Patent Publication Number: US-2023155598-A1

Title: Matrix Processor Generating SAR-Searched Input Delay Adjustments to Calibrate Timing Skews in a Multi-Channel Interleaved Analog-to-Digital Converter (ADC)

Description:
RELATED APPLICATION 
     This invention is a Continuation-in-Part (CIP) of “Calibration of Timing Skews in a Multi-Channel Interleaved Analog-to-Digital Converter (ADC) by Auto-Correlation of Muxed-Together Channels in Binary Output Tree”, U.S. Ser. No. 17/455,471, filed Nov. 18, 2021. 
    
    
     FIELD OF THE INVENTION 
     This invention relates to Analog-to-Digital Converters (ADC), and more particularly to calibration of interleaved ADCs. 
     BACKGROUND OF THE INVENTION 
     Analog-to-Digital Converters (ADCs) are widely used to convert analog signals to digital values. Multi-bit ADCs have a high resolution, and its accuracy can be improved by calibration. Higher sampling rates can be achieved by interleaving two ADCs that each operate at half the sampling rate. 
       FIG.  1    shows a prior-art interleaved ADC. ADC  10  and ADC  12  are interleaved, with ADC  10  sampling analog input AIN when clock CLK closes switch  20 , and ADC  12  sampling analog input AIN when inverse clock CLKB closes switch  22 . Mux  18  selects digital output Y1 from ADC  10  when CLK is high, when ADC  10  has had sufficient time to sample and hold AIN and convert it to a digital value. The digital output DOUT is Y2 when CLK is low to mux  18 . Thus, each of ADC  10 ,  12  can operate at half the data rate of the final output DOUT. 
       FIG.  2    is a graph of analog sampling and clock skew. AIN is sampled into ADC  10  to generate Y1[K−1] and Y1[K] on the falling edges of CLK, while AIN is sampled into ADC  12  to generate Y2[K−1] and Y2[K] on the falling edges of CLKB, where K is the sample or time-index number. The sampling time or period of AIN is Ts. Ideally there is no clock skew in CLK, and all samples are separated by Ts. However, CLK may not have a pulse width that is exactly 50% of period 2*Ts, introducing sampling pulse-width mismatch and non-linearities. Sampling of channel Y2 may be delayed relative to sampling of channel Y1 by Ts+ΔT/2, while Sampling of channel Y1 may be delayed relative to sampling of channel Y2 by Ts−ΔT/2. Ideally ΔT=0 without mismatch on sampling. However, in reality ΔT is finite. It is desired to reduce ΔT to a minimal acceptable level for more than 2 channels. 
     ADC  10 ,  12  and switches  20 ,  22  may not be exactly matched, introducing finite bandwidth mismatches among the two channels Y1, Y2. Thus, both sampling-pulse mismatches and ADC component mismatches may contribute to nonlinearities. 
       FIG.  3    is a graph of spurious tones in a spectrum for a prior-art interleaved ADC. Sampling pulse-width mismatches and component mismatches may introduce non-linearities or errors that cause spurious tones  302 . These spurious tones can occur at integer multiples of Fs/N, K*Fs/N±F, wherein K is an integer, where Fs is the sampling frequency (period Ts=1/Fs), and N is the number of channels interleaved together. These spurious tones are undesirable since they can restrict the dynamic range of high-speed ADCs and are proportional to analog input signal amplitude and frequency. 
     What is desired is a highly-interleaved ADC with at least 3 ADC channels interleaved together for operation at higher sampling rates. It is desired to introduce a variable, programmable delay to each of the channel inputs to correct for timing skews caused by sampling pulse-width, clock, and component mismatches among the 3 or more channels interleaved together. A calibration method is desired to test various values of these delays to program these delays to minimize the skew among the multiple channels. Both a rapid foreground calibration method and a background calibration method to adjust for gradual temperature skews are desired. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG.  1    shows a prior-art interleaved ADC. 
         FIG.  2    is a graph of analog sampling and clock skew. 
         FIG.  3    is a graph of spurious tones in a spectrum for a prior-art interleaved ADC. 
         FIG.  4    is a block diagram of a 4-channel interleaved ADC with product derivative correlators and a matrix processor for calibration of programmable input delays. 
         FIG.  5    is a diagram of a SAR delay element with binary-weighted capacitors. 
         FIG.  6    is a block diagram of a 3-channel interleaved ADC with details of the product derivative correlators that drive a matrix processor for calibration of programmable input delays. 
         FIGS.  7 A- 7 B  is a flowchart of foreground calibration of N ADC channels interleaved together. 
         FIG.  8    is a graph of reduced spurious tones in a spectrum for an interleaved ADC with channel input delays calibrated by product derivative correlators and matrix processing. 
         FIG.  9    shows a backend processor triggering background calibration of an interleaved ADC. 
         FIGS.  10 A- 10 B  show a flowchart of a background calibration process. 
         FIG.  11    shows a non-binary number of ADCs interleaved and calibrated with product derivative correlators and a matrix processor. 
         FIG.  12    shows a non-binary odd number of ADCs interleaved and calibrated with product derivative correlators and a matrix processor. 
         FIG.  13    shows a model of an interleaved ADC. 
     
    
    
     DETAILED DESCRIPTION 
     The present invention relates to an improvement in interleaved ADC calibration. The following description is presented to enable one of ordinary skill in the art to make and use the invention as provided in the context of a particular application and its requirements. Various modifications to the preferred embodiment will be apparent to those with skill in the art, and the general principles defined herein may be applied to other embodiments. Therefore, the present invention is not intended to be limited to the particular embodiments shown and described but is to be accorded the widest scope consistent with the principles and novel features herein disclosed. 
       FIG.  4    is a block diagram of a 4-channel interleaved ADC with product derivative correlators and a matrix processor for calibration of programmable input delays. Analog input AIN is buffered by analog buffer  30  and sampled by switches  20 ,  22 ,  23 ,  24  into ADC  10 ,  12 ,  13 ,  14  that generate digital values D1, D2, D3, D4, respectively. Mux  64  alternately selects D1, D2, D3, D4 to generate the final data output DOUT. Final mux  64  operates with clock TSX4 that has four times the frequency of sampling clock TS1. 
     Sampling clocks TS1, TS2, TS3, TS4 can be a four-phase clock all operating at the same frequency but with phase shifts of 0, 90, 180, and 270 degrees. Sampling clocks TS1, TS2, TS3, TS4 are delayed by variable delays to generate clocks T1, T2, T3, T4 that control switches  20 ,  22 ,  23 ,  24 , respectively, that sample AIN to ADC  10 ,  12 ,  13 ,  14  that generate channel digital outputs D1, D2, D3, D4, having phases of 0, 90, 180, and 270 degrees. These variable delays are programmed during calibration with digital values that are stored in a Successive-Approximation-Register (SAR) that enable and disable binary-weighted capacitor delay elements. Calibration uses a Successive-Approximation method testing larger, Most-Significant Bit (MSB) capacitors first, then testing successively smaller capacitors until a Least-Significant Bit (LSB) capacitor is tested. 
     SAR delay  40  delays sampling clock TS1 to generate T1 to switch  20 , while SAR delays  42 ,  43 ,  44  delay sampling clocks TS2, TS3, TS4 to generate T2, T3, T4 to switches  22 ,  23 ,  24 , respectively. By adjusting the delay values programmed into SAR delays  40 ,  42 ,  43 ,  44 , the timing skews between channels D1, D2, D3, D4 can be compensated for and matched to within the delay of 1 LSB capacitor in SAR delay  40 ,  42 ,  43 ,  44 . 
     Finite-Impulse-Response (FIR) filters  17  filter digital outputs D1, D2, D3, D4 from ADC  10 ,  12 ,  13 ,  14  to generate filtered digital values Y1, Y2, Y3, Y4. FIR filters  17  can act as lowpass or bandpass filters for calibration. 
     Product derivative correlators  52  receive filtered digital values Y1, Y2, Y3, Y4 and generate product derivative factors F1, F2, F3, F4. The product derivative factor F2 is a function of the current channel filtered digital value Y2, and the adjacent channels Y1, Y3. In general, the product derivative factor F(X) for a channel X has inputs Y(X), Y(X−1), and Y(X+1), where X−1 and X+1 are modulo N, where N is the number of interleaved channels. Each product derivative correlator  52  generates a correlation factor of the current channel compared to the two adjacent channels. Product derivative correlator  52  can be a mid-point correlator such as shown by the logic implemented of  FIG.  6   . 
     Matrix processor  50  receives product derivative factors F1, F2, F3, F4 from product derivative correlators  52 , and forms a matrix from F1, F2, F3, F4 that is multiplied by a correlation matrix to generate sign bits. The correlation matrix is constant matrix that is fixed for a fixed number of channels N. 
     During calibration, the delay in SAR delay  40  is fixed and acts as a timing reference for the other N-1 channels. Therefore, the sign bit for channel 1 is not generated by matrix processor  50 . 
     For a constant or slow-moving analog input AIN, all channels should generate the same filtered digital values Y1, Y2, Y3, Y4. Differences in these values among channels can indicate skews or timing differences. 
     Product derivative factors F1, F2, F3, F4 each indicate a digital value or timing difference between a channel and its two adjacent channels. These timing differences are combined with all other timing differences by matrix processor  50  to generate the sign bits. The sign bits indicate which channels have larger delays, and which channels have smaller delays. 
     Calibrator  55  uses these sign bits during a successive-approximation sequence to decide when to keep a test bit set in SAR delay  42 ,  43 ,  44  and when to reset the test bit, as successively smaller bit-positions are tested. 
       FIG.  5    is a diagram of a SAR delay element with binary-weighted capacitors. SAR delay  40  has input inverter  92  that inverts input sampling clock TS1 to drive delay node D, and output inverter  94  that drives local sampling clock T1 for channel 1. 
     A series of binary-weighted capacitors  110 ,  108 ,  106 ,  104 , . . .  102  have capacitance values or weights of 64, 32, 16, 8, 4, 2, and 1 times a minimum capacitor value of C, C1 capacitor  102 . All of binary-weighted capacitors  110 ,  108 ,  106 ,  104 , . . .  102  have one terminal that connects to delay node D between inverters  92 ,  94 , and another terminal connected to ground through enabling transistors  120 ,  118 ,  116 ,  114 , . . .  112 . Bits in SAR register  130  drive the gates of enabling transistors  120 ,  118 ,  116 ,  114 , . . .  112 . When the bit in SAR register  130  is high, the enabling transistor is turned on, connecting the lower terminal of the capacitor to ground, enabling the capacitor and increasing the capacitance and delay of delay node D. 
     For example, the value 1010000 programmed into SAR register  130  enables transistors  120 ,  116 , and capacitors  110 ,  106 , increasing the capacitance on delay node D by 64 C+16 C, or 80 C. MSB capacitor  110  adds a 64 C delay while capacitor  106  adds a 16 C delay. Other SAR delays  42 ,  43 ,  44  operate in a similar manner and can each be programmed with a different SAR delay value. 
       FIG.  6    is a block diagram of a 3-channel interleaved ADC with details of the product derivative correlators that drive a matrix processor for calibration of programmable input delays. In this simplified 3-channel ADC, there are only three ADC  10 ,  12 ,  13 , three sampling switches  20 ,  22 ,  23 , three SAR delays  40 ,  42 ,  43 , and three inputs D1, D2, D3 to mux  64 , which operates with clock TSX3. 
     The three FIR filters  17  generate filtered digital values Y1, Y2, Y3 that are input to product derivative correlators  52 - 1 ,  52 - 2 , and  52 - 3  for channels 1, 2, 3 that generate product derivative factors F1, F2, F3, respectively. 
     Channel 2 product derivative correlator  52 - 2  has sample delay  11  delay Y2 before being input to multiplier  142 , where its is multiplied by the sum from adder  140 , which is Y3 delayed by sample delay  148  minus Y1 delayed by sample delay  146 . Averager  54  then generates the average over M analog-input samples to generate product derivative factor F2. Product derivative correlator  52 - 2  generates the function: 
     F2=Average{Y2[n−1]*(Y3 [n−1]−Y1 [n−1])} 
     where n is the sample number, from  1  to N, and the average is over N samples. F2 is the average correlation of the current channel Y2 with its adjacent channels Y1, Y3. F2 is a product derivative correlation factor. 
     In general, for any channel X in an N-channel interleaved ADC, 
     F(X)=Average{Y(X)[n−1]*(Y(X+1)[n−1]−Y(X−1)[n-1])} 
     where X+1 and X−1 are performed modulo N, so that first channel 1 has Y(X−1)=Y(N)=Y3 input to sample delay  146 , which delays by 2 samples (Z −2 ) in product derivative correlator  52 - 1 , while last channel 3 has Y(X+1)=Y(1)=Y1 input directly to adder  140 , bypassing sample delay  148  in product derivative correlator  52 - 3 . When modulo wrapping, the sample number is also increased or decreased by 1. 
     First channel (X=1) product derivative correlator  52 - 1  generates: 
     F(1)=Average{Y(1)[n−1]*(Y(2)[n−1]−Y(N)[n−2])} 
     since X−1 modulo N wraps back to N, the last channel. 
     Last channel (X=N) product derivative correlator  52 - 3  generates: 
     F(N)=Average{Y(N)[n−1]*(Y(1)[n]−Y(N−1)[n−1])} 
     since X+1 modulo N wraps forward to channel 1. 
     product derivative factors F1, F2, F3, F4 are combined and arranged into a matrix F: 
     
       
         
           
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     The correlation matrix is based on the shift matrix C, which is a square N×N matrix with N rows and N columns: 
     
       
         
           
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     The correlation matrix is the transpose of C, C T , divided by the product of the transpose of C, C T , and C, or: 
     
       
         
           
             
               
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     Matrix processor  50  generates the sign bits for channels 2 through N by multiplying matrix F by the correlation matrix: 
     
       
         
           
             
               
                 
                   
                     
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     Since the shift matrix C is a constant that only depends on its size N, the adjacent channel correlation results from product derivative factors [F1, F2, F3, FN] is a variable that is multiplied by a constant to generate the sign bits. Sign bits sign(2), sign(3), . . . sign(N) form a 1-dimensional matrix or a vector, and can be transposed from a vertical 1×N matrix to a horizontal N×1 matrix for easier processing or data storage. 
     At the start of calibration SAR delay  40  is set with a fixed delay value, such as a mid-point value of 10000 . . . 0, while SAR delays  42 ,  43  have bits set and cleared by calibrator  55  during a SAR routine. When sign bit sign(2) generated by matrix processor  50  is high, calibrator  55  sets high the bit in SAR delay  42 , but when sign bit sign(2) is low, calibrator  55  clears low the bit in SAR delay  42 . Likewise, when sign bit sign(3) generated by matrix processor  50  is high, calibrator  55  sets high the bit in SAR delay  43 , but when sign bit sign(3) is low, calibrator  55  clears low the bit in SAR delay  43 . 
       FIGS.  7 A- 7 B  is a flowchart of foreground calibration of N ADC channels interleaved together. This process can be used to initially calibrate the 4-way interleaved ADC of  FIG.  4   , the 3-way interleaved ADC of  FIG.  6   , the 6-way interleaved ADC of  FIG.  11   , the 7-way interleaved ADC of  FIG.  12   , or any arbitrary N-way interleaved ADC, where N is a whole number of at least 3. 
     All bits in all of SAR delay  40 ,  42 ,  43 ,  44  are cleared for initialization for all arbitrary channels J, SAR(J), step  202 . SAR delay  40  for the first channel, SAR(1), is set to the midpoint value of 1000 . . . 0, step  204 . The first channel acts as a fixed timing reference for correlation with the other channels, so SAR(1) remains fixed for the rest of foreground calibration. The other channels have bits in their SAR(J) set or cleared during calibration. 
     The current bit-position Q in SAR delays  42 ,  42 ,  44 , SAR(J), is set to the MSB of the variable delay or variable capacitor, step  206 . This MSB bit Q=MSB is set in SAR(J=2..N) for each of the remaining N-1 channels, step  208 . 
     In  FIG.  7 B , product derivative correlators  52  generate product derivative factors F1, F2, F3, F4, . . . FN that are each the average correlation to adjacent channels over M samples, step  210 . Matrix processor  50  arranges product derivative factors F1, F2, F3, F4, . . . FN into a matrix F and multiples it by the correlation matrix, C T /(C T *C), step  212 . The result from matrix processor  50  is sign vector sign(2 . . . N). This sign vector has one sign bit for each of channels 2 to N. Each sign bit indicates the sign of the correlated delay difference for that channel relative to the first channel. 
     When the sign bit for a channel X is 1, step  214 , then calibrator  55  keeps SAR(Q) high for that channel. The added delay in SAR delay  42 ,  43 ,  44  was not large enough. SAR(Q) is the corresponding bit in SAR delay  42 ,  43 ,  44  for bit-position Q. 
     When the sign bit for a channel X is 0, step  216 , then calibrator  55  clears SAR(Q) to 0 for that channel. The added delay in SAR delay  42 ,  43 ,  44  was too large. 
     When Q has not yet reached the LSB, step  222 , then Q is decremented, step  224 , and the next significant bit Q in SAR delay  42 ,  43 ,  44  is set, step  218 . Next in step  210 , product derivative correlators  52  generate new values of product derivative factors F1, F2, F3, F4, . . . Fn using the new delay values in SAR delay  42 ,  43 ,  44  set by calibrator  55  in step  218 . Matrix processor  50  generates new sign bits, step  212 , and calibrator  55  clears any bits that have zero sign bits, step  216 . 
     This process repeats for successively smaller Q bit-positions and smaller added capacitances in steps  210 - 224  until Q reaches the LSB, step  222 , when calibration ends. 
       FIG.  8    is a graph of reduced spurious tones in a spectrum for an interleaved ADC with channel input delays calibrated by product derivative correlators and matrix processing. Sampling pulse-width mismatches and component mismatches that introduce non-linearities or errors are compensated for by the calibration routine of  FIGS.  7 A- 7 B  adjusting the programmable delays in SAR delay  40 ,  42 ,  43 ,  44  in the interleaved ADC shown in  FIG.  4   . This calibration reduces the amplitude of spurious tones  303 . These spurious tones still occur at integer multiples of Fs/N, K*Fs/N±Fin, where Fs is the sampling frequency (period Ts=1/Fs) and N is the number of channels interleaved together. However, the amplitude of spurious tones  303  are reduced when compared with spurious tones  302  of the prior-art of  FIG.  3   . 
       FIG.  9    shows a backend processor triggering background calibration of an interleaved ADC. Foreground calibration, such as shown in  FIGS.  7 A- 7 B , can be triggered on power-up, initialization, or a reset. This foreground calibration uses product derivative correlators and a matrix processor with calibrator  55  to perform Successive-Approximation searches that load delay values into SAR delay registers  744  in interleaved ADC  730  to compensate for timing skews among ADC channels in interleaved ADC  730 . 
     Temperature and voltage conditions can drift over time. Switches, delays, ADCs, and other components and their errors can be temperature and voltage dependent. The circuitry in interleaved ADC  730  is sensitive to temperature and voltage. Over time, as the system heats up or as the environment changes, temperature and supply-voltage changes may cause increased timing skews in interleaved ADC  730 . As conditions drift, the calibrated delays may need to be updated to compensate for this drift. Background calibration can be triggered periodically to compensate for these drifts. 
     Downstream device  732  could be a baseband modem, a Digital Signal Processor (DSP), a Field-Programmable Logic Array (FPGA), or other device that uses the digital output DOUT from interleaved ADC  730 . Downstream device  732  may include logic that detects when temperature, voltage, or other conditions have changed and trigger interleaved ADC  730  to perform background calibration. Downstream device  732  could have a timer and trigger background calibration after some period of time, such as hourly or daily. Downstream device  732  could detect idle times or times when the analog input AIN has a suitable signal strength and frequency for calibration. Pattern-generation logic integrated with interleaved ADC  730  may also be enabled to generate suitable analog input AIN signals for calibration. 
       FIGS.  10 A- 10 B  show a flowchart of a background calibration process. Background calibration is performed after foreground calibration has completed and calibrated delays are loaded into all of SAR delay  40 ,  42 ,  43 ,  44 , . . . SAR(J) and re-used, step  252 . 
     Product derivative correlators  52  generate product derivative factors F1, F2, F3, F4, . . . FN that are each the average correlation to adjacent channels over M samples, step  260 . Matrix processor  50  arranges product derivative factors F1, F2, F3, F4, . . . FN into a matrix F and multiples it by the correlation matrix, C T /(C T *C), step  262 . The result from matrix processor  50  is sign vector sign(2 . . . N). This sign vector has one sign bit for each of channels 2 to N. 
     When the sign bit for a channel X is 1, step  254 , then calibrator  55  adds one LSB to the delay value in SAR delay  42 ,  43 ,  44  for that channel X. The added delay in SAR delay  42 ,  43 ,  44  was not large enough. 
     When the sign bit for a channel X is 0, step  256 , then calibrator  55  subtracts one LSB from the SAR delay value SAR(X) for that channel. The added delay in SAR delay  42 ,  43 ,  44  was too large. 
     In  FIG.  10 B , the sign vector sign(2. . N) is stored as the old_sign vector old_sign(2. . N), step  258 . Product derivative correlators  52  again generate product derivative factors F1, F2, F3, F4, . . . FN that are each the average correlation to adjacent channels over M samples, step  270 . Matrix processor  50  once again arranges product derivative factors F1, F2, F3, F4, . . . FN into a matrix F and multiples it by the correlation matrix, C T /(C T *C), step  272 . The result from matrix processor  50  is the new sign vector sign(2 . . . N). 
     When the sign bit for a channel X has changed, so that sign(X) is not equal to old_sign(X), step  274 , then the LSB either added in step  254  or subtracted in step  256  is removed, such as by subtracting one LSB or adding one LSB, or by re-loading the prior value in SAR(X) before the last loop. A flip indicator for that channel X is incremented when the sign bit changes. 
     When a flip limit is not yet reached, step  276 , then another loop begins. When the sign bit for a channel X is 1, and the old_sign bit old_sign(X) was also 1, step  264 , then calibrator  55  adds one LSB to the delay value in SAR delay  42 ,  43 ,  44  for that channel X. 
     When the sign bit for a channel X is 0, and the old_sign bit old_sign(X) was also 0, step  266 , then calibrator  55  subtracts one LSB from the SAR delay value SAR(X) for that channel. Then steps  258 ,  270 ,  272 ,  274  are repeated. 
     When a flip limit is reached, step  276 , the background calibration process ends. The flip limit can be when each of the channels has flipped in each of the last FL loops or can require that the majority of the channels have flipped in each of the last FL loop, or to some other endpoint condition. For example, FL could be 20. When the endpoint is reached, each loop iteration successively adds and removes the LSB in each channel&#39;s SAR, indicating that a steady-state has been reached and further iterations do not result in better skew matching. The flip limit FL can be set to as little as 1, or to a larger value for better robustness. 
     The MSB is used for foreground calibration when large changes in delay are desirable to converge on the calibration result more rapidly. However the LSB is used for background calibration, since the drift is not expected to be large, so slower adjustments are more desirable for background calibration. 
       FIG.  11    shows a non-binary number of ADCs interleaved and calibrated with product derivative correlators and a matrix processor. While a binary number of channels is very efficient, there may be a non-binary number of ADC channels N. For example, there may be N=6 ADC channels that generate filtered outputs Y1, Y2, Y3, . . . Y6 from FIR filters  17 . Switches  20 - 26  sample the analog input to ADC  10 - 16 , which are filtered by FIR filters  17  and drive six product derivative correlators  52 . Six product derivative correlators  52  generate product derivative factors F1, F2, F3, F4, F5, F5, F6 that form a matrix that is multiplied by the correlation matrix in matrix processor  50  to generate sign bit vector sign(2..6) that controls calibrator  55  adjusting delay values in SAR delay  40 . 
     For N=6 channels, F is: 
     
       
         
           
             F 
             = 
             
               { 
               
                 
                   
                     
                       E 
                       [ 
                       
                         Y 
                         ⁢ 
                         
                           1 
                           [ 
                           
                             n 
                             - 
                             1 
                           
                           ] 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               Y 
                               ⁢ 
                               
                                 2 
                                 [ 
                                 
                                   n 
                                   - 
                                   1 
                                 
                                 ] 
                               
                             
                             - 
                             
                               Y 
                               ⁢ 
                               
                                 6 
                                 [ 
                                 
                                   n 
                                   - 
                                   2 
                                 
                                 ] 
                               
                             
                           
                           ) 
                         
                       
                       ] 
                     
                   
                 
                 
                   
                     
                       
                         E 
                         [ 
                         
                           Y 
                           ⁢ 
                           
                             2 
                             [ 
                             
                               n 
                               - 
                               1 
                             
                             ] 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 Y 
                                 ⁢ 
                                 
                                   3 
                                   [ 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                   ] 
                                 
                               
                               - 
                               
                                 Y 
                                 ⁢ 
                                 
                                   1 
                                   [ 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                   ] 
                                 
                               
                             
                             ) 
                           
                         
                         ] 
                       
                       ] 
                     
                   
                 
                 
                   
                     
                       
                         E 
                         [ 
                         
                           Y 
                           ⁢ 
                           
                             3 
                             [ 
                             
                               n 
                               - 
                               1 
                             
                             ] 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 Y 
                                 ⁢ 
                                 
                                   4 
                                   [ 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                   ] 
                                 
                               
                               - 
                               
                                 Y 
                                 ⁢ 
                                 
                                   2 
                                   [ 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                   ] 
                                 
                               
                             
                             ) 
                           
                         
                         ] 
                       
                       ] 
                     
                   
                 
                 
                   
                     
                       
                         E 
                         [ 
                         
                           Y 
                           ⁢ 
                           
                             4 
                             [ 
                             
                               n 
                               - 
                               1 
                             
                             ] 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 Y 
                                 ⁢ 
                                 
                                   5 
                                   [ 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                   ] 
                                 
                               
                               - 
                               
                                 Y 
                                 ⁢ 
                                 
                                   3 
                                   [ 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                   ] 
                                 
                               
                             
                             ) 
                           
                         
                         ] 
                       
                       ] 
                     
                   
                 
                 
                   
                     
                       
                         E 
                         [ 
                         
                           Y 
                           ⁢ 
                           
                             5 
                             [ 
                             
                               n 
                               - 
                               1 
                             
                             ] 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 Y 
                                 ⁢ 
                                 
                                   6 
                                   [ 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                   ] 
                                 
                               
                               - 
                               
                                 Y 
                                 ⁢ 
                                 
                                   4 
                                   [ 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                   ] 
                                 
                               
                             
                             ) 
                           
                         
                         ] 
                       
                       ] 
                     
                   
                 
                 
                   
                     
                       
                         E 
                         [ 
                         
                           Y 
                           ⁢ 
                           
                             6 
                             [ 
                             
                               n 
                               - 
                               1 
                             
                             ] 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 Y 
                                 ⁢ 
                                 
                                   1 
                                   [ 
                                   n 
                                   ] 
                                 
                               
                               - 
                               
                                 Y 
                                 ⁢ 
                                 
                                   5 
                                   [ 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                   ] 
                                 
                               
                             
                             ) 
                           
                         
                         ] 
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     The correlation matrix is C T /(C T *C), where C is: 
     
       
         
           
             C 
             = 
             
               [ 
               
                 
                   
                     
                       - 
                       1 
                     
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     
                       - 
                       1 
                     
                   
                 
                 
                   
                     2 
                   
                   
                     
                       - 
                       1 
                     
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                   
                   
                     2 
                   
                   
                     
                       - 
                       1 
                     
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       - 
                       1 
                     
                   
                   
                     2 
                   
                   
                     
                       - 
                       1 
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     
                       - 
                       1 
                     
                   
                   
                     2 
                   
                   
                     
                       - 
                       1 
                     
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     
                       - 
                       1 
                     
                   
                   
                     2 
                   
                 
               
               ] 
             
           
         
       
     
     Other non-binary even values of N could be substituted and the system adjusted as needed. 
       FIG.  12    shows a non-binary odd number of ADCs interleaved and calibrated with product derivative correlators and a matrix processor. In this alternative, there are and odd number N=7 ADC channels that generate filtered outputs Y1, Y2, Y3, . . . Y7. Switches  20 - 27  sample the analog input to ADC  10 - 16 ,  19 , which are filtered by FIR filters  17  and drive seven product derivative correlators  52 . 
     Seven product derivative correlators  52  generate product derivative factors F1, F2, F3, F4, F5, F5, F6, F7 that form a matrix that is multiplied by the correlation matrix in matrix processor  50  to generate sign bit vector sign(2..7) that controls calibrator  55  adjusting delay values in SAR delay  40 . 
     For N=7 channels, F is: 
     
       
         
           
             F 
             = 
             
               { 
               
                 
                   
                     
                       E 
                       [ 
                       
                         Y 
                         ⁢ 
                         
                           1 
                           [ 
                           
                             n 
                             - 
                             1 
                           
                           ] 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               Y 
                               ⁢ 
                               
                                 2 
                                 [ 
                                 
                                   n 
                                   - 
                                   1 
                                 
                                 ] 
                               
                             
                             - 
                             
                               Y 
                               ⁢ 
                               
                                 7 
                                 [ 
                                 
                                   n 
                                   - 
                                   2 
                                 
                                 ] 
                               
                             
                           
                           ) 
                         
                       
                       ] 
                     
                   
                 
                 
                   
                     
                       
                         E 
                         [ 
                         
                           Y 
                           ⁢ 
                           
                             2 
                             [ 
                             
                               n 
                               - 
                               1 
                             
                             ] 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 Y 
                                 ⁢ 
                                 
                                   3 
                                   [ 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                   ] 
                                 
                               
                               - 
                               
                                 Y 
                                 ⁢ 
                                 
                                   1 
                                   [ 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                   ] 
                                 
                               
                             
                             ) 
                           
                         
                         ] 
                       
                       ] 
                     
                   
                 
                 
                   
                     
                       
                         E 
                         [ 
                         
                           Y 
                           ⁢ 
                           
                             3 
                             [ 
                             
                               n 
                               - 
                               1 
                             
                             ] 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 Y 
                                 ⁢ 
                                 
                                   4 
                                   [ 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                   ] 
                                 
                               
                               - 
                               
                                 Y 
                                 ⁢ 
                                 
                                   2 
                                   [ 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                   ] 
                                 
                               
                             
                             ) 
                           
                         
                         ] 
                       
                       ] 
                     
                   
                 
                 
                   
                     
                       
                         E 
                         [ 
                         
                           Y 
                           ⁢ 
                           
                             4 
                             [ 
                             
                               n 
                               - 
                               1 
                             
                             ] 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 Y 
                                 ⁢ 
                                 
                                   5 
                                   [ 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                   ] 
                                 
                               
                               - 
                               
                                 Y 
                                 ⁢ 
                                 
                                   3 
                                   [ 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                   ] 
                                 
                               
                             
                             ) 
                           
                         
                         ] 
                       
                       ] 
                     
                   
                 
                 
                   
                     
                       
                         E 
                         [ 
                         
                           Y 
                           ⁢ 
                           
                             5 
                             [ 
                             
                               n 
                               - 
                               1 
                             
                             ] 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 Y 
                                 ⁢ 
                                 
                                   6 
                                   [ 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                   ] 
                                 
                               
                               - 
                               
                                 Y 
                                 ⁢ 
                                 
                                   4 
                                   [ 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                   ] 
                                 
                               
                             
                             ) 
                           
                         
                         ] 
                       
                       ] 
                     
                   
                 
                 
                   
                     
                       
                         E 
                         [ 
                         
                           Y 
                           ⁢ 
                           
                             6 
                             [ 
                             
                               n 
                               - 
                               1 
                             
                             ] 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 Y 
                                 ⁢ 
                                 
                                   7 
                                   [ 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                   ] 
                                 
                               
                               - 
                               
                                 Y 
                                 ⁢ 
                                 
                                   5 
                                   [ 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                   ] 
                                 
                               
                             
                             ) 
                           
                         
                         ] 
                       
                       ] 
                     
                   
                 
                 
                   
                     
                       
                         E 
                         [ 
                         
                           Y 
                           ⁢ 
                           
                             7 
                             [ 
                             
                               n 
                               - 
                               1 
                             
                             ] 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 Y 
                                 ⁢ 
                                 
                                   1 
                                   [ 
                                   n 
                                   ] 
                                 
                               
                               - 
                               
                                 Y 
                                 ⁢ 
                                 
                                   6 
                                   [ 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                   ] 
                                 
                               
                             
                             ) 
                           
                         
                         ] 
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     the correlation matrix is C T /(C T *C), where C is: 
     
       
         
           
             C 
             = 
             
               [ 
               
                 
                   
                     
                       - 
                       1 
                     
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     
                       - 
                       1 
                     
                   
                 
                 
                   
                     2 
                   
                   
                     
                       - 
                       1 
                     
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                   
                   
                     2 
                   
                   
                     
                       - 
                       1 
                     
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       - 
                       1 
                     
                   
                   
                     2 
                   
                   
                     
                       - 
                       1 
                     
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     
                       - 
                       1 
                     
                   
                   
                     2 
                   
                   
                     
                       - 
                       1 
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     
                       - 
                       1 
                     
                   
                   
                     2 
                   
                   
                     
                       - 
                       1 
                     
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     
                       - 
                       1 
                     
                   
                   
                     2 
                   
                 
               
               ] 
             
           
         
       
     
     Other odd non-binary values of N could be substituted and the system adjusted as needed. 
     Mathematical Derivation of Matrix Formulas 
     Shown below is a mathematical derivation of the formulas implemented by product derivative correlators  52  and matrix processor  50  to generate the sign bit vector used for the SAR routine decisions. 
     An autocorrelation function is defined as: 
     R=E[Yi[n] Yj[n]] 
     Herein ΔT i  denotes the timing skew mismatch in a channel i relative to a reference channel. If ΔT 1 , ΔT 2 , ΔT 3 , ΔT 4  are small compared to Ts, then 1 st  order Taylor Series can be approximated as 
     
       
         
           
             
               E 
               [ 
               
                 Y 
                 ⁢ 
                 
                   1 
                   [ 
                   n 
                   ] 
                 
                 ⁢ 
                 Y 
                 ⁢ 
                 
                   
                     1 
                     ′ 
                   
                   [ 
                   n 
                   ] 
                 
               
               ] 
             
             = 
             
               E 
               [ 
               
                 Y 
                 ⁢ 
                 
                   1 
                   [ 
                   
                     nTs 
                     + 
                     
                       Δ 
                       ⁢ 
                       
                         T 
                         1 
                       
                     
                   
                   ] 
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       Y 
                       ⁢ 
                       
                         2 
                         [ 
                         
                           nTs 
                           + 
                           
                             Δ 
                             ⁢ 
                             
                               T 
                               2 
                             
                           
                         
                         ] 
                       
                     
                     - 
                     
                       Y 
                       ⁢ 
                       
                         4 
                         [ 
                         
                           
                             
                               ( 
                               
                                 n 
                                 - 
                                 1 
                               
                               ) 
                             
                             ⁢ 
                             nTs 
                           
                           + 
                           
                             Δ 
                             ⁢ 
                             
                               T 
                               4 
                             
                           
                         
                         ] 
                       
                     
                   
                   ) 
                 
               
               ] 
             
           
         
       
       
         
           
             	 
             
               ≈ 
               
                 
                   
                     R 
                     Y 
                   
                   ( 
                   
                     Ts 
                     + 
                     
                       Δ 
                       ⁢ 
                       
                         T 
                         2 
                       
                     
                     - 
                     
                       Δ 
                       ⁢ 
                       
                         T 
                         1 
                       
                     
                   
                   ) 
                 
                 - 
                 
                   
                     R 
                     Y 
                   
                   ( 
                   
                     Ts 
                     - 
                     
                       Δ 
                       ⁢ 
                       
                         T 
                         4 
                       
                     
                     + 
                     
                       Δ 
                       ⁢ 
                       
                         T 
                         1 
                       
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             	 
             
               ≈ 
               
                 
                   
                     ( 
                     
                       
                         Δ 
                         ⁢ 
                         
                           T 
                           2 
                         
                       
                       - 
                       
                         Δ 
                         ⁢ 
                         
                           T 
                           1 
                         
                       
                     
                     ) 
                   
                   ⁢ 
                   
                     
                       ∂ 
                         
                       
                         R 
                         Y 
                       
                     
                     
                       ∂ 
                       τ 
                     
                   
                 
                 - 
                 
                   
                     ( 
                     
                       
                         - 
                         Δ 
                         ⁢ 
                         
                           T 
                           4 
                         
                       
                       + 
                       
                         Δ 
                         ⁢ 
                         
                           T 
                           1 
                         
                       
                     
                     ) 
                   
                   ⁢ 
                   
                     
                       ∂ 
                       
                         R 
                         Y 
                       
                     
                     
                       ∂ 
                       τ 
                     
                   
                 
               
             
           
         
       
       
         
           
             	 
             
               = 
               
                 
                   ( 
                   
                     
                       - 
                       2 
                       ⁢ 
                       Δ 
                       ⁢ 
                       
                         T 
                         1 
                       
                     
                     + 
                     
                       Δ 
                       ⁢ 
                       
                         T 
                         2 
                       
                     
                     + 
                     
                       Δ 
                       ⁢ 
                       
                         T 
                         4 
                       
                     
                   
                   ) 
                 
                 ⁢ 
                 
                   
                     ∂ 
                     
                       R 
                       Y 
                     
                   
                   
                     ∂ 
                     τ 
                   
                 
               
             
           
         
       
       
         
           
             	 
             
               = 
               
                 - 
                 
                   
                     
                       
                         
                           
                             [ 
                             2 
                           
                         
                         
                           
                             - 
                             1 
                           
                         
                         
                           0 
                         
                         
                           
                             
                               - 
                               1 
                             
                             ] 
                           
                         
                       
                     
                     [ 
                     
                       
                         
                           
                             Δ 
                             ⁢ 
                             
                               T 
                               1 
                             
                           
                         
                         
                           
                             Δ 
                             ⁢ 
                             
                               T 
                               2 
                             
                           
                         
                         
                           
                             Δ 
                             ⁢ 
                             
                               T 
                               3 
                             
                           
                         
                         
                           
                             Δ 
                             ⁢ 
                             
                               T 
                               4 
                             
                           
                         
                       
                     
                     ] 
                   
                   T 
                 
                 ⁢ 
                 
                   
                     ∂ 
                     
                       R 
                       Y 
                     
                   
                   
                     ∂ 
                     τ 
                   
                 
               
             
           
         
       
     
     For 4 channel, we can generalize the following product derivative vectors 
     
       
         
           
             F 
             = 
             
               { 
               
                 
                   
                     
                       E 
                       [ 
                       
                         Y 
                         ⁢ 
                         
                           1 
                           [ 
                           n 
                           ] 
                         
                         ⁢ 
                         Y 
                         ⁢ 
                         
                           
                             1 
                             ′ 
                           
                           [ 
                           n 
                           ] 
                         
                       
                       ] 
                     
                   
                 
                 
                   
                     
                       E 
                       [ 
                       
                         Y 
                         ⁢ 
                         
                           2 
                           [ 
                           n 
                           ] 
                         
                         ⁢ 
                         Y 
                         ⁢ 
                         
                           
                             2 
                             ′ 
                           
                           [ 
                           n 
                           ] 
                         
                       
                       ] 
                     
                   
                 
                 
                   
                     
                       E 
                       [ 
                       
                         Y 
                         ⁢ 
                         
                           3 
                           [ 
                           n 
                           ] 
                         
                         ⁢ 
                         Y 
                         ⁢ 
                         
                           
                             3 
                             ′ 
                           
                           [ 
                           n 
                           ] 
                         
                       
                       ] 
                     
                   
                 
                 
                   
                     
                       E 
                       [ 
                       
                         Y 
                         ⁢ 
                         
                           4 
                           [ 
                           n 
                           ] 
                         
                         ⁢ 
                         Y 
                         ⁢ 
                         
                           
                             4 
                             ′ 
                           
                           [ 
                           n 
                           ] 
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     An Matrix C for timing skew error is generalized as 
     
       
         
           
             C 
             = 
             
               [ 
               
                 
                   
                     2 
                   
                   
                     
                       - 
                       1 
                     
                   
                   
                     0 
                   
                   
                     
                       - 
                       1 
                     
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                   
                   
                     2 
                   
                   
                     
                       - 
                       1 
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       - 
                       1 
                     
                   
                   
                     2 
                   
                   
                     
                       - 
                       1 
                     
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                   
                   
                     0 
                   
                   
                     
                       - 
                       1 
                     
                   
                   
                     2 
                   
                 
               
               ] 
             
           
         
       
     
     Therefore, all related to this equation 
     
       
         
           
             F 
             = 
             
               - 
               
                 
                   C 
                   [ 
                   
                     
                       
                         
                           Δ 
                           ⁢ 
                           
                             T 
                             1 
                           
                         
                       
                       
                         
                           Δ 
                           ⁢ 
                           
                             T 
                             2 
                           
                         
                       
                       
                         
                           Δ 
                           ⁢ 
                           
                             T 
                             3 
                           
                         
                       
                       
                         
                           Δ 
                           ⁢ 
                           
                             T 
                             4 
                           
                         
                       
                     
                   
                   ] 
                 
                 T 
               
               ⁢ 
               
                 
                   ∂ 
                   
                     R 
                     Y 
                   
                 
                 
                   ∂ 
                   τ 
                 
               
             
           
         
       
     
     To solve this equation, we need to fix one timing skew error to be zero by removing first column of C 
     
       
         
           
             C 
             = 
             
               [ 
               
                 
                   
                     
                       - 
                       1 
                     
                   
                   
                     0 
                   
                   
                     
                       - 
                       1 
                     
                   
                 
                 
                   
                     2 
                   
                   
                     
                       - 
                       1 
                     
                   
                   
                     0 
                   
                 
                 
                   
                     
                       - 
                       1 
                     
                   
                   
                     2 
                   
                   
                     
                       - 
                       1 
                     
                   
                 
                 
                   
                     0 
                   
                   
                     
                       - 
                       1 
                     
                   
                   
                     2 
                   
                 
               
               ] 
             
           
         
       
     
     Without loss of generality, vector ΔT can be resolved as 
     
       
         
           
             
               
                 [ 
                 
                   
                     
                       
                         Δ 
                         ⁢ 
                         
                           T 
                           2 
                         
                       
                     
                     
                       
                         Δ 
                         ⁢ 
                         
                           T 
                           3 
                         
                       
                     
                     
                       
                         Δ 
                         ⁢ 
                         
                           T 
                           4 
                         
                       
                     
                   
                 
                 ] 
               
               T 
             
             = 
             
               - 
               
                 1 
                 
                   
                     ∂ 
                     
                       R 
                       Y 
                     
                   
                   
                     ∂ 
                     τ 
                   
                 
               
               ⁢ 
               
                 
                   C 
                   T 
                 
                 
                   
                     C 
                     T 
                   
                   ⁢ 
                   C 
                 
               
               ⁢ 
               F 
             
           
         
       
     
     The sign value of the above equation provides the calibration direction to converge min. vector ΔT 
     
       
         
           
             
               sign 
               ⁡ 
               ( 
               
                 
                   [ 
                   
                     
                       
                         
                           Δ 
                           ⁢ 
                           
                             T 
                             2 
                           
                         
                       
                       
                         
                           Δ 
                           ⁢ 
                           
                             T 
                             3 
                           
                         
                       
                       
                         
                           Δ 
                           ⁢ 
                           
                             T 
                             4 
                           
                         
                       
                     
                   
                   ] 
                 
                 T 
               
               ) 
             
             = 
             
               - 
               
                 sign 
                 ( 
                 
                   
                     1 
                     
                       
                         ∂ 
                         
                           R 
                           Y 
                         
                       
                       
                         ∂ 
                         τ 
                       
                     
                   
                   ⁢ 
                   
                     
                       C 
                       T 
                     
                     
                       
                         C 
                         T 
                       
                       ⁢ 
                       C 
                     
                   
                   ⁢ 
                   F 
                 
                 ) 
               
             
           
         
       
     
     Since derivative of autocorrelation has fixed polarity over Nyquist bandwidth, then we can simply the calibration equation into 
     
       
         
           
             
               sign 
               ⁡ 
               ( 
               
                 
                   [ 
                   
                     
                       
                         
                           Δ 
                           ⁢ 
                           
                             T 
                             2 
                           
                         
                       
                       
                         
                           Δ 
                           ⁢ 
                           
                             T 
                             3 
                           
                         
                       
                       
                         
                           Δ 
                           ⁢ 
                           
                             T 
                             4 
                           
                         
                       
                     
                   
                   ] 
                 
                 T 
               
               ) 
             
             = 
             
               - 
               
                 sign 
                 ( 
                 
                   
                     
                       C 
                       T 
                     
                     
                       
                         C 
                         T 
                       
                       ⁢ 
                       C 
                     
                   
                   ⁢ 
                   F 
                 
                 ) 
               
             
           
         
       
     
     Theoretical Model of Interleaved ADC 
       FIG.  13    shows a model of an interleaved ADC. An analog input X(S) is sampled by first sampler  152  as function H 1 (S), which is multiplied by first multiplier  158  with first delay  156  D 1 (T) to generate X S1 (S) that is input to first ADC  154 . In the second channel, analog input X(S) is also sampled by second sampler  162  as function H 2 (S), which is multiplied by second multiplier  168  with second delay  166  D 2 (T) to generate X S2 (S) that is input to second ADC  164 . The digital outputs of first ADC  154  and second ADC  164  are muxed together by mux  150  to generate output Y(S). 
     The following equations and discussion are presented as a theoretical background and is not intended to limit the invention or its claims but are presented for edification. 
     Assuming a single-pole sampling system, sampled analog signal (odd) and (even) are given by: 
     
       
         
           
             	 
             
               
                 
                   X 
                   
                     S 
                     ⁢ 
                     1 
                       
                   
                 
                 ( 
                 s 
                 ) 
               
               = 
               
                 
                   
                     H 
                     1 
                   
                   ( 
                   s 
                   ) 
                 
                 × 
                 
                   [ 
                   
                     
                       1 
                       
                         T 
                         
                           ? 
                         
                       
                     
                     ⁢ 
                     
                       
                         ∑ 
                         ∞ 
                       
                       
                         
                           ? 
                         
                           
                         
                           X 
                           ( 
                           
                             j 
                             ( 
                             
                               ω 
                               - 
                               
                                 
                                   2 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   k 
                                 
                                 
                                   T 
                                   
                                     ? 
                                   
                                 
                               
                             
                             ) 
                           
                           ) 
                         
                       
                     
                   
                   ] 
                 
               
             
           
         
       
       
         
           
             	 
             
               
                 
                   X 
                   
                     S 
                     ⁢ 
                     2 
                   
                 
                 ( 
                 s 
                 ) 
               
               = 
               
                 
                   
                     
                       H 
                       2 
                     
                     ( 
                     s 
                     ) 
                   
                   · 
                   
                     e 
                     
                       - 
                       j 
                       ⁢ 
                       2 
                       ⁢ 
                       π 
                       ⁢ 
                       k 
                     
                   
                 
                 
                   ? 
                 
                 × 
                 
                   [ 
                   
                     
                       1 
                       
                         T 
                         
                           ? 
                         
                       
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           
                             - 
                             ∞ 
                           
                         
                         ∞ 
                       
                       
                         X 
                         ( 
                         
                           j 
                           ( 
                           
                             ω 
                             - 
                             
                               
                                 2 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 k 
                               
                               
                                 T 
                                 s 
                               
                             
                           
                           ) 
                         
                         ) 
                       
                     
                   
                   ] 
                 
                 ⁢ 
                 
                   e 
                   
                     - 
                     jk 
                   
                 
                 
                   ? 
                 
               
             
           
         
       
       
         
           
             
               ? 
             
             indicates text missing or illegible when filed 
           
         
       
     
     Therefore, digital output Y(s) is given by: 
     
       
         
           
             	 
             
               
                 Y 
                 ⁡ 
                 ( 
                 s 
                 ) 
               
               = 
               
                 
                   
                     X 
                     
                       S 
                       ⁢ 
                       1 
                     
                   
                   ( 
                   s 
                   ) 
                 
                 + 
                 
                   
                     X 
                     
                       S 
                       ⁢ 
                       2 
                     
                   
                   ( 
                   s 
                   ) 
                 
               
             
           
         
       
       
         
           
             	 
             
               = 
               
                 
                   [ 
                   
                     
                       1 
                       
                         T 
                         
                           ? 
                         
                       
                     
                     ⁢ 
                     
                       
                         ∑ 
                         ∞ 
                       
                         
                       
                         
                           ? 
                         
                         
                           X 
                           ( 
                           
                             j 
                             ( 
                             
                               ω 
                               - 
                               
                                 
                                   2 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   k 
                                 
                                 
                                   T 
                                   
                                     ? 
                                   
                                 
                               
                             
                             ) 
                           
                           ) 
                         
                       
                     
                   
                   ] 
                 
                 × 
                 
                   [ 
                   
                     
                       
                         H 
                         1 
                       
                       ( 
                       s 
                       ) 
                     
                     + 
                     
                       
                         e 
                         
                           - 
                           jk 
                           ⁢ 
                           π 
                         
                       
                       ⁢ 
                       
                         
                           
                             H 
                             2 
                           
                           ( 
                           s 
                           ) 
                         
                         · 
                         
                           e 
                           
                             - 
                             j 
                             ⁢ 
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             k 
                           
                         
                       
                       
                         ? 
                       
                     
                   
                   ] 
                 
               
             
           
         
       
       
         
           
             
               ? 
             
             indicates text missing or illegible when filed 
           
         
       
     
     Ideally, if H 1 (s)=H 2 (s)=H(s) and ΔT=0, then 
     
       
         
           
             
               
                 Y 
                 ideal 
               
               ( 
               s 
               ) 
             
             = 
             
               
                 
                   1 
                   
                     T 
                     S 
                     ′ 
                   
                 
                 ⁢ 
                 
                   
                     ∑ 
                     
                       n 
                       = 
                       
                         - 
                         ∞ 
                       
                     
                     ∞ 
                   
                     
                   
                     
                       X 
                       ( 
                       
                         j 
                         ( 
                         
                           ω 
                           - 
                           
                             
                               2 
                               ⁢ 
                               π 
                               ⁢ 
                               n 
                             
                             
                               T 
                               S 
                               ′ 
                             
                           
                         
                         ) 
                       
                       ) 
                     
                     ⁢ 
                     where 
                     ⁢ 
                         
                     
                       T 
                       S 
                       ′ 
                     
                   
                 
               
               = 
               
                 
                   T 
                   S 
                 
                 2 
               
             
           
         
       
     
     Odd image replica from H 2 (s) is well-cancelled with H 1 (s), in other words, bandwidth of X(s) could be double due to ping-pong sampling system as expected. However, error image (i.e. uncorrected odd image replica) exists in reality due to H 1 (s) H 2 (s) if ΔT≠0. 
     Herein, we would like to approximate the error due to finite bandwidth mismatch and timing skew as below: 
     
       
         
           
             	 
             
               
                 Y 
                 ⁡ 
                 ( 
                 s 
                 ) 
               
               = 
               
                 
                   
                     Y 
                     ideal 
                   
                   ( 
                   s 
                   ) 
                 
                 + 
                 
                   
                     E 
                     Y 
                   
                   ( 
                   s 
                   ) 
                 
               
             
           
         
       
       
         
           
             	 
             
               = 
               
                 
                   
                     Y 
                     ideal 
                   
                   ( 
                   s 
                   ) 
                 
                 + 
                 
                   
                     
                       Y 
                       ideal 
                     
                     ( 
                     s 
                     ) 
                   
                   × 
                   
                     [ 
                     
                       
                         
                           H 
                           1 
                         
                         ( 
                         s 
                         ) 
                       
                       - 
                       
                         
                           
                             
                               H 
                               2 
                             
                             ( 
                             s 
                             ) 
                           
                           · 
                           
                             e 
                             
                               - 
                               j 
                               ⁢ 
                               2 
                               ⁢ 
                               π 
                               ⁢ 
                               m 
                             
                           
                         
                         
                           ? 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
       
         
           
             
               ? 
             
             indicates text missing or illegible when filed 
           
         
       
     
     where m is odd #. 
     This provides the basis analysis on the effect of “dynamic error” in the proposed ADC system. Our target is to evaluate the frequency response of E(s), basically, exponential term can be generalized into: 
     
       
         
           
             
               e 
               
                 - 
                 j 
                 ⁢ 
                 2 
                 ⁢ 
                 π 
                 ⁢ 
                 
                   
                     Δ 
                     ⁢ 
                     T 
                   
                   
                     T 
                     z 
                   
                 
               
             
             
               → 
               &#34;\[Rule]&#34; 
             
             
               e 
               
                 - 
                 s 
                 ⁢ 
                 Δ 
                 ⁢ 
                 T 
               
             
           
         
       
     
     Consequently, equivalent frequency response of two sampling systems converges into 
     E(s)=H 1 ′(s)−H 2 ′(s) 
     Consider the case in ping-pong sampling with mismatches, we have the following: 
     
       
         
           
             
               
                 H 
                 1 
                 ′ 
               
               ( 
               s 
               ) 
             
             = 
             
               1 
               
                 1 
                 + 
                 
                   s 
                   
                     ω 
                     n 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 H 
                 2 
                 ′ 
               
               ( 
               s 
               ) 
             
             = 
             
               
                 e 
                 
                   - 
                   s 
                   ⁢ 
                   Δ 
                   ⁢ 
                   T 
                 
               
               
                 1 
                 + 
                 
                   s 
                   
                     
                       ω 
                       n 
                     
                     + 
                     Δω 
                   
                 
               
             
           
         
       
     
     where Δt denotes the timing skew mismatch in channel 2 relative to channel 1, Δω denotes the finite bandwidth mismatch in channel 2 relative to channel 1. 
     Equivalently, it becomes: 
     
       
         
           
             
               
                 H 
                 1 
                 ′ 
               
               ( 
               s 
               ) 
             
             = 
             
               1 
               
                 1 
                 + 
                 
                   s 
                   
                     ω 
                     n 
                   
                 
               
             
           
         
       
       
         
           and 
         
       
       
         
           
             
               
                 H 
                 2 
                 ′ 
               
               ( 
               s 
               ) 
             
             = 
             
               
                 e 
                 
                   - 
                   s 
                   ⁢ 
                   Δ 
                   ⁢ 
                   T 
                 
               
               
                 1 
                 + 
                 
                   s 
                   
                     
                       ω 
                       o 
                     
                     ( 
                     
                       1 
                       + 
                       
                         Δω 
                         
                           ω 
                           n 
                         
                       
                     
                     ) 
                   
                 
               
             
           
         
       
     
     Herein, we exclude the effect of gain mismatches on H 1 ′(s) and H 2 ′(s) because it is not input freq. dependent if cross products terms are negligible compared to fundamental terms. 
     Consider the “Dynamic Error” E(s), i.e. 
     E(s)=H 1 ′(s)−H 2 ′(s) 
     Equivalently, neglecting cross product terms, it becomes: 
     
       
         
           
             
               E 
               ⁡ 
               ( 
               s 
               ) 
             
             ≈ 
             
               - 
               
                 
                   ( 
                   
                     
                       ω 
                       o 
                     
                     
                       s 
                       + 
                       
                         ω 
                         n 
                       
                     
                   
                   ) 
                 
                 [ 
                 
                   
                     
                       s 
                       
                         s 
                         + 
                         
                           ω 
                           n 
                         
                       
                     
                     ⁢ 
                     
                       Δω 
                       
                         ω 
                         n 
                       
                     
                   
                   - 
                   
                     
                       ( 
                       
                         1 
                         + 
                         
                           Δω 
                           
                             ω 
                             n 
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     s 
                     ⁢ 
                     Δ 
                     ⁢ 
                     T 
                   
                 
                 ] 
               
             
           
         
       
       
         
           
             ≈ 
             
               - 
               
                 
                   ( 
                   
                     1 
                     
                       1 
                       + 
                       
                         s 
                         
                           ω 
                           o 
                         
                       
                     
                   
                   ) 
                 
                 [ 
                 
                   s 
                   ⁢ 
                   
                     
                       
                         
                           
                             
                               ( 
                               
                                 1 
                                 
                                   1 
                                   + 
                                   
                                     s 
                                     
                                       ω 
                                       o 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                           
                             
                               
                                 ( 
                                 
                                   1 
                                   
                                     ω 
                                     o 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   Δω 
                                   
                                     ω 
                                     o 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                       ︸ 
                     
                     
                       
                         E 
                         bw 
                       
                       ( 
                       s 
                       ) 
                     
                   
                   ⁢ 
                   
                     
                       
                         
                           
                             - 
                           
                           
                             
                               s 
                               ⁢ 
                               Δ 
                               ⁢ 
                               T 
                             
                           
                         
                       
                       ︸ 
                     
                     
                       
                         E 
                         sk 
                       
                       ( 
                       s 
                       ) 
                     
                   
                 
                 ] 
               
             
           
         
       
     
     E bw (s) and E sk (s) account the effect of finite bandwidth and timing-skew mismatches, respectively. 
     Clearly, if both Δω=0 and ΔT=0, dynamic errors caused by finite bandwidth mismatch and timing skew is zero, respectively. 
     However, dynamic error always exists and therefore calibration is required to fulfill the target performance. As a result, the time derivatives for the dynamic error is of interest in our case. Consider the following: 
     
       
         
           
             	 
             
               
                 
                   E 
                   
                     ? 
                   
                   
                     ( 
                     s 
                     ) 
                   
                 
                 ≈ 
                 
                   
                     s 
                     ⁢ 
                     Δ 
                     ⁢ 
                     t 
                   
                   
                     1 
                     + 
                     
                       
                         ? 
                       
                       
                         ω 
                         o 
                       
                     
                   
                 
               
               = 
               
                 
                   [ 
                   
                     s 
                     - 
                     
                       s 
                       ( 
                       
                         
                           ? 
                         
                         
                           ω 
                           o 
                         
                       
                       ) 
                     
                     + 
                     
                       
                         s 
                         ( 
                         
                           
                             ? 
                           
                           
                             ω 
                             o 
                           
                         
                         ) 
                       
                       2 
                     
                     - 
                     
                       
                         s 
                         ( 
                         
                           
                             ? 
                           
                           
                             ω 
                             o 
                           
                         
                         ) 
                       
                       3 
                     
                     + 
                     … 
                   
                       
                   ] 
                 
                 ⁢ 
                 Δ 
                 ⁢ 
                 T 
               
             
           
         
       
       
         
           
             	 
             
               
                 
                   E 
                   bw 
                 
                 ( 
                 s 
                 ) 
               
               ≈ 
               
                 - 
                 
                   
                     s 
                     ( 
                     
                       1 
                       
                         1 
                         + 
                         
                           s 
                           
                             ω 
                             o 
                           
                         
                       
                     
                     ) 
                   
                   2 
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       Δ 
                       ⁢ 
                       ω 
                     
                     
                       ω 
                       o 
                     
                   
                   ) 
                 
                 ⁢ 
                 
                   ( 
                   
                     1 
                     
                       ω 
                       o 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             	 
             
               = 
               
                 
                   - 
                   [ 
                   
                     
                       ( 
                       
                         s 
                         
                           ω 
                           o 
                         
                       
                       ) 
                     
                     - 
                     
                       2 
                       ⁢ 
                       
                         
                           ( 
                           
                             s 
                             
                               ω 
                               o 
                             
                           
                           ) 
                         
                         2 
                       
                     
                     + 
                     
                       3 
                       ⁢ 
                       
                         
                           ( 
                           
                             s 
                             
                               ω 
                               o 
                             
                           
                           ) 
                         
                         3 
                       
                     
                     - 
                     
                       4 
                       ⁢ 
                       
                         
                           ( 
                           
                             s 
                             
                               ω 
                               o 
                             
                           
                           ) 
                         
                         4 
                       
                     
                     + 
                     … 
                   
                       
                   ] 
                 
                 ⁢ 
                 
                   ( 
                   
                     Δω 
                     
                       ω 
                       o 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               ? 
             
             indicates text missing or illegible when filed 
           
         
       
     
     By taking “Inverse Laplace Transform” back to time domain, we have: 
     
       
         
           
             	 
             
               
                 e 
                 
                   ? 
                 
                 
                   ( 
                   t 
                   ) 
                 
               
               ≈ 
               
                 
                   ( 
                   
                     
                       
                         ∂ 
                         Y 
                       
                       
                         ∂ 
                         t 
                       
                     
                     - 
                     
                       
                         1 
                         
                           ω 
                           o 
                         
                       
                       ⁢ 
                       
                         
                           
                             ∂ 
                             2 
                           
                           Y 
                         
                         
                           ∂ 
                           
                             ? 
                           
                         
                       
                     
                     + 
                     
                       
                         1 
                         
                           ω 
                           
                             ? 
                           
                         
                       
                       ⁢ 
                       
                         
                           
                             ∂ 
                             
                               ? 
                             
                           
                           ⁢ 
                           Y 
                         
                         
                           ∂ 
                           
                             ? 
                           
                         
                       
                     
                     - 
                     
                       
                         1 
                         
                           ω 
                           
                             ? 
                           
                         
                       
                       ⁢ 
                       
                         
                           
                             ∂ 
                             
                               ? 
                             
                           
                           ⁢ 
                           Y 
                         
                         
                           ∂ 
                           
                             ? 
                           
                         
                       
                     
                     + 
                     … 
                   
                       
                   ) 
                 
                 ⁢ 
                 Δ 
                 ⁢ 
                 T 
               
             
           
         
       
       
         
           
             	 
             
               
                 
                   e 
                   bw 
                 
                 ( 
                 t 
                 ) 
               
               ≈ 
               
                 - 
                 
                   ( 
                   
                     
                       
                         1 
                         
                           ω 
                           o 
                         
                       
                       ⁢ 
                       
                         
                           ∂ 
                           Y 
                         
                         
                           ∂ 
                           t 
                         
                       
                     
                     - 
                     
                       
                         2 
                         
                           ω 
                           o 
                           2 
                         
                       
                       ⁢ 
                       
                         
                           
                             ∂ 
                             2 
                           
                           Y 
                         
                         
                           ∂ 
                           
                             t 
                             2 
                           
                         
                       
                     
                     + 
                     
                       
                         3 
                         
                           ω 
                           o 
                           3 
                         
                       
                       ⁢ 
                       
                         
                           
                             ∂ 
                             3 
                           
                           Y 
                         
                         
                           ∂ 
                           
                             t 
                             3 
                           
                         
                       
                     
                     - 
                     
                       
                         4 
                         
                           ω 
                           o 
                           4 
                         
                       
                       ⁢ 
                       
                         
                           
                             ∂ 
                             4 
                           
                           Y 
                         
                         
                           ∂ 
                           
                             t 
                             4 
                           
                         
                       
                     
                     + 
                     … 
                   
                       
                   ) 
                 
                 ⁢ 
                 
                   ( 
                   
                     Δω 
                     
                       ω 
                       o 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               ? 
             
             indicates text missing or illegible when filed 
           
         
       
     
     From equations above, Y(s)=Y ideal (s)·H(s)=X(s)·H(s) (i.e. filtered analog input signal). 
     Since H(s) is a desired response of loop filter common to ping-pong channels, the above equation is also identical to: 
     
       
         
           
             	 
             
               
                 
                   e 
                   sk 
                 
                 ( 
                 t 
                 ) 
               
               ≈ 
               
                 
                   ( 
                   
                     
                       
                         ∂ 
                         x 
                       
                       
                         ∂ 
                         t 
                       
                     
                     - 
                     
                       
                         1 
                         
                           ω 
                           o 
                         
                       
                       ⁢ 
                       
                         
                           
                             ∂ 
                             2 
                           
                           x 
                         
                         
                           ∂ 
                           
                             t 
                             2 
                           
                         
                       
                     
                     + 
                     
                       
                         1 
                         
                           ω 
                           o 
                           2 
                         
                       
                       ⁢ 
                       
                         
                           
                             ∂ 
                             3 
                           
                           x 
                         
                         
                           ∂ 
                           
                             t 
                             3 
                           
                         
                       
                     
                     - 
                     
                       
                         1 
                         
                           ω 
                           o 
                           3 
                         
                       
                       ⁢ 
                       
                         
                           
                             ∂ 
                             4 
                           
                           x 
                         
                         
                           ∂ 
                           
                             t 
                             4 
                           
                         
                       
                     
                     + 
                     … 
                   
                        
                   ) 
                 
                 ⁢ 
                 Δ 
                 ⁢ 
                 T 
               
             
           
         
       
       
         
           
             
               
                 e 
                 bw 
               
               ( 
               t 
               ) 
             
             ≈ 
             
               - 
               
                 ( 
                 
                   
                     
                       1 
                       
                         ω 
                         o 
                       
                     
                     ⁢ 
                     
                       
                         ∂ 
                         x 
                       
                       
                         ∂ 
                         t 
                       
                     
                   
                   - 
                   
                     
                       2 
                       
                         
                           ω 
                           o 
                         
                         
                           ? 
                         
                       
                     
                     ⁢ 
                     
                       
                         
                           ∂ 
                           
                             ? 
                           
                         
                         ⁢ 
                         x 
                       
                       
                         
                           ∂ 
                           t 
                         
                         
                           ? 
                         
                       
                     
                   
                   + 
                   
                     
                       3 
                       
                         
                           ω 
                           o 
                         
                         
                           ? 
                         
                       
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ? 
                         
                       
                       
                         
                           ∂ 
                           t 
                         
                         
                           ? 
                         
                       
                     
                   
                   - 
                   
                     
                       4 
                       
                         
                           ω 
                           o 
                         
                         
                           ? 
                         
                       
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           ? 
                         
                       
                       
                         
                           ∂ 
                           t 
                         
                         
                           ? 
                         
                       
                     
                   
                   + 
                   … 
                 
                     
                 ) 
               
               ⁢ 
               
                 ( 
                 
                   Δω 
                   
                     ω 
                     o 
                   
                 
                 ) 
               
             
           
         
       
       
         
           
             
               ? 
             
             indicates text missing or illegible when filed 
           
         
       
     
     Notice that the effect of H(s) introduces finite gain error and phase shift according to input signal freq. that doesn&#39;t contribute odd image replica but the mismatch on two sampling systems. 
     In time domain, an earlier equation is converged linearly into: 
     y(t)=y ideal (t)+e(t)=x(t)+e bw (t)+e sk (t) 
     Consider the absolute dynamic error signal e(t), 
     
       
         
           
             
               e 
               ⁡ 
               ( 
               t 
               ) 
             
             ≈ 
             
               - 
               [ 
               
                 
                   
                     ( 
                     
                       
                         Δ 
                         ⁢ 
                         T 
                       
                       + 
                       
                         
                           τ 
                           SAMP 
                         
                         ⁢ 
                         
                           Δω 
                           
                             ω 
                             o 
                           
                         
                       
                     
                     ) 
                   
                   ⁢ 
                   
                     
                       ∂ 
                       x 
                     
                     
                       ∂ 
                       t 
                     
                   
                 
                 - 
                 
                   
                     ( 
                     
                       
                         Δ 
                         ⁢ 
                         T 
                       
                       + 
                       
                         2 
                         ⁢ 
                         
                           τ 
                           SAMP 
                         
                         ⁢ 
                         
                           Δω 
                           
                             ω 
                             o 
                           
                         
                       
                     
                     ) 
                   
                   ⁢ 
                   
                     τ 
                     SAMP 
                   
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     For instance, if ΔT=+/−1 ps, τ SAMP =1/(2*π*5G)=32 ps and Δω/ω o =1%, first order error term is bounded by +/−1.064 ps (in modern multi-GS/s ADC). 
     From the equation above, both ΔT and Δω/ω o  are somewhat related to signal/clock/component mismatches that could be reduced by properly layout and physical component sizing whereas τ SAMP  is circuit design parameter (PVT var.). 
     To mitigate the effect of dynamic errors, linearity performance (interleaving spur) can be reduced by design of a wider sampling bandwidth (lower τ SAMP ) that tradeoff fundamentally with noise, power and other circuit design physical constraints. 
     From the above equation, it can be generalized into a Taylor Series Expansion error signal: 
     
       
         
           
             	 
             
               
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     Consider a single carrier signal x(t)=Asin(ωt), odd and even time derivatives are given by: 
     
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       x 
                     
                     
                       ∂ 
                       t 
                     
                   
                   = 
                   
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                         t 
                       
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     Error signal is proportional to amplitude A and Frequency of signal. 
     Error signals appear at 90deg, 180deg, 270deg and so on (i.e. integer*Fs/2±Fin)−&gt;spurious tones. Both sampling pulse and finite bandwidth mismatches appear as pulse width difference error among channels. Therefore, it is possible to cancel it by adjusting the sampling pulse width via variable delay elements before samplers to align their effective sampling pulse width. 
     Alternate Embodiments 
     Several other embodiments are contemplated by the inventors. For example clocks may be derived from other clocks and thus synchronized. Clocks may be buffered, enabled, and qualified by logic. The analog input signal may be buffered in a variety of ways and buffer arrangements or trees. While initial clearing of all bits in all SARs has been described, the SAR bits could all be initially cleared to a high state rather than to a low state. Active-low rather than active-high bit could be used. 
     All matrixes could be rotated or transposed, or otherwise operated upon so that columns become rows and rows become columns. Thus the terms row and column could be interchanged. 
     Binary-weighted capacitors  110 ,  108 ,  106 ,  104 , . . .  102  could be connected to the power-supply or to some other voltage rather than to ground. These capacitors may be enabled by p-channel or n-channel transistors, and the bits stored in SAR delay  40  may be active-high or active-low, and these bits may be encoded in various ways and need to be decoded by a decoder in SAR delay  40 . While binary-weighted capacitors  110 ,  108 ,  106 ,  104 , . . .  102  have been shown, these capacitors could have other weight sequences, such as 1 C, 1 C, 2 C, 5 C, 11 C, 15 C, etc., and the Successive-Approximation-Register (SAR) programming could be adjusted for these non-binary sequence of weights. Rather than having binary-weighted capacitors  110 ,  108 ,  106 ,  104 , . . .  102 , SAR delay  40  could employ other weighted delay elements, such as resistors, transistors or buffers of various sizes or weights. While binary-weighted capacitors have been described, other weightings could be substituted, such as decimally-weighted, prime-weighted, or linearly-weighted, or octal-weighted. The digital delay value in the SAR could be in these other number systems, such as octal numbers rather than binary numbers. Other kinds of delay elements could be substituted, such as parallel current sources, resistors, or various combinations, and in parallel, serial, or combined network arrangements. Values may be shifted, transformed, or processed in a variety of ways. 
     While product derivative correlator  52  has been shown for a particular midpoint correlation, other correlation functions could be substituted and product derivative correlator  52  adjusted to perform these substitute correlation functions. Inversions and complements may be added at various locations. Switches  20 ,  22 ,  23 ,  24  may be simple transistor switches, pass transistors, transmission gates, or other kinds of switches. Sample delay  11 ,  146 ,  148  may be implemented as a latch or other storage element or as a delay with combinational logic including NAND, NOR, XOR, XNOR gates. Rather than use capacitors for delay elements, MOSFETs, FinFETs, or other devices, either p-channel or n-channel, driven to power or ground, may be used as delay elements. 
     While the first channel&#39;s SAR delay  40  has been described in step  204  as being initialized to the midpoint value of 1000..0, a different channel could be initialized, or the initial value could be another value, such as 0100..0, 0010..0, etc. Any channel could act as the fixed timing reference, and the timing delay of that fixed reference could be any value. 
     Matrix processor  50  may use a Digital Signal Processor (DSP) or other processor that is efficient when performing matrix operations. Product derivative correlators  52  may be implemented in hardware and in parallel for high-speed calibration. Various combinations of hardware, firmware, and software may be used in these implementations and for calibrator  55 . 
     FIR filters  17  can act as lowpass or bandpass filters for calibration, as long as the polarity of its correlation derivatives is known. FIR filtering can help define the polarity of its correlation derivatives to a well-defined value or specification to calibrate the interleaved ADC for a known frequency range. Since correlation derivatives are frequency dependent, FIR filtering can prevent any potential convergence problems during calibration. 
     Sample delays  11 ,  146 ,  148  each could be reduced by 1 sample delay period for very fast logic that does not require pipelining to meet timing requirements. Alternately, these sample delays could be increased by 1 or more samples for very fast pipelined systems with slow logic delays. 
     While some operations have been described in a parallel manner for faster processing, serial operation may be used. When performed serially, a single instance of product derivative correlator  52  could be used rather than separate instances of product derivative correlator  52  in the hardware. While a bank of N product derivative correlators  52  have been shown, product derivative correlators  52  could be re-used or operate in various series and parallel arrangements. 
     The process steps could be performed serially, or some steps may be performed in parallel. Various sequences may be adjusted or modified. Higher-level operations may be performed in software or firmware, such as SAR testing and decision logic, while lower-level functions may be performed in hardware, such as using product derivative correlator  52  to generate product derivative factors F1, F2, F3, F4. Some or all of the calibration routine could be replaced with hardware such as programmable logic, FPGA, or other logic gates on an Integrated Circuit (IC) or another chip. Various combinations of hardware, software, firmware, etc. may be substituted. 
     Product derivative correlators could operate upon more than 3 inputs as another alternative. Analog input buffers could be rearranged, so that one analog input buffer drives  4  or  2  ADCs, or there may be a tree structure of analog input buffers with multiple levels. 
     The number of samples averaged M could be different for foreground and background calibration and could even differ for different capacitor bit-positions, such as more samples for LSB&#39;s that are more sensitive and fewer samples for MSBs. M could also differ for other reasons such as varying voltage or temperature conditions. 
     The analog input signal AIN does not have to be a sine wave, but could be other forms of AC signals, such as a triangular wave, sine waves of different frequencies that are superimposed, or any wireless baseband signal. When the polarity of the correlation derivative can be determined, these signals may be used as input signals for calibration. 
     Averaging of the product derivative factors F1, F2, F3, F4 could be performed by setting a flip-flop when the sign bit is 1 and clearing the flip-flop when the sign bit is 0 for the current sum. 
     A flip limit FL may be used for ending background calibration when incrementing the LSB keeps flipping sign bits more than the FL times. Alternately, background calibration can end when the sign bits first flip. 
     The number of channels N can be binary, non-binary, even or odd. While 4 channel interleaving has been shown in detail, 8-channel, 7-channel, 6-channel, 16-channel, 32-channel, or N-channel interleaved ADC&#39;s may be substituted. The interleave order of the channels may be changed. Interleaving may be nested or may be one long loop at level 1. 
     Additional components may be added at various nodes, such as resistors, capacitors, inductors, transistors, etc., and parasitic components may also be present. Enabling and disabling the circuit could be accomplished with additional transistors or in other ways. Pass-gate transistors or transmission gates could be added for isolation. Inversions may be added, or extra buffering. Capacitors may be connected together in parallel to create larger capacitors that have the same fringing or perimeter effects across several capacitor sizes. Switches could be n-channel transistors, p-channel transistors, or transmission gates with parallel n-channel and p-channel transistors, or more complex circuits, either passive or active, amplifying or non-amplifying. 
     The number of ADC digital bits may be adjusted. For example, a 15 bit ADC could be used, or an 8-bit, 6-bit, 22-bit, or 18-bit. A different number of bits could be substituted for a different precision, and the number of bits could be fixed or could be variable. 
     The background of the invention section may contain background information about the problem or environment of the invention rather than describe prior art by others. Thus inclusion of material in the background section is not an admission of prior art by the Applicant. 
     Any methods or processes described herein are machine-implemented or computer-implemented and are intended to be performed by machine, computer, or other device and are not intended to be performed solely by humans without such machine assistance. Tangible results generated may include reports or other machine-generated displays on display devices such as computer monitors, projection devices, audio-generating devices, and related media devices, and may include hardcopy printouts that are also machine-generated. Computer control of other machines is another tangible result. 
     Any advantages and benefits described may not apply to all embodiments of the invention. When the word “means” is recited in a claim element, Applicant intends for the claim element to fall under 35 USC Sect. 112, paragraph 6. Often a label of one or more words precedes the word “means”. The word or words preceding the word “means” is a label intended to ease referencing of claim elements and is not intended to convey a structural limitation. Such means-plus-function claims are intended to cover not only the structures described herein for performing the function and their structural equivalents, but also equivalent structures. For example, although a nail and a screw have different structures, they are equivalent structures since they both perform the function of fastening. Claims that do not use the word “means” are not intended to fall under 35 USC Sect. 112, paragraph 6. Signals are typically electronic signals but may be optical signals such as can be carried over a fiber optic line. 
     The foregoing description of the embodiments of the invention has been presented for the purposes of illustration and description. It is not intended to be exhaustive or to limit the invention to the precise form disclosed. Many modifications and variations are possible in light of the above teaching. It is intended that the scope of the invention be limited not by this detailed description, but rather by the claims appended hereto.