PATENT DOCUMENT

Abstract:
A digital correction circuit for a pipelined analog-to-digital converter (ADC) is disclosed. Compared to the conventional digital correction circuit which uses adders to perform operations in ADC digital correction part and hence needs a rather long operation time, the digital correction circuit of this invention can reduce the time needed in operations in the finial digital correction circuits and thus can optimize operation time, by allocating the operations to a plurality of pipeline stages of second sub-circuits configured to synchronize digital codes, each of which can perform part of the operations only with NAND gates, NOR gates, phase inverters and D-type flip-flops, without needing to use adders.

Full Description:
CROSS-REFERENCES TO RELATED APPLICATIONS 
       [0001]    This application claims the priority of Chinese patent application number 201210088427.2, filed on Mar. 30, 2012, the entire contents of which are incorporated herein by reference. 
       TECHNICAL FIELD 
       [0002]    This invention relates in general to semiconductor integrated circuits, and more particularly, to a digital correction circuit for a pipelined analog-to-digital converter (ADC). 
       BACKGROUND 
       [0003]    A typical pipelined analog-to-digital converter (ADC) includes a plurality of cascaded sub-circuit stages each of which converts a portion of an analog input signal into one or more digital bits. In a pipelined ADC constructed based on 1.5-bit multiplying analog-to-digital converters (MADCs), except for the sub-circuit of a last stage which includes a 2-bit parallel ADC without a redundancy bit, each of the rest sub-circuit stages includes a 1.5-bit MADC which outputs a 2-bit code that may be a significant code “00”, “01” or “10”, or a redundancy code “11”. For this reason, conventional technologies generally further employ a correction circuit for eliminating outputs of redundancy code(s) and thus correcting outputs of the 1.5-bit MADCs. 
         [0004]      FIG. 1  shows a redundancy elimination algorithm for correcting outputs of pipelined 1.5-bit MADCs. As illustrated, a totaling of 9 stages of codes are output from the sub-circuits of the pipelined ADC, including a code consisting of bits D 18  and D 17  of an upmost stage labeled “stage 9”, a code consisting of bits D 16  and D 15  of a next stage labeled “stage 8”, . . . , and a code consisting of bits D 2  and D 1  of a bottom stage labeled “stage 1”. After performing shift-and-add operations to these codes, 10 quantized bits Q 1 , Q 2 , . . . , and Q 10  can be obtained, wherein C 1 , C 2 , . . . , and C 7  are carry bits of the shift-and-add operations. 
         [0005]      FIG. 2  is a schematic illustration of a conventional digital correction circuit for eliminating redundancy bits. As illustrated, this digital correction circuit includes a data alignment circuit formed by interconnecting a number of D-type flip-flops and a shift-and-add circuit consisting of adders. The data alignment circuit can perform different delays to the codes from different stages so as to synchronize them upon their arrival at the shift-and-add circuit. In addition, the adders can perform shift-and-add operations to the codes from different stages and thereby output quantized bits. 
         [0006]    In the conventional digital correction circuit for eliminating redundancy bits, as the adders generally require the involvement of a rather great number of gate stages, the operations always require a long operation time. 
       SUMMARY OF THE INVENTION 
       [0007]    The present invention is to provide a digital correction circuit for a pipelined ADC to optimize the operation time. 
         [0008]    To achieve the above objective, the present invention provides a digital correction circuit for a pipelined ADC, the pipelined ADC including N stages of first sub-circuits, each of which outputting a 2-bit code, the 2-bit code output by the n-th-stage of first sub-circuit being expressed as D 2(N+1-n)  and D 2(N+1-n)-1 , where N is an integer greater than 3 and n is an integer satisfying 1≦n≦N, the digital correction circuit comprising N stages of second sub-circuits, among which: the first-stage of second sub-circuit is configured to receive the 2-bit code D 2N  and D 2N−1  output by the first-stage of first sub-circuit; the i-th-stage of second sub-circuit is configured to receive both the 2-bit code D 2(N+1-i)  and D 2(N+1-i)-1  output by the i-th-stage of first sub-circuit and an output from the (i−1)-th-stage of second sub-circuit, where is an integer satisfying 2≦i≦N−1; and the N-th-stage of second sub-circuit is configured to receive D 2 ; wherein the digital correction circuit outputs N+1 quantized bits, including a first quantized bit Q 1  that equals to D 1  and second to (N+1)-th quantized bits Q 2  to Q N+1  that are output by the N-th-stage of second sub-circuit, among which: 
         [0009]    the quantized bit Q N+1  is expressed as Q N+1 =A N+1,2 +B N+1,2 D 2 , where A N+1,2  and B N+1,2  are coefficients of an (N−1)-th-stage of intermediate value that is output by the (N−1)-th-stage of second sub-circuit and corresponds to the quantized bit Q N+1 , coefficients of the respective stages of intermediate values that are output by the respective stages of second sub-circuits and correspond to the quantized bit Q N+1  satisfying the following recurrence relations: 
         [0000]    
       
         
           
             
               A 
               
                 
                   N 
                   + 
                   1 
                 
                 , 
                 m 
               
             
             = 
             
               { 
               
                 
                   
                     
                       
                         
                           
                             A 
                             
                               
                                 N 
                                 + 
                                 1 
                               
                               , 
                               
                                 m 
                                 + 
                                 1 
                               
                             
                           
                           + 
                           
                             
                               B 
                               
                                 
                                   N 
                                   + 
                                   1 
                                 
                                 , 
                                 
                                   m 
                                   + 
                                   1 
                                 
                               
                             
                              
                             
                               D 
                               
                                 2 
                                  
                                 m 
                               
                             
                           
                         
                       
                       
                         
                           ( 
                           
                             
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                               = 
                               2 
                             
                             , 
                             3 
                             , 
                             … 
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                             , 
                             
                               N 
                               - 
                               2 
                             
                           
                           ) 
                         
                       
                     
                     
                       
                         
                           
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                                   2 
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                                   N 
                                 
                                 - 
                                 2 
                               
                             
                           
                         
                       
                       
                         
                           ( 
                           
                             m 
                             = 
                             
                               N 
                               - 
                               1 
                             
                           
                           ) 
                         
                       
                     
                   
                    
                   
                     
 
                   
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                     B 
                     
                       
                         N 
                         + 
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                       , 
                       m 
                     
                   
                 
                 = 
                 
                   { 
                   
                     
                       
                         
                           
                             B 
                             
                               
                                 N 
                                 + 
                                 1 
                               
                               , 
                               
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                                 + 
                                 1 
                               
                             
                           
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                               - 
                               1 
                             
                           
                         
                       
                       
                         
                           ( 
                           
                             
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                               = 
                               3 
                             
                             , 
                             4 
                             , 
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                             , 
                             
                               N 
                               - 
                               2 
                             
                           
                           ) 
                         
                       
                     
                     
                       
                         
                           
                             B 
                             
                               
                                 N 
                                 + 
                                 1 
                               
                               , 
                               3 
                             
                           
                            
                           
                             D 
                             3 
                           
                         
                       
                       
                         
                           ( 
                           
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                             = 
                             2 
                           
                           ) 
                         
                       
                     
                     
                       
                         
                           
                             D 
                             
                               
                                 2 
                                  
                                 N 
                               
                               - 
                               1 
                             
                           
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                                 2 
                               
                             
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                            
                           
                             D 
                             
                               
                                 2 
                                  
                                 N 
                               
                               - 
                               3 
                             
                           
                         
                       
                       
                         
                           
                             ( 
                             
                               m 
                               = 
                               
                                 N 
                                 - 
                                 1 
                               
                             
                             ) 
                           
                           ; 
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0010]    the quantized bit Q k  is expressed as Q k =A k,2   D   2 +B k,2 +E k,2 D 2 , where k is an integer satisfying 3≦k≦N, A k,2 , B k,2 , and E k,2  are coefficients of an (N−1)-th-stage of intermediate value that is output by the (N−1)-th-stage of second sub-circuit and corresponds to the quantized bit Q k , coefficients of the respective stages of intermediate values that are output by the respective stages of second sub-circuits and correspond to the quantized bit Q k  satisfying the following recurrence relations: 
         [0000]    
       
         
           
             
               A 
               
                 k 
                 , 
                 m 
               
             
             = 
             
               { 
               
                 
                   
                     
                       
                         
                           
                             D 
                             
                               
                                 2 
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                                 m 
                               
                               + 
                               1 
                             
                           
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                               D 
                               
                                 2 
                                  
                                 m 
                               
                             
                             _ 
                           
                         
                       
                       
                         
                           ( 
                           
                             m 
                             = 
                             
                               k 
                               - 
                               1 
                             
                           
                           ) 
                         
                       
                     
                     
                       
                         
                           
                             A 
                             
                               k 
                               , 
                               
                                 m 
                                 + 
                                 1 
                               
                             
                           
                            
                           
                             
                               D 
                               
                                 2 
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                                 m 
                               
                             
                             _ 
                           
                         
                       
                       
                         
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                               m 
                               = 
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                             , 
                             3 
                             , 
                             
                               
                                 … 
                                  
                                 
                                     
                                 
                                  
                                 k 
                               
                               - 
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                    
                   
                     
 
                   
                    
                   
                     B 
                     
                       k 
                       , 
                       m 
                     
                   
                 
                 = 
                 
                   { 
                   
                     
                       
                         
                           
                             
                               
                                 
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                                       2 
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                                     1 
                                   
                                 
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                                     D 
                                     
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                                       m 
                                     
                                   
                                   _ 
                                 
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                                         m 
                                       
                                       - 
                                       1 
                                     
                                   
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                               + 
                               
                                 
                                   
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                                         2 
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                                         m 
                                       
                                       + 
                                       1 
                                     
                                   
                                   _ 
                                 
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                                   D 
                                   
                                     2 
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                                     m 
                                   
                                 
                               
                             
                           
                           
                             
                               ( 
                               
                                 m 
                                 = 
                                 
                                   k 
                                   - 
                                   1 
                                 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   A 
                                   
                                     k 
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                                       m 
                                       + 
                                       1 
                                     
                                   
                                 
                                  
                                 
                                   
                                     D 
                                     
                                       2 
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                                       m 
                                     
                                   
                                   _ 
                                 
                                  
                                 
                                   
                                     D 
                                     
                                       
                                         2 
                                          
                                         m 
                                       
                                       - 
                                       1 
                                     
                                   
                                   _ 
                                 
                               
                               + 
                               
                                 B 
                                 
                                   k 
                                   , 
                                   
                                     m 
                                     + 
                                     1 
                                   
                                 
                               
                               + 
                               
                                 
                                   E 
                                   
                                     k 
                                     , 
                                     
                                       m 
                                       + 
                                       1 
                                     
                                   
                                 
                                  
                                 
                                   D 
                                   
                                     2 
                                      
                                     m 
                                   
                                 
                               
                             
                           
                           
                             
                               ( 
                               
                                 
                                   m 
                                   = 
                                   2 
                                 
                                 , 
                                 3 
                                 , 
                                 
                                   
                                     … 
                                      
                                     
                                         
                                     
                                      
                                     k 
                                   
                                   - 
                                   2 
                                 
                               
                               ) 
                             
                           
                         
                       
                        
                       
                         
 
                       
                        
                       
                         E 
                         
                           k 
                           , 
                           m 
                         
                       
                     
                     = 
                     
                       { 
                       
                         
                           
                             
                               
                                 
                                   D 
                                   
                                     
                                       2 
                                        
                                       m 
                                     
                                     + 
                                     1 
                                   
                                 
                                 _ 
                               
                                
                               
                                 
                                   D 
                                   
                                     2 
                                      
                                     m 
                                   
                                 
                                 _ 
                               
                                
                               
                                 D 
                                 
                                   
                                     2 
                                      
                                     m 
                                   
                                   - 
                                   1 
                                 
                               
                             
                           
                           
                             
                               ( 
                               
                                 m 
                                 = 
                                 
                                   k 
                                   - 
                                   1 
                                 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 E 
                                 
                                   k 
                                   , 
                                   
                                     m 
                                     + 
                                     1 
                                   
                                 
                               
                                
                               
                                 
                                   D 
                                   
                                     2 
                                      
                                     m 
                                   
                                 
                                 _ 
                               
                                
                               
                                 D 
                                 
                                   
                                     2 
                                      
                                     m 
                                   
                                   - 
                                   1 
                                 
                               
                             
                           
                           
                             
                               ( 
                               
                                 
                                   m 
                                   = 
                                   3 
                                 
                                 , 
                                 4 
                                 , 
                                 
                                   
                                     … 
                                      
                                     
                                         
                                     
                                      
                                     k 
                                   
                                   - 
                                   2 
                                 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 E 
                                 
                                   k 
                                   , 
                                   3 
                                 
                               
                                
                               
                                 D 
                                 3 
                               
                             
                           
                           
                             
                               ( 
                               
                                 m 
                                 = 
                                 2 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0000]    and 
         [0011]    the quantized bit Q 2  is expressed as Q 2 =D 3   D   2 +  D   3 D 2 . 
         [0012]    In one embodiment, N is 9. 
         [0013]    In one embodiment, each stage of second sub-circuit is consisted of NAND gates, NOR gates, phase inverters and D-type flip-flops. 
         [0014]    Compared to the conventional digital correction circuit that uses adders to perform the shift-and-add operations and hence needs a rather long operation time, the digital correction circuit according to the present invention is able to reduce the operation time and thus optimize the operation time, by allocating the operations to a plurality of stages of second sub-circuits configured to synchronize digital codes, wherein each of the second sub-circuits can perform part of the operations only with NAND gates, NOR gates, phase inverters and D-type flip-flops, without needing to use adders. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0015]    To provide a more complete understanding of the present invention, reference is made to the following description on exemplary embodiments, taken in conjunction with the accompanying drawings, in which: 
           [0016]      FIG. 1  shows a redundancy elimination algorithm for correcting outputs of pipelined 1.5-bit MADCs; 
           [0017]      FIG. 2  is a schematic illustration of a conventional digital correction circuit for eliminating redundancy bits; 
           [0018]      FIG. 3  is a schematic diagram illustrating a digital correction circuit for a pipelined ADC according to an embodiment of the present invention; and 
           [0019]      FIG. 4  shows curves representing the results of an analog-to-digital conversion performed by a pipelined ADC incorporating the digital correction circuit according to an embodiment of the present invention. 
       
    
    
     DETAILED DESCRIPTION 
       [0020]    A digital correction circuit for a pipelined ADC employing the present invention will be described in details below with reference to an exemplary embodiment in which the pipelined ADC includes 9 stages of first sub-circuits and the digital correction circuit includes 9 stages of second sub-circuits. 
         [0021]      FIG. 3  is a schematic diagram illustrating a digital correction circuit for a pipelined ADC according to an embodiment of the present invention. The pipelined ADC may include N (N is an integer and equals to 9 in this exemplary embodiment) stages of first sub-circuits, among which, the last stage of first sub-circuit includes a 2-bit parallel ADC that outputs a 2-bit code, D 2  and D 1 , without any redundancy bits; each of the rest stages of first sub-circuits includes a 1.5-bit MADC which outputs a 2-bit code that may be a significant code “00”, “01” or “10”, or a redundancy code “11”. Bits of a code output by the n-th-stage of first sub-circuit can be expressed as D 2(N+1-n)  and D 2(N+1-n)-1 , wherein n is an integer that satisfies 1≦n≦N. That is, as illustrated in  FIG. 3 , the first-stage of first sub-circuit outputs a code consisting of bits D 18  and D 17 ; the second-stage of first sub-circuit outputs a code consisting of bits D 16  and D 15 ; . . . ; and the ninth-stage of first sub-circuit outputs a code consisting of bits D 2  and D 1 . 
         [0022]    Moreover, the digital correction circuit may include N (N is an integer and equals to 9 in this exemplary embodiment) stages of second sub-circuits. As illustrated in  FIG. 3 , in the exemplary embodiment, these second sub-circuits are indicated as a first-stage of second sub-circuit, a second-stage of second sub-circuit, . . . , and a ninth-stage of second sub-circuit. Each of these second sub-circuits may include one or more NAND gates, one or more NOR gates, one or more phase inverters and one or more D-type flip-flops, but does not include adders. 
         [0023]    The first-stage of second sub-circuit is configured to receive a code consisting of two bits D 2N  and D 2N−1  (i.e., D 18  and D 17  in the exemplary embodiment) from the first-stage of first sub-circuit. Additionally, the i-th-stage of second sub-circuit is configured to both receive a 2-bit code consisting of bits D 2(N+1-i)  and D 2(N+1-i)-1  output by the i-th-stage of first sub-circuit and receive an output of the (i−1)-th-stage of second sub-circuit, where i is an integer that satisfies 2≦i≦N−1. Moreover, the N-th-stage of second sub-circuit is configured to receive a code D 2 . 
         [0024]    The digital correction circuit outputs N+1 (i.e., 10 in the exemplary embodiment) quantized bits Q 1 , Q 2 , . . . , and Q N+1 , among which the first quantized bit Q 1  equals to D 1  and the second to the (N+1)-th bits Q 2  to Q N+1  are output by the N-th-stage of second sub-circuit. 
         [0025]    More specifically, the quantized bit Q N+1  is expressed as Q N+1 =A N+1,2 +B N+1,2 D 2 , where A N+1,2  and B N+1,2  are coefficients of an (N−1)-th-stage of intermediate value that is output by the (N−1)-th-stage of second sub-circuit and corresponds to the quantized bit Q N+1 . Assume the coefficients of an (N−2)-th-stage of intermediate value that is output by the (N−2)-th-stage of second circuit and corresponds to the quantized bit Q N+1  are expressed as A N+1,3  and B N+1,3 , the coefficients of an (N−3)-th-stage of intermediate value that is output by the (N−3)-th-stage of second circuit and corresponds to the quantized bit Q N+1  are expressed as A N+1,4  and B N+1,4 , . . . , and the coefficients of a second-stage of intermediate value that is output by the second-stage of second circuit and corresponds to the quantized bit Q N+1  are expressed as A N+1,N−1  and B N+1,N−1 , coefficients of the respective stages of intermediate values that are output by the respective stages of second sub-circuits and correspond to the quantized bit Q N+1  satisfy the following recurrence relations: 
         [0000]    
       
         
           
             
               A 
               
                 
                   N 
                   + 
                   1 
                 
                 , 
                 m 
               
             
             = 
             
               { 
               
                 
                   
                     
                       
                         
                           
                             A 
                             
                               
                                 N 
                                 + 
                                 1 
                               
                               , 
                               
                                 m 
                                 + 
                                 1 
                               
                             
                           
                           + 
                           
                             
                               B 
                               
                                 
                                   N 
                                   + 
                                   1 
                                 
                                 , 
                                 
                                   m 
                                   + 
                                   1 
                                 
                               
                             
                              
                             
                               D 
                               
                                 2 
                                  
                                 m 
                               
                             
                           
                         
                       
                       
                         
                           ( 
                           
                             
                               m 
                               = 
                               2 
                             
                             , 
                             3 
                             , 
                             … 
                              
                             
                                 
                             
                             , 
                             
                               N 
                               - 
                               2 
                             
                           
                           ) 
                         
                       
                     
                     
                       
                         
                           
                             D 
                             
                               2 
                                
                               N 
                             
                           
                           + 
                           
                             
                               D 
                               
                                 
                                   2 
                                    
                                   N 
                                 
                                 - 
                                 1 
                               
                             
                              
                             
                               D 
                               
                                 
                                   2 
                                    
                                   N 
                                 
                                 - 
                                 2 
                               
                             
                           
                         
                       
                       
                         
                           ( 
                           
                             m 
                             = 
                             
                               N 
                               - 
                               1 
                             
                           
                           ) 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     B 
                     
                       
                         N 
                         + 
                         1 
                       
                       , 
                       m 
                     
                   
                 
                 = 
                 
                   { 
                   
                     
                       
                         
                           
                             B 
                             
                               
                                 N 
                                 + 
                                 1 
                               
                               , 
                               
                                 m 
                                 + 
                                 1 
                               
                             
                           
                            
                           
                             
                               D 
                               
                                 2 
                                  
                                 m 
                               
                             
                             _ 
                           
                            
                           
                             D 
                             
                               
                                 2 
                                  
                                 m 
                               
                               - 
                               1 
                             
                           
                         
                       
                       
                         
                           ( 
                           
                             
                               m 
                               = 
                               3 
                             
                             , 
                             4 
                             , 
                             … 
                              
                             
                                 
                             
                             , 
                             
                               N 
                               - 
                               2 
                             
                           
                           ) 
                         
                       
                     
                     
                       
                         
                           
                             B 
                             
                               
                                 N 
                                 + 
                                 1 
                               
                               , 
                               3 
                             
                           
                            
                           
                             D 
                             3 
                           
                         
                       
                       
                         
                           ( 
                           
                             m 
                             = 
                             2 
                           
                           ) 
                         
                       
                     
                     
                       
                         
                           
                             D 
                             
                               
                                 2 
                                  
                                 N 
                               
                               - 
                               1 
                             
                           
                            
                           
                             
                               D 
                               
                                 
                                   2 
                                    
                                   N 
                                 
                                 - 
                                 2 
                               
                             
                             _ 
                           
                            
                           
                             D 
                             
                               
                                 2 
                                  
                                 N 
                               
                               - 
                               3 
                             
                           
                         
                       
                       
                         
                           
                             ( 
                             
                               m 
                               = 
                               
                                 N 
                                 - 
                                 1 
                               
                             
                             ) 
                           
                           , 
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0000]    where, A N+1,m  and B N+1,m  are coefficients of the (N+1−m)-th-stage of intermediate value that is output by the (N+1−m)-th-stage of second sub-circuit and corresponds to the quantized bit Q N+1 . 
         [0026]    Moreover, the quantized bit Q k  is expressed as Q k =A k,2   D   2 +B k,2 +E k,2 D 2 , where, k is an integer satisfying 3≦k≦N; A k,2 , B k,2 , and E k,2  are coefficients of an (N−1)-th-stage of intermediate value that is output by the (N−1)-th-stage of second sub-circuit and corresponds to the quantized bit Q k ; and coefficients of the respective stages of intermediate values that are output by the respective stages of second sub-circuits and correspond to the quantized bit Q k  satisfy the following recurrence relations: 
         [0000]    
       
         
           
             
                 
             
              
             
               
                 A 
                 
                   k 
                   , 
                   m 
                 
               
               = 
               
                 { 
                 
                   
                     
                       
                         
                           
                             
                               D 
                               
                                 
                                   2 
                                    
                                   m 
                                 
                                 + 
                                 1 
                               
                             
                              
                             
                               
                                 D 
                                 
                                   2 
                                    
                                   m 
                                 
                               
                               _ 
                             
                           
                         
                         
                           
                             ( 
                             
                               m 
                               = 
                               
                                 k 
                                 - 
                                 1 
                               
                             
                             ) 
                           
                         
                       
                       
                         
                           
                             
                               A 
                               
                                 k 
                                 , 
                                 
                                   m 
                                   + 
                                   1 
                                 
                               
                             
                              
                             
                               
                                 D 
                                 
                                   2 
                                    
                                   m 
                                 
                               
                               _ 
                             
                           
                         
                         
                           
                             ( 
                             
                               
                                 m 
                                 = 
                                 2 
                               
                               , 
                               3 
                               , 
                               
                                 
                                   … 
                                    
                                   
                                       
                                   
                                    
                                   k 
                                 
                                 - 
                                 2 
                               
                             
                             ) 
                           
                         
                       
                     
                      
                     
                       
 
                     
                      
                     
                       B 
                       
                         k 
                         , 
                         m 
                       
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 
                                   
                                     D 
                                     
                                       
                                         2 
                                          
                                         m 
                                       
                                       + 
                                       1 
                                     
                                   
                                    
                                   
                                     
                                       D 
                                       
                                         2 
                                          
                                         m 
                                       
                                     
                                     _ 
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       D 
                                       
                                         
                                           2 
                                            
                                           m 
                                         
                                         - 
                                         1 
                                       
                                     
                                     _ 
                                   
                                 
                                 + 
                                 
                                   
                                     
                                       D 
                                       
                                         
                                           2 
                                            
                                           m 
                                         
                                         + 
                                         1 
                                       
                                     
                                     _ 
                                   
                                    
                                   
                                     D 
                                     
                                       2 
                                        
                                       m 
                                     
                                   
                                 
                               
                             
                             
                               
                                 ( 
                                 
                                   m 
                                   = 
                                   
                                     k 
                                     - 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     A 
                                     
                                       k 
                                       , 
                                       
                                         m 
                                         + 
                                         1 
                                       
                                     
                                   
                                    
                                   
                                     
                                       D 
                                       
                                         2 
                                          
                                         m 
                                       
                                     
                                     _ 
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       D 
                                       
                                         
                                           2 
                                            
                                           m 
                                         
                                         - 
                                         1 
                                       
                                     
                                     _ 
                                   
                                 
                                 + 
                                 
                                   B 
                                   
                                     k 
                                     , 
                                     
                                       m 
                                       + 
                                       1 
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     E 
                                     
                                       k 
                                       , 
                                       
                                         m 
                                         + 
                                         1 
                                       
                                     
                                   
                                    
                                   
                                     D 
                                     
                                       2 
                                        
                                       m 
                                     
                                   
                                 
                               
                             
                             
                               
                                 ( 
                                 
                                   
                                     m 
                                     = 
                                     2 
                                   
                                   , 
                                   3 
                                   , 
                                   
                                     
                                       … 
                                        
                                       
                                           
                                       
                                        
                                       k 
                                     
                                     - 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                          
                         
                           
 
                         
                          
                         
                             
                         
                          
                         
                           E 
                           
                             k 
                             , 
                             m 
                           
                         
                       
                       = 
                       
                         { 
                         
                           
                             
                               
                                 
                                   
                                     D 
                                     
                                       
                                         2 
                                          
                                         m 
                                       
                                       + 
                                       1 
                                     
                                   
                                   _ 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     D 
                                     
                                       2 
                                        
                                       m 
                                     
                                   
                                   _ 
                                 
                                  
                                 
                                   D 
                                   
                                     
                                       2 
                                        
                                       m 
                                     
                                     - 
                                     1 
                                   
                                 
                               
                             
                             
                               
                                 ( 
                                 
                                   m 
                                   = 
                                   
                                     k 
                                     - 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 
                                   E 
                                   
                                     k 
                                     , 
                                     
                                       m 
                                       + 
                                       1 
                                     
                                   
                                 
                                  
                                 
                                   
                                     D 
                                     
                                       2 
                                        
                                       m 
                                     
                                   
                                   _ 
                                 
                                  
                                 
                                   D 
                                   
                                     
                                       2 
                                        
                                       m 
                                     
                                     - 
                                     1 
                                   
                                 
                               
                             
                             
                               
                                 ( 
                                 
                                   
                                     m 
                                     = 
                                     3 
                                   
                                   , 
                                   4 
                                   , 
                                   
                                     
                                       … 
                                        
                                       
                                           
                                       
                                        
                                       k 
                                     
                                     - 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 
                                   E 
                                   
                                     k 
                                     , 
                                     3 
                                   
                                 
                                  
                                 
                                   D 
                                   3 
                                 
                               
                             
                             
                               
                                 
                                   ( 
                                   
                                     m 
                                     = 
                                     2 
                                   
                                   ) 
                                 
                                 , 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0000]    where, A k,m , B k,m  and E k,m  are coefficients of the (N+1−m)-th-stage of intermediate value that is output by the (N+1−m)-th-stage of second sub-circuit and corresponds to the quantized bit Q k . 
         [0027]    Furthermore, the quantized bit Q 2  is expressed as Q 2 =D 3   D   2 +  D   3 D 2 . 
         [0028]    It is obvious from the foregoing description that, compared to the conventional digital correction circuit, the digital correction circuit according to the above exemplary embodiment of the present invention is able to optimize operation time without using any adders or sacrificing its function of eliminating the redundancy code “11”, by allocating the operations to a plurality of stages of second sub-circuits configured to synchronize digital codes. 
         [0029]    The aforementioned coefficients may be calculated according to a method described below. 
         [0030]    As possible digital outputs for a 1.5-bit MADC are “00”, “10” and “01” only and do not include “11”, in the shift-and-add operations shown in  FIG. 1 , the carry bit C i  (i=1, 2, . . . , 7) will definitely be “0” in the case that the bit D 2i+2  is “1”. That is, it is never a case that the bits C i  and D 2i+2  are “1” at the same time. Further, C i  will be “1” only when the code “D 2i+2 D 2i+1 ” is “01” and at the same one of the bits D 2i  and C i-1  is “1”. Therefore, the carry bit C i  can be expressed as 
         [0000]    
       
         
           
             
               C 
               i 
             
             = 
             
               { 
               
                 
                   
                     
                       
                         
                           D 
                           _ 
                         
                         
                           
                             2 
                              
                             i 
                           
                           + 
                           2 
                         
                       
                        
                       
                           
                       
                        
                       
                         
                           D 
                           
                             
                               2 
                                
                               i 
                             
                             + 
                             1 
                           
                         
                          
                         
                           ( 
                           
                             
                               D 
                               
                                 2 
                                  
                                 i 
                               
                             
                             + 
                             
                               C 
                               
                                 i 
                                 - 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   
                     
                       ( 
                       
                         
                           i 
                           = 
                           2 
                         
                         , 
                         3 
                         , 
                         … 
                          
                         
                             
                         
                         , 
                         7 
                       
                       ) 
                     
                   
                 
                 
                   
                     
                       
                         D 
                         3 
                       
                        
                       
                         D 
                         2 
                       
                     
                   
                   
                     
                       
                         ( 
                         
                           i 
                           = 
                           1 
                         
                         ) 
                       
                       . 
                     
                   
                 
               
             
           
         
       
     
         [0031]    In the exemplary embodiment, as the quantized bit Q 10  output by the ninth stage of second sub-circuit is expressed by the following equation (1): 
         [0000]        Q   10   =D   18   +D   17 ( D   16   +C   7 )  (1),
 
         [0000]    where D 18  and D 17  are the 2-bit code input into the first-stage of second sub-circuit, the following equation (2) can be obtained by putting the expression of the carry bit C 7  into the equation (1): 
         [0000]        Q   10   =D   18   +D   17   D   16   +D   17     D     16   D   15 ( D   14   +C   6 )  (2).
 
         [0032]    In this equation, as D 16  and D 15  are the 2-bit code input into the second-stage of second sub-circuit, the values of D 18 +D 17 D 16  and D 17   D   16 D 15  can be obtained from operations performed therein. Thus, the bit Q 10 -related coefficients A 10,8  and B 10,8  output from this stage can be obtained: 
         [0000]    
       
         
           
               
             
               { 
               
                 
                   
                     
                       
                         A 
                         
                           10 
                           , 
                           8 
                         
                       
                       = 
                       
                         
                           D 
                           18 
                         
                         + 
                         
                           
                             D 
                             17 
                           
                            
                           
                             D 
                             16 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         B 
                         
                           10 
                           , 
                           8 
                         
                       
                       = 
                       
                         
                           D 
                           17 
                         
                          
                         
                           
                             D 
                             _ 
                           
                           16 
                         
                          
                         
                           
                             D 
                             15 
                           
                           . 
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0033]    Further, when the coefficients A 10,8  and B 10,8  are put into the above equation (2), there can be obtained the following equation (3): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           Q 
                           10 
                         
                         = 
                           
                          
                         
                           
                             A 
                             
                               10 
                               , 
                               8 
                             
                           
                           + 
                           
                             
                               B 
                               
                                 10 
                                 , 
                                 8 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   D 
                                   14 
                                 
                                 + 
                                 
                                   C 
                                   6 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             A 
                             
                               10 
                               , 
                               8 
                             
                           
                           + 
                           
                             
                               B 
                               
                                 10 
                                 , 
                                 8 
                               
                             
                              
                             
                               D 
                               14 
                             
                           
                           + 
                           
                             
                               B 
                               
                                 10 
                                 , 
                                 8 
                               
                             
                              
                             
                               
                                 D 
                                 _ 
                               
                               14 
                             
                              
                             
                               
                                 
                                   D 
                                   13 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       D 
                                       12 
                                     
                                     + 
                                     
                                       C 
                                       5 
                                     
                                   
                                   ) 
                                 
                               
                               . 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
         [0034]    Similarly to the above, as D 14  and D 13  are the 2-bit code input into the third-stage of second sub-circuit, the values of A 10,8 +B 10,8 D 14  and B 10,8   D   14 D 13  in this equation can be obtained from operations performed therein. Thus, the bit Q 10 -related coefficient A 10,7  and B 10,7  output from this stage can be obtained: 
         [0000]    
       
         
           
               
             
               { 
               
                 
                   
                     
                       
                         A 
                         
                           10 
                           , 
                           7 
                         
                       
                       = 
                       
                         
                           A 
                           
                             10 
                             , 
                             8 
                           
                         
                         + 
                         
                           
                             B 
                             
                               10 
                               , 
                               8 
                             
                           
                            
                           
                             D 
                             14 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         B 
                         
                           10 
                           , 
                           7 
                         
                       
                       = 
                       
                         
                           B 
                           
                             10 
                             , 
                             8 
                           
                         
                          
                         
                           
                             D 
                             14 
                           
                           _ 
                         
                          
                         
                           
                             D 
                             13 
                           
                           . 
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0035]    In this way, when m is defined as (N+1−n), the bit Q 10 -related coefficients A 10,m  and B 10,m  output from the respective stages of second sub-circuits can be calculated as 
         [0000]    
       
         
           
             
               A 
               
                 10 
                 , 
                 m 
               
             
             = 
             
               
                 A 
                 
                   10 
                   , 
                   
                     m 
                     + 
                     1 
                   
                 
               
               + 
               
                 
                   B 
                   
                     10 
                     , 
                     
                       m 
                       + 
                       1 
                     
                   
                 
                  
                 
                   D 
                   
                     2 
                      
                     m 
                   
                 
                  
                 
                     
                 
                  
                 
                   ( 
                   
                     
                       m 
                       = 
                       2 
                     
                     , 
                     3 
                     , 
                     
                       … 
                        
                       
                           
                       
                        
                       7 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               B 
               
                 10 
                 , 
                 m 
               
             
             = 
             
               { 
               
                 
                   
                     
                       
                         B 
                         
                           10 
                           , 
                           
                             m 
                             + 
                             1 
                           
                         
                       
                        
                       
                           
                       
                        
                       
                         
                           D 
                           
                             2 
                              
                             m 
                           
                         
                         _ 
                       
                        
                       
                         D 
                         
                           
                             2 
                              
                             m 
                           
                           + 
                           1 
                         
                       
                     
                   
                   
                     
                       ( 
                       
                         
                           m 
                           = 
                           3 
                         
                         , 
                         4 
                         , 
                         
                           … 
                            
                           
                               
                           
                            
                           7 
                         
                       
                       ) 
                     
                   
                 
                 
                   
                     
                       
                         B 
                         
                           10 
                           , 
                           3 
                         
                       
                        
                       
                         D 
                         3 
                       
                     
                   
                   
                     
                       
                         ( 
                         
                           m 
                           = 
                           2 
                         
                         ) 
                       
                       . 
                     
                   
                 
               
             
           
         
       
     
         [0036]    Accordingly, the bit Q 10  can be calculated as 
         [0000]        Q   10   =A   10,2   +B   10,2   D   2 . 
         [0037]    In addition, as the carry bit C i  (i=1, 2, . . . , 7) and the bit D 2i+2  will never be “1” at the same time, the quantized bit Q 9  output by the ninth stage of second sub-circuit can be expressed as the following equation (4): 
         [0000]    
       
         
           
             
               
                 
                     
                   
                     
                       
                         
                           
                             Q 
                             9 
                           
                           = 
                             
                            
                           
                             
                               
                                 D 
                                 17 
                               
                                
                               
                                 
                                   ( 
                                   
                                     
                                       D 
                                       16 
                                     
                                     + 
                                     
                                       C 
                                       7 
                                     
                                   
                                   ) 
                                 
                                 _ 
                               
                             
                             + 
                             
                               
                                 
                                   D 
                                   17 
                                 
                                 _ 
                               
                                
                               
                                 ( 
                                 
                                   
                                     D 
                                     16 
                                   
                                   + 
                                   
                                     C 
                                     7 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                            
                           
                             
                               
                                 D 
                                 17 
                               
                                
                               
                                 
                                   D 
                                   16 
                                 
                                 _ 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   D 
                                   14 
                                 
                                 _ 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   C 
                                   6 
                                 
                                 _ 
                               
                             
                             + 
                             
                               
                                 D 
                                 17 
                               
                                
                               
                                 
                                   D 
                                   16 
                                 
                                 _ 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   D 
                                   15 
                                 
                                 _ 
                               
                             
                             + 
                             
                               
                                 
                                   D 
                                   17 
                                 
                                 _ 
                               
                                
                               
                                 D 
                                 16 
                               
                             
                             + 
                           
                         
                       
                     
                     
                       
                         
                             
                            
                           
                             
                               
                                 D 
                                 
                                   17 
                                    
                                   
                                       
                                   
                                 
                               
                               _ 
                             
                              
                             
                                 
                             
                              
                             
                               
                                 D 
                                 16 
                               
                               _ 
                             
                              
                             
                               
                                 
                                   D 
                                   15 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       D 
                                       14 
                                     
                                     + 
                                     
                                       C 
                                       6 
                                     
                                   
                                   ) 
                                 
                               
                               . 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
         [0038]    In this equation, as D 16  and D 15  are the 2-bit code input into the second-stage of second sub-circuit, the values of D 17   D 16   , D 17   D 16     D 15   +  D 17   D 16  and  D 17     D 16   D 15  can be obtained from operations performed therein. Thus, the bit Q 9 -related coefficients A 9,8 , B 9,8  and E 9,8  output from this stage can be obtained: 
         [0000]    
       
         
           
               
             
               { 
               
                 
                   
                     
                       
                         A 
                         
                           9 
                           , 
                           8 
                         
                       
                       = 
                       
                         
                           D 
                           17 
                         
                          
                         
                           
                             D 
                             16 
                           
                           _ 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         B 
                         
                           9 
                           , 
                           8 
                         
                       
                       = 
                       
                         
                           
                             D 
                             17 
                           
                            
                           
                             
                               D 
                               16 
                             
                             _ 
                           
                            
                           
                               
                           
                            
                           
                             
                               D 
                               15 
                             
                             _ 
                           
                         
                         + 
                         
                           
                             
                               D 
                               17 
                             
                             _ 
                           
                            
                           
                             D 
                             16 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         E 
                         
                           9 
                           , 
                           8 
                         
                       
                       = 
                       
                         
                           
                             D 
                             17 
                           
                           _ 
                         
                          
                         
                             
                         
                          
                         
                           
                             D 
                             16 
                           
                           _ 
                         
                          
                         
                           
                             D 
                             15 
                           
                           . 
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0039]    Further, after putting the coefficient A 9,8 , B 9,8  and E 9,8  into the above equation (4), there can be obtained: 
         [0000]    
       
         
           
               
             
               
                 
                   
                     
                       Q 
                       9 
                     
                     = 
                       
                      
                     
                       
                         
                           A 
                           
                             9 
                             , 
                             8 
                           
                         
                          
                         
                           
                             D 
                             14 
                           
                           _ 
                         
                          
                         
                             
                         
                          
                         
                           
                             C 
                             6 
                           
                           _ 
                         
                       
                       + 
                       
                         B 
                         
                           9 
                           , 
                           8 
                         
                       
                       + 
                       
                         
                           E 
                           
                             9 
                             , 
                             8 
                           
                         
                          
                         
                           ( 
                           
                             
                               D 
                               14 
                             
                             + 
                             
                               C 
                               6 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   
                     = 
                       
                      
                     
                       
                         
                           A 
                           
                             9 
                             , 
                             8 
                           
                         
                          
                         
                           
                             D 
                             14 
                           
                           _ 
                         
                          
                         
                             
                         
                          
                         
                           
                             D 
                             12 
                           
                           _ 
                         
                          
                         
                             
                         
                          
                         
                           
                             C 
                             5 
                           
                           _ 
                         
                       
                       + 
                       
                         
                           A 
                           
                             9 
                             , 
                             8 
                           
                         
                          
                         
                           
                             D 
                             14 
                           
                           _ 
                         
                          
                         
                             
                         
                          
                         
                           
                             D 
                             13 
                           
                           _ 
                         
                       
                       + 
                       
                         B 
                         
                           9 
                           , 
                           8 
                         
                       
                       + 
                       
                         
                           E 
                           
                             9 
                             , 
                             8 
                           
                         
                          
                         
                           D 
                           14 
                         
                       
                       + 
                     
                   
                 
               
               
                 
                   
                       
                      
                     
                       
                         E 
                         
                           9 
                           , 
                           8 
                         
                       
                        
                       
                         
                           D 
                           14 
                         
                         _ 
                       
                        
                       
                         
                           
                             D 
                             13 
                           
                            
                           
                             ( 
                             
                               
                                 D 
                                 12 
                               
                               + 
                               
                                 C 
                                 5 
                               
                             
                             ) 
                           
                         
                         . 
                       
                     
                   
                 
               
             
           
         
       
     
         [0040]    Similarly to the above, as D 14  and D 13  are the 2-bit code input into the third-stage of second sub-circuit, the values of A 9,8   D 14   , A 9,8   D 14     D 13   +B 9,8 +E 9,8 D 14  and E 9,8   D 14   D 13  in this equation can be obtained from operations performed therein. Thus, the bit Q 9 -related coefficients A 9,7 , B 9,7  and E 9,7  output from this stage can be obtained: 
         [0000]    
       
         
           
               
             
               { 
               
                 
                   
                     
                       
                         A 
                         
                           9 
                           , 
                           7 
                         
                       
                       = 
                       
                         
                           A 
                           
                             9 
                             , 
                             8 
                           
                         
                          
                         
                           
                             D 
                             14 
                           
                           _ 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         B 
                         
                           9 
                           , 
                           7 
                         
                       
                       = 
                       
                         
                           
                             A 
                             
                               9 
                               , 
                               8 
                             
                           
                            
                           
                             
                               D 
                               14 
                             
                             _ 
                           
                            
                           
                               
                           
                            
                           
                             
                               D 
                               13 
                             
                             _ 
                           
                         
                         + 
                         
                           B 
                           
                             9 
                             , 
                             8 
                           
                         
                         + 
                         
                           
                             E 
                             
                               9 
                               , 
                               8 
                             
                           
                            
                           
                             D 
                             14 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         E 
                         
                           9 
                           , 
                           7 
                         
                       
                       = 
                       
                         
                           E 
                           
                             9 
                             , 
                             8 
                           
                         
                          
                         
                           
                             D 
                             14 
                           
                           _ 
                         
                          
                         
                             
                         
                          
                         
                           
                             D 
                             13 
                           
                           . 
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0041]    In this way, when m is defined as m=N+1−n, the bit Q 9 -related coefficients A 9,m , B 9,m  and E 9,m  output by the n-th-stage of second sub-circuit can be calculated as 
         [0000]    
       
         
           
             
               A 
               
                 9 
                 , 
                 m 
               
             
             = 
             
               
                 A 
                 
                   9 
                   , 
                   
                     m 
                     + 
                     1 
                   
                 
               
                
               
                 
                   D 
                   
                     2 
                      
                     m 
                   
                 
                 _ 
               
                
               
                   
               
                
               
                 ( 
                 
                   
                     m 
                     = 
                     2 
                   
                   , 
                   3 
                   , 
                   
                     … 
                      
                     
                         
                     
                      
                     7 
                   
                 
                 ) 
               
             
           
         
       
       
         
           
             
               B 
               
                 9 
                 , 
                 m 
               
             
             = 
             
               
                 
                   A 
                   
                     9 
                     , 
                     
                       m 
                       + 
                       1 
                     
                   
                 
                  
                 
                   
                     D 
                     
                       2 
                        
                       m 
                     
                   
                   _ 
                 
                  
                 
                     
                 
                  
                 
                   
                     D 
                     
                       
                         2 
                          
                         m 
                       
                       - 
                       1 
                     
                   
                   _ 
                 
               
               + 
               
                 B 
                 
                   9 
                   , 
                   
                     m 
                     + 
                     1 
                   
                 
               
               + 
               
                 
                   E 
                   
                     9 
                     , 
                     
                       m 
                       + 
                       1 
                     
                   
                 
                  
                 
                   D 
                   
                     2 
                      
                     m 
                   
                 
                  
                 
                     
                 
                  
                 
                   ( 
                   
                     
                       m 
                       = 
                       2 
                     
                     , 
                     3 
                     , 
                     
                       … 
                        
                       
                           
                       
                        
                       7 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               E 
               
                 9 
                 , 
                 m 
               
             
             = 
             
               { 
               
                 
                   
                     
                       
                         E 
                         
                           9 
                           , 
                           
                             m 
                             + 
                             1 
                           
                         
                       
                        
                       
                           
                       
                        
                       
                         
                           D 
                           
                             2 
                              
                             m 
                           
                         
                         _ 
                       
                        
                       
                         D 
                         
                           
                             2 
                              
                             m 
                           
                           - 
                           1 
                         
                       
                     
                   
                   
                     
                       ( 
                       
                         
                           m 
                           = 
                           3 
                         
                         , 
                         4 
                         , 
                         
                           … 
                            
                           
                               
                           
                            
                           7 
                         
                       
                       ) 
                     
                   
                 
                 
                   
                     
                       
                         E 
                         
                           9 
                           , 
                           3 
                         
                       
                        
                       
                         D 
                         3 
                       
                     
                   
                   
                     
                       
                         ( 
                         
                           m 
                           = 
                           2 
                         
                         ) 
                       
                       ; 
                     
                   
                 
               
             
           
         
       
     
         [0042]    Accordingly, the quantized bit Q 9  output by the ninth-stage of second sub-circuit can be calculated as 
         [0000]        Q   9   =A   9,2     D   2   + B   9,2   +E   9,2   D   2 . 
         [0043]    Following the same process for calculating the quantized bit Q 9 , when m is defined as m=N+1−n, the bit Q k -related (k=3, 4, . . . , 8) coefficients A k,m , B k,m  and E k,m  output by the n-th-stage of second sub-circuit can be calculated as 
         [0000]    
       
         
           
             
                 
             
              
             
               
                 A 
                 
                   k 
                   , 
                   m 
                 
               
               = 
               
                 { 
                 
                   
                     
                       
                         
                           
                             
                               D 
                               
                                 
                                   2 
                                    
                                   m 
                                 
                                 + 
                                 1 
                               
                             
                              
                             
                               
                                 D 
                                 
                                   2 
                                    
                                   m 
                                 
                               
                               _ 
                             
                           
                         
                         
                           
                             ( 
                             
                               m 
                               = 
                               
                                 k 
                                 - 
                                 1 
                               
                             
                             ) 
                           
                         
                       
                       
                         
                           
                             
                               A 
                               
                                 k 
                                 , 
                                 
                                   m 
                                   + 
                                   1 
                                 
                               
                             
                              
                             
                               
                                 D 
                                 
                                   2 
                                    
                                   m 
                                 
                               
                               _ 
                             
                           
                         
                         
                           
                             ( 
                             
                               
                                 m 
                                 = 
                                 2 
                               
                               , 
                               3 
                               , 
                               
                                 
                                   … 
                                    
                                   
                                       
                                   
                                    
                                   k 
                                 
                                 - 
                                 2 
                               
                             
                             ) 
                           
                         
                       
                     
                      
                     
                       
 
                     
                      
                     
                       B 
                       
                         k 
                         , 
                         m 
                       
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 
                                   
                                     D 
                                     
                                       
                                         2 
                                          
                                         m 
                                       
                                       + 
                                       1 
                                     
                                   
                                    
                                   
                                     
                                       D 
                                       
                                         2 
                                          
                                         m 
                                       
                                     
                                     _ 
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       D 
                                       
                                         
                                           2 
                                            
                                           m 
                                         
                                         - 
                                         1 
                                       
                                     
                                     _ 
                                   
                                 
                                 + 
                                 
                                   
                                     
                                       D 
                                       
                                         
                                           2 
                                            
                                           m 
                                         
                                         + 
                                         1 
                                       
                                     
                                     _ 
                                   
                                    
                                   
                                     D 
                                     
                                       2 
                                        
                                       m 
                                     
                                   
                                 
                               
                             
                             
                               
                                 ( 
                                 
                                   m 
                                   = 
                                   
                                     k 
                                     - 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     A 
                                     
                                       k 
                                       , 
                                       
                                         m 
                                         + 
                                         1 
                                       
                                     
                                   
                                    
                                   
                                     
                                       D 
                                       
                                         2 
                                          
                                         m 
                                       
                                     
                                     _ 
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       D 
                                       
                                         
                                           2 
                                            
                                           m 
                                         
                                         - 
                                         1 
                                       
                                     
                                     _ 
                                   
                                 
                                 + 
                                 
                                   B 
                                   
                                     k 
                                     , 
                                     
                                       m 
                                       + 
                                       1 
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     E 
                                     
                                       k 
                                       , 
                                       
                                         m 
                                         + 
                                         1 
                                       
                                     
                                   
                                    
                                   
                                     D 
                                     
                                       2 
                                        
                                       m 
                                     
                                   
                                 
                               
                             
                             
                               
                                 ( 
                                 
                                   
                                     m 
                                     = 
                                     2 
                                   
                                   , 
                                   3 
                                   , 
                                   
                                     
                                       … 
                                        
                                       
                                           
                                       
                                        
                                       k 
                                     
                                     - 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                          
                         
                           
 
                         
                          
                         
                             
                         
                          
                         
                           E 
                           
                             k 
                             , 
                             m 
                           
                         
                       
                       = 
                       
                         { 
                         
                           
                             
                               
                                 
                                   
                                     D 
                                     
                                       
                                         2 
                                          
                                         m 
                                       
                                       + 
                                       1 
                                     
                                   
                                   _ 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     D 
                                     
                                       2 
                                        
                                       m 
                                     
                                   
                                   _ 
                                 
                                  
                                 
                                   D 
                                   
                                     
                                       2 
                                        
                                       m 
                                     
                                     - 
                                     1 
                                   
                                 
                               
                             
                             
                               
                                 ( 
                                 
                                   m 
                                   = 
                                   
                                     k 
                                     - 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 
                                   E 
                                   
                                     k 
                                     , 
                                     
                                       m 
                                       + 
                                       1 
                                     
                                   
                                 
                                  
                                 
                                   
                                     D 
                                     
                                       2 
                                        
                                       m 
                                     
                                   
                                   _ 
                                 
                                  
                                 
                                   D 
                                   
                                     
                                       2 
                                        
                                       m 
                                     
                                     - 
                                     1 
                                   
                                 
                               
                             
                             
                               
                                 ( 
                                 
                                   
                                     m 
                                     = 
                                     3 
                                   
                                   , 
                                   4 
                                   , 
                                   
                                     
                                       … 
                                        
                                       
                                           
                                       
                                        
                                       k 
                                     
                                     - 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 
                                   E 
                                   
                                     k 
                                     , 
                                     3 
                                   
                                 
                                  
                                 
                                   D 
                                   3 
                                 
                               
                             
                             
                               
                                 
                                   ( 
                                   
                                     m 
                                     = 
                                     2 
                                   
                                   ) 
                                 
                                 , 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0044]    and thus the quantized bit Q k  output by the ninth stage of second sub-circuit can be calculated as 
         [0000]        Q   k   =A   k,2     D     2   +B   k,2   +E   k,2   D   2 . 
         [0045]    Furthermore, the quantized bit Q 2  output by the ninth stage of second sub-circuit can be calculated as 
         [0000]        Q   2   =D   3     D     2   +  D     3   D   2 , 
         [0046]    and the quantized bit Q 1  is given value D 1 . 
         [0047]      FIG. 4  shows curves representing the results of an analog-to-digital conversion performed by a pipelined ADC incorporating the digital correction circuit of the exemplary embodiment. As illustrated, when an input is a 1.3215 MHz sine signal sampled at a frequency of 192 MHz, the digital correction circuit can correctly digitalize the input signal to 10 quantized bits expressed by the upper 10 curves collectively indicated by reference number  1  in  FIG. 4 . As the sine signal can be recovered from the quantized bits using an ideal digital-to-analog behavioral model, the 10 quantized bits were converted to a sine signal expressed by the curve indicated by reference number  2  using an appropriate digital-to-analog converter in order to verify the correctness of them. As illustrated, the conversion is correct and without issues such as non-monotonicity. 
         [0048]    While specific embodiments have been presented in the foregoing description, they are not intended to limit the invention in any way. Those skilled in the art can make various modifications and variations without departing from the scope of the invention. Thus, it is intended that the present invention covers all such modifications and variations.