Abstract:
A SAR ADC is provided. A DAC provides an intermediate analog signal according to an analog input signal, a most significant bit capacitance and a plurality of significant bit capacitances smaller than the most significant bit capacitance. A first switched capacitor array selectively provides the most significant bit capacitance or the significant bit capacitances according to a select signal. Sum of the significant bit capacitances is equal to the most significant bit capacitance. The second switched capacitor array provides the significant bit capacitances when the first switched capacitor array provides the most significant bit capacitance, and provides the most significant bit capacitance when the first switched capacitor array provides the significant bit capacitances. A comparator provides a comparison result according to the intermediate analog signal. A SAR logic provides an digital output signal according to the comparison result.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     This application claims the benefit of U.S. Provisional Application No. 62/105,418, filed on Jan. 20, 2015, the contents of which are incorporated herein by reference. 
    
    
     BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The invention relates to an analog to digital converter (ADC), and more particularly to an ADC that uses successive approximation techniques. 
     2. Description of the Related Art 
     Analog to digital converters (ADCs) are widely used in a variety of applications, such as medical systems, audio systems, test and measurement equipment, communication systems, and image and video systems, etc. The most common ADC construction comprises flash ADCs, pipeline ADCs and successive approximation register (SAR) ADCs. The power consumption of the SAR ADC is smaller than the flash ADC and the pipeline ADC. Thus the systems with limited power supply, such as portable devices, usually use SAR ADCs. 
     The conversion accuracy of the SAR ADCs is mainly subject to the DAC mismatch and offset errors. Particularly, for the switched-capacitor SAR ADCs, the capacitor mismatch is the dominant one. 
     BRIEF SUMMARY OF THE INVENTION 
     Thus, it is desired to provide a successive approximation register circuit which can correct error caused by the capacitor mismatch. 
     An exemplary embodiment of a successive approximation register (SAR) analog to digital converter (ADC) circuit is provided. The SAR ADC circuit receives an analog input signal and operates in a sample phase and a conversion phase following the sample phase to generate a digital output signal. The SAR ADC circuit comprises a plurality of capacitors, a comparator, and a logic unit. The plurality of capacitors are coupled to a summing node. Before the conversion phase, a target capacitor among the plurality of capacitors is coupled to a direct current (DC) voltage and the other capacitors among the plurality of capacitors are coupled to the analog input signal. The comparator has an input terminal coupled to the summing node. In the conversion phase, the comparator performs a comparison operation to a summing voltage at the summing node The logic unit has a plurality of weighting values corresponding to the plurality of capacitors respectively and generates the digital output signal according to the weighting values and a comparison result of the comparison operation. The DC voltage has a first voltage level or a second voltage level different from the first voltage level according to a random sequence. The weighting value of the target capacitor is calibrated according to the digital output signal and the random sequence. 
     An exemplary embodiment of a successive approximation register (SAR) analog to digital converter (ADC) circuit is provided. The SAR ADC circuit receives an analog input signal and operates in a sample phase and a conversion phase following the sample phase to generate a digital output signal. The SAR ADC circuit comprises a first switch, a plurality of capacitors, a plurality of second switches, a plurality of switch circuits, a comparator, a logic unit, and an extraction and compensation unit. The first switch is coupled between a first voltage and a summing node. Each capacitor has a first terminal coupled to the summing node and further has a second terminal. Each second switch is coupled between the second terminal of one of the plurality of capacitors and a second voltage. The plurality of switch circuits receives the analog input signal, Each switch circuits is coupled to the second terminal of one of the plurality of capacitors and, before the conversion phase, provides a DC voltage or the analog input signal to the corresponding capacitor. The comparator has an input terminal coupled to the summing node. In the conversion phase, the comparator performs a comparison operation to a summing voltage at the summing node. The logic unit has a plurality of weighting values corresponding to the plurality of capacitors respectively and generates the digital output signal according to the weighting values and a comparison result of the comparison operation. The extraction and compensation unit receives the digital output signal. When a target capacitor among the plurality of switch circuits receive the DC voltage from the corresponding switch circuit, the DC voltage has a first voltage level or a second voltage level different from the first voltage level based on a random sequence,. The extraction and compensation unit obtains a calibrated weighting value according to the digital output signal and the random sequence, and the calibrated weighting value serves as the weighting value of the target capacitor. 
     A detailed description is given in the following embodiments with reference to the accompanying drawings. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
       The invention can be more fully understood by reading the subsequent detailed description and examples with references made to the accompanying drawings, wherein: 
         FIG. 1  shows an exemplary embodiment of a successive approximation register (SAR) analog to digital converter (ADC) circuit; 
         FIG. 2  shows an exemplary embodiment of an SAR DAC of the SAR ADC circuit in  FIG. 1 ; 
         FIG. 3  shows timing of a sample phase and a conversion phase of the SAR ADC circuit in  FIG. 1 ; 
         FIG. 4  shows an exemplary embodiment of an extraction and compensation unit of the SAR ADC circuit in  FIG. 1 ; 
         FIG. 5  shows an exemplary embodiment of switch circuits of the SAR ADC in  FIG. 2 ; 
         FIGS. 6A and 6B  show another exemplary embodiment of an SAR DAC of the SAR ADC circuit in  FIG. 1 ; and 
         FIG. 7  shows an exemplary embodiment of a low-pass filter. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     The following description is of the best-contemplated mode of carrying out the invention. This description is made for the purpose of illustrating the general principles of the invention and should not be taken in a limiting sense. The scope of the invention is best determined by reference to the appended claims. 
       FIG. 1  shows an exemplary embodiment of a successive approximation register (SAR) analog to digital converter (ADC) circuit. As shown in  FIG. 1  an SAR ADC circuit  1  comprises an SAR DAC  10 , and an extraction and compensation unit  11 . The SAR ADC circuit  1  operates in a sample phase and a conversion phase following the sample phase in several cycles. The SAR ADC  10  receives an analog input signal V in  and generates a digital output signal D o  according to the analog input signal V in , weighting values of capacitors (shown in  FIG. 2 ) , a direct current (DC) voltage, and a random signal q·V r , wherein q represents is a binary value random sequence which is uncorrelated with the analog input signal V in , and each value of the random sequence is equal to 1 or −1. Thus, the voltage level of the random signal q·V r  is the level of −V r  or V r . In the embodiment, the random signal q·V r  is injected to at least one capacitor (shown in  FIG. 2 ) to be calibrated in the SAR ADC  10 . Thus, the digital output signal D o  comprises one term related to the random signal q·V r . The extraction and compensation unit  11  extracts the real weighting value of the capacitor to be calibrated according to the digital output signal D o  and the random signal q·V r . Then, the extraction and compensation unit  11  corrects the digital output signal D o  based on the real weighting value of the capacitor to be calibrated 
     The detailed structure of the SAR DAC  10  is shown in  FIG. 2 . As shown in  FIG. 2 , the SAR DAC  10  comprises N capacitors C 0 ˜C N−1 , a sample switch SWS, N switches SW 0 ˜SW N−1 , N switch circuits SWC 0 ˜SWC N−1 , a comparator  20 , and a logic unit  21 , wherein N is an positive integer. The sample switch SWS is coupled to a summing node N 20  and a signal. In the embodiment, the signal which is coupled to the sample switch SWS is a ground voltage GND. One terminal of the comparator  20  is coupled to the summing node N 20 , and the other terminal thereof is coupled to the ground voltage GND. Each of the capacitors C 0 ˜C N−1  has two terminals. A first terminal of each capacitor is coupled to the summing node N 20 . In the embodiment, a second terminal of each capacitor is coupled to one switch SW j  and one switch circuit SWC j , wherein 0≦j≦N−1. For example, the other terminal of the capacitor C N−1  is coupled to the switch SW N−1  and the switch circuit SWC N−1 , and the other terminal of the capacitor C 0  is coupled to the switch SW 0  and the switch circuit SWC 0 . The sample switch SWS, the N switches SW 0 ˜SW N−1 , and the N switch circuits SWC 0 ˜SWC N−1  are controlled by the logic unit  21 . 
     The each of the switch circuits SWC 0 ˜SWC N−1  receives the analog input signal V in  and the random signal q·V r . Each switch circuit is controlled by the logic unit  21  to provide the analog input signal V in  or a DC voltage to the corresponding capacitor. The level of the DC voltage is determined by the random signal q·V r , that is the level of the DC voltage is the level of −V r  or V r .  FIG. 3  shows the timing of the sample phase and the conversion phase of the SAR ADC circuit  10  In  FIG. 3 , φ 1e  represents the timing of the sample phase, φ SAR  represents the timing of the conversion phase, and φ 1  represents the timing of the operation of the switch circuits. In the sample phase between a time point T 1  and a time point T 2 , the sample switch SWS is turned on. In the case, it is desired to calibrate the weighting value of one capacitor C j  (also referred to as “target capacitor”) of the capacitor C 0 ˜C N−1 , before the occurrence of the conversion phase (that is before the time point T 4 ), the corresponding switch circuit SWC j  provides the DC voltage to the one capacitor C j , and the other switch circuits provide the analog input signal V in  to the corresponding capacitors. Atthe sampling time, the charge stored at the summing node N 20  is equal to:
 
 Q   x,l   c   =−V   in   ×C   tot   +V   in   ×C   j   −q·V   r   ×C   j   (1)
 
where C tot =Σ i=0   N−1 C i .
 
     In the conversion phase between the time point T 4  and a time point T 5 , the switches SW N−1 ˜SW 0  are sequentially turned on to provide the voltage Vr to the respective capacitors SW N−1 ˜SW 0 . The comparator  20  determines the binary code word from MSB b N−1  to the LSB b 0  by examining the polarity of the voltage at the summing node N 20  sequentially. At the end of the conversion phase, the charged stored at the summing node N 20  is given by: 
                     Q     x   ,   2       =       ∑     i   =   0       N   -   1       ⁢       (       V   x     -       b   i     ·     V   r         )     ⨯     C   i                 (   2   )               
where V x  represents the voltage at the summing node N 20 .
 
     According to the charge conservation at the summing node N 20 , Q x,1   C  is equal to Q x,2 , and the voltage at the summing node N 20  is express as: 
     
       
         
           
             
               
                 
                   
                     V 
                     x 
                   
                   = 
                   
                     
                       - 
                       
                         V 
                         in 
                       
                     
                     + 
                     
                       
                         V 
                         in 
                       
                       ⨯ 
                       
                         
                           C 
                           j 
                         
                         
                           C 
                           tot 
                         
                       
                     
                     - 
                     
                       q 
                       · 
                       
                         
                           V 
                           r 
                         
                         ⨯ 
                         
                           
                             C 
                             j 
                           
                           
                             C 
                             tot 
                           
                         
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           0 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         
                           b 
                           i 
                         
                         · 
                         
                           
                             V 
                             r 
                           
                           ⨯ 
                           
                             
                               C 
                               i 
                             
                             
                               C 
                               tot 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     Ideally, since the voltage V x  approaches zero at the end of the conversion phase, D o =[b N−1 , b N−2 , . . . b 0 ] is the best quantized representation of the analog input signal V in , and the corresponding digital value of the digital output signal D o  is given by: 
                       D   o     ≈       V   in     -       V   in     ⨯       C   j       C   tot         +     q   ·       V   r     ⨯       C   j       C   tot               =     S   +     q   ·     R   j                 (   4   )               
where S=V in −V in ×(C j /C tot ) and R j =V r ×(C j /C tot ).
 
     According to the above description, the weighting value of capacitor C j  is desired to be calibrated. As shown in Equation (4), the R j =V r ×(C j /C tot ) represents the weighting value of capacitor C j  and determines the real weighting value of the capacitor C j . In order to extract the parameter R j , the extraction and compensation unit  11  performs a correlation operation to the digital output signal D o  with the random sequence q and further performs a low-pass-filtering operation to the digital output signal D o  to generate a calibrated weighting value Ŵ j  of the capacitor C j . In other words, the digital output signal D o  is correlated with the random sequence q and then low-pass filtered to obtain the weighting value Ŵ j . The extraction and compensation unit  11  then corrects the digital output signal D o  according to the calibrated weighting value Ŵ j  to generate the output signal D o   C    
       FIG. 4  shows an exemplary embodiment of an extraction circuit  110  in the extraction and compensation unit  11 . For detailed illustration,  FIG. 4  also shows the SAR DAC  10 . As shown in  FIG. 4 , the extraction circuit  11  comprises a multiplier  40  and a low-pass filter (LPF)  41 . The multiplier  40  receives the digital output signal D o  and the random sequence q to achieve the correlation operation to the digital output signal D o  and the random sequence q. The low-pass filter  41  is coupled to the multiplier  40  and performs the low-pass-filtering operation to the digital output signal D o  to generate the calibrated weighting value Ŵ j . According to the operations of the multiplier  40  and the low-pass filter  41 , the term R j  is retained, and the calibrated weighting value Ŵ j  is the digital expression of the term R j . Thus, the calibrated weighting value Ŵ j  related to the real weighting value of the capacitor C j  is obtained and used to correct the digital output signal D o . 
     Finally, the digital output signal D o  is corrected, and the corrected output signal is represented by: 
     
       
         
           
             
               
                 
                   
                     D 
                     o 
                     c 
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           0 
                         
                         
                           j 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         
                           b 
                           i 
                         
                         · 
                         
                           W 
                           i 
                         
                       
                     
                     + 
                     
                       
                         b 
                         j 
                       
                       · 
                       
                         
                           W 
                           ^ 
                         
                         j 
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           
                             j 
                             + 
                             1 
                           
                         
                         
                           N 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         
                           b 
                           i 
                         
                         · 
                         
                           W 
                           i 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     If several weighting values of the capacitors required to be calibrated, the above operations performed to obtain the calibrated weighting value Ŵ j  related to the real weighting value of the capacitor C j  are also performed for the other capacitors required to be calibrated. The related description is omitted. For example, in the case where it is desired to calibrate the weighting values of the capacitors C 0 ˜C j  among the capacitor C 0 ˜C N−1 , the corresponding calibrated weighting values Ŵ 0 ˜Ŵ j  the corrected output signal is represented by: 
     
       
         
           
             
               
                 
                   
                     D 
                     o 
                     c 
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           0 
                         
                         j 
                       
                       ⁢ 
                       
                         
                           b 
                           i 
                         
                         · 
                         
                           
                             W 
                             ^ 
                           
                           i 
                         
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           
                             j 
                             + 
                             1 
                           
                         
                         
                           N 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         
                           b 
                           i 
                         
                         · 
                         
                           W 
                           i 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     According to Equation (5) and Equation (6), the weighting values of the capacitors can be calibrated. Even though the capacitor mismatch occurs, the digital output signal D o  can more approach the accurate digital value of the analog input signal V in . 
       FIG. 5  shows an exemplary embodiment of the switch circuits SWC 0 ˜SWC N−1 . In  FIG. 5 , only the switch circuit SW j  is shown. The structures of the other switch circuits are the same as the structure of the switch circuit SW j , thus, omitting the related description here. The switch circuit SWC j  comprises three switches  50 ˜ 52 . The switch  50  is coupled between the capacitor C j  and the voltage Vr. The switch  51  is coupled between the capacitor C j  and the voltage −Vr. The switch  52  is coupled between the capacitor C j  and the analog input signal V in . The switches  50 - 52  are controlled by the logic unit  21  and not turned on at the same time. Thus, one of the three voltages Vr, −Vr, and V in  serves as the DC voltage provided the capacitor C j . The logic unit  21  controls the switches  50  and  51  according to the random sequence q. Thus, in equivalent, the switch circuit SWC j  receives the random signal q·V r , which determines the voltage Vr or −Vr to be provided to the capacitor C j  before the conversion phase. 
     In  FIG. 3 , the DC voltage (Vr or −Vr) is provided to the capacitor C j  to be calibrated in the period between the time point T 1  and a time point T 3 , as shown by φ 1 , however, without limitation. The DC voltage can be provided to the capacitor C j  in any period which just occurs before the conversion phase occurs (that is before the time point T 4 ). 
       FIG. 6A  shows another exemplary embodiment of the SAR ADC  10 . The SAR ADC shown in  FIG. 6A  is a differential-type, 12-bit ADC. The signals which are coupled to the sample switches SWS are analog input signals V ip  and V in . In  FIG. 6A , a switch group are coupled to the capacitors C 1 ˜C 11 . The switch group  60  comprises several switches and switch circuits as shown in  FIG. 2 . Each set of one switch SW j  and one switch circuit SWC j  is coupled to the corresponding capacitor and performs the same operations of the embodiment of  FIG. 2 . Moreover, V rp  replaces the V r  to be provided to capacitors coupled to the positive input (+) of the comparator  20 , while V rn  replaces the V r  to be provided to capacitors coupled to the negative input (−) of the comparator  20 . 
     In order to calibrate the weighting values of the capacitors C 8 ˜C 11 , the larger capacitance of the capacitors C 9 ˜C 11  is divided to small capacitance to relax the lost of dynamic range, as shown in  FIG. 6B , that is:
 
 C   9   =C   9,1   +C   9,0  
 
 C   10   =C   10,3   +C   10,2   +C   10,1   +C   10,0  
 
 C   11   =C   11,7   +C   11,6   +C   11,5   +C   11,4   +C   11,3   +C   11,2   ++C   11,1   +C   11,0  
 
     Thus, the weighting values of the capacitors C 8 , C 9,0 ˜C 9,1 , C 10,0 ˜C 10,3 , and C 11,0 ˜C 11,7  are required to be calibrated. The calibration process of the capacitor C 8  is the similar to the above process for calibrating the weighting value of the capacitor C j . Thus, before the conversion phase, the charge at the input of the comparator  01  is equal to: 
     
       
         
           
             
               
                 
                   
                     Q 
                     ϕ1 
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         11 
                       
                       ⁢ 
                       
                         
                           C 
                           k 
                         
                         ⨯ 
                         
                           ( 
                           
                             
                               V 
                               ip 
                             
                             - 
                             
                               V 
                               in 
                             
                           
                           ) 
                         
                       
                     
                     - 
                     
                       
                         C 
                         8 
                       
                       ⨯ 
                       
                         ( 
                         
                           
                             V 
                             ip 
                           
                           - 
                           
                             V 
                             
                               i 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               n 
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         
                           C 
                           8 
                         
                         ⨯ 
                         q 
                       
                       · 
                       
                         ( 
                         
                           
                             V 
                             rp 
                           
                           - 
                           
                             V 
                             rn 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     At the end of the conversion phase, the charge at the input of the comparator  20  is given by: 
     
       
         
           
             
               
                 
                   
                     Q 
                     
                       ϕ 
                       SAR 
                     
                   
                   = 
                   
                     - 
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         11 
                       
                       ⁢ 
                       
                         
                           C 
                           k 
                         
                         ⨯ 
                         
                           [ 
                           
                             
                               ( 
                               
                                 
                                   V 
                                   xp 
                                 
                                 - 
                                 
                                   V 
                                   xn 
                                 
                               
                               ) 
                             
                             - 
                             
                               
                                 b 
                                 k 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     V 
                                     rp 
                                   
                                   - 
                                   
                                     V 
                                     rn 
                                   
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     According to the charge conservation, Q φ1 =Q φ     SAR   , and the voltage at the input of the comparator  20  is expressed as: 
                       V   xp     -     V   xn       =       (       V   ip     -     V   in       )     -         C   8       C   tot       ⨯     (       V   ip     -     V   in       )       +           C   8       C   tot       ⨯   q     ·     (       V   rp     -     V   rn       )       -       ∑     k   =   11     11     ⁢           C   k       C   tot       ⨯     b   k       ·     (       V   rp     -     V   rn       )                   (   9   )               
where,
 
     
       
         
           
             
               C 
               tot 
             
             = 
             
               
                 ∑ 
                 
                   k 
                   = 
                   1 
                 
                 11 
               
               ⁢ 
               
                 C 
                 k 
               
             
           
         
       
     
     At the end of the conversion phase, (V xp −V xn ) approaches to zero, and Equation (9) is rewritten as: 
     
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         
                           V 
                           ip 
                         
                         - 
                         
                           V 
                           in 
                         
                       
                       ) 
                     
                     - 
                     
                       
                         
                           C 
                           8 
                         
                         
                           C 
                           tot 
                         
                       
                       ⨯ 
                       
                         ( 
                         
                           
                             V 
                             ip 
                           
                           - 
                           
                             V 
                             
                               i 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               n 
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         
                           
                             C 
                             8 
                           
                           
                             C 
                             tot 
                           
                         
                         ⨯ 
                         q 
                       
                       · 
                       
                         ( 
                         
                           
                             V 
                             rp 
                           
                           - 
                           
                             V 
                             rn 
                           
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         k 
                         = 
                         1 
                       
                       11 
                     
                     ⁢ 
                     
                       
                         
                           
                             C 
                             k 
                           
                           
                             C 
                             tot 
                           
                         
                         ⨯ 
                         
                           b 
                           k 
                         
                       
                       · 
                       
                         ( 
                         
                           
                             V 
                             rp 
                           
                           - 
                           
                             V 
                             rn 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     To extract actual weighting value of the capacitor C 8 , the digital output signal D o  is correlated with the random q and then low-pass filtered. In the embodiment, the low-pass filtering is achieved by integration and average operations. That is, the digital output signal D o  is further integrated and divided by M cycles. As shown in  FIG. 7 , in this embodiment the low-pass filter  41  is implemented by a digital accumulator (Digital Accumu.)  70  and a divider (1/M)  71 . Thus, we have: 
     
       
         
           
             
               
                 
                   
                     
                       1 
                       M 
                     
                     ⨯ 
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           0 
                         
                         
                           M 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         q 
                         · 
                         
                           D 
                           O 
                         
                       
                     
                   
                   ≈ 
                   
                     
                       1 
                       M 
                     
                     ⨯ 
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           0 
                         
                         
                           M 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         q 
                         · 
                         
                           [ 
                           
                             
                               ( 
                               
                                 
                                   V 
                                   ip 
                                 
                                 - 
                                 
                                   V 
                                   
                                     i 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     n 
                                   
                                 
                               
                               ) 
                             
                             - 
                             
                               
                                 
                                   C 
                                   8 
                                 
                                 
                                   C 
                                   tot 
                                 
                               
                               ⨯ 
                               
                                 ( 
                                 
                                   
                                     V 
                                     ip 
                                   
                                   - 
                                   
                                     V 
                                     
                                       i 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       n 
                                     
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               
                                 
                                   
                                     C 
                                     8 
                                   
                                   
                                     C 
                                     tot 
                                   
                                 
                                 ⨯ 
                                 q 
                               
                               · 
                               
                                 ( 
                                 
                                   
                                     V 
                                     rp 
                                   
                                   - 
                                   
                                     V 
                                     rn 
                                   
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     Assume that the random sequence q is uncorrelated with the analog input signal V in . The calibrated weighting value of the capacitor C 8  is obtained as: 
     
       
         
           
             
               
                 
                   
                     
                       W 
                       ^ 
                     
                     8 
                   
                   ≈ 
                   
                     
                       
                         C 
                         8 
                       
                       
                         C 
                         tot 
                       
                     
                     ⨯ 
                     
                       ( 
                       
                         
                           V 
                           rp 
                         
                         - 
                         
                           V 
                           rn 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     While the calibrated weighting value Ŵ 8  is obtained, the digital output signal D o  can be corrected according to the calibrated weighting value Ŵ 8  as: 
     
       
         
           
             
               
                 
                   
                     D 
                     
                       o 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       c 
                     
                   
                   = 
                   
                     
                       
                         b 
                         11 
                       
                       · 
                       
                         W 
                         11 
                       
                     
                     + 
                     
                       
                         b 
                         10 
                       
                       · 
                       
                         W 
                         10 
                       
                     
                     + 
                     
                       
                         b 
                         9 
                       
                       · 
                       
                         W 
                         9 
                       
                     
                     + 
                     
                       
                         b 
                         8 
                       
                       · 
                       
                         
                           W 
                           ^ 
                         
                         8 
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           0 
                         
                         7 
                       
                       ⁢ 
                       
                         
                           b 
                           k 
                         
                         · 
                         
                           W 
                           k 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     The above calibration process is also performed to the C 9,0 ˜C 9,1 , C 10,0 ˜C 10,3 , and C 11,0 ˜C 11,7 . The corresponding calibrated weighting values are given by: 
     
       
         
           
             
               
                 
                   
                     
                       W 
                       ^ 
                     
                     9 
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           0 
                         
                         1 
                       
                       ⁢ 
                       
                         
                           W 
                           ^ 
                         
                         
                           9 
                           , 
                           i 
                         
                       
                     
                     ≈ 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           0 
                         
                         1 
                       
                       ⁢ 
                       
                         
                           
                             C 
                             
                               9 
                               , 
                               i 
                             
                           
                           
                             C 
                             tot 
                           
                         
                         ⨯ 
                         
                           ( 
                           
                             
                               V 
                               rp 
                             
                             - 
                             
                               V 
                               rn 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     
       
         
           
             
               
                 
                   
                     
                       W 
                       ^ 
                     
                     10 
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           0 
                         
                         3 
                       
                       ⁢ 
                       
                         
                           W 
                           ^ 
                         
                         
                           10 
                           , 
                           i 
                         
                       
                     
                     ≈ 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           0 
                         
                         3 
                       
                       ⁢ 
                       
                         
                           
                             C 
                             
                               10 
                               , 
                               i 
                             
                           
                           
                             C 
                             tot 
                           
                         
                         ⨯ 
                         
                           ( 
                           
                             
                               V 
                               rp 
                             
                             - 
                             
                               V 
                               rn 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     
       
         
           
             
               
                 
                   
                     
                       W 
                       ^ 
                     
                     11 
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           0 
                         
                         7 
                       
                       ⁢ 
                       
                         
                           W 
                           ^ 
                         
                         
                           11 
                           , 
                           i 
                         
                       
                     
                     ≈ 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           0 
                         
                         7 
                       
                       ⁢ 
                       
                         
                           
                             C 
                             
                               11 
                               , 
                               i 
                             
                           
                           
                             C 
                             tot 
                           
                         
                         ⨯ 
                         
                           ( 
                           
                             
                               V 
                               rp 
                             
                             - 
                             
                               V 
                               rn 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     After the calibration process of the weighting values of C 8 , C 9,0 ˜C 9,1 , C 10,0 ˜C 10,3 , and C 11,0 ˜C 11,7 , the digital output signal D o  is corrected as: 
     
       
         
           
             
               
                 
                   
                     D 
                     oc 
                   
                   = 
                   
                     
                       
                         b 
                         11 
                       
                       · 
                       
                         
                           W 
                           ^ 
                         
                         11 
                       
                     
                     + 
                     
                       
                         b 
                         10 
                       
                       · 
                       
                         
                           W 
                           ^ 
                         
                         10 
                       
                     
                     + 
                     
                       
                         b 
                         9 
                       
                       · 
                       
                         
                           W 
                           ^ 
                         
                         9 
                       
                     
                     + 
                     
                       
                         b 
                         8 
                       
                       · 
                       
                         
                           W 
                           ^ 
                         
                         8 
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           0 
                         
                         7 
                       
                       ⁢ 
                       
                         
                           b 
                           k 
                         
                         · 
                         
                           W 
                           k 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
     While the invention has been described by way of example and in terms of the preferred embodiments, it is to be understood that the invention is not limited to the disclosed embodiments. To the contrary, it is intended to cover various modifications and similar arrangements (as would be apparent to those skilled in the art). Therefore, the scope of the appended claims should be accorded the broadest interpretation so as to encompass all such modifications and similar arrangements.