Patent Publication Number: US-10333538-B1

Title: Time interleaved ADC adaptive filtering

Description:
BACKGROUND 
     Technical Field 
     The present disclosure relates to a time interleaved analog-to-digital converter (ADC) and to an adaptive filtering technique of a time interleaved analog-to-digital converter. 
     Description of the Related Art 
     Time interleaved analog-digital converters (TI ADC) have been developed in the context of high speed interfaces, such as serial/deserializers (SerDes), in communication applications. A TI ADC is normally placed close to the radio frequency (RF) components and is used to provide high sampling rate and high resolution to digitize the high speed SerDes signals or RF signals. The speed demand over band limited channels, such as board and backplane traces, is such that two level analog transceivers are no longer sufficient to sustain the Inter Symbolic Interference (ISI) and digital based transceivers in conjunction with multilevel modulations will gradually replace the analog ones. The TI ADC is a typical structure used in this case to address a high sampling rate, e.g., tens of gigasamples per second, and a relatively high number of bits per sample, e.g., 8 or 10 bits. In a TI ADC, a large number N of ADC circuits (“sub-converters”) are used to sample the signal at N equally spaced time instants, so that the sampling rate of each ADC is reduced by a factor of N. The overall sampling quality deteriorates as the bank of sub-converters presents mismatching between/among ADC instances. 
       FIG. 1  illustrates an example time interleaved ADC comprising four sub-converters ADC 1  to ADC 4 . Each of the sub-converters is coupled to an input line  102  via a corresponding switch  104  to  107  controlled by a respective timing signal ϕ 1  to ϕ 4  having respective phase offsets. Thus each one of sub-converters ADC 1  to ADC 4  samples an analog input signal A in  on the input line  102  at a different time, and provides a corresponding output signal (referred to herein as “ADC instance”) D 1  to D 4  to a corresponding input of plural inputs of a multiplexer (MUX)  108 . Multiplexer  108  generates an output data signal D out  on a line  110  by periodically selecting each of the output signals D 1  to D 4  in turn. Thus, by providing the four time-interleaved sub-converters ADC 1  to ADC 4 , the input signal A in  can be sampled at four times the rate of a single ADC, and thus the sampling frequency F s  can be four times as high. 
     The use of a time interleaved ADC enables the analog-to-digital conversion be performed at very high speed. On the other hand, because the signal is sampled sequentially by different sub-converters, many unwanted effect are introduced. In particular, the multiple sub-converters suffer from offset, gain and skew mismatches, i.e., samples taken by different sub-converters will suffer from different mismatches. Solutions have been adopted to calibrate each sub-converter to try resolving the mismatch issues. Analog calibration structures are costly and might be limited themselves by process mismatches and thermal or aging drifts. For each mismatch a different calibration structure and procedure would be needed. 
     BRIEF SUMMARY 
     The current technique solves the TI ADC sub-converter calibration tasks in the digital domain where highly precise structures and self-adaptive and process-independent mechanisms can be easily implemented. The technique uses a digital finite impulse response (FIR) equalization filtering unit coupled to outputs of the ADC instances, referred to herein as “sub-converters”, in a TI ADC device. The FIR filtering unit includes a different digital FIR sub-filter for each sub-converter of the TI ADC device. Namely, the FIR filtering coefficients are adapted specifically for each sub-converter to achieve a compensation for sub-converter mismatches and inter-symbol interference (ISI) equalization. Hence the present technique addresses the mismatch problem by compensating the mismatches in the digital domain rather than trying to calibrate every sub-converter to make them as much similar as possible to one another. 
     For high speed implementation, e.g., tens of gigasamples per second, digital FIR processing may be instantiated multiple times to operate in parallel and deliver processed samples at the desired speed. The current technique instantiates a FIR sub-filter for each ADC instance, i.e., each sampling taken and processed by a sub-converter is filtered using FIR filtering coefficients adapted specifically for the respective sub-converter. The FIR filter will produce N outputs, with N being the number of sub-converters that sample the analog signal, which are provided to a multiplexer that recreates the equalized stream at the nominal data rate. In the current technique, gain and timing skew mismatches are compensated by adapting the filtering coefficients of each FIR filtering taps of each sub-filter specifically with respect to the respective sub-converter via recursive least mean squares (LMS) adaptation process. That is, the FIR filter includes multiple separate sets of FIR filtering coefficients, each set specifically adapted for each sub-converter. So, the FIR filter includes multiple filters (referred to as “sub-filters”), namely, multiple sets of filtering coefficients. Within each set of filtering coefficients, a specific filtering coefficient of a filtering tap is adapted to apply to sampling instance of a specific sub-converter. Gain mismatch compensation is achieved through a combination of weighted average gain correction and sub-converter specific mismatch compensation. Timing skew compensation is achieved by sub-converter specific mismatch compensation. 
     According to one embodiment, a sub-filter for a sub-converter includes two groups of filtering coefficients. A filtering coefficient in the first group is determined based on a time domain error(s) associated with the specific sub-converter only. The time domain error (“error”) is defined as the difference between the filter output level and the corresponding detector decision level. The detection is obtained through any suitable approaches, none of which limits the scope of the disclosure. A filtering coefficient in the second group is determined based on the error(s) associated with the specific sub-converter only and based on a weighted average of the time domain errors associated with all sub-converters that sample an analog signal. 
    
    
     
       BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS 
       The foregoing and other purposes, features, aspects and advantages of embodiments of the present disclosure will become apparent from the following detailed description of embodiments, given by way of illustration and not limitation with reference to the accompanying drawings, in which: 
         FIG. 1  illustrates an example conventional interleaved ADC; 
         FIG. 2  illustrates an ADC converting device according to an example embodiment of the present disclosure; 
         FIG. 3  illustrates an example allocation of FIR filtering coefficient according to an example embodiment of the present disclosure; 
         FIG. 4  illustrates an example ADC method process according to an example embodiment of the present disclosure; and 
         FIGS. 5A-5E  illustrate the example ADC method process of  FIG. 4  with example ADC sampling instances. 
     
    
    
     DETAILED DESCRIPTION 
     Throughout the following description, only those elements useful for an understanding of the various embodiments will be described in detail. Other aspects, such as the particular type and form of the analog to digital converter circuitry implementations, have not been described in detail. The following embodiments applying to a wide range of converter types, such as pipeline converters or successive approximation register (SAR) ADCs. 
       FIG. 2  illustrates time interleaved ADC device  200  according to one embodiment. As an illustrative example, the TI ADC  200  includes a switch circuit  205 , a converter block  210  that includes multiple sub-converters  212 , three (ADC 1 , ADC 2 , ADC 3 ) shown as an illustrative example, each coupled to an analog input terminal A in  through the switch circuit  205 . Sub-converters  212  operate to sequentially sample an analog input signal at input terminal A in , with a time interval T s  through the timing control of the switch circuit  205 . Note that switch circuit  205  may include multiple switches for the multiple sub-converters  212  or may include a single switch configured to sequentially connect each sub-converter  212  to the analog input terminal A in , which are all included in the disclosure. Sub-converters  212  then process the analog samplings into ADC instance digital outputs y (y 1 , y 2 , y 3 ). The digital outputs y 1 , y 2 , y 3  of sub-converters ADC 1 , ADC 2 , ADC 3  are then processed by a finite impulse response (FIR) digital filter unit  220  before entering a multiplexer (MUX)  230 . As appreciated, digital outputs y 1 , y 2 , y 3  are affected by mismatches among sub-converters  212  in respective gains G 1 , G 2 , G 3 , offsets O 1 , O 2 , O 3 , and timing skews τ 1 , τ 2 , τ 3  (T s  is sampling interval). FIR digital filter unit  220  includes separate sub-filters  222 ( 1 ),  222 ( 2 ) and  222 ( 3 ) shown as illustrative examples, for each sub-converter  212 , namely the FIR filtering coefficients of FIR filter unit  220  include multiple separate sets of FIR filtering coefficients for the multiple sub-converters  212 , each set dedicated to and specifically adapted for each sub-converter  212 , ADC 1 , ADC 2 , ADC 3 . 
       FIG. 3  illustrates example FIR filtering operation architecture  300  of the FIR digital filter unit  220 . Referring to  FIG. 3 , in the example FIR architecture  300 , three sub-filters  222 ( 1 ),  222 ( 2 ) and  222 ( 3 ) are each dedicated to the respective sub-converter  212 , in the sense that each sub-filter  222 ( 1 ),  222 ( 2 ) and  222 ( 3 ) receives the current sampling instance only from the respective sub-converter  212 . Each sub-filter  222 ( 1 ),  222 ( 2 ) and  222 ( 3 ) also receive sampling instances prior to the current sampling instance of the respective sub-converter  212  and have filtering coefficients for the previous sampling instances. The previous sampling instances include sampling instances obtained by multiple sub-converters  2121  at different sampling time points. Each filtering coefficient of a sub-filter other filtering  222 ( 1 ),  222 ( 2 ) and  222 ( 3 ) is specifically adapted to be applied to a sampling instance obtained by a specific respective sub-converter  212 . Specifically, sub-filter  222 ( 1 ) is dedicated to sub-converter ADC 1 , sub-filter  222 ( 2 ) is dedicated to sub-converter ADC 2  and sub-filter  222 ( 3 ) is dedicated to sub-converter ADC 3 . It should be appreciated that each sub-filter  222 ( 1 ),  222 ( 2 ) and  222 ( 3 ) is not necessarily a separate filter unit and may just refer to a separate set of filtering coefficients dedicated to filtering the digital output y 1 , y 2 , y 3  of a specific sub-converter  212 . Each sub-filter  222 ( 1 ),  222 ( 2 ) and  222 ( 3 ) may include multiple filtering terms (taps) each with its own filtering coefficient. In an example, at each filtering tap of a sub-filter  222 ( 1 ),  222 ( 2 ),  222 ( 3 ), a delayed digital sampling value of the digital output y 1 , y 2 , y 3  is combined, e.g., multiplied, with the respective filtering coefficient to generate a filtering product. The filtering products of the multiple filtering taps of the respective sub-filter  222 ( 1 ),  222 ( 2 ),  222 ( 3 ) are combined, e.g., added, in the respective adder  224 ( 1 ),  224 ( 2 ),  224 ( 3 ) to generate the respective filtering results D 1 , D 2 , D 3 . 
     Note that the filtering coefficient for a filtering tap of a sub-filter  222 ( 1 ),  222 ( 2 ) and  222 ( 3 ) is specifically adapted for the respective sub-converter  212 . As shown in  FIG. 3  as an illustrative example, sub-filter  222 ( 1 ) includes filtering coefficient f 1,1 , f 1,2 , f 1,3 , f 1,4 , f 1,5 , f 1,6  for the six filtering taps t 1,1 , t 1,2 , t 1,3 , t 1,4 , t 1,6 . Sub-filter  222 ( 2 ) includes filtering coefficient f 2,1 , f 2,2 , f 2,3 , f 2,4 , f 2,5 , f 2,6  for their respective filtering taps t 2,1 , t 2,2 , t 2,3 , t 2,4 , t 2,5 , t 2,6  and sub-filter  222 ( 3 ) includes filtering coefficient f 3,1 , f 3,2 , f 3,3 , f 3,4 , f 3,5 , f 3,6  for their respective filtering taps t 3,1 , t 3,2 , t 3,3 , t 3,5 , t 3,6 . For example, the filtering coefficient f 1,1  for the first filtering tap t 1,1  of sub-filter  222 ( 1 ) is different from the filtering coefficient f 2,1  for the first filtering tap t 2,1  of sub-filter  222 ( 2 ). 
     Among the filtering taps of each sub-filter  222 ( 1 ),  222 ( 2 ),  222 ( 3 ), the respective first filtering taps, t 1,1 , t 2,1 , t 3,1 , apply to a current digital sampling y 1 , y 2 , y 3 , and the rest filtering taps, e.g., t 1,2 , t 1,3 , t 1,4 , t 1,5 , t 1,6  of sub-filter  222 ( 1 ), apply to some previous digital samplings obtained by the three sub-converters  212 , namely “delayed digital samplings”. In the description herein, a current digital sampling is also referred to as a “delayed digital sampling” in a general sense. As illustratively shown in  FIG. 2 , sub-converters ADC 1 , ADC 2  and ADC 3  sequentially sample analog input A in  in the listed order. In  FIG. 3 , for descriptive purposes, delayed digital samplings of a sub-converter ADC 1 , ADC 2 , ADC 3  are referred to a d x,y , where x indicate the sub-converter and y indicates the delay position of the digital sampling in the overall sequence of the sampling by all the sub-converters  212 . Note that the three sub-converters ADC 1 , ADC 2 , ADC 3  sequentially sample the input A in . For example, sub-converter ADC 1  includes sampling instances d 1,2 , d 1,3 , d 1,4 , d 1,5 , d 1,6 , where d 1,1  is the current digital sampling. After ADC 1 , the second sub-converter ADC 2  sequentially obtains its current digital sampling, the previously digital sampling d 1,1  will become d 1,2 , indicting its second delay position in the overall sampling sequence; after ADC 2 , the third sub-converter ADC 3  sequentially obtains its current digital sampling, the digital sampling d 1,2  will become d 1,3 , i.e., in the third delay position in the overall sampling sequence; after ADC 3 , the first sub-converter ADC 1  sequentially obtains its new current digital sampling, i.e., a new d 1,1 , the previously digital sampling d 1,3  will become d 1,4 , i.e., in the fourth delay position in the overall sampling sequence; after ADC 1 , the second sub-converter ADC 2  sequentially obtains its new current digital sampling, the previously digital sampling d 1,1  will become d 1,2  and the previously digital sampling d 1,4  will become d 1,5 , indicating their respective second and fifth delay positions in the overall sampling sequence; and again after ADC 2 , the third sub-converter ADC 3  sequentially obtains its new current digital sampling, the previously digital sampling d 1,2  will become d 1,3  and the previously digital sampling d 1,5  will become d 1,6 , indicating their respective third and sixth delay positions in the overall sampling sequence. 
     Similarly, delayed digital samplings of sub-converter ADC 2  includes d 2,1 , d 2,2 , d 2,3 , d 2,4 , d 2,5 , d 2,6 , where d 2,1  is the current digital sampling and d 2,2 , d 2,3 , d 2,4 , d 2,5 , d 2,6  are previous sampling values obtained by sub-converter ADC 2  in the overall sampling sequence. Delayed digital samplings of sub-converter ADC 3  includes d 3,1 , d 3,2 , d 3,3 , d 3,4 , d 3,5 , d 3,6 , where d 3,1  is the current digital sampling and d 3,2 , d 3,3 , d 3,4 , d 3,5 , d 3,6  are previous sampling values of ADC 3  in the overall sampling sequence. 
     As sub-converters ADC 1 , ADC 2  and ADC 3  sequentially sample analog input A in  in an interleaved manner, the retrospective time sequence (i.e., from present time point to previous time point) includes digital samplings obtained by/from different sub-converters ADC 1 , ADC 2  and ADC 3  in an interleaved manner. For example, when sub-converter ADC 1  is currently sampling analog input A in , the retrospective time sequence is d 1,1 , d 3,2 , d 2,3 , d 1,4 , d 3,5 , d 2,6 , where d 1,1  is the current sampling among all sub-converter ADC 1 , ADC 2  and ADC 3 , d 3,2  is the immediately previous sampling, i.e., one sampling interval T s  before d 1,1 , d 2,3  is two sampling interval T s  before, d 1,4  is three sampling interval T s  before, d 3,5  is four sampling interval T s  before, and d 2,6  is five sampling interval T s  before. When sub-converter ADC 2  is currently sampling analog input A in , the retrospective time sequence is d 2,1 , d 1,2 , d 3,3 , d 2,4 , d 1,5 , d 3,6 , where d 2,1  is the current sampling among all sub-converter ADC 1 , ADC 2  and ADC 3 . When sub-converter ADC 3  is currently sampling analog input A in , the retrospective time sequence is d 3,1 , d 2,2 , d 1,3 , d 3,4 , d 2,5 , d 1,6 , where d 3,1  is the current sampling among all sub-converter ADC 1 , ADC 2  and ADC 3 . 
     Filtering taps of each sub-filter  222 ( 1 ),  222 ( 2 ),  222 ( 3 ) apply to the current digital sampling of the respective sub-converter  212 , and the retrospective sequence of the delayed digital samplings including the current digital sampling. For example, for sub-filter  222 ( 1 ), the first filtering tap t 1,1  applies to the current digital sampling d 1,1  of sub-converter ADC 1 , the second filtering tap t 1,2  applies to the immediately previous digital sampling d 3,2 . And filtering taps t 1,3 , t 1,4 , t 1,5 , t 1,6  apply to further previous digital samplings d 2,3 , d 1,4 , d 3,5 , d 2,6 , respectively. Similarly, filtering taps t 2,1 , t 2,2 , t 2,3 , t 2,4 , t 2,5 , t 2,6  of sub-filter  212 ( 2 ) apply to the retrospective sequence of digital samplings d 2,1 , d 1,2 , d 3,3 , d 2,4 , d 1,5 , d 3,6 , with first filtering tap t 2,1  applying to digital sampling d 2,1  of ADC 2  as the current digital sampling. Filtering taps t 3,1 , t 3,2 , t 3,3 , t 3,4 , t 3,5 , t 3,6  of sub-filter  212 ( 3 ) apply to the retrospective sequence of digital samplings d 3,1 , d 2,2 , d 1,3 , d 3,4 , d 2,5 , d 1,6 , with first filtering tap t 3,1  applying to digital sampling d 3,1  of ADC 3  as the current digital sampling. Note that, d 1,1  and d 1,2 , for example, are the same sampling instance obtained sub-converter ADC 1 , and a current sampling d 1,1  will become d 1,2 , i.e., second delay position in the overall sampling sequence, when next sampling of analog input A in  by sub-converter ADC 2  is obtained. In the illustrative example of three sub-converters  212 , d 1,1  and d 1,4  are the sequential sampling instances obtained by sub-converter ADC 1 , and d 1,4  is obtained prior to d 1,1 . 
     Further, a FIR filtering coefficient of each sub-filter  222 ( 1 ),  222 ( 2 ),  222 ( 3 ) is specifically adapted with respect to the respective sub-converter  212 . For example, for the first filtering taps of sub-filters  222 ( 1 ),  222 ( 2 ) and  222 ( 3 ), three separate filtering coefficient f 1,1 , f 2,1  and f 3,1  are specifically adapted to be applied to the digital samplings d 1,1 , d 2,1 , d 3,1  (i.e., the current sampling values of y 1 , y 2 , y 3 ) obtained by the sub-converters ADC 1 , ADC 2 , ADC 3 , respectively. Similarly, for the third filtering taps of sub-filters  222 ( 1 ),  222 ( 2 ) and  222 ( 3 ), three different filtering coefficients f 1,3 , f 2,3  and f 3,3  are specifically adapted for the sub-converters ADC 1 , ADC 2 , ADC 3 , respectively. Note that different filtering coefficients, e.g., f 1,1 , f 2,1 , are different in the sense that they are adapted differently, which does not necessarily mean that they have different values. 
       FIG. 3  shows, as an illustrative example, how previous digital samplings are used in the filtering, which does not limit the scope of the disclosure. Other patterns of applying filtering coefficients to delayed digital samplings are also possible and included in the disclosure. 
     Within each sub-filter  222 , the processing results of the respective filtering taps will be combined by respective adders  224 ( 1 ),  224 ( 2 ),  224 ( 3 ), that separately generate the respective filtering results D 1 , D 2 , D 3 . 
     In an example, the set of filtering coefficients of each sub-filter  222 ( 1 ),  222 ( 2 ) and  222 ( 3 ), i.e., specifically for a respective sub-converter  212 , e.g., filtering coefficient f 1,1 , f 1,2 , f 1,3 , f 1,4 , f 1,5 , f 1,6  of sub-filter  222 ( 1 ) for sub-converter ADC 1 , may include a first group of filtering coefficients and a second group of filtering coefficients. For the first group of filtering coefficients of the sub-filter  222 , e.g., sub-filter  222 ( 1 ), the coefficient values are adapted based on the time domain error (referred herein as “error”) associated only with the corresponding sub-converter  212 , here e.g., ADC 1 , and errors of other sub-converters  212 , here e.g., ADC 2  and ADC 3 , are not factored in. For the second group of filtering coefficients of sub-filter  222 ( 1 ), the coefficient values are adapted based on the error associated with the corresponding sub-converter  212 , here ADC 1 , and a weighted average of errors of all sub-converters  212 , here ADC 1 , ADC 2  and ADC 3 . In an example, the first group includes more elements than the second group. In an example, the second group includes only the filtering coefficient(s) for the center filtering tap, t 1,3  and/or t 1,4 , of the sub-filter  222 ( 1 ) for sub-converter ADC 1 . In the example six filtering taps of  FIG. 3 , the second group of filtering coefficient of each sub-filter  222  may include the respective filtering coefficients for the respective center taps, namely one or both of the third filtering tap or the fourth filtering tap of each sub-filter  222 , e.g., one or both of f 1,3  and f 1,4  of sub-filter  222 ( 1 ) for sub-converter ADC 1 . 
     As the first group of FIR filtering coefficients of each sub-filter  222  are adapted using only the error(s) associated with the respective same sub-converter  212 , the skew mismatches among various sub-converters  212  are compensated for. As the second group of FIR filtering coefficients of each sub-filter  222  for the respective sub-converter  212  are adapted using the error(s) associated with the respective sub-converter  212  and a weighted average of errors of all sub-converters  212 , the gain mismatches among different sub-converters  212  are effectively compensated for. 
     According to an example, the second group of FIR filtering coefficients of a sub-filter  222  may be adapted following the below algorithms: 
     
       
         
           
             
               
                 
                   
                     
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                           ∑     n   =   1     N     ⁢     f   nj       N     =       M   j     ⁢           ⁢   constraint       ,           (   2   )               
where, N indicates the number of sub-converters  212 , here for the example of  FIG. 3 , N=3; L indicates the number of filtering taps of a sub-filter  222 , here for the example of  FIG. 3 , L=6; f ni  indicates a filtering coefficient of the ith filtering tap of the sub-filter  222  for nth sub-converter  212 ; y ni  indicates the delayed ADC instance input to the ith filtering tap for the nth sub-converter  212 , here for the example of  FIG. 3 , y 1,2  of the 2nd tap of sub-filter  212 ( 1 ) for sub-converter ADC 1  is the delayed sampling data from ADC 3 ; and d n  indicates the decision output of the sub-filter  222  for the specific sub-nth converter  212 , e.g., for example of  FIG. 3 , d 1  is filtered output of sub-converter ADC 1 .
 
     In the algorithm (2), constant m j  is determined as a constraint on the jth FIR filtering taps within the second group of filtering coefficients for each of the sub-converters  212 . This value constraint functions to compensate for the gain mismatches among different sub-converters  212 . As provided by algorithm (2), the average value of the jth FIR filtering taps of all the sub-filters  222  respectively for all the sub-converters  212  shall be equal to the constraint value m j . 
     With the m j  value set, the below algorithms may be used to determine a filtering coefficient in the second group 
     A Lagrange optimization function is: 
     
       
         
           
             
               
                 
                   
                     
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     With FIR adaptation procedures applied: 
                         ∂   Λ       ∂     f     n   ⁢           ⁢   j           =         2   ·     err   n     ·     y   nj       +     λ   N       =   0       ,           (   4   )               
where, n=1, . . . N; j=a filtering tap in the second group, for example, the central tap(s). And,
 
                         ∂   Λ       ∂   λ       =         (         ∑     n   =   1     N     ⁢     f     n   ⁢           ⁢   j         N     )     -     m   j       =   0       ,           (   5   )               
With algorithms (3), (4) and (5) applied together, it can be obtained that:
 
     
       
         
           
             
               
                 
                   
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     Therefore, a filtering coefficient in the second group might be determined as: 
                       f     n   ⁢           ⁢   j       =       f     n   ⁢           ⁢   j       -     γ   ·     (         err   n     ·     y     n   ⁢           ⁢   j         -         ∑     n   =   1     N     ⁢       err   n     ·     y     n   ⁢           ⁢   j           N       )           ,           (   7   )               
where γ is a constant. Note that in the adaptation of coefficient f nj , it is factored in both the error that is associated only to the specific sub-converter  212 , i.e., err n  for nth sub-converter  212 , and the weighted average of errors of all sub-converters  212 , i.e., (Σ n=1   N  err n ·y nj )/N.
 
     For the value adaptation of a filtering coefficient in the first group, only the error associated to the specific nth sub-converter  212  is used:
 
 f   ni   =f   ni −γ·err n   ·y   ni  where,  n= 1, . . .  N,i= 1, . . .  L,i≠j   (8)
 
     For a filtering coefficient in the second group, the constraint value m j  may be setup as a predetermined value or may be dynamically adapted. In an example, the value m j  may be used as a constraint in the following procedures:
 
 f   nj   =f   nj −γ(2·err n   ·y   nj   +λ/N )  (9),
 
                           ∑     n   =   1     N     ⁢     [       f     n   ⁢           ⁢   j       -     γ   ⁡     (       2   ·     err   n     ·     y     n   ⁢           ⁢   j         +     λ   N       )         ]       N     =     m   j       ,           (   10   )               Σ n=1   N   f   nj −Σ n=1   N γ(2·err n   ·y   nj )−γλ= m   j   N   (11),
 
     
       
         
           
             
               
                 
                   
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                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     Based on algorithms (9)-(12), an updated FIR coefficient adaptation algorithm could be obtained for a filtering tap in the second group: 
                       f     n   ⁢           ⁢   j       =       f     n   ⁢           ⁢   j       -     γ   ·     err   n     ·     y     n   ⁢           ⁢   j         -     (           ∑     n   =   1     N     ⁢     f     n   ⁢           ⁢   j         N     -     m   j     -     γ   ·         ∑     n   =   1     N     ⁢     (     2   ·     err   n     ·     y     n   ⁢           ⁢   j         )       N         )         ,           (   13   )               
where n=1, . . . N; and where j=a filtering tap in the second group.
 
     With the example algorithm (13) for adapting/determining the FIR filtering coefficient of a filtering tap in the second group of a sub-filter  222 , the constraint value m j  may be dynamically adapted to a desired value, which could be different than the initial value. Note that if the term 
                   ∑     n   =   1     N     ⁢     f     n   ⁢           ⁢   j         N     -     m   j           
is omitted from algorithm (13), the algorithm (13) becomes the same as algorithm (7) described herein.
 
     In an example, the initial value of f ni  in algorithm (8) for a filtering tap in the first group may be set up as “0” and the initial value f nj  in algorithm (7) or (13) for a filtering tap in the second group may be set up as “1”. 
       FIG. 4  illustrates an example process  400  of converting an analog signal to a digital signal. Referring to  FIG. 4 , in example operation  410 , an analog signal is sampled sequentially by sub-converters  212  of a TI ADC  210  as shown in  FIG. 2  as an example embodiment, to generate ADC instance digital outputs (“ADC instances”) y, each ADC instance y 1 , y 2 , y 3  being a digital output of the respective sub-converter  212 . 
     In example operation  420 , the ADC instances of the sub-converters  212  are fed into a FIR filtering unit  220  for a digital FIR filtering processing. The digital FIR filtering unit  220  has multiple sub-filters  222  each dedicated to a respective sub-converter  212 . A sub-filter  222  includes FIR filtering coefficients specifically adapted for the respective sub-converter  212 . Specifically, the filtering coefficients of a sub-filter  222  for a sub-converter  212  includes a first group of filtering coefficients for a first group of filtering taps of the sub-filter  222 , which are determined based on errors of the respective sub-converter  212  only and a second group of filtering coefficients for a second group of filtering taps of the sub-filter  222 , which are determined based on errors of the respective sub-converter  212  and a weighted average of errors of all the sub-converters  212  that sample the analog signal. 
     In an example, the filtering coefficients of the first group of filtering taps may be determined using example algorithm (8) described herein. In an example, the filtering coefficients of the second group of filtering taps may be determined using one of example algorithms (7) or (13) described herein. 
     In example operation  430 , the results of the sub-filters  222  of FIR filtering unit  220  are fed into the multiplexer  230  to be combined to generate a digital signal. 
       FIGS. 5A-5E  illustrates the operation of the example process  400  and/or the system  200  using illustrative example samplings. 
     Referring to  FIG. 5A , the example ADC samplings are sequential ADC sampling instances y 1 , y 2 , y 3 , y 4 , y 5 , y 6 , y 7 , y 8 , y 9 . The sequential samplings are obtained by multiple sub-converters  212 , ADC 1 , ADC 2 , ADC 3  sequentially. Specifically, y 1 , y 4 , y 7  are obtained by ADC 1 , y 2 , y 5 , y 8  are obtained by ADC 2 , and y 3 , y 6 , y 9  are obtained by ADC 3 . As shown in  FIG. 4A , until a new sampling instance y 4  is obtained by the sub-converter ADC 1 , sampling instance y 1  is the delay sampling instances that the sub-converter ADC 1  provides to the filtering taps of the sub-filters  222 ( 1 ),  222 ( 2 ),  222 ( 3 ) (not specifically shown in  FIG. 4A  for simplicity). Similarly, until a new sampling instance y 7  is obtained by the sub-converter ADC 1 , sampling instance y 4  is the delay sampling instances that the sub-converter ADC 1  provides to the filtering taps of the sub-filters  222 ( 1 ),  222 ( 2 ),  222 ( 3 ). 
     The example sampling instances y 1 , y 2 , y 3 , y 4 , y 5 , y 6 , y 7 , y 8 , y 9  are obtained at sampling time points t 1 , t 2 , t 3 , t 4 , t 5 , t 6 , t 7 , t 8 , t 9 , respectively. 
     Referring to  FIG. 5B , at the time point t 6 , the current sampling instance y 6  is obtained by the sub-converter ADC 3  and is fed into the first filtering tap of the respective sub-filter  222 ( 3 ), which has a filtering coefficient f 3,1 . Previous sampling instances y 5 , y 4 , y 3 , y 2 , y 1  are also received by the sub-filter  222 ( 3 ) and are processed with the filtering coefficients f 3,2 , f 3,3 , f 3,4 , f 3,5 , f 3,6  respectively. The result D 3  of the sub-filter  222 ( 3 ) will be output as the digital output D out . 
     Referring to  FIG. 5C , at the time point t 7 , the current sampling instance y 7  is obtained by the sub-converter ADC 1  and is fed into the first filtering tap of the respective sub-filter  222 ( 1 ), which has a filtering coefficient f 1,1 . Sampling instance y 6  now becomes a previous sampling instance. Previous sampling instances y 6 , y 5 , y 4 , y 3 , y 2  are also received by the sub-filter  222 ( 1 ) and are processed with the filtering coefficients f 1,2 , f 1,3 , f 1,4 , f 1,5 , f 1,6  respectively. The result D 1  of the sub-filter  222 ( 1 ) will be output as the digital output D out .  FIG. 4C  shows that for the sub-filters  222 ( 2 ) and  222 ( 3 ), the filtering coefficients are shown as applied to previous sampling instances, which indicates that the procedures in each sub-filters  222  may be conducted partially in parallel with one another. 
     Referring to  FIG. 5D , at the time point t 8 , the current sampling instance y 8  is obtained by the sub-converter ADC 2  and is fed into the first filtering tap of the respective sub-filter  222 ( 2 ), which has a filtering coefficient f 2,1 . Sampling instance y 7  now becomes a previous sampling instance. Previous sampling instances y 7 , y 6 , y 5 , y 4 , y 3  are also received by the sub-filter  222 ( 2 ) and are processed with the filtering coefficients f 2,2 , f 2,3 , f 2,4 , f 2,5 , f 2,6  respectively. The result D 2  of the sub-filter  222 ( 1 ) will be output as the digital output D out . 
     Referring to  FIG. 5E , at the time point t 9 , the current sampling instance y 9  is obtained by the sub-converter ADC 3  and is fed into the first filtering tap of the respective sub-filter  222 ( 3 ), which has a filtering coefficient f 3,1 . Sampling instance y 8  now becomes a previous sampling instance. Previous sampling instances y 8 , y 7 , y 6 , y 5 , y 4  are also received by the sub-filter  222 ( 3 ) and are processed with the filtering coefficients f 3,2 , f 3,3 , f 3,4 , f 3,5 , f 3,6  respectively. The result D 3  of the sub-filter  222 ( 3 ) will be output as the digital output D put . 
     Referring to  FIGS. 5B and 5E  together, it show that the filtering coefficients f 3,1 , f 3,2 , f 3,3 , f 3,4 , f 3,5 , f 3,6  of sub-filter  222 ( 3 ) each is applied to a sampling instance obtained by a respective same sub-converter  212 . The first filtering coefficient f 3,1  is always applied to a current sampling instance obtained by the respective sub-converter ADC 3 . The second filtering coefficient f 3,2  is always applied to a delayed sampling instance obtained by the sub-converter ADC 2 . The third filtering coefficient f 3,3  is always applied to a delayed sampling instance obtained by the sub-converter ADC 1 . The fourth filtering coefficient f 3,4  is always applied to a delayed sampling instance obtained by the sub-converter ADC 3 . The fifth filtering coefficient f 3,5  is always applied to a delayed sampling instance obtained by the sub-converter ADC 2 . The sixed filtering coefficient f 3,6  is always applied to a delayed sampling instance obtained by the sub-converter ADC 1 . At the same time, the filtering coefficients f 3,1 , f 3,2 , f 3,3 , f 3,4 , f 3,5 , f 3,6  are adapted using the time domain error of the respective sub-converter ADC 3 . Similar descriptions also apply to sub-filters  222 ( 1 ) and  222 ( 2 ). 
     Having thus described at least one illustrative embodiment of the disclosure, various alterations, modifications and improvements will readily occur to those skilled in the art. Such alterations, modifications, and improvements are intended to be part of this disclosure, and are intended to be within the spirit and the scope of the present disclosure. Accordingly, the foregoing description is by way of example only and is not intended to be limiting. 
     The various embodiments described above can be combined to provide further embodiments. These and other changes can be made to the embodiments in light of the above-detailed description. In general, in the following claims, the terms used should not be construed to limit the claims to the specific embodiments disclosed in the specification and the claims, but should be construed to include all possible embodiments along with the full scope of equivalents to which such claims are entitled. Accordingly, the claims are not limited by the disclosure.