Abstract:
With high speed, high resolution time-interleaved (TI) analog-to-digital converters (ADCs), bandwidth mismatches between the various ADC branches can pose a significant problem. Previously, though, no adequate solution has been found. Here, a method and apparatus are provided that can calculate and compensate for bandwidth mismatches in a TI ADC, enabling a high speed, high resolution TI ADC to be produced.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
       [0001]    This application is a continuation of U.S. patent application Ser. No. 12/572,831, entitled “BANDWIDTH MISMATCH ESTIMATION IN TIME-INTERLEAVED ANALOG-TO-DIGITAL CONVERTERS,” filed on Oct. 2, 2009, which is hereby incorporated by reference for all purposes. 
     
    
     TECHNICAL FIELD 
       [0002]    The invention relates generally to analog-to-digital converters (ADCs) and, more particularly, to time-interleaved (TI) ADCs. 
       BACKGROUND 
       [0003]    Referring to  FIG. 1  of the drawings, the reference numeral  100  generally designates a conventional analog-to-digital converter (ADC). ADC  100  generally comprises a track-and-hold (T/H) circuit  102  and a sub-ADC  104  so that, in operation, the ADC  100  can sample an analog input signal X(t) at a plurality of sampling instants and convert the sampled signal into a digital signal Y[n]. As is shown in  FIG. 1 , though, the T/H circuit  104  generally comprises switches and capacitors, which causes the T/H circuit  102  to function as a filter (typically a single pole filter). 
         [0004]    Turning to  FIG. 2 , a model  200  of the ADC  100  is shown. In model  200 , the filter aspects of the ADC  100  are represented by filter  202 , while the remainder of the functionality of the ADC  100  is represented by ideal ADC  204 . Filter  202  has a transfer function in the time-domain of h a (t), which can, in turn, be represented in the frequency-domain as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         H 
                         a 
                       
                        
                       
                         ( 
                         ω 
                         ) 
                       
                     
                     = 
                     
                       
                         
                           g 
                           a 
                         
                          
                         
                            
                           
                             ωΔ 
                              
                             
                                 
                             
                              
                             t 
                           
                         
                       
                       
                         1 
                         + 
                         
                            
                            
                           
                             ( 
                             
                               ω 
                               
                                 ω 
                                 a 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where g a  is the gain of ADC  100 , Δt a  is the time delay relative to a reference, and ω a  is the cutoff frequency (bandwidth). This model  200  can be useful when determining mismatches for time-interleaved (TI) ADCs. 
         [0005]    In  FIG. 3A , an example of a TI ADC  300  can be seen. TI ADC  300  generally comprises ADCs  100 - 0  to  100 -(M−1) (where each of ADCs  100 - 0  to  100 -(M−1) generally has the same structure as ADC  100  from  FIG. 1 ) that are clocked by divider  302  so that the outputs from ADCs  100 - 0  to  100 -(M−1) can be multiplexed by multiplexer  304  to produce digital signal Y[n]. Yet, when building TI ADC  300 , ADCs  100 - 0  to  100 -(M−1) are not identical to each other; there are slight structural and operational variations. These slight variations result in Direct Current (DC) offset mismatches, timing skew, gain mismatches, and bandwidth mismatches between ADCs  100 - 0  to  100 -(M−1). 
         [0006]    Of the different types of mismatches listed, bandwidth mismatches are the weakest, and, to date, have largely been ignored, but, in order to build a high accuracy (generally greater than 6 bits), high speed (generally greater than 1 GS/s) TI ADCs, bandwidth mismatches between interleaved ADC branches need to be corrected. Looking to TI ADC  300 , the output spectrum when the input signal is a tone with frequency ω *  can be represented as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     Y 
                      
                     
                       ( 
                       
                          
                         ω 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         k 
                         = 
                         0 
                       
                       
                         M 
                         - 
                         1 
                       
                     
                      
                     
                         
                     
                      
                     
                       
                         ( 
                         
                           
                             1 
                             M 
                           
                            
                           
                             
                               ∑ 
                               
                                 a 
                                 = 
                                 0 
                               
                               
                                 M 
                                 - 
                                 1 
                               
                             
                              
                             
                                 
                             
                              
                             
                               
                                 
                                   H 
                                   a 
                                 
                                  
                                 
                                   ( 
                                   
                                     ω 
                                     * 
                                   
                                   ) 
                                 
                               
                                
                               
                                  
                                 
                                   
                                     - 
                                      
                                   
                                    
                                   
                                     
                                       2 
                                        
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       k 
                                     
                                     M 
                                   
                                    
                                   a 
                                 
                               
                             
                           
                         
                         ) 
                       
                        
                       
                         
                           δ 
                            
                           
                             ( 
                             
                               ω 
                               - 
                               
                                 ω 
                                 * 
                               
                               - 
                               
                                 
                                   2 
                                    
                                   π 
                                    
                                   
                                       
                                   
                                    
                                   k 
                                 
                                 M 
                               
                             
                             ) 
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Assuming a 2-way TI ADC (M=2), which generally represents the upper-bound or worst-case for bandwidth mismatch, equation (2) can be reduced to: 
         [0000]    
       
         
           
             
               
                 
                   
                     Y 
                      
                     
                       ( 
                       
                          
                         ω 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             
                               
                                 H 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   ω 
                                   0 
                                 
                                 ) 
                               
                             
                             + 
                             
                               
                                 H 
                                 1 
                               
                                
                               
                                 ( 
                                 
                                   ω 
                                   0 
                                 
                                 ) 
                               
                             
                           
                           2 
                         
                         ) 
                       
                        
                       
                         X 
                          
                         
                           ( 
                           
                              
                             ω 
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         ( 
                         
                           
                             
                               
                                 H 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   ω 
                                   0 
                                 
                                 ) 
                               
                             
                             - 
                             
                               
                                 H 
                                 1 
                               
                                
                               
                                 ( 
                                 
                                   ω 
                                   0 
                                 
                                 ) 
                               
                             
                           
                           2 
                         
                         ) 
                       
                        
                       
                         X 
                          
                         
                           ( 
                           
                              
                             
                                
                                
                               
                                 ( 
                                 
                                   ω 
                                   - 
                                   π 
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
         [0000]    with a Spurious-Free Dynamic Range (SFDR) of 
         [0000]    
       
         
           
             
               
                 
                   SFDR 
                   = 
                   
                     20 
                      
                     
                         
                     
                      
                     
                       
                         log 
                         10 
                       
                        
                       
                         ( 
                         
                           
                             
                               
                                 H 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   ω 
                                   0 
                                 
                                 ) 
                               
                             
                             + 
                             
                               
                                 H 
                                 1 
                               
                                
                               
                                 ( 
                                 
                                   ω 
                                   0 
                                 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 H 
                                 0 
                               
                                
                               
                                 ( 
                                 
                                   ω 
                                   0 
                                 
                                 ) 
                               
                             
                             - 
                             
                               
                                 H 
                                 1 
                               
                                
                               
                                 ( 
                                 
                                   ω 
                                   0 
                                 
                                 ) 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The SFDR for an M-way interleaved TI ADC, therefore, can then be determined to be: 
         [0000]    
       
         
           
             
               
                 
                   
                     SFDR 
                     = 
                     
                       
                         max 
                         k 
                       
                        
                       
                         ( 
                         
                           20 
                            
                           
                               
                           
                            
                           
                             
                               log 
                               10 
                             
                              
                             
                               ( 
                               
                                 
                                   A 
                                    
                                   
                                     [ 
                                     0 
                                     ] 
                                   
                                 
                                 
                                   A 
                                    
                                   
                                     [ 
                                     k 
                                     ] 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   where 
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
             
               
                 
                   
                     A 
                      
                     
                       [ 
                       k 
                       ] 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         a 
                         = 
                         0 
                       
                       
                         M 
                         - 
                         1 
                       
                     
                      
                     
                         
                     
                      
                     
                       
                         
                           H 
                           a 
                         
                          
                         
                           ( 
                           
                             ω 
                             0 
                           
                           ) 
                         
                       
                        
                       
                          
                         
                           
                             - 
                              
                           
                            
                           
                             
                               2 
                                
                               π 
                                
                               
                                   
                               
                                
                               k 
                             
                             M 
                           
                            
                           a 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Now, equation (1) can be applied to TI ADC  300  for the purposes of simulation so 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         H 
                         a 
                       
                        
                       
                         ( 
                         
                           ω 
                           0 
                         
                         ) 
                       
                     
                     = 
                     
                       1 
                       
                         1 
                         + 
                         
                           
                             τ 
                             a 
                           
                            
                           
                             ω 
                             0 
                           
                         
                       
                     
                   
                   , 
                   
                     
                       
                         for 
                          
                         
                             
                         
                          
                         
                           T 
                           S 
                         
                       
                       &gt; 
                       
                         τ 
                         a 
                       
                     
                     = 
                     
                       1 
                       
                         ω 
                         a 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where T S  is the period of clock signal CLK. Such a simulation yields that variations in bandwidth mismatches are dependent on gain mismatches and timing skews and that (with high accuracy, high speed TI ADCs) bandwidth mismatch can significantly affect performance. An example of a simulation of the effect bandwidth mismatch can be seen in  FIG. 3B  for different gain and skew compensations. Thus, to achieve the desired SFDR (i.e., greater than 70 dB) for a TI ADC, the bandwidths of ADCs within the TI ADC should be matched to be within 0.1% to 0.25%. 
         [0007]    To date, however, no estimation algorithm or circuit exists to blindly determine bandwidth mismatches. The two most relevant conventional circuits, though, are described in the following: Satarzadeh et al., “Bandwidth Mismatch Correction for a Two-Channel Time-Interleaved A/D Converter,”  Proceedings of  2007  IEEE International Symposium on Circuits and Systems,  2007; and Tsai et al., “Bandwidth Mismatch and Its Correction in Time-Interleaved Analog-to-Digital Converters,”  IEEE Transactions on Circuits and Systems II: Express Briefs , Vol. 53, No. 10, pp. 1133-1137, Oct. 23, 2006. Neither of these circuits, though, adequately addresses blind bandwidth mismatch estimation, indicating a need for an apparatus and/or method to determine and compensate for bandwidth mismatches. 
       SUMMARY 
       [0008]    A preferred embodiment of the present invention, accordingly, provides an apparatus. The apparatus comprises a plurality of analog-to-digital converter (ADC) branches, wherein each ADC branch receives an analog input signal, and wherein each ADC branch has an autocorrelation sequence that is estimated by an autocorrelation estimator, and wherein each ADC branch has an ADC that includes: a track-and-hold (T/H) circuit that receives the analog input signal and that has filter characteristics, wherein the filter characteristics are adjustable; and a sub-ADC that is coupled to the T/H circuit; a multiplexer that is coupled to each of the ADC branches; and a mismatch estimation circuit that is coupled to each T/H circuit and that receives an output signal from each ADC branch, wherein the mismatch estimation circuit adjusts the filter characteristics for each T/H circuit to determine a desired range for a cost function over a plurality of samples of the analog input signal. 
         [0009]    In accordance with a preferred embodiment of the present invention, each ADC branch includes an amplifier that is coupled to its corresponding ADC. 
         [0010]    In accordance with a preferred embodiment of the present invention, the mismatch estimation circuit adjusts the gain for each amplifier to determine the desired range for the cost function over the plurality of samples of the analog input signal. 
         [0011]    In accordance with a preferred embodiment of the present invention, the cost function (V) is: 
         [0000]    
       
         
           
             
               V 
               = 
               
                 
                   ∑ 
                   l 
                 
                  
                 
                     
                 
                  
                 
                    
                   
                     
                       
                         
                           R 
                           ^ 
                         
                         aa 
                       
                        
                       
                         [ 
                         
                           lMT 
                           S 
                         
                         ] 
                       
                     
                     - 
                     
                       
                         
                           R 
                           ^ 
                         
                         00 
                       
                        
                       
                         [ 
                         
                           lMT 
                           S 
                         
                         ] 
                       
                     
                   
                    
                 
               
             
             , 
           
         
       
     
         [0000]    wherein {circumflex over (R)} aa  denotes the estimate for the autocorrelation sequence for the a th  ADC branch, M denotes the number of ADC branches, and T S  is a sampling clock period. 
         [0012]    In accordance with a preferred embodiment of the present invention, the apparatus further comprises a divider that receives a clock signal and that is coupled to each ADC branch. 
         [0013]    In accordance with a preferred embodiment of the present invention, each ADC branch further comprises an adjustable delay element that is coupled to each ADC, to the divider, and to the mismatch estimation circuit, wherein the mismatch estimation circuit adjusts each delay element to compensate for timing skews. 
         [0014]    In accordance with a preferred embodiment of the present invention, the cost function (V) is: 
         [0000]    
       
         
           
             
               V 
               = 
               
                 
                   ∑ 
                   l 
                 
                  
                 
                     
                 
                  
                 
                   
                     ( 
                     
                       
                         
                           
                             
                               R 
                               ^ 
                             
                             aa 
                           
                            
                           
                             [ 
                             
                               lMT 
                               S 
                             
                             ] 
                           
                         
                         
                           
                             
                               R 
                               ^ 
                             
                             aa 
                           
                            
                           
                             [ 
                             
                               fMT 
                               S 
                             
                             ] 
                           
                         
                       
                       - 
                       
                         
                           
                             
                               R 
                               ^ 
                             
                             00 
                           
                            
                           
                             [ 
                             
                               lMT 
                               S 
                             
                             ] 
                           
                         
                         
                           
                             
                               R 
                               ^ 
                             
                             00 
                           
                            
                           
                             [ 
                             
                               fMT 
                               S 
                             
                             ] 
                           
                         
                       
                     
                     ) 
                   
                   2 
                 
               
             
             , 
           
         
       
     
         [0000]    wherein {circumflex over (R)} aa  denotes the estimate for the autocorrelation sequence for the a th  ADC branch, M denotes the number of ADC branches, f is an arbitrary delay, and T S  is a sampling clock period. 
         [0015]    In accordance with a preferred embodiment of the present invention, an apparatus is provided. The apparatus comprises a plurality of ADC branches, wherein each ADC branch receives an analog input signal, and wherein each ADC branch has an autocorrelation sequence that is estimated by an autocorrelation estimator, and wherein each ADC branch has an ADC that includes: a T/H circuit that receives the analog input signal and that has filter characteristics, wherein the filter characteristics are adjustable; and a sub-ADC that is coupled to the T/H circuit; a multiplexer that is coupled to each of the ADC branches; a mismatch estimation circuit that is coupled to each T/H circuit and that receives an output signal from each ADC branch, wherein the mismatch estimation circuit has a computer program embodied thereon that includes: computer code for squaring an error between the estimate for the autocorrelation sequence of a first ADC branch of the plurality of ADC branches and the estimates for the autocorrelation sequences for the remainder of the ADC branches as a cost function; and computer code for adjusting each of the T/H circuits to compensate for bandwidth mismatches. 
         [0016]    In accordance with a preferred embodiment of the present invention, each ADC branch further comprises a Direct Current (DC) offset circuit that is coupled to its corresponding ADC. 
         [0017]    In accordance with a preferred embodiment of the present invention, a method for bandwidth matching a plurality of ADC branches in a time-interleaved (TI) ADC is provided. The method comprises sampling an analog input signal at a plurality of sampling instants by the plurality of ADC branches; calculating an autocorrelation sequence for each ADC branch, wherein each ADC branch includes an T/H circuit; calculating a cost function to determine bandwidth mismatches for the plurality of ADC branches, wherein the cost function is a function of the autocorrelation sequences of the plurality of ADC branches; and adjusting at least one of the T/H circuits from the plurality of ADC branches based at least in part on the cost function to substantially bandwidth match the plurality of ADC branches. 
         [0018]    In accordance with a preferred embodiment of the present invention, each autocorrelation sequence estimate is 
         [0000]    
       
         
           
             
               
                 
                   R 
                   ^ 
                 
                 aa 
               
                
               
                 [ 
                 
                   lMT 
                   S 
                 
                 ] 
               
             
             = 
             
               
                 1 
                 L 
               
                
               
                 
                   ∑ 
                   
                     k 
                     = 
                     0 
                   
                   
                     L 
                     - 
                     1 
                   
                 
                  
                 
                     
                 
                  
                 
                   
                     x 
                      
                     
                       [ 
                       
                         
                           kMT 
                           S 
                         
                         + 
                         a 
                       
                       ] 
                     
                   
                    
                   
                     x 
                      
                     
                       [ 
                       
                         
                           
                             ( 
                             
                               k 
                               - 
                               l 
                             
                             ) 
                           
                            
                           
                             MT 
                             S 
                           
                         
                         + 
                         a 
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
         [0000]    for the a th  ADC branch, wherein M denotes the number of ADC branches and T S  is a sampling clock period. 
         [0019]    In accordance with a preferred embodiment of the present invention, the cost function (V) is a distance metric defined on the sequences {circumflex over (R)} aa [lMT S ] and {circumflex over (R)} 00 [lMT S ]. 
         [0020]    In accordance with a preferred embodiment of the present invention, the cost function (V) is a distance metric defined on the sequences 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       R 
                       ^ 
                     
                     aa 
                   
                    
                   
                     [ 
                     
                       lMT 
                       S 
                     
                     ] 
                   
                 
                 
                   
                     
                       R 
                       ^ 
                     
                     aa 
                   
                    
                   
                     [ 
                     
                       fMT 
                       S 
                     
                     ] 
                   
                 
               
                
               
                   
               
                
               and 
                
               
                   
               
                
               
                 
                   
                     
                       R 
                       ^ 
                     
                     00 
                   
                    
                   
                     [ 
                     
                       lMT 
                       S 
                     
                     ] 
                   
                 
                 
                   
                     
                       R 
                       ^ 
                     
                     00 
                   
                    
                   
                     [ 
                     
                       fMT 
                       S 
                     
                     ] 
                   
                 
               
             
             , 
           
         
       
     
         [0000]    for an arbitrary delay f. 
         [0021]    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 
         [0022]    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: 
           [0023]      FIG. 1  is a circuit diagram of a conventional analog-to-digital converter (ADC); 
           [0024]      FIG. 2  is a block diagram of a model of the ADC of  FIG. 1 ; 
           [0025]      FIG. 3A  is a circuit diagram of a convention time-interleaved (TI) ADC using the ADC of  FIG. 1 ; 
           [0026]      FIG. 3B  is an example of a simulation showing the effect of bandwidth mismatch on the Spurious-Free Dynamic Range (SFDR) of a TI ADC; 
           [0027]      FIG. 4  is a circuit diagram of a TI ADC in accordance with a preferred embodiment of the present invention; 
           [0028]      FIG. 5  is a circuit diagram of the autocorrelation estimator of the TI ADC of  FIG. 4 ; and 
           [0029]      FIGS. 6A through 6D  are graphs depicting the operation of the TI ADC of  FIG. 4 . 
       
    
    
     DETAILED DESCRIPTION 
       [0030]    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. 
         [0031]    Referring to  FIG. 4  of the drawings, the reference numeral  400  generally designates a time-interleaved (TI) analog-to-digital converter (ADC) in accordance with a preferred embodiment of the present invention. ADC  400  generally comprises ADC branches  406 - 0  to  406 -(M−1), divider  402 , multiplexer or mux  408 , and a mismatch estimation circuit  410 . Each ADC branch  406 - 0  to  406 -(M−1) also generally comprises (respectively) ADC  412 - 0  to  412 -(M−1), DC offset circuit  414 - 0  to  414 -(M−1), amplifier  416 - 0  to  416 -(M−1), adjustable delays element  404 - 0  to  404 -(M−1), and autocorrelation estimator  422 - 0  to  422 -(M−1). Additionally, each ADC  412 - 0  to  412 -(M−1) generally comprises (respectively) a track-and-hold (T/H) circuit  418 - 0  to  418 -(M−1) and a sub-ADC  420 - 0  to  420 -(M−1). 
         [0032]    In operation, TI ADC  400  converts analog input signal X(t) to a digital signal Y[n]. To accomplish this, divider  402  divides a clock signal CLK (with a frequency of F S  or period of T S ) into M clock signals (each with a frequency of F S /M) that are staggered by delay elements  404 - 0  to  404 -(M−1) and provided to ADCs  412 - 0  to  412 -(M−1). This allows each of ADCs  412 - 0  to  412 -(M−1) to convert the analog signal X(t) to digital signals X 0 (k) to X M-1 (k). The gain and DC offset adjustments are applied to digital signals X 0 (k) to X M-1 (k) by DC offset circuits  414 - 0  to  414 -(M−1) and amplifiers  416 - 0  to  416 -(M−1) to generate digital signals Y[ 0 ] to Y[M−1], which can then be multiplexed by mux  408  to generate a digital signal Y[N]. 
         [0033]    To generally ensure that signals Y[ 0 ] to Y[M−1] are matched, mismatch estimation circuit  410  calculates and compensates for gain mismatches, DC offset mismatches, timing skews, and bandwidth mismatches. The mismatch estimation circuit  410  is generally a digital signals processor (DSP) or dedicated hardware, which determines the gain mismatches, DC offset mismatches, timing skews, and bandwidth mismatches and which can provide adjustments for gain, DC offset, timing skew, and bandwidth to amplifiers  402 - 0  to  402 -(M−1), DC offset circuit  404 - 0  to  404 -(M−1), delays  408 - 0  to  408 -(M−1), and T/H circuits  410 - 0  to  410 -(M−1), respectively. 
         [0034]    In general communications systems, signals are generally wide-sense stationary (WSS), which is primarily due to the generally random nature of the transmitted signals. Thus, input signal X(t) can generally be thought of as a WSS signal. As a result, an autocorrelation sequence R aa  exists for each of ADC branches  401 - 0  to  401 -(M−1). This autocorrelation sequence R aa  for branch “a” is generally a function of the number of ADC branches (M) and the period T S  of clock signal CLK, which can be represented as follows: 
         [0000]        R   aa   [lMT   S ]=( R   ha,ha   *R   x,x )[ lMT   S ]  (8)
 
         [0000]    Calculation of an estimate ({circumflex over (R)} aa ) of autocorrelation sequence R aa  for each branch  406 - 0  to  406 -(M−1) is performed by autocorrelation estimators  422 - 0  to  422 -(M−1), respectively, which is discussed in greater detail below. If the bandwidths of the T/H circuits  410 - 0  to  410 -(M−1) are matched exactly, the autocorrelation sequences R aa  for each ADC branch output Y[ 0 ] to Y[M−1] should be the equal. To ensure that the bandwidths of the T/H circuits  410 - 0  to  410 -(M−1), the mismatch estimation circuit  410  receives estimations for autocorrelation sequence {circumflex over (R)} aa  for each ADC branch  401 - 0  to  401 -(M−1) from autocorrelation estimators  422 - 0  to  422 -(M−1) and computes a cost function based on these estimates {circumflex over (R)} aa . For this arrangement, the cost function is generally “bowl-shaped” or has positive concavity at every point except at a point where the bandwidths are matched (where the concavity approximately equals zero). The expression for the autocorrelation estimate is: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         R 
                         ^ 
                       
                       aa 
                     
                      
                     
                       [ 
                       
                         lMT 
                         S 
                       
                       ] 
                     
                   
                   = 
                   
                     
                       1 
                       L 
                     
                      
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           0 
                         
                         
                           L 
                           - 
                           1 
                         
                       
                        
                       
                           
                       
                        
                       
                         
                           x 
                            
                           
                             [ 
                             
                               
                                 kMT 
                                 S 
                               
                               + 
                               a 
                             
                             ] 
                           
                         
                          
                         
                           x 
                            
                           
                             [ 
                             
                               
                                 
                                   ( 
                                   
                                     k 
                                     - 
                                     l 
                                   
                                   ) 
                                 
                                  
                                 
                                   MT 
                                   S 
                                 
                               
                               + 
                               a 
                             
                             ] 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
         [0000]    wherein L is the number of products accumulated to estimate the autocorrelation. 
         [0035]    One cost function that can be employed is based on normalization and, effectively, eliminates gain from the calculation. This “normalization” cost function uses the squared error between the estimated autocorrelation sequences {circumflex over (R)} aa  of the first ADC branch  406 - 0  and the estimated autocorrelation sequences {circumflex over (R)} aa  for the remaining ADC branches  406 - 1  to  406 -(M−1) that quantifies the extent of the bandwidth mismatch. In particular, the “normalization” cost function is represented as follows: 
         [0000]    
       
         
           
             
               
                 
                   V 
                   = 
                   
                     
                       
                         ∑ 
                         l 
                       
                        
                       
                           
                       
                        
                       
                         
                           ( 
                           
                             
                               
                                 
                                   
                                     R 
                                     ^ 
                                   
                                   aa 
                                 
                                  
                                 
                                   [ 
                                   
                                     lMT 
                                     S 
                                   
                                   ] 
                                 
                               
                               
                                 
                                   
                                     R 
                                     ^ 
                                   
                                   aa 
                                 
                                  
                                 
                                   [ 
                                   
                                     fMT 
                                     S 
                                   
                                   ] 
                                 
                               
                             
                             - 
                             
                               
                                 
                                   
                                     R 
                                     ^ 
                                   
                                   00 
                                 
                                  
                                 
                                   [ 
                                   
                                     lMT 
                                     S 
                                   
                                   ] 
                                 
                               
                               
                                 
                                   
                                     R 
                                     ^ 
                                   
                                   00 
                                 
                                  
                                 
                                   [ 
                                   
                                     fMT 
                                     S 
                                   
                                   ] 
                                 
                               
                             
                           
                           ) 
                         
                         2 
                       
                     
                     ∝ 
                     
                       
                         ∑ 
                         l 
                       
                        
                       
                         
                           ( 
                           
                             
                               
                                 
                                   
                                     R 
                                     ^ 
                                   
                                   aa 
                                 
                                  
                                 
                                   [ 
                                   
                                     lMT 
                                     S 
                                   
                                   ] 
                                 
                               
                                
                               
                                 
                                   
                                     R 
                                     ^ 
                                   
                                   00 
                                 
                                  
                                 
                                   [ 
                                   
                                     fMT 
                                     S 
                                   
                                   ] 
                                 
                               
                             
                             - 
                             
                               
                                 
                                   
                                     R 
                                     ^ 
                                   
                                   00 
                                 
                                  
                                 
                                   [ 
                                   
                                     lMT 
                                     S 
                                   
                                   ] 
                                 
                               
                                
                               
                                 
                                   
                                     R 
                                     ^ 
                                   
                                   aa 
                                 
                                  
                                 
                                   [ 
                                   
                                     fMT 
                                     S 
                                   
                                   ] 
                                 
                               
                             
                           
                           ) 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where f is an arbitrary delay. Therefore, the mismatch estimation circuit  410  can search for a desired bandwidth match by varying the filter characteristics (for example, resistance and capacitance) of each of the T/H circuits  418 - 0  to  418 -(M−1) and looking for a desired range or “minimum” for the cost function of equation (10), essentially looking for the point where the concavity of the cost function of equation (10) is approximately zero. 
         [0036]    Alternatively, a cost function that simultaneously compensates for both gain and bandwidth may be employed. In particular, this cost function can be represented as: 
         [0000]    
       
         
           
             
               
                 
                   V 
                   = 
                   
                     
                       ∑ 
                       l 
                     
                      
                     
                         
                     
                      
                     
                        
                       
                         
                           
                             
                               R 
                               ^ 
                             
                             aa 
                           
                            
                           
                             [ 
                             
                               lMT 
                               S 
                             
                             ] 
                           
                         
                         - 
                         
                           
                             
                               R 
                               ^ 
                             
                             00 
                           
                            
                           
                             [ 
                             
                               lMT 
                               S 
                             
                             ] 
                           
                         
                       
                        
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Here, the mismatch estimation circuit  410  can search for a desired bandwidth match by varying the filter characteristics of each of the T/H circuits  418 - 0  to  418 -(M−1) and the gains of amplifiers  416 - 0  to  416 -(M−1) and looking for a desired range or “minimum” for the cost function of equation (11), essentially looking for the point where the concavity of the cost function of equation (11) is approximately zero. Typically, convergence for equation (11) usually requires more samples than equation (10) because there is increased complexity, but bandwidth matching and gain matching for equation (11) would be greater than equation (10), improving performance. 
         [0037]    In the most general case, the cost function can be a distance metric defined on either the estimated autocorrelation sequences directly, or on the sequences generated when all the elements of the autocorrelation sequences are divided by the same element of each sequence. In the case of equation (10), the division is performed using element f of the sequence. 
         [0038]    Turning now to  FIG. 5 , an example of the construction of autocorrelation estimators  422 - 0  to  422 -(M−1) can be seen and which is denoted by reference numeral  422 . Autocorrelation estimator  422  generally includes several branches (“p” branches as shown). Each branch calculates an estimation {circumflex over (R)} aa  (for the a th  ADC branch  412 - a ) of the autocorrelation sequence for a sample (0 to pMT S ), and each branch is generally comprised of a multiplier  506 - 0  to  506 - p , an adder  508 - 0  to  508 - p , and a delay element  504 - 0  to  504 - p . Delay elements  502 - 1  to  502 - p  are generally coupled in series with one another so that samples X[kMT S -a] to X[(k−p)MT S -a] are available for the branches. Each branch uses its multiplier  506 - 0  to  506 - p  to multiply its sample X[kMT S -a] to X[(k−p)MT S -a] with the current sample X[kMT S -a]. The output of multiplier  506 - 0  to  506 - p  is then added to its previous sum (from delay element  508 - 0  to  508 - p ) by adder  508 - 0  to  508 - p . Looking to branch “1” for example, multiplier  506 - 1  multiples sample X[kMT S -a] with sample X[(k—1)MT S -a], and adder  508 - 1  adds the output of multiplier (X[kMT S -a] X[(k−1)MT S -a]) with the previous sum from adder  508 - 1  to generate the autocorrelation sequence estimate {circumflex over (R)} aa [MT S ] for branch “1.” 
         [0039]    Referring now to  FIGS. 6A through 6D , an example of the operation of TI ADC  400  can be seen. In particular,  FIGS. 6A through 6D  show the convergence for bandwidth, timing skew, gain, and DC offset (respectively) substantially simultaneously at about 600 data blocks (with a data block sample size 2 18  samples) for a three tone signal at 0.27F S , 0.35F S , and −0.27F S . Clearly, the TI ADC  400  now provides a structure that allows for the construction of a high speed, high resolution ADC that was previously unachievable. 
         [0040]    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.