
                               
                               - 
                               
                                 V 
                                 inQ 
                               
                             
                             ) 
                           
                           
                             R 
                             1 
                           
                         
                         + 
                         
                           
                             V 
                             3 
                           
                           
                             R 
                             2 
                           
                         
                         + 
                         
                           
                             V 
                             3 
                           
                           · 
                           
                             sC 
                             1 
                           
                         
                         + 
                         
                           
                             V 
                             3 
                           
                           
                             R 
                             4 
                           
                         
                       
                       = 
                       0 
                     
                   
                 
                 
                   
                     
                       
                         
                           - 
                           
                             
                               V 
                               3 
                             
                             
                               R 
                               2 
                             
                           
                         
                         - 
                         
                           
                             V 
                             4 
                           
                           · 
                           
                             sC 
                             2 
                           
                         
                         + 
                         
                           
                             V 
                             2 
                           
                           
                             R 
                             
                               f 
                                
                               
                                   
                               
                                
                               2 
                             
                           
                         
                       
                       = 
                       0 
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             V 
                             4 
                           
                           · 
                           
                             sC 
                             2 
                           
                         
                         + 
                         
                           
                             V 
                             4 
                           
                           
                             R 
                             3 
                           
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 V 
                                 4 
                               
                               - 
                               
                                 V 
                                 outQ 
                               
                             
                             ) 
                           
                           · 
                           
                             sC 
                             3 
                           
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 V 
                                 4 
                               
                               + 
                               
                                 V 
                                 outI 
                               
                             
                             ) 
                           
                           
                             R 
                             f 
                           
                         
                         + 
                         
                           
                             V 
                             4 
                           
                           
                             R 
                             
                               f 
                                
                               
                                   
                               
                                
                               2 
                             
                           
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 V 
                                 4 
                               
                               - 
                               
                                 V 
                                 inQ 
                               
                             
                             ) 
                           
                           · 
                           
                             ( 
                             
                               
                                 1 
                                 
                                   R 
                                   5 
                                 
                               
                               + 
                               
                                 sC 
                                 4 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 V 
                                 4 
                               
                               + 
                               
                                 V 
                                 inI 
                               
                             
                             ) 
                           
                           
                             R 
                             
                               f 
                                
                               
                                   
                               
                                
                               3 
                             
                           
                         
                       
                       = 
                       0 
                     
                   
                 
               
             
           
         
       
     
         [0029]    Assuming that, in each of the second-order integrators  100   I  and  100   Q , the overall admittance where the elements connected to the node  101  are in parallel connection is equal to the overall admittance where the elements connected to the node  102  are in parallel connection, e.g., C 1 =C 2 +C 3 +C 4 , 1/R 3 =1/R 1 +1/R 2 , and 1/R 4 =1/R f1 +1/R f2 +1/R f3 +1/R 5 , and moreover assuming that R f2 =C 3 /C 2 ×R f1 , R f3 =C 3 /C 4 ×R f1 , for simplicity, the following transfer function is derived for the complex second-order integrator  100 . 
         [0000]    
       
         
           
             
               
                 
                   V 
                   outI 
                 
                 + 
                 
                   j 
                    
                   
                       
                   
                    
                   
                     V 
                     outQ 
                   
                 
               
               
                 
                   V 
                   inI 
                 
                 + 
                 
                   j 
                    
                   
                       
                   
                    
                   
                     V 
                     inQ 
                   
                 
               
             
             = 
             
               
                 - 
                 
                   { 
                   
                     
                       
                         
                           
                             
                               
                                 C 
                                 4 
                               
                               
                                 C 
                                 3 
                               
                             
                             · 
                             
                               
                                 ( 
                                 
                                   s 
                                   - 
                                   
                                     j 
                                     
                                       
                                         C 
                                         3 
                                       
                                        
                                       
                                         R 
                                         
                                           f 
                                            
                                           
                                               
                                           
                                            
                                           1 
                                         
                                       
                                     
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                           + 
                           
                             
                               1 
                               
                                 
                                   C 
                                   3 
                                 
                                  
                                 
                                   R 
                                   5 
                                 
                               
                             
                             · 
                           
                         
                       
                     
                     
                       
                         
                           
                             ( 
                             
                               s 
                               - 
                               
                                 j 
                                 
                                   
                                     C 
                                     3 
                                   
                                    
                                   
                                     R 
                                     
                                       f 
                                        
                                       
                                           
                                       
                                        
                                       1 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                           + 
                           
                             1 
                             
                               
                                 R 
                                 1 
                               
                                
                               
                                 R 
                                 2 
                               
                                
                               
                                 C 
                                 2 
                               
                                
                               
                                 C 
                                 3 
                               
                             
                           
                         
                       
                     
                   
                   } 
                 
               
               
                 
                   ( 
                   
                     s 
                     - 
                     
                       j 
                       
                         
                           C 
                           3 
                         
                          
                         
                           R 
                           
                             f 
                              
                             
                                 
                             
                              
                             1 
                           
                         
                       
                     
                   
                   ) 
                 
                 2 
               
             
           
         
       
     
         [0030]    As is evident from the above transfer function, the frequency characteristic of the complex second-order integrator  100  is equal to that obtained by shifting the following transfer function H(s) of a general second-order integrator by 1/C 3 R f1  in the positive direction. 
         [0000]    
       
         
           
             
               H 
                
               
                 ( 
                 s 
                 ) 
               
             
             = 
             
               
                 - 
                 
                   { 
                   
                     
                       
                         
                           C 
                           4 
                         
                         
                           C 
                           3 
                         
                       
                       · 
                       
                         s 
                         2 
                       
                     
                     + 
                     
                       s 
                       
                         
                           C 
                           3 
                         
                          
                         
                           R 
                           5 
                         
                       
                     
                     + 
                     
                       1 
                       
                         
                           R 
                           1 
                         
                          
                         
                           R 
                           2 
                         
                          
                         
                           C 
                           2 
                         
                          
                         
                           C 
                           3 
                         
                       
                     
                   
                   } 
                 
               
               
                 s 
                 2 
               
             
           
         
       
     
         [0031]    In other words, the complex second-order integrator  100  has a transfer characteristic asymmetric with respect to the positive and negative frequencies, and exhibits a complex integral characteristic equivalent to the conventional complex second-order integrator that uses operational amplifiers heavily. Moreover, the terms of the numerator of the transfer function can be changed freely and independently from one another. For example, only the second-order term can be changed by changing the capacitance value C 4 , only the first-order term can be changed by changing the resistance value R 5 , and only the zero-order term can be changed by changing any one of the resistance values R 1  and R 2  and the capacitance value C 2 . 
         [0032]    Note that the complex second-order integrators  100  of the first and second embodiments can be transformed into complex second-order resonators by providing a resistive path between the node  101  and the output terminal of the operational amplifier  10  in each of the second-order integrators  100   I  and  100   Q . Note however that, since the complex second-order integrator  100  virtually acts as a resonator by setting the element values so that the overall admittance where the elements connected to the node  101  are in parallel connection is equal to the overall admittance where the elements connected to the node  102  are in parallel connection, it is unnecessary to take the trouble to provide a resistive path. 
       Third Embodiment  
       [0033]      FIG. 3  shows a configuration of a CTDS-ADC of the third embodiment. In  FIG. 3 , the reference character  110  denotes a general complex first-order integrator,  120  denotes an adder,  130  denotes a quantizer, and  140  denotes D/A converters (voltage-current converters). Each of complex second-order integrators  100  may be the one of the first embodiment or the second embodiment. By the cascade connection of the complex first-order integrator  110  and the two complex second-order integrators  100 , the CTDS-ADC of this embodiment exhibits the fifth-order complex integral characteristic.  FIG. 4  shows a quantization noise transfer characteristic of the CTDS-ADC of this embodiment. The CTDS-ADC of this embodiment has an asymmetric transfer characteristic shifted in the positive direction. Thus, in this embodiment, a CTDS-ADC having a high-order complex integral characteristic can be implemented with a reduced number of operational amplifiers.