Patent Publication Number: US-8525711-B2

Title: Method and apparatus for performing nonlinearity calibration

Description:
BACKGROUND 
     The present invention relates to access to performance control of a nonlinear system, and more particularly, to a method for performing nonlinearity calibration, and to an associated apparatus. 
     Nonlinearity of a component such as an analog-to-digital converter (ADC) is an issue in some applications. Although there are some solutions proposed by the related art in response to the problem of nonlinearity, some other problems may occur. For example, the related art algorithm may be too complicated, causing the chip area to be greatly increased. In another example, after power-on of a system comprising a conventional ADC, the conventional ADC cannot rapidly get ready for being used by the system. It seems unlikely that the related art provides a real solution having no side effect. Therefore, a novel method is required for performing nonlinearity calibration on a nonlinear system, in order to remove the influence of nonlinearity. 
     SUMMARY 
     It is therefore an objective of the claimed invention to provide a method for performing nonlinearity calibration, and to provide an associated apparatus, in order to solve the above-mentioned problems. 
     An exemplary embodiment of a method for performing nonlinearity calibration comprises the steps of: obtaining temporarily values of a plurality of compensation parameters by performing a perturbation-based calibration process on a nonlinear system with at least one predetermined input being applied to the nonlinear system; and updating the compensation parameters by performing the perturbation-based calibration process in an online manner, wherein the temporarily values are utilized as initial values of the compensation parameters for the step of updating the compensation parameters. In addition, the compensation parameters are utilized for controlling a compensation response of the perturbation-based calibration process. 
     An exemplary embodiment of an apparatus for performing nonlinearity calibration comprises a calibration loop and an input selector. The calibration loop is arranged to perform a perturbation-based calibration process, the calibration loop comprising a nonlinear system to be calibrated, wherein during performing compensation for the nonlinear system in the perturbation-based calibration process, a plurality of compensation parameters are utilized for controlling a compensation response. The input selector is arranged to select at least one input to be applied to the nonlinear system from at least one ordinary input and at least one predetermined input. In addition, the calibration loop is arranged to obtain temporarily values of the compensation parameters by performing the perturbation-based calibration process for the nonlinear system with the at least one predetermined input being applied to the nonlinear system. Additionally, the calibration loop is arranged to determine latest values of the compensation parameters by performing the perturbation-based calibration process in an online manner, wherein the temporarily values are utilized as initial values of the compensation parameters for online calibration. 
     These and other objectives of the present invention will no doubt become obvious to those of ordinary skill in the art after reading the following detailed description of the preferred embodiment that is illustrated in the various figures and drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1A  is a diagram of an apparatus for performing nonlinearity calibration according to a first embodiment of the present invention. 
         FIG. 1B  illustrates a data available time (DAT) involved with the apparatus shown in  FIG. 1A  according to an embodiment of the present invention. 
         FIG. 2  is a flowchart of a method for performing nonlinearity calibration according to an embodiment of the present invention. 
         FIG. 3  illustrates some implementation details of the apparatus shown in  FIG. 1A  according to an embodiment of the present invention. 
         FIG. 4  illustrates some implementation details of the compensation module shown in  FIG. 3  according to an embodiment of the present invention. 
         FIG. 5A  illustrates some implementation details of the estimation module shown in  FIG. 3  according to an embodiment of the present invention. 
         FIG. 5B  illustrates some implementation details of the multiplexer shown in  FIG. 5A  according to an embodiment of the present invention. 
         FIG. 5C  illustrates some implementation details involved with the Lyapunov-based estimator shown in  FIG. 5A  according to an embodiment of the present invention. 
     
    
    
     DETAILED DESCRIPTION 
     Certain terms are used throughout the following description and claims, which refer to particular components. As one skilled in the art will appreciate, electronic equipment manufacturers may refer to a component by different names. This document does not intend to distinguish between components that differ in name but not in function. In the following description and in the claims, the terms “include” and “comprise” are used in an open-ended fashion, and thus should be interpreted to mean “include, but not limited to . . . ”. Also, the term “couple” is intended to mean either an indirect or direct electrical connection. Accordingly, if one device is coupled to another device, that connection may be through a direct electrical connection, or through an indirect electrical connection via other devices and connections. 
     Please refer to  FIG. 1A , which illustrates a diagram of an apparatus  100  for performing nonlinearity calibration according to a first embodiment of the present invention. The apparatus  100  comprises a nonlinear system  100 A and a compensation system  100 B. The nonlinear system  100 A performs an operation of a function f(x) on an input x to generate an output y, where the function f(x) is typically a nonlinear function. In order to correct the nonlinearity of the nonlinear system  100 A, the compensation system  100 B is arranged to perform an operation of a correction function f c (y) on the output y to generate a corrected output y c . According to a first embodiment, in a situation where the compensation system  100 B is properly designed, the nonlinear effect due to the nonlinearity of the nonlinear system  100 A can be absent in the corrected output y c . For example, the nonlinear function f(x) can be expressed in an approximation form as follows:
 
 y=f ( x )≈Σ i1=0, 1, . . . , l ( a   i1   x   i1 );
 
where the notation a i1  represents the coefficient of the term x i1  in the above equation. In addition, the correction function f c (y) can be expressed in an approximation form as follows:
 
 y   c   =f   c ( y )≈Σ i=0, 1, . . . , k ( b   i   y   i )≈ x;  
 
where the notation b i  represents the coefficient of the term y i  in the above equation, and can be regarded as a compensation parameter of the compensation system  100 B.
 
       FIG. 1B  illustrates a data available time (DAT) involved with the apparatus  100  shown in  FIG. 1A  according to an embodiment of the present invention. The horizontal axis represents time, while the vertical axis represents a difference (y c −x) between the corrected output y c  and the input x. In this embodiment, the curve shown in  FIG. 1B  indicates that the difference (y c −x) approaches 0 within the DAT, where the DAT typically represents the time to make the corrected output y c  (e.g. the corrected output data) approach the input x after power-on or waking up from a standby status. 
     In practice, the DAT can be determined by detecting the time for the absolute value of the difference (y c −x) to fall within a predetermined upper limit, where the upper limit may correspond to some standards that the apparatus  100  complies with. This is for illustrative purposes only, and is not meant to be a limitation of the present invention. According to some variations of this embodiment, in a situation where the apparatus  100  is an analog-to-digital converter (ADC), the DAT can be determined by detecting the time for the effective number of bits (ENOB) of the ADC to reach a predetermined value corresponding to some standards that the ADC complies with. For example, the apparatus  100  can be a 12 bit ADC, and the DAT can be determined by detecting the time for the ENOB of the 12 bit ADC to reach a predetermined value 11.5. Referring to  FIG. 2 , further details for rapidly obtaining correct values of the compensation parameters {b i } are described as follows. 
       FIG. 2  is a flowchart of a method  910  for performing nonlinearity calibration according to an embodiment of the present invention. The method  910  can be applied to the apparatus  100  shown in  FIG. 1A , and more particularly, to the compensation system  100 B mentioned above. For example, referring to  FIG. 3 , the apparatus  100  comprises an input selector  110  and a calibration loop arranged to perform a perturbation-based calibration process, with the calibration loop comprising an arithmetic unit  120  (e.g. an adder), a nonlinear system  130  to be calibrated, a compensation module  140 , an estimation module  150 , and a perturbation generator  160 , where the input selector  110 , the arithmetic unit  120 , and the nonlinear system  130  can be regarded as the nonlinear system  100 A shown in  FIG. 1A , and the compensation module  140 , the estimation module  150 , and the perturbation generator  160  can be regarded as the compensation system  100 B shown in  FIG. 1A . The method is described as follows. 
     In Step  912 , the calibration loop mentioned above obtains temporarily values of a plurality of compensation parameters {b i } by performing the perturbation-based calibration process on the nonlinear system  130  with at least one predetermined input x 0  being applied to the nonlinear system  130 . In particular, during performing compensation on the nonlinear system  130  in the perturbation-based calibration process, the compensation parameters {b i } are utilized for controlling the compensation response of the perturbation-based calibration process, and more particularly, the compensation response of the compensation module  140 . 
     As shown in  FIG. 3 , the input selector  110  is arranged to select at least one input to be applied to the nonlinear system  130  from at least one ordinary input x and the aforementioned at least one predetermined input x 0 . Please note that the aforementioned at least one ordinary input x typically represents the input that the apparatus  100  shown in  FIG. 3  handles in its normal operation, and therefore, the aforementioned at least one ordinary input x can be at least one arbitrary input such as at least one non-predetermined input. According to this embodiment, in a foreground calibration mode, the input selector  110  selects the predetermined input x 0  on the path P Foreground  as the input to be applied to the nonlinear system  130 . In addition, in a background calibration mode, the input selector  110  selects the ordinary input x on the path P Background  as the input to be applied to the nonlinear system  130 . Thus, Step  912  is performed in the foreground calibration mode. 
     In Step  914 , the calibration loop utilizes the temporarily values (i.e. those mentioned in Step  912 ) as initial values of the compensation parameters {b i } for online calibration, which means the calibration loop utilizes the temporarily values mentioned above as the initial values of the compensation parameters {b i } in the background calibration mode. More particularly, when it is detected that the absolute value of the difference (y c −x) falls within the aforementioned predetermined upper limit, Step  914  is entered. This is for illustrative purposes only, and is not meant to be a limitation of the present invention. According to some variations of the embodiment shown in  FIG. 2 , in a situation where the apparatus  100  is the ADC mentioned in the variations of the embodiment shown in  FIG. 1B , when it is detected that the ENOB of the ADC reaches the aforementioned predetermined value corresponding to some standards that the ADC complies with, Step  914  is entered. For example, the apparatus  100  can be the 12 bit ADC mentioned above, and when it is detected that the ENOB of the 12 bit ADC reaches the predetermined value 11.5, Step  914  is entered. 
     In Step  916 , the calibration loop updates the compensation parameters {b i } by performing the perturbation-based calibration process in an online manner. More particularly, the calibration loop determines the latest values of the compensation parameters {b i } by performing the perturbation-based calibration process in the online manner, having no need to apply any predetermined input (e.g. the aforementioned at least one predetermined input x 0 ) to the nonlinear system after a DAT, such as the aforementioned DAT of the embodiment shown in  FIG. 1B . Thus, Step  916  is performed in the background calibration mode mentioned above. Please note that, when Step  916  is entered, applying a predetermined input such as the aforementioned at least one predetermined input x 0  to the nonlinear system is not required. This is for illustrative purposes only, and is not meant to be a limitation of the present invention. According to some variations of this embodiment, after the perturbation-based calibration process is performed in the online manner for a while, a predetermined input such as the aforementioned at least one predetermined input x 0  may intentionally be applied to the nonlinear system when needed. According to some variations of this embodiment, no predetermined input is applied to the nonlinear system once the background calibration mode mentioned above is entered. 
     According to the embodiment shown in  FIG. 2 , with the proposed scheme/architecture shown in  FIG. 3  being applied, the perturbation generator  160  is arranged to generate a set of perturbation values {Δ q }, where Δ q =(q·Δ), and the value q sent from the estimation module  150  is selected from the set {−1, 0, 1} in this embodiment. Please note that the set {−1, 0, 1} is taken as an example of the set from which the value q is selected. This is for illustrative purposes only, and is not meant to be a limitation of the present invention. According to variations of this embodiment, the set from which the value q is selected may be formed with other combinations of values. For example, the set may comprise three or more values, each of which may be different from any of the others within the three or more values. 
     In the embodiment shown in  FIG. 2 , with the proposed scheme/architecture shown in  FIG. 3  being applied, the arithmetic unit  120  is arranged to combine the set of perturbation values {Δ q } into the aforementioned at least one input to be applied to the nonlinear system  130  (e.g. the aforementioned at least one predetermined input x 0 , or the aforementioned at least one ordinary input x). As shown in  FIG. 3 , the notation x′ is utilized for representing the combined input, i.e. the input carrying a perturbation value Δ q  of the set of perturbation values {Δ q }. The arithmetic unit  120  is arranged to apply (or input) the input carrying the set of perturbation values {Δ q } (more particularly, the combined input {x′}) to the nonlinear system  130 , in order to obtain a set of outputs {y} of the nonlinear system  130 , respectively. In addition, the compensation module  140  is arranged to perform compensation on the set of outputs {y} according to the compensation parameters {b i }, in order to obtain a set of compensated results, such as the superset {{y c   Δ }, {y c }} of the compensated results {y c   Δ } corresponding to the set of perturbation values {Δ q } and the compensated results {y c } carrying no influence of the set of perturbation values {Δ q }. Additionally, the estimation module  150  is arranged to perform estimation according to the compensated results {y c   Δ } corresponding to the set of perturbation values {Δ q } within the set of compensated results {{y c   Δ }, {y c }}, in order to determine/update the compensation parameters {b i } for performing compensation. 
     Please note that, in the compensated results {y c } of the embodiment shown in  FIG. 2 , influence of the set of perturbation values {Δ q } is removed. On the contrary, in at least a portion (e.g. a portion or all) of the compensated results {y c   Δ } corresponding to the set of perturbation values {Δ q }, influence of the set of perturbation values {Δ q } is not removed. For example, in a situation where Δ q =(q·Δ) and the value q sent from the estimation module  150  is selected from the set {−1, 0, 1}, at least a portion of the compensated results {y c   Δ }, and more particularly, those corresponding to the perturbation values {−1, 1}, carry influence of the perturbation values {−1, 1}, respectively. In addition, another portion of the compensated results {y c   Δ }, and more particularly, that corresponding to the perturbation value {0}, intrinsically carries no influence of the perturbation value {0} since there should be no influence of any perturbation value that is zero. This is for illustrative purposes only, and is not meant to be a limitation of the present invention. According to some variations of this embodiment, in a situation where none of the set of perturbation values {Δ q } is zero, the compensated results {y c   Δ } surely carry influence of the perturbation values {Δ q }, respectively. 
     Regarding Step  912  performed in the foreground calibration mode mentioned above, in a situation where the aforementioned at least one predetermined input x 0  is applied to the nonlinear system  130 , the arithmetic unit  120  is arranged to combine the set of perturbation values {Δ q } into the aforementioned at least one predetermined input x 0 , and is arranged to apply (or input) the predetermined input carrying the set of perturbation values {Δ q } (more particularly, the combined predetermined input {x 0 ′} in this situation) to the nonlinear system  130 , in order to determine the temporarily values of the compensation parameters {b i }. Regarding Step  916  performed in the background calibration mode mentioned above, in a situation where the aforementioned at least one ordinary input x is applied to the nonlinear system  130 , the arithmetic unit  120  is arranged to combine the same set of perturbation values {Δ q } into the aforementioned at least one ordinary input x, and is arranged to apply (or input) the ordinary input carrying the set of perturbation values {Δ q } (more particularly, the combined ordinary input {x′} in this situation) to the nonlinear system  130 , in order to determine the latest values of the compensation parameters {b i }. This is for illustrative purposes only, and is not meant to be a limitation of the present invention. According to some variations of this embodiment, the calibration loop can utilize different sets of perturbation values, such as a first set of perturbation values {Δ q     —     Foreground } and a second set of perturbation values {Δ q     —     Background }, in the foreground calibration mode and the background calibration mode, respectively. 
     As both a 0  of {a i1 } and b 0  of {b i } can be zero in the embodiment shown in  FIG. 2 , the equivalent functions of the nonlinear system  130  and the compensation module  140  can be written as f(x′, a 1 , a 2 , . . . , a l } and f c (y, b 1 , b 2 , . . . , b k ), respectively. In addition, as illustrated in  FIG. 3 , only a portion of compensation parameters {b 1 , b 2 , . . . , b k } are taken as examples of the compensation parameters {b i } sent from the estimation module  150  to the compensation module  140 . This is for illustrative purposes only, and is not meant to be a limitation of the present invention. According to some variations of this embodiment, in a situation where none of {b i } is zero, it can be illustrated in  FIG. 3  that the compensation parameters {b i } sent from the estimation module  150  to the compensation module  140  comprises all of the compensation parameters {b 0 , b 1 , b 2 , . . . , b k }. 
     According to an embodiment, such as a variation of the embodiment shown in  FIG. 2 , the estimation module  150  can perform estimation such as Lyapunov-based estimation according to the compensated results {y c   Δ } in order to determine/update the compensation parameters {b i }, where the Lyapunov-based estimation can be performed according to the following equations:
 
 b   i   [n+ 1 ]=L   i ( b   i   [n],x   0 ), for the foreground calibration mode; and
 
 b   i   [n+ 1 ]=L   i ( b   i   [n],E[x] ), for the background calibration mode;
 
where the notation n represents an iteration index, and the notations L i (•) and E[•] respectively represent an estimation function and an operation of mean calculation (e.g. a moving average operation).
 
       FIG. 4  illustrates some implementation details of the compensation module  140  shown in  FIG. 3  according to an embodiment of the present invention. The compensation module  140  comprises a plurality of first arithmetic units  142 - 2 ,  142 - 3 , . . . , and  142 - k  (e.g. power calculation units), a plurality of second arithmetic units  144 - 1 ,  144 - 2 ,  144 - 3 , . . . , and  144 - k  (e.g. multipliers), a third arithmetic unit  146  (e.g. an adder), and a fourth arithmetic unit  148  (e.g. a subtraction unit). The first arithmetic units  142 - 2 ,  142 - 3 , . . . , and  142 - k  calculate y 2 , y 3 , . . . , and y k , respectively. In addition, the second arithmetic units  144 - 1 ,  144 - 2 ,  144 - 3 , . . . , and  144 - k  calculate (b 1 ·y), (b 2 ·y 2 ), (b 3 ·y 3 ), . . . , and (b k ·y k ), respectively. As a result of the arrangement shown in  FIG. 4 , the compensation module  140  can obtain any compensated result y c   Δ  or any compensated result y c  within the set of compensated results {{y c   Δ }, {y c }} mentioned above. 
     More specifically, the compensation module  140  can calculate the compensated result y c   Δ  and the compensated result y, according to the following equations:
 
 y   c   Δ =Σ i=1, 2, . . . , k ( b   i   y   i ); and
 
 y   c   =y   c   Δ −Δ q .
 
     This is for illustrative purposes only, and is not meant to be a limitation of the present invention. According to some variations of this embodiment, in general, the compensated result y c   Δ  can be expressed as follows:
 
 y   c   Δ   =f   c ( y, b   1   , b   2   , . . . , b   k );
 
where the format of the function f c (y, b 1 , b 2 , . . . , b k ) may be arbitrary.
 
       FIG. 5A  illustrates some implementation details of the estimation module  150  shown in  FIG. 3  according to an embodiment of the present invention. The estimation module  150  comprises a multiplexer  150 M (labeled “MUX” in  FIG. 5A ), a dispatch unit  152  (labeled “Dispatch” in  FIG. 5A ), a first set of mean calculation units  154 - 0 ,  154 - 1 , and  154 - 2 , a plurality of fourth arithmetic units  156 - 1  and  156 - 2  (e.g. subtraction units), and a Lyapunov-based estimator  158 , and further comprises a fourth arithmetic unit  151  (e.g. a subtraction unit), a first arithmetic unit  153  (e.g. a power calculation unit such as a square calculation unit), and a second set of mean calculation units  155 - 1  and  155 - 2 . 
     Please note that, in the above embodiments such as those shown in  FIGS. 2-4 , the compensated results {y c   Δ } may comprise those obtained by utilizing the aforementioned at least one ordinary input x and comprise those obtained by utilizing the aforementioned at least one predetermined input x 0 . According to this embodiment, for better comprehension, a portion of compensated results within the compensated results {y c   Δ } can be re-written as the compensated results {y c0   Δ } in a situation where they are obtained by utilizing the aforementioned at least one predetermined input x 0 , rather than the aforementioned at least one ordinary input x. As shown in  FIG. 5A , the multiplexer  150 M is arranged to select a compensated result from the compensated result y c   Δ  and the compensated result y c0   Δ  according to the selection signal S Mode . More particularly, referring to  FIG. 5B , in a situation where the selection signal S Mode  is in an active state and the inverted signal S Mode     —     INV  thereof is in an inactive state, the multiplexer  150 M selects the compensated result y c   Δ . On the contrary, in a situation where the selection signal S Mode  is in an inactive state and the inverted signal S Mode     —     INV  thereof is in an active state, the multiplexer  150 M selects the compensated result y c0   Δ . 
     Regarding the upper paths shown in  FIG. 5A , as the value q sent from the estimation module  150  is selected from the set {−1, 0, 1} in this embodiment, the dispatch unit  152  dispatches the output of the multiplexer  150 M to three output terminals respectively corresponding to the values in the set {−1, 0, 1} (more particularly, the output terminals respectively labeled “−1”, “0”, and “+1” in the dispatch unit  152  shown in  FIG. 5A ) according to the value q. For example, the dispatch unit  152  may dispatch the output of the multiplexer  150 M to the output terminal labeled “−1” in a situation where q=−1, or dispatch the output of the multiplexer  150 M to the output terminal labeled “0” in a situation where q=0, or dispatch the output of the multiplexer  150 M to the output terminal labeled “+1” in a situation where q=1. In addition, the first set of mean calculation units  154 - 0 ,  154 - 1 , and  154 - 2  perform operations of mean calculation (e.g. moving average operations) on the compensated results received from the output terminals labeled “0”, “+1”, and “−1”, respectively. Additionally, the fourth arithmetic units  156 - 1  and  156 - 2  calculate some linear combinations H 1  and H 2  of the means output from the first set of mean calculation units  154 - 0 ,  154 - 1 , and  154 - 2 , respectively. More particularly, the fourth arithmetic unit  156 - 1  obtains the linear combination H 1  by calculating the difference between the means respectively output from the mean calculation units  154 - 1  and  154 - 0 , and the fourth arithmetic unit  156 - 2  obtains the linear combination H 2  by calculating the difference between the means respectively output from the mean calculation units  154 - 0  and  154 - 2 . This is for illustrative purposes only, and is not meant to be a limitation of the present invention. According to some variations of this embodiment, in a situation where the number of the perturbation values {Δ q } (or the number of possible values of q) is increased and the number of associated output terminals of the dispatch unit  152  is correspondingly increased, the number of linear combinations to be sent to the Lyapunov-based estimator  158  can be increased. For example, the aforementioned linear combinations {H 1 , H 2 } can be extended to be {H 1 , H 2 , . . . , H m } in these variations. 
     Regarding the lower paths shown in  FIG. 5A , the fourth arithmetic unit  151  can calculate the compensated result y c  by subtracting the perturbation value Δ q  from the compensated result y c   Δ  (and/or y c0   Δ ). In addition, the first arithmetic unit  153  calculates y c   2 , and the second set of mean calculation units  155 - 1  and  155 - 2  perform operations of mean calculation (e.g. moving average operations) on the outputs of the fourth arithmetic unit  151  and the first arithmetic unit  153 , in order to obtain the means E[y c ] and E[y c   2 ], respectively. As a result of the arrangement shown in  FIG. 5A , the Lyapunov-based estimator  158  can perform Lyapunov-based estimation according to derivatives of some compensated results {y c   Δ } (and/or {y c0   Δ }) obtained by performing compensation on the nonlinear system  130 , such as the linear combinations {H 1 , H 2 } and the means E[y c ] and E[y c   2 ], in order to determine/update the compensation parameters {b i } mentioned in Step  912  or Step  916 . 
     According to a variation of the embodiment shown in  FIG. 2 , such as the embodiment shown in  FIG. 5C , the aforementioned derivatives of the compensated results {y c   Δ } (and/or {y c   Δ }) comprise linear combinations {H j } of means of different portions of the compensated results {y c   Δ } (and/or {y c0   Δ }), respectively. Regarding the linear combinations {H j }, the suffix j represents an index of measurement for the Lyapunov-based estimation. For example, the index j may vary from 1 to m, and the linear combinations {H j } comprise {H 1 , H 2 , . . . , H m }, which can be regarded as m measurement results. Thus, the Lyapunov-based estimator  158  can perform Lyapunov-based estimation according to derivatives of some compensated results {y c   Δ } (and/or {y c0   Δ }) obtained by performing compensation on the nonlinear system  130 , such as the linear combinations {H 1 , H 2 , . . . , H m } in order to determine/update the compensation parameters {b i } mentioned in Step  912  or Step  916 . 
     More particularly, in a situation where notations Δ, n, and μ i  of this embodiment respectively represent a perturbation value, an iteration index, and an updating factor, the Lyapunov-based estimator  158  is arranged to perform the Lyapunov-based estimation according to the following equations:
 
 E   j   =H   j −Δ; and
 
 b   i   [n+ 1 ]=b   i   [n] −(μ i   ·L   i );
 
where the estimation function L i  of this embodiment is a summation of products associated to both {E j } and the compensation parameter b i . In practice, the estimation function L can be expressed as follows:
 
 L   i =Σ j1=1, 2, . . . m ( E   j1 ·(∂ E   j1   /∂b   i ));
 
where the original suffix j of the aforementioned E j  may be replaced by another notation j1 in the above equation, in order to prevent this equation from clashing with the aforementioned E j .
 
     According to some embodiments, such as the embodiments/variations disclosed above, as the difference (y c −x) such as that shown in  FIG. 1B  usually approaches 0 after the operations disclosed in Step  912  are completed, the DAT of the present invention is typically the overall time for performing the operations disclosed in Step  912  in the foreground calibration mode, and is much shorter than that of the related art. For example, the ratio of the DAT of some embodiments of the present invention to that of a correlation-based calibration method of the related art is less than 1/10000. In another example, the ratio of the DAT of some embodiments of the present invention to that of another correlation-based calibration method of the related art is less than 1/1000. 
     It is an advantage of the embodiments of the present invention that, after power-on of a system comprising a component (e.g. an ADC) implemented, the component can rapidly get ready for being used by the system. The embodiments can properly remove the influence of nonlinearity and greatly reduce the DAT. 
     Those skilled in the art will readily observe that numerous modifications and alterations of the device and method may be made while retaining the teachings of the invention. Accordingly, the above disclosure should be construed as limited only by the metes and bounds of the appended claims.