Abstract:
Traditionally, pipelined continuous-time (CT) sigma-delta modulators (SDM) have been difficult to build due at least in part to the difficulties in calibrating the pipeline. Here, however, a pipelined CT SDM is provided that has an architecture that is conducing to being calibrated. Namely, the system includes a digital filter and other features that can be adjusted to account for input imbalance errors and well as quantization leakage noise.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
       [0001]    This application is related to co-pending U.S. patent application Ser. No. ______, entitled “METHOD FOR CALIBRATING A PIPELINED CONTINUOUS-TIME SIGMA DELTA MODULATOR,” filed herewith, which is hereby incorporated by reference for all purposes. 
     
    
     TECHNICAL FIELD 
       [0002]    The invention relates generally to data converters and, more particularly, to continuous-time (CT) sigma-delta modulators (SDMs) or sigma-delta analog-to-digital converters (ADCs). 
       BACKGROUND 
       [0003]    Referring to  FIG. 1  of the drawings, the reference numeral  100  generally designates a pipelined discrete-time (DT) SDM. With a DT data converter, an analog input signal (such as signal AIN) is sampled by a sample-and-hold (S/H) circuit (such as S/H circuit  102 ) at discrete points in time or sampling instants, and the samples are converted to digital. Here, two SDM stages  104 - 1  and  104 - 2  are used in a pipeline configuration to perform the conversion for each of the samples. Each of stages  104 - 1  and  104 - 2  respectively comprise summing circuits  116 - 1 / 118 - 1  or  116 - 2 / 118 - 2 , a delay  120 - 1  or  120 - 2 , quantizer  122 - 1  or  122 - 2 , digital low pass filter (LPF)  124 - 1  or  124 - 2 , and digital-to-analog converter (DAC)  128 - 1  or  128 - 2 . Additionally, stage  104 - 1  also includes digital filter  126 . Between stages  104 - 1  and  104 - 2 , there are several other components that enable stages  104 - 1  and  104 - 2  to operate as a pipeline; namely, these components are delay  108 , summing circuit  110 , amplifiers  112  and  114 , analog LPF  113 , and digital output circuit  106 . 
         [0004]    In operation, DT SDM  100  converts the analog input signal AIN to digital output signal DOUT. To accomplish this, a sample of the analog input signal AIN is provided to stage  104 - 1  (by S/H circuit  102 ), where the sample is converted to digital using conventional sigma-delta modulation. The same sample is provided to delay  108  so as to provide stage  104 - 1  with sufficient time to perform the data conversion. The difference analog representation of the data conversion (from DAC  128 - 1 ) and the sampled analog input signal AIN (from delay  108 ) or residue is determined by summing circuit  110 . This residue is amplified and filtered by amplifiers  112  and  114  and analog LPF  113  and provided to stage  104 - 2 . Stage  104 - 2  can then convert the residue to digital using conventional sigma-delta modulation. The digital output circuit  106  then generates the digital output signal DOUT based the output from each pipeline  104 - 1  and  104 - 2 . 
         [0005]    This architecture, however, is incompatible with CT sigma-delta modulation. With DT sigma-delta modulation, the input to the stages (i.e., stages  104 - 1  and  104 - 2 ) is constant during conversion because the S/H circuit  102  holds the sampled analog input signal AIN. In contrast, an input to stages of a pipeline would be varying. Looking to DT SDM  100 , it specifically employs a delay  108  so that stages  104 - 1  and  104 - 2  perform sigma-delta modulation on the same sample. If one were to remove the S/H circuit  102  so as to provide a continuously varying signal (i.e., analog input signal AIN) directly to stage  104 - 1  and delay  108 , DT SDM  100  would not function. 
         [0006]    Some other conventional circuits are: U.S. Pat. No. 5,729,230; U.S. Pat. No. 6,788,232; U.S. Pat. No. 7,460,046; U.S. Pat. No. 7,486,214. 
       SUMMARY 
       [0007]    A preferred embodiment of the present invention, accordingly, provides an apparatus. The apparatus comprises a first continuous-time (CT) sigma-delta modulator (SDM) that receives an analog input signal; a digital-to-analog converter (DAC) that is coupled to the first CT SDM; a first summing circuit that receives the analog input signal and that is coupled to the DAC, wherein the first summing circuit determines a difference between the analog input signal and an output from the DAC; an amplifier that is coupled to the summing circuit, wherein the amplifier has a first gain, and wherein the amplifier includes a filter; a second CT SDM that is coupled to the amplifier; a digital gain circuit that is coupled to the second CT SDM, wherein the digital gain circuit has a second gain, and wherein the second gain is substantially the inverse of the first gain, and wherein the amplifier, the second CTSDM, and the DAC collectively have a first transfer function; a digital filter that is coupled to the first CT SDM, wherein the digital filter has a second transfer function, wherein the second transfer function substantially matches the first transfer function; and a second summing circuit that is coupled to the digital filter and the digital gain circuit. 
         [0008]    In accordance with a preferred embodiment of the present invention, the DAC further comprises a first DAC having a third gain, and wherein the digital filter has a fourth gain, and wherein the second CT SDM further comprises: a third summing circuit that is coupled to the amplifier; an SDM filter that is coupled to the third summing circuit; a quantizer that is coupled to the SDM filter; and a second DAC that is coupled to the quantizer and the third summing circuit, wherein the third summing circuit determines a difference between an output of the amplifier and an output of the second DAC, and wherein the second DAC has a fifth gain, and wherein the ratio of the third gain to the fifth gain is approximately equal to the fourth gain. 
         [0009]    In accordance with a preferred embodiment of the present invention, the SDM filter and the quantizer further comprise a first SDM filter and a first quantizer, and wherein the first CT SDM further comprises: a fourth summing circuit that receives the analog input signal; a second SDM filter that is coupled to the fourth summing circuit; a second quantizer that is coupled to the second SDM filter; and a second DAC that is coupled to the second quantizer and the fourth summing circuit, wherein the fourth summing circuit determines a difference between the analog input signal and an output of the second DAC. 
         [0010]    In accordance with a preferred embodiment of the present invention, the apparatus further comprises: an analog delay line that receives the analog input signal is coupled to the first summing circuit; and a digital predictor that is coupled between the first CT SDM and the first DAC. 
         [0011]    In accordance with a preferred embodiment of the present invention, the apparatus further comprises an analog predictor that receives the analog input signal and that is coupled to the fourth summing circuit. 
         [0012]    In accordance with a preferred embodiment of the present invention, the amplifier further comprises a first amplifier, and wherein the apparatus further comprises a second amplifier that is coupled to the first summing circuit and that receives the analog input signal. 
         [0013]    In accordance with a preferred embodiment of the present invention, the second amplifier has a third gain, and wherein the third gain is dimensioned to minimize an autocorrelation of an output of the second CT SDM. 
         [0014]    In accordance with a preferred embodiment of the present invention, the apparatus further comprises an output circuit that is coupled to the second circuit and that provides a digital output signal. 
         [0015]    In accordance with a preferred embodiment of the present invention, an apparatus is provided. The apparatus comprises an input terminal; a first stage of a pipeline including: a first CT SDM that is coupled to the input terminal; and a digital filter that is coupled to the first CT SDM, wherein the digital filter has a first transfer function; a second stage of a pipeline including: a first summing circuit that is coupled to the input terminal, wherein the first summing circuit is adapted to determine a difference; an amplifier that is coupled to the first summing circuit, wherein the amplifier has a first gain, and wherein the amplifier includes a filter; a second CT SDM that is coupled to the first amplifier; and a digital gain circuit that is coupled to the second CT SDM, wherein the digital gain circuit has a second gain that is an inverse of the first gain; a DAC that is coupled between the first CT SDM and the first summing circuit, wherein the amplifier, the DAC, and the second CT SDM collectively have a second transfer function; and a second summing circuit that is coupled to each stage of the pipeline, wherein the first transfer function is adjusted to substantially match the second transfer function. 
         [0016]    In accordance with a preferred embodiment of the present invention, the DAC further comprises a first DAC having a third gain, and wherein the digital filter has a fourth gain, and wherein the second CT SDM further comprises: a third summing circuit that is coupled to the amplifier; an SDM filter that is coupled to the third summing circuit; a quantizer that is coupled to the SDM filter; and a second DAC that is coupled to the quantizer and the third summing circuit, wherein the third summing circuit determines a difference between an output of the amplifier and an output of the second DAC, and wherein the second DAC has a fifth gain, and wherein the ratio of the third gain to the fifth gain is approximately equal to the fourth gain. 
         [0017]    In accordance with a preferred embodiment of the present invention, the SDM filter and the quantizer further comprise a first SDM filter and a first quantizer, and wherein the first CT SDM further comprises: a fourth summing circuit that receives the analog input signal; a second SDM filter that is coupled to the fourth summing circuit; a second quantizer that is coupled to the second SDM filter; and a second DAC that is coupled to the second quantizer and the fourth summing circuit, wherein the fourth summing circuit determines a difference between the analog input signal and an output of the second DAC. 
         [0018]    In accordance with a preferred embodiment of the present invention, the amplifier further comprises a first amplifier, and wherein the apparatus further comprises a second amplifier that is coupled to the first summing circuit and that receives the analog input signal. 
         [0019]    In accordance with a preferred embodiment of the present invention, the second amplifier has a third gain, and wherein the third gain is adjusted by the controller to minimize an autocorrelation of an output of the second CT SDM. 
         [0020]    In accordance with a preferred embodiment of the present invention, the apparatus further comprises an output circuit that is coupled to the second circuit and that provides a digital output signal. 
         [0021]    In accordance with a preferred embodiment of the present invention, an apparatus is provided. The apparatus comprises an input terminal that receives an analog input signal; a first stage of a pipeline including: a first CT SDM including: a first summing circuit that is coupled to the input terminal so as to receive the analog input signal; a first SDM filter that is coupled to the first summing circuit; a first quantizer that is coupled to the first SDM filter; and a first DAC that is coupled to the first quantizer and the first summing circuit, wherein the first summing circuit determines a difference between the analog input signal and an output of the second DAC; and a digital filter that is coupled to the first CT SDM, wherein the digital filter has a first transfer function; a second stage of a pipeline including: a first amplifier that is coupled to the input terminal so as to receive the analog input signal, wherein the first amplifier has a first gain; a second summing circuit that is coupled to the first amplifier, wherein the second summing circuit is adapted to determine a difference; a second amplifier that is coupled to the second summing circuit, wherein the second amplifier has a second gain, wherein the second amplifier includes a filter; a second CT SDM having: a third summing circuit that is coupled to the second amplifier; a second SDM filter that is coupled to the third summing circuit; a second quantizer that is coupled to the second SDM filter; and a second DAC that is coupled to the second quantizer and the third summing circuit, wherein the third summing circuit determines a difference between an output of the second amplifier and an output of the second DAC; and a digital gain circuit that is coupled to the second CT SDM, wherein the third amplifier has a third gain that is an inverse of the second gain; a third DAC that is coupled between the first CT SDM and the second summing circuit, wherein the third DAC, the second CT SDM, and the second amplifier collectively have a second transfer function; a fourth summing circuit that is coupled to each stage of the pipeline, wherein the first transfer function is adjusted to substantially match the second transfer function, and wherein the first gain is adjusted to minimize an autocorrelation of an output of the second CT SDM, and wherein a gain of the digital filter to be approximately equal to a ratio of the gains of the second and third DACs; and an output circuit that is coupled to the fourth summing circuit and that provides a digital output signal. 
         [0022]    In accordance with a preferred embodiment of the present invention, a method for calibrating at least a portion of a pipelined continuous-time (CT) sigma-delta modulator (SDM) is provided, The CT SDM includes a first stage, a second stage, and a first digital-to-analog converter (DAC) coupled between the first and second stages, and a digital filter that is coupled to the first and second stages, and wherein the second stage includes a second DAC. The method comprises determining a ratio of a gain of the first DAC to a gain of the second DAC; adjusting a gain of a digital filter to be approximately equal to the ratio of the gain of the first DAC to the gain of the second DAC; and adjusting the digital filter to maximize a cross-correlation between an output of the digital filter and the output of the second stage. 
         [0023]    In accordance with a preferred embodiment of the present invention, the method further comprises: disabling the first DAC, wherein the first DAC is located between a first stage and a second stage of the pipelined CT SDM; applying a predetermined input signal to the second stage while the first DAC is disabled; enabling the first DAC; disabling the second DAC within the second stage; and applying the predetermined input signal to the second stage while the using the first DAC as a feedback DAC for the second stage. 
         [0024]    In accordance with a preferred embodiment of the present invention, the method further comprises determining a gain of an amplifier located in the second stage that minimizes an autocorrelation of an output of the second stage. 
         [0025]    In accordance with a preferred embodiment of the present invention, a method for calibrating at least a portion of a pipelined CT SDM is provided. The method comprises disabling a first DAC, wherein the first DAC is located between the first stage and the second stage of the pipelined CT SDM; applying a predetermined input signal to the second stage while the first DAC is disabled; enabling the first DAC; disabling a second DAC within the second stage; applying the predetermined input signal to the second stage while the using the first DAC as a feedback DAC for the second stage; determining gains of the first and second DACs; and adjusting a gain of a digital filter to be a function of the gains of the first and second DACs. 
         [0026]    In accordance with a preferred embodiment of the present invention, the digital filter is coupled to the first and second stages. 
         [0027]    In accordance with a preferred embodiment of the present invention, the function is a ratio of the gains of the first and second DACs. 
         [0028]    In accordance with a preferred embodiment of the present invention, the method further comprises determining a gain of an amplifier located in the second stage that minimizes an autocorrelation of an output of the second stage. 
         [0029]    In accordance with a preferred embodiment of the present invention, the method further comprises adjusting the digital filter to maximize a cross-correlation between an output of the digital filter and the output of the second stage. 
         [0030]    The foregoing has outlined rather broadly the features and technical advantages of the present invention in order that the detailed description of the invention that follows may be better understood. Additional features and advantages of the invention will be described hereinafter which form the subject of the claims of the invention. It should be appreciated by those skilled in the art that the conception and the specific embodiment disclosed may be readily utilized as a basis for modifying or designing other structures for carrying out the same purposes of the present invention. It should also be realized by those skilled in the art that such equivalent constructions do not depart from the spirit and scope of the invention as set forth in the appended claims. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0031]    For a more complete understanding of the present invention, and the advantages thereof, reference is now made to the following descriptions taken in conjunction with the accompanying drawings, in which: 
           [0032]      FIG. 1  is a block diagram of a conventional pipelined DT SDM; 
           [0033]      FIG. 2  is a block diagram of an example of a pipelined CT SDM in accordance with a preferred embodiment of the present invention; and 
           [0034]      FIGS. 3 and 4  are block diagrams of examples of the sub-CT SDMs of  FIG. 2 . 
       
    
    
     DETAILED DESCRIPTION 
       [0035]    Refer now to the drawings wherein depicted elements are, for the sake of clarity, not necessarily shown to scale and wherein like or similar elements are designated by the same reference numeral through the several views. 
         [0036]    Turning to  FIGS. 2-4 , an example of a pipelined CT SDM  200  in accordance with a preferred embodiment of the present invention can be seen. As an example, CT SDM  200  is a two-stage pipeline; however, CT SDM  200  can be scaled to include more stages. In this example, CT SDM  200  generally comprises CT SDMs (or sub-CT SDMs)  202  and  212 , DAC  204 , digital filter  206 , amplifiers  220  and  210 , digital gain circuit  214 , summing circuits  208  and  216 , output circuit  218 , and an adjustable delay  222 . CT SDM  202  (which can be seen in  FIG. 3 ) generally comprises summing circuit  302 , SDM filter  304 , quantizer  306 , and DAC  308 , and CT SDM  212  generally comprises summing circuit  402 , SDM filter  404 , quantizer  405 , and DAC  408 . Also, CT SDM  202  can be a lower order modulator (i.e., order of 1 or 2), while CT SDM  212  can be a higher order modulator (i.e., order greater than 3) with aggressive noise shaping. Moreover, amplifier  210  also includes a filter. Amplifiers  220  and  210  can take on many forms, including but not limited to voltage-to-voltage amplifiers (i.e., operational amplifiers), voltage-to-current amplifiers with a current gain (i.e., transconductance amplifiers or variable resistors), current-to-voltage amplifiers (i.e., transimpedance amplifiers), or current-to-current amplifiers (i.e., current mode amplifiers having a topology that depends on the input signal AIN). 
         [0037]    In order for CT SDM to function, CT SDM  200  is generally calibrated to compensate for system mismatches, but to make any calibrations, the sources of mismatch and error should be identified. Each of DACs  204 ,  308 , and  408  each have gains of g 4 , g 1 , and g 2 , while amplifiers  220  and  210  and digital gain circuit  214  have gains of g 3 , g 5 , and g 6 . Gain g 6  can be adjusted to be approximately equal to be the inverse of gain g 5  (or g 6 =1/g 5 ), which generally eliminates the effect of amplifiers  210 . Alternatively, the gain g 5  can be 1 so as to use the filter incorporated therein. Additionally, each of SDM filters  304  and  306  include a gain of 1/g 1  and 1/g 2 , respectively, to compensate for the gains of their respective DACs  308  and  408 , and digital filter  206  has a gain of g F  and a transfer function of C F (z). As a result, the output Y 1 (z) (which is in the frequency domain or z-domain) from CT SDMs  202  is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         Y 
                         1 
                       
                        
                       
                         ( 
                         z 
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                     = 
                     
                       
                         
                           
                             
                               S 
                               1 
                             
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                               ( 
                               z 
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         [0000]    where, S 1 (z) is the Signal Transfer Function (STF) and N 1 (z) is the Noise Transfer Function (NTF) of an equivalent Discrete-Time Sigma Delta Modulator to which the CT SDM  200  is mapped for purposes of analysis using techniques well known to those skilled in the art, Q 1 (z) is the quantization error of the quantizer and X(z) is the discrete time equivalent of the continuous-time input. This leads to the output Y 1,N (z) from digital filter  206  being: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
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         [0000]    The input R(z) into CT SDM  212  is a combination of the output from amplifier  220  and output Y 1 (z) yielding: 
         [0000]    
       
         
           
             
               
                 
                   
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         [0000]    Now using the same rationale applied to CT SDM  202  (because the structures of CT SDMs  202  and  212  are similar), the output Y 2 (z) for CT SDM  212  is: 
         [0000]    
       
         
           
             
               
                 
                   
                     
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                             1 
                           
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         [0000]    Thus, the output Y(z) of CT SDM  200  should be: 
         [0000]    
       
         
           
             
               
                 
                   
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                               2 
                             
                           
                           ) 
                         
                          
                         
                           
                             S 
                             2 
                           
                            
                           
                             ( 
                             z 
                             ) 
                           
                         
                          
                         
                           X 
                            
                           
                             ( 
                             z 
                             ) 
                           
                         
                       
                       - 
                       
                         
                           ( 
                           
                             
                               g 
                               4 
                             
                             
                               
                                 g 
                                 1 
                               
                                
                               
                                 g 
                                 2 
                               
                             
                           
                           ) 
                         
                          
                         
                           
                             S 
                             2 
                           
                            
                           
                             ( 
                             z 
                             ) 
                           
                         
                          
                         
                           
                             S 
                             1 
                           
                            
                           
                             ( 
                             z 
                             ) 
                           
                         
                          
                         
                           X 
                            
                           
                             ( 
                             z 
                             ) 
                           
                         
                       
                       + 
                       
                         
                           g 
                           F 
                         
                          
                         
                           
                             N 
                             1 
                           
                            
                           
                             ( 
                             z 
                             ) 
                           
                         
                          
                         
                           
                             Q 
                             1 
                           
                            
                           
                             ( 
                             z 
                             ) 
                           
                         
                          
                         
                           
                             C 
                             F 
                           
                            
                           
                             ( 
                             z 
                             ) 
                           
                         
                       
                       - 
                       
                         
                           ( 
                           
                             
                               g 
                               4 
                             
                             
                               g 
                               2 
                             
                           
                           ) 
                         
                          
                         
                           
                             S 
                             2 
                           
                            
                           
                             ( 
                             z 
                             ) 
                           
                         
                          
                         
                           
                             N 
                             1 
                           
                            
                           
                             ( 
                             z 
                             ) 
                           
                         
                          
                         
                           
                             Q 
                             1 
                           
                            
                           
                             ( 
                             z 
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Equation (5) can then be reduced as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     Y 
                      
                     
                       ( 
                       z 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             g 
                             F 
                           
                           
                             g 
                             1 
                           
                         
                         ) 
                       
                        
                       
                         
                           S 
                           1 
                         
                          
                         
                           ( 
                           z 
                           ) 
                         
                       
                        
                       
                         X 
                          
                         
                           ( 
                           z 
                           ) 
                         
                       
                        
                       
                         
                           C 
                           F 
                         
                          
                         
                           ( 
                           z 
                           ) 
                         
                       
                     
                     + 
                     
                       
                         
                           N 
                           2 
                         
                          
                         
                           ( 
                           z 
                           ) 
                         
                       
                        
                       
                         
                           Q 
                           2 
                         
                          
                         
                           ( 
                           z 
                           ) 
                         
                       
                     
                     + 
                     
                       
                         ( 
                         
                           
                             
                               
                                 g 
                                 1 
                               
                                
                               
                                 g 
                                 3 
                               
                             
                             - 
                             
                               
                                 g 
                                 4 
                               
                                
                               
                                 
                                   S 
                                   1 
                                 
                                  
                                 
                                   ( 
                                   z 
                                   ) 
                                 
                               
                             
                           
                           
                             
                               g 
                               1 
                             
                              
                             
                               g 
                               2 
                             
                           
                         
                         ) 
                       
                        
                       
                         
                           S 
                           2 
                         
                          
                         
                           ( 
                           z 
                           ) 
                         
                       
                        
                       
                         X 
                          
                         
                           ( 
                           z 
                           ) 
                         
                       
                     
                     + 
                     
                       
                         ( 
                         
                           
                             
                               g 
                               F 
                             
                              
                             
                               
                                 C 
                                 F 
                               
                                
                               
                                 ( 
                                 z 
                                 ) 
                               
                             
                           
                           - 
                           
                             
                               ( 
                               
                                 
                                   g 
                                   4 
                                 
                                 
                                   g 
                                   2 
                                 
                               
                               ) 
                             
                              
                             
                               
                                 S 
                                 2 
                               
                                
                               
                                 ( 
                                 z 
                                 ) 
                               
                             
                           
                         
                         ) 
                       
                        
                       
                         
                           N 
                           1 
                         
                          
                         
                           ( 
                           z 
                           ) 
                         
                       
                        
                       
                         
                           Q 
                           1 
                         
                          
                         
                           ( 
                           z 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Therefore, it can be easily observed that output Y(z) is a combination of the desired output Y DES (z), the input phase imbalance Y PI (z), and the quantization noise leakage Y QNL (z), which are as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         Y 
                         DES 
                       
                        
                       
                         ( 
                         z 
                         ) 
                       
                     
                     = 
                     
                       
                         
                           ( 
                           
                             
                               g 
                               F 
                             
                             
                               g 
                               1 
                             
                           
                           ) 
                         
                          
                         
                           
                             S 
                             1 
                           
                            
                           
                             ( 
                             z 
                             ) 
                           
                         
                          
                         
                           X 
                            
                           
                             ( 
                             z 
                             ) 
                           
                         
                          
                         
                           
                             C 
                             F 
                           
                            
                           
                             ( 
                             z 
                             ) 
                           
                         
                       
                       + 
                       
                         
                           
                             N 
                             2 
                           
                            
                           
                             ( 
                             z 
                             ) 
                           
                         
                          
                         
                           
                             Q 
                             2 
                           
                            
                           
                             ( 
                             z 
                             ) 
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         Y 
                         PI 
                       
                        
                       
                         ( 
                         z 
                         ) 
                       
                     
                     = 
                     
                       
                         ( 
                         
                           
                             
                               
                                 g 
                                 1 
                               
                                
                               
                                 g 
                                 3 
                               
                             
                             - 
                             
                               
                                 g 
                                 4 
                               
                                
                               
                                 
                                   S 
                                   1 
                                 
                                  
                                 
                                   ( 
                                   z 
                                   ) 
                                 
                               
                             
                           
                           
                             
                               g 
                               1 
                             
                              
                             
                               g 
                               2 
                             
                           
                         
                         ) 
                       
                        
                       
                         
                           S 
                           2 
                         
                          
                         
                           ( 
                           z 
                           ) 
                         
                       
                        
                       
                         X 
                          
                         
                           ( 
                           z 
                           ) 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         Y 
                         QNL 
                       
                        
                       
                         ( 
                         z 
                         ) 
                       
                     
                     = 
                     
                       
                         ( 
                         
                           
                             
                               g 
                               F 
                             
                              
                             
                               
                                 C 
                                 F 
                               
                                
                               
                                 ( 
                                 z 
                                 ) 
                               
                             
                           
                           - 
                           
                             
                               ( 
                               
                                 
                                   g 
                                   4 
                                 
                                 
                                   g 
                                   2 
                                 
                               
                               ) 
                             
                              
                             
                               
                                 S 
                                 2 
                               
                                
                               
                                 ( 
                                 z 
                                 ) 
                               
                             
                           
                         
                         ) 
                       
                        
                       
                         
                           N 
                           1 
                         
                          
                         
                           ( 
                           z 
                           ) 
                         
                       
                        
                       
                         
                           Q 
                           1 
                         
                          
                         
                           ( 
                           z 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
         [0038]    Looking first to the quantization noise leakage Y QNL (z), this error is related to the gains g 4 , g 2 , and g F  and transfer functions C F (z) and S 2 (z). If one were to set the ratio of gains g 4  and g 2  to be approximately equal to gain g F   
         [0000]    
       
         
           
             
               ( 
               
                 
                   
                     g 
                     4 
                   
                   
                     g 
                     2 
                   
                 
                 = 
                 
                   g 
                   F 
                 
               
               ) 
             
             , 
           
         
       
     
         [0000]    then a matching of the transfer functions C F (z) and S 2 (z) would result in elimination of this quantization noise leakage Y QNL (z). Since gain g F  and C F (z) transfer function is adjustable (as being part of digital filter  206 ), adjustment can be based on determinations of the gains g 4  and g 2  and transfer function. 
         [0039]    To determine the gains g 4  and g 2 , DACs  204  and  408  can be selectively deactivated. Initially, a test signal (of any magnitude) can be applied to the CT SDM  200  with DAC  204  in a deactivated state and the gain g 3  set to 1 so that the output Y 2 (z) of CT SDM  212  can be measured. Under these circumstances, the gain g 4  is effectively 0, allowing equation (4) to be reduced to become output Y 2C1 (z) as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       Y 
                       
                         2 
                          
                         
                             
                         
                          
                         C 
                          
                         
                             
                         
                          
                         1 
                       
                     
                      
                     
                       ( 
                       z 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         ( 
                         
                           1 
                           
                             g 
                             2 
                           
                         
                         ) 
                       
                        
                       
                         
                           S 
                           2 
                         
                          
                         
                           ( 
                           z 
                           ) 
                         
                       
                        
                       
                         X 
                          
                         
                           ( 
                           z 
                           ) 
                         
                       
                     
                     + 
                     
                       
                         
                           N 
                           2 
                         
                          
                         
                           ( 
                           z 
                           ) 
                         
                       
                        
                       
                         
                           Q 
                           2 
                         
                          
                         
                           ( 
                           z 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Then, the same test signal can be applied to CT SDM  212  with DAC  408  in a deactivated state and with DAC  204  as a feedback DAC for CT SDM  212 . This changes the output Y 2 (z) to become output Y 2C2 (z) as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       Y 
                       
                         2 
                          
                         
                             
                         
                          
                         C 
                          
                         
                             
                         
                          
                         2 
                       
                     
                      
                     
                       ( 
                       z 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         1 
                         
                           g 
                           4 
                         
                       
                        
                       
                         
                           S 
                           2 
                         
                          
                         
                           ( 
                           z 
                           ) 
                         
                       
                        
                       
                         X 
                          
                         
                           ( 
                           z 
                           ) 
                         
                       
                     
                     + 
                     
                       
                         
                           N 
                           2 
                         
                          
                         
                           ( 
                           z 
                           ) 
                         
                       
                        
                       
                         
                           Q 
                           2 
                         
                          
                         
                           ( 
                           z 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Each of outputs Y 2C1 (z) and Y 2C2 (z) can be measured. By dividing the outputs Y 2C1 (z) and Y 2C2 (z) and noting that for a small band-width around the signal of interest the term N 2 (z)Q 2 (z) is negligible yields: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         Y 
                         
                           2 
                            
                           
                               
                           
                            
                           C 
                            
                           
                               
                           
                            
                           1 
                         
                       
                        
                       
                         ( 
                         z 
                         ) 
                       
                     
                     
                       
                         Y 
                         
                           2 
                            
                           
                               
                           
                            
                           C 
                            
                           
                               
                           
                            
                           2 
                         
                       
                        
                       
                         ( 
                         z 
                         ) 
                       
                     
                   
                   = 
                   
                     
                       g 
                       4 
                     
                     
                       g 
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Thus, a simple analysis of the system (which depends on the structures of the SDM filter  404 ) can yield the ratio 
         [0000]    
       
         
           
             
               
                 g 
                 4 
               
               
                 g 
                 2 
               
             
             . 
           
         
       
     
         [0000]    Typically, CT SDM  212  can be a higher order modulator (i.e., greater than 3) so the SDM filter  404  be, accordingly, a higher order filter. Gain g F  can then be adjusted to be proximately equal to the ratio 
         [0000]    
       
         
           
             
               
                 g 
                 4 
               
               
                 g 
                 2 
               
             
             . 
           
         
       
     
         [0040]    With gain g F  set, the transfer function C F (z) can be adjusted to substantially match the transfer function S 2 (z). To do this, an error function E that is a cross-correlation of an output Y 1,N (z) of digital filter and output Y 2 (z) of CT SDM  212  is used, where error function E is as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     E 
                      
                     
                       { 
                       
                         
                           
                             Y 
                             
                               1 
                               , 
                               N 
                             
                           
                            
                           
                             ( 
                             k 
                             ) 
                           
                         
                         , 
                         
                           
                             Y 
                             2 
                           
                            
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                       } 
                     
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             Y 
                             
                               1 
                               , 
                               N 
                             
                           
                           * 
                           
                             Y 
                             2 
                           
                         
                         ) 
                       
                        
                       
                         ( 
                         k 
                         ) 
                       
                     
                     = 
                     
                       ∑ 
                       
                         
                           
                             
                               Y 
                               _ 
                             
                             
                               1 
                               , 
                               N 
                             
                           
                            
                           
                             ( 
                             i 
                             ) 
                           
                         
                          
                         
                           
                             Y 
                             2 
                           
                            
                           
                             ( 
                             
                               i 
                               + 
                               k 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
         [0000]    This error function E is maximized when the transfer functions C F (z) and S 2 (z) are matched. Thus, digital filter  206  can be adjusted until the error function E is substantially maximized. Additionally, because the Q 1 (z) are common terms between outputs Y 1,N (z) and Y 2 (z), digital filter  206  can be blindly adjusted or calibrated. 
         [0041]    Now, turning to the gain imbalance, the output Y 2 (z) is generally comprised shaped of Q-noise Y 2Q (z) and phase/gain imbalance Y 2PI (z), which are as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         Y 
                         
                           2 
                            
                           
                               
                           
                            
                           PI 
                         
                       
                        
                       
                         ( 
                         z 
                         ) 
                       
                     
                     = 
                     
                       
                         X 
                          
                         
                           ( 
                           z 
                           ) 
                         
                       
                        
                       
                         ( 
                         
                           
                             g 
                             3 
                           
                           - 
                           
                             
                               ( 
                               
                                 
                                   g 
                                   4 
                                 
                                 
                                   g 
                                   1 
                                 
                               
                               ) 
                             
                              
                             
                               
                                 S 
                                 1 
                               
                                
                               
                                 ( 
                                 z 
                                 ) 
                               
                             
                           
                         
                         ) 
                       
                        
                       
                         ( 
                         
                           
                             
                               S 
                               2 
                             
                              
                             
                               ( 
                               z 
                               ) 
                             
                           
                           
                             g 
                             2 
                           
                         
                         ) 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         Y 
                         
                           2 
                            
                           Q 
                         
                       
                        
                       
                         ( 
                         z 
                         ) 
                       
                     
                     = 
                     
                       
                         
                           
                             N 
                             2 
                           
                            
                           
                             ( 
                             z 
                             ) 
                           
                         
                          
                         
                           
                             Q 
                             2 
                           
                            
                           
                             ( 
                             z 
                             ) 
                           
                         
                       
                       - 
                       
                         
                           ( 
                           
                             
                               g 
                               4 
                             
                             
                               g 
                               2 
                             
                           
                           ) 
                         
                          
                         
                           
                             S 
                             2 
                           
                            
                           
                             ( 
                             z 
                             ) 
                           
                         
                          
                         
                           
                             N 
                             1 
                           
                            
                           
                             ( 
                             z 
                             ) 
                           
                         
                          
                         
                           
                             Q 
                             1 
                           
                            
                           
                             ( 
                             z 
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Because there can be a delay associated with amplifier  220  and DAC  204 , gains g 3  and g 4  can be represented as g 3 A d (z) and g 4 D d (z), and from equation (11) above, it is clear that the following condition should substantially eliminate the gain imbalance Y 2PI (z): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         g 
                         3 
                       
                        
                       
                         
                           A 
                           d 
                         
                          
                         
                           ( 
                           z 
                           ) 
                         
                       
                     
                     - 
                     
                       
                         ( 
                         
                           
                             g 
                             4 
                           
                           
                             g 
                             1 
                           
                         
                         ) 
                       
                        
                       
                         
                           D 
                           d 
                         
                          
                         
                           ( 
                           z 
                           ) 
                         
                       
                        
                       
                         
                           S 
                           1 
                         
                          
                         
                           ( 
                           z 
                           ) 
                         
                       
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
         [0000]    This would mean that the gain imbalance of Y 2PI (z) would be substantially eliminated when the when the autocorrelation of Y 2 (z) (with gains g 3  and g 4  represented as g 3 A d (z) and g 4 D d (z)) is approximately equal to zero. Thus, by adjusting delay  222  and the gain of  220 , the gain imbalance Y 2PI (z) cab be substantially eliminated. 
         [0042]    Alternatively, the ratio 
         [0000]    
       
         
           
             
               g 
               4 
             
             
               g 
               1 
             
           
         
       
     
         [0000]    can be determined by selectively deactivating DACs  204  and  408 , similar to the method described above to determine the ratio 
         [0000]    
       
         
           
             
               
                 g 
                 4 
               
               
                 g 
                 2 
               
             
             . 
           
         
       
     
         [0000]    Initially, a test signal (of any magnitude) can be applied to the CT SDM  200  with DAC  204  in a deactivated state and the gain g 3  set to the ratio 
         [0000]    
       
         
           
             
               g 
               4 
             
             
               g 
               1 
             
           
         
       
     
         [0000]    so that the output Y 2 (z) of CT SDM  212  can be measured. Under these circumstances, equation (4) can be reduced to become output Y 2D1 (z) as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       Y 
                       
                         2 
                          
                         
                             
                         
                          
                         D 
                          
                         
                             
                         
                          
                         1 
                       
                     
                      
                     
                       ( 
                       z 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         ( 
                         
                           1 
                           
                             g 
                             1 
                           
                         
                         ) 
                       
                        
                       
                         
                           S 
                           2 
                         
                          
                         
                           ( 
                           z 
                           ) 
                         
                       
                        
                       
                         X 
                          
                         
                           ( 
                           z 
                           ) 
                         
                       
                     
                     + 
                     
                       
                         
                           N 
                           2 
                         
                          
                         
                           ( 
                           z 
                           ) 
                         
                       
                        
                       
                         
                           Q 
                           2 
                         
                          
                         
                           ( 
                           z 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Then, the same test signal can be applied to CT SDM  212  with DAC  408  in a deactivated state and with DAC  204  as a feedback DAC for CT SDM  212 . This changes the output Y 2 (z) to become output Y 2C2 (z) as denoted in equation (9) above. Each of outputs Y 2D1 (z) and Y 2C2 (z) can be measured. By dividing the outputs Y 2D1 (z) and Y 2C2 (z) and noting that, for a small band-width around the signal of interest, the term N 2 (z)Q 2 (z) is negligible yields: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         Y 
                         
                           2 
                            
                           
                               
                           
                            
                           D 
                            
                           
                               
                           
                            
                           1 
                         
                       
                        
                       
                         ( 
                         z 
                         ) 
                       
                     
                     
                       
                         Y 
                         
                           2 
                            
                           
                               
                           
                            
                           C 
                            
                           
                               
                           
                            
                           2 
                         
                       
                        
                       
                         ( 
                         z 
                         ) 
                       
                     
                   
                   = 
                   
                     
                       g 
                       4 
                     
                     
                       g 
                       1 
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Thus, a simple analysis of the system (which depends on the structures of the SDM filter  404 ) can yield the ratio 
         [0000]    
       
         
           
             
               
                 g 
                 4 
               
               
                 g 
                 1 
               
             
             . 
           
         
       
     
         [0000]    Thus, by adjusting gain g 3  to be approximately equal to the ratio 
         [0000]    
       
         
           
             
               g 
               4 
             
             
               g 
               1 
             
           
         
       
     
         [0000]    with this foreground calibration scheme (as opposed to the background scheme described above), gain imbalance can be substantially eliminated. 
         [0043]    To generally eliminate phase imbalance, either a digital predictor  220  or digital predictor with an analog delay line  222  can be employed. The tuning of either the digital predictor  220  or the analog delay line  222  can be done by minimizing the autocorrelation (similar to the scheme described above). For the digital predictor  220 , for example, an analog delay line  222  can be introduced such that the delay through delay line  222  is greater than that through the CT SDM  202  so as to allow digital predictor  220  to be tuned such that the auto-correlation is minimized. 
         [0044]    Having thus described the present invention by reference to certain of its preferred embodiments, it is noted that the embodiments disclosed are illustrative rather than limiting in nature and that a wide range of variations, modifications, changes, and substitutions are contemplated in the foregoing disclosure and, in some instances, some features of the present invention may be employed without a corresponding use of the other features. Accordingly, it is appropriate that the appended claims be construed broadly and in a manner consistent with the scope of the invention.