Sar adc comparator noise suppression method based on kalman filtering

Abstract

The present application relates to the field of analog and mixed signal integrated circuit design, and particularly to a SAR ADC comparator noise suppression method based on Kalman filtering. After the charge redistribution and successive comparison process of the SAR ADC, a small DC error which is not quantized will be left in the differential input end due to the noise interference of the comparator itself. Through innovative algorithm fusion, the local linearization of the iterative weighted least squares algorithm is combined with the optimal recursive estimation framework of Kalman filtering to construct an efficient estimation model suitable for binary observation sequences. The residual voltage can be quickly and accurately estimated with much fewer comparison times than traditional statistical methods, and the original output code value is calibrated accordingly to achieve effective compensation of the comparator noise. Thus, efficient and high-speed comparator noise suppression is achieved without increasing the complexity and power consumption of the analog front-end circuit, and the overall conversion speed and dynamic performance of the SAR ADC are significantly improved.

Description

Technical Field

[0001] This invention relates to the field of analog and mixed-signal integrated circuit design, specifically to a noise suppression method for a SAR ADC comparator based on Kalman filtering. Background Technology

[0002] Successive approximation register analog-to-digital converters (SAR ADCs) have become core components for low-power, high-precision applications such as wireless sensing and biomedical electronics due to their simple structure and high power efficiency. However, with the increasing demands on ADC resolution (e.g., exceeding 10 effective bits), SAR ADC design faces significant challenges: to achieve a high signal-to-noise ratio, comparator noise must be significantly suppressed. The traditional approach of "analog scaling," which proportionally increases the size of the comparator input transistor to reduce thermal noise, results in a fourth-order increase in power consumption, severely compromising its core energy efficiency advantage.

[0003] Therefore, a technical approach based on digital back-end calibration has emerged. This approach utilizes the characteristic that there is a residual voltage at the comparator input that is not quantized due to comparator noise after the SAR ADC conversion cycle ends. By estimating this residual voltage and subtracting it from the original output code value, a comparator with higher noise and lower power consumption can be used physically, ultimately achieving high-precision output at the system level.

[0004] To improve ADC accuracy through residual voltage estimation, some papers propose repeated least squares (LSB) comparisons of the residual voltage using maximum likelihood estimation. This method leverages the probability of the comparator returning "1" during repeated LSB comparisons to estimate the residual voltage in reverse. This approach utilizes prior knowledge that the comparator noise follows a Gaussian distribution to reduce estimation errors. However, because it relies on statistical knowledge, it requires a large number of statistical samples for accurate estimation, which is unfavorable for medium- and high-speed SAR ADCs. Similarly, other papers employ Bayesian estimation, using the Gaussian distribution of the residual voltage and prior knowledge of comparator noise to estimate the residual voltage. Like maximum likelihood estimation, this method reduces the speed of the SAR ADC due to the numerous LSB comparisons.

[0005] Therefore, existing technologies in high-precision SAR ADC design generally suffer from a contradiction between calibration accuracy, calibration speed, and hardware overhead, making it difficult to achieve a coordinated optimization. There is an urgent need for a new method capable of achieving high-precision residual voltage estimation with a limited number of LSB comparisons. Summary of the Invention

[0006] To address the aforementioned problems and shortcomings, and to resolve the difficulty in balancing calibration accuracy, calibration speed, and hardware overhead in existing high-precision SAR ADC designs, this invention provides a SAR ADC comparator noise suppression method based on Kalman filtering. After the charge redistribution and successive comparison processes of the SAR ADC, a small, unquantized DC error (residual voltage) remains at the differential input due to the comparator's own noise interference. Through innovative algorithm fusion, combining the Kalman filtering algorithm with iterative weighted least squares linearization, the residual voltage is estimated quickly and accurately with far fewer comparisons than traditional statistical methods. This residual voltage is then used to calibrate the original output code value, effectively compensating for comparator noise. Thus, without increasing the complexity and power consumption of the analog front-end circuitry, efficient and high-speed comparator noise suppression is achieved, significantly improving the overall conversion speed and dynamic performance of the SAR ADC.

[0007] A SAR ADC comparator noise suppression method based on Kalman filtering includes the following steps:

[0008] Step 1: Control the comparator to adjust the residual voltage. The codeword sequence obtained by performing N LSB repeated comparisons. It is latched by SAR digital logic and output to an external device.

[0009] The residual voltage This refers to the SAR ADC completing successive approximation quantization, and the capacitor-to-analog converter (CDAC) array outputting the codewords. The voltage presented at the comparator input after all switches have been switched is complete.

[0010] Step 2: In the off-chip processing unit, process the codeword sequence of the N comparison results output in Step 1. The Kalman filter method is used for recursive optimal estimation to finally obtain the residual voltage estimate. By iteratively performing the "prediction-update" step using the Kalman filter method, the Kalman filter can converge to the true residual voltage with minimal mean square error accuracy with a small number of repeated comparisons, thus achieving efficient calibration.

[0011] First, the codeword sequence output by the comparator Represented in sequence as Since the comparator output codeword consists of 0 or 1, the codeword sequence Given a binary observation sequence, then each element... The value of is in the set {0,1}, and k∈[1,N].

[0012] Then, the state equation and the original observation equation are constructed. The original observation equation introduces the concept of working response from the iterative weighted least squares method. The original observation equation is locally linearized at the k-th iteration point. Based on the local linear relationship at the k-th prediction point, it is generalized to a general linear model about the true state, resulting in a linear pseudo-observation equation applicable to the Kalman filter update step.

[0013] Finally, using binary observation sequences Based on the constructed state equations and linear pseudo-observation equations, the Kalman filter method is used to recursively estimate the corresponding residual voltage state sequence. The optimal estimate sequence of residual voltage is obtained. This allows us to obtain the optimal estimate of the final residual voltage. , This is the estimated residual voltage value. .

[0014] Among them, binary observations The residual voltage at the same time k in the state sequence to be estimated A unique correspondence exists, forming a time-aligned observation-state pair, which forms the basis for the recursive estimation algorithm; and the state variables of the Kalman filter are used as the foundation. Corresponding to the residual voltage to be estimated (The true value of the residual voltage to be estimated), Observations Corresponding binary observations ; This is the optimal state estimate for the k-th iteration point of the Kalman filter.

[0015] Step 3: Take the estimated residual voltage obtained in Step 2. Used for digital calibration and compensation of the raw output code value of SAR ADC.

[0016] Furthermore, step two specifically involves:

[0017] Step 1: Parameter relationship mapping;

[0018] In the SAR ADC residual voltage estimation scenario of this invention, the state variables of the Kalman filter... Corresponding to the residual voltage to be estimated Observation The codeword sequence output by the comparator during N LSB repeated comparisons .

[0019] To facilitate the subsequent model building and description, the codeword sequence output by the comparator is defined. Represented in sequence as Given that the comparator output codeword consists of either "0" or "1", the codeword sequence Given a binary observation sequence, then we have The value of is in the set {0,1}, and k∈[1,N].

[0020] The binary observations are the output of the comparator. For the observations of the Kalman filter, the true value of the residual voltage to be estimated. For the state variables of the Kalman filter, It is the state prediction value at the k-th iteration point of the Kalman filter. It is the optimal state estimate at the k-th iteration point of the Kalman filter.

[0021] Step 2: System modeling, constructing state equations and original observation equations;

[0022] 1. Construction of state equations (describing the dynamic characteristics of residual voltage)

[0023] During the LSB repetitive comparison phase, the switching state of the CDAC array remains unchanged. Therefore, the residual voltage is theoretically constant, only slightly affected by environmental disturbances (such as capacitive coupling and power supply noise). The state quantity is defined as the residual voltage. Its state equation is:

[0024] (1)

[0025] in Process noise reflects the slight changes in residual voltage caused by environmental disturbances (such as capacitive coupling, power supply noise, and other non-ideal factors). Since the variation in residual voltage is extremely small, the variance Q of the process noise is on the same order of magnitude as the sampling noise power in the SAR ADC.

[0026] The inherent thermal noise introduced by the sampling network in a SAR ADC is called sampling noise, and its noise power is KT / C, where K is the Boltzmann constant, T is the absolute temperature, and C is the sampling capacitance.

[0027] 2. Construction of the original observation equations (describing the input-output characteristics of the comparator)

[0028] Ignoring the comparator offset voltage, the comparator input This is the superposition of the residual voltage and the comparator noise, i.e.:

[0029] (2)

[0030] in, The comparator noise has a mean of 0 and a variance of . The Gaussian distribution.

[0031] The comparator's output is a binary codeword, determined by the comparison result between its input voltage and 0:

[0032] (3)

[0033] When the residual voltage is When, the comparator output The probability can be derived using the cumulative distribution function (CDF) of a Gaussian distribution:

[0034] (4)

[0035] in The cumulative distribution function (CDF) of the standard normal distribution is defined as:

[0036]

[0037] Therefore, the observed values In a given state The time follows a Bernoulli distribution:

[0038] (5)

[0039] This equation is nonlinear and does not meet the linear observation requirements of standard Kalman filtering.

[0040] Step 3: Linearization of the original observation equation: To solve the nonlinear problem, this invention introduces the concept of working response from the Iterative Weighted Least Squares (IWLS) method, and performs local linearization of the observation equation at the k-th iteration point.

[0041] 3-1. Define normalized state variables Then the observation probability is: .

[0042] 3-2. Constructing pseudo-observations At the k-th iteration point, based on the state prediction value Calculate normalized state predictions and the corresponding probability estimates and probability density function Resulting in pseudo-observations The expression is:

[0043] (6)

[0044] in, It is the probability density function PDF of the standard normal distribution.

[0045] 3-3. Determining Observation Weights: The reliability of the linearized approximation of the original observation equation is determined by the weights. measure;

[0046] (7)

[0047] 3-4. The linear pseudo-observation equation is obtained:

[0048] Constructing pseudo-observations And calculate its weights The next crucial step is to establish a formal linear observation equation. The core of this derivation lies in transforming the predictions based on a single state. The local linear relationship can be generalized to the relationship with respect to the true value of the state. The general linear model is adapted to fit the update framework of the standard Kalman filter.

[0049] To construct a general observation equation, the normalized state prediction value at time k is... Move to the unknown normalized state true value According to the theory of generalized linear models, the linear relationship equation (6) can be generalized as:

[0050] (8)

[0051] in and , Represents higher-order linearization error; Substitution yields:

[0052] (9)

[0053] The term is a random variable whose statistical properties depend on binary observations. To shape the equations into the standard form of Kalman filtering; ,Will The term and its linearization error are combined and defined as pseudo-observation noise. :

[0054] (10)

[0055] right Statistical analysis can prove the existence of spurious observation noise. The expected value is zero, and its variance can be expressed as: ;

[0056] Although the pseudo-observation equation is theoretically about the true state value Yes, but in the k-th step of the Kalman filter method, the true state value It is unknown (otherwise there would be no need to estimate); in statistical estimation theory, the standard method for dealing with unknown parameters is to use a first-order approximation: that is, the best estimate currently available—the state prediction. At this point, the theoretical variance is calculated locally;

[0057] Therefore, at the algorithm implementation level, the variance of false observation noise is... The value is assigned to the state prediction value. First-order approximation at:

[0058] (11)

[0059] Therefore, it is assumed Follows a pattern with a mean of zero and a variance of . If the Gaussian distribution is true, then we have ;

[0060] The simultaneous equations (9)-(11) and the spurious observation noise satisfy the following conditions: Based on the assumptions made, we obtain the linear pseudo-observation equation applicable to the Kalman filter update step:

[0061] (12)

[0062] Where the observation matrix Thus, the nonlinear Bernoulli observation problem has been successfully transformed into a linear observation problem with (approximate) Gaussian noise. This process ensures the application of the Kalman filter framework while guaranteeing that the noise variance used in each iteration is constant. It is a statistical measure based on the current best information, which drives the algorithm to converge toward the optimal estimate as a whole.

[0063] Step 4: Kalman filter recursive estimation

[0064] Substitute the state equation constructed in step 2 and the linear pseudo-observation equation constructed in step 3 into the standard Kalman filter framework to perform recursive optimal estimation. It is important to note that the observations in the standard Kalman filter are pseudo-observations under the current conditions. Instead of the original comparator output observations The observation noise is the pseudo-observation noise of the pseudo-observation equation. .

[0065] Furthermore, the Kalman filter recursive estimation in step 4 specifically involves:

[0066] 1. Input known parameters (initialization): comparator noise standard deviation The planned number of repeated comparisons N, and the binary observation sequence generated by repeated LSB comparisons. Boltzmann constant K, total capacitance of CDAC C, and absolute temperature T.

[0067] Assume the process noise variance Q = KT / C, and the initial residual voltage estimate is a typical value. The reason is that the residual voltage itself follows a Gaussian distribution independent of the input, with a statistical mean of 0. Initial estimation of state error covariance. The empirical range is This approach aims to balance convergence speed and estimation stability. In practical applications, the process noise variance Q and the initial estimation state error covariance are considered. It can be further adjusted and optimized according to specific noise characteristics and performance requirements.

[0068] 2. Iterative processing: For k=1,2,...,N, execute the loop:

[0069] a. Based on time k-1 (time 0 represents the initial time, initial time parameters) and (Defined by the initialization step), predict the current state at time k: ; Calculate the covariance of the predicted state error at time k: ;

[0070] b. Linearization: computation , , , , ;

[0071] c. Combine with the currently observed codewords Calculate Kalman gain And update to obtain the optimal estimate. and estimated state error covariance ;

[0072] Calculate the Kalman gain: Update state estimate: Update the estimated state error covariance: ;

[0073] 3. Output and Calibration: After iteration, the final residual voltage estimate is output. .

[0074] 4. The SAR ADC comparator noise suppression method based on Kalman filtering as described in claim 1, characterized in that, the digital calibration in step three refers to: converting the original output code value of the SAR ADC... minus .

[0075] In summary, this invention addresses the issue that after the charge redistribution and successive comparison processes of a SAR ADC, a small, unquantized DC error (residual voltage) remains at the differential input due to the comparator's own noise interference. Through innovative algorithm fusion, combining Kalman filtering and iterative weighted least squares linearization, the residual voltage is estimated quickly and accurately with far fewer comparisons than traditional statistical methods. This residual voltage is then used to calibrate the original output code value, effectively compensating for comparator noise. Thus, without increasing the complexity and power consumption of the analog front-end circuitry, this invention achieves efficient and high-speed comparator noise suppression, significantly improving the overall conversion speed and dynamic performance of the SAR ADC.

[0076] The core idea of ​​this invention is to combine the local linearization of the iterative weighted least squares algorithm with the optimal recursive estimation framework of Kalman filtering to construct an efficient estimation model suitable for binary observation sequences. Compared with existing technologies, this invention has the following significant advantages:

[0077] 1. Extremely simple hardware implementation and high integration: The entire high-precision estimation algorithm is implemented in the off-chip digital domain (such as DSP or FPGA). Modifications to the prototype SAR ADC chip only require adding a simple circuit to the SAR digital logic to generate the LSB repeated comparison timing. The analog front-end core circuits (such as comparators and CDAC arrays) remain completely unchanged, greatly reducing design risk, additional area and power consumption.

[0078] 2. Significantly Improved Calibration Speed ​​and Efficiency: This invention achieves a leap in sample efficiency through the optimal recursive information fusion mechanism of Kalman filtering. As shown in the simulation results of the embodiment, when achieving the same signal-to-noise ratio (SNDR) improvement target (e.g., 5dB), the number of LSB repetitions required by this invention is only 1 / 2 of that of Bayesian estimation and 1 / 3 of that of maximum likelihood estimation (e.g., 10 vs 20 vs 30 times), a reduction of 50% to 67% compared to Bayesian estimation and maximum likelihood estimation. This directly translates into a year-on-year reduction in calibration time, significantly improving the effective conversion rate of the ADC and enabling its application in higher-speed scenarios.

[0079] 3. High estimation accuracy and robustness: This invention theoretically provides a minimum mean square error estimate under linear Gaussian conditions. Simultaneously, it provides accurate estimation of key parameters (comparator noise standard deviation). The estimation error exhibits good robustness. Simulations show that even when... Even with a mismatch of ±10%, the algorithm's performance degradation is limited, and it can be corrected in real time through simple foreground noise estimation, ensuring the system's practicality and reliability. Attached Figure Description

[0080] Figure 1This is a schematic block diagram illustrating the system-level implementation of the present invention.

[0081] Figure 2 This is a flowchart of the residual voltage estimation method of the present invention.

[0082] Figure 3 This is a schematic diagram of the 16-bit SAR ADC architecture used in the embodiment.

[0083] Figure 4 This is a simulation diagram showing the relationship between the signal-to-noise ratio (SNDR) improvement and the number of LSB comparisons under different comparator noise levels according to the present invention.

[0084] Figure 5 for =0.8, simulation graph comparing the performance of the embodiment with existing technologies (Bayesian estimation, maximum likelihood estimation).

[0085] Figure 6 for =1, a simulation diagram comparing the performance of the embodiment with existing technologies (Bayesian estimation, maximum likelihood estimation).

[0086] Figure 7 for =1.2, Simulation diagram comparing the performance of the embodiment with existing technologies (Bayesian estimation, maximum likelihood estimation).

[0087] Figure 8 This is a simulation diagram showing the robustness of the present invention when there is an estimation error in the comparator noise parameters. Detailed Implementation

[0088] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. This embodiment aims to verify the effectiveness, performance advantages, and robustness of the proposed method through algorithmic modeling and system-level simulation. The demonstration is a performance verification process in an application scenario, rather than a specific transistor-level circuit simulation.

[0089] Example: Performance Verification Based on Algorithm Modeling

[0090] To quantitatively evaluate the performance of the Kalman filter residual voltage estimation method proposed in this invention, this embodiment constructs a complete 16-bit high-precision SAR ADC system based on a behavioral model in mathematical modeling and simulation software (MATLAB). This model accurately simulates the successive approximation quantization process of the SAR ADC, the behavior of the CDAC array with non-binary weights, the Gaussian noise characteristics of the sampling noise, and the Gaussian noise characteristics of the comparator. The focus of this embodiment is to verify the digital calibration method itself; therefore, the comparator noise standard deviation is... In the model, it is a configurable input parameter that represents the equivalent noise from the preamplifier to the latch, referred back to the input.

[0091] A schematic diagram of the implementation of the present invention is attached. Figure 1 As shown. After the SAR ADC completes conventional successive approximation quantization, the capacitor-to-digital converter (CDAC) array calculates the original output codeword. After all switches have been switched, the voltage presented at the comparator input is the residual voltage to be estimated. At this point, the comparator performs N LSB repeated comparisons on the residual voltage. The resulting N comparison results (i.e., the comparator output codeword sequence) The codeword sequence is latched by SAR digital logic and output to an off-chip processing unit. In the off-chip processing unit, the Kalman filtering algorithm described in this invention performs recursive optimal estimation on the codeword sequence, ultimately obtaining a high-precision residual voltage estimate. This estimate is used to digitally calibrate and compensate the raw output code value of the SAR ADC, thereby improving the overall signal-to-noise ratio and effective number of bits of the system without significantly increasing the power consumption and circuit complexity of the comparator.

[0092] First, a brief introduction to the Kalman filter algorithm is given. The Kalman filter is an algorithm used to perform optimal recursive estimation of the state of a dynamic system in the presence of noise. Its core idea is to continuously update the minimum mean square error estimate of the system state in the time domain by fusing the system prediction model with noisy observation data. The standard Kalman filter algorithm requires the system to satisfy the Gaussian linearity condition, and its core is described by two linear equations:

[0093] State equations describe the evolution of the system state over time; observation equations describe the linear relationship between observed data and the system state. Their mathematical forms are as follows:

[0094] (13)

[0095] in: This represents the state variables of the system at time k; The noise is a process noise that follows a Gaussian distribution. ; Represents the observation at time k; It is observation noise, which follows a Gaussian distribution. F is the state transition matrix, describing the evolution of the state variables from time k−1 to k; B is the control matrix, describing the control variables. The impact on state variables; H is the observation matrix, used to resolve state variables. With observation The issue of inconsistent dimensions is addressed by correcting only the observable portion, without affecting other state variables.

[0096] Based on the known state equation and observation equation, Kalman filtering achieves recursive estimation through the following steps:

[0097] 1. Forecasting Phase (Time Update)

[0098] The optimal state estimate based on time k-1 Predict the state at time k and its error covariance :

[0099] (14)

[0100] in: The predicted state value at time k; The covariance matrix of the predicted state error; This represents the transpose of the state transition matrix F.

[0101] 2. Update Phase (Measurement Update)

[0102] Using the observations at time k The predicted state values ​​are then corrected to obtain the optimal state estimate.

[0103] (15)

[0104] in: The optimal state estimate at time k; To estimate the covariance matrix of the state error; Kalman gain is used to balance the reliability of predictions and observations. H represents the transpose of the observation matrix H; I represents the identity matrix.

[0105] In this embodiment, the estimation of the residual voltage of the off-chip SAR ADC is as follows:

[0106] Step 1: Parameter relationship mapping;

[0107] In the scenario of SAR ADC residual voltage estimation, the state variables of the Kalman filter Corresponding to the residual voltage to be estimated Observation The codeword sequence output by the comparator during N LSB repeated comparisons .

[0108] The codeword sequence output by the comparator The corresponding representations are as follows: Codeword sequence Given a binary observation sequence, The value of is in the set {0,1}, and k∈[1,N].

[0109] Using binary observations For the observations of the Kalman filter, the true value of the residual voltage to be estimated. For the state variables of the Kalman filter, It is the state prediction value at the k-th iteration point of the Kalman filter. It is the optimal state estimate at the k-th iteration point of the Kalman filter.

[0110] Step 2: System modeling, constructing state equations and original observation equations (laying the foundation for Kalman filtering);

[0111] Kalman filtering requires the system to satisfy the Gaussian linearity condition, and its core is the linearity of the state equation and observation equation. The modeling objective of this invention is to utilize the binary observation sequence generated by repeated LSB comparisons. Recursive estimation of residual voltage state sequence The optimal estimate sequence of residual voltage is obtained. This allows us to obtain the optimal estimate of the final residual voltage. The optimal residual voltage estimate is the desired residual voltage estimate. Among them, the binary observations in the observation sequence The residual voltage at the same time k in the state sequence to be estimated The unique correspondences together form time-aligned observation-state pairs, which form the basis for the recursive estimation algorithm.

[0112] 1. Construction of state equations (describing the dynamic characteristics of residual voltage)

[0113] During the LSB repetitive comparison phase, the switching state of the CDAC array remains unchanged. Therefore, the residual voltage is theoretically constant, only slightly affected by environmental disturbances (such as capacitive coupling and power supply noise). The state quantity is defined as the residual voltage. Its state equation is:

[0114] (1)

[0115] in The process noise reflects the slight changes in residual voltage caused by environmental disturbances (such as capacitive coupling, power supply noise, and other non-ideal factors). Since the variance Q of the process noise can be on the same order of magnitude as the sampling noise power in the SAR ADC, we have Q = KT / C.

[0116] 2. Construction of the original observation equations (describing the input-output characteristics of the comparator);

[0117] Ignoring the comparator offset voltage, the comparator input This is the superposition of the residual voltage and the comparator noise, i.e.:

[0118] (2)

[0119] in, The comparator noise has a mean of 0 and a variance of . The Gaussian distribution.

[0120] The comparator's output is a binary codeword, determined by the comparison result between its input voltage and 0:

[0121] (3)

[0122] When the residual voltage is When, the comparator output The probability can be derived using the cumulative distribution function (CDF) of a Gaussian distribution:

[0123] (4)

[0124] in The cumulative distribution function (CDF) of the standard normal distribution is defined as:

[0125]

[0126] Therefore, the observed values In a given state The time follows a Bernoulli distribution:

[0127] (5)

[0128] This equation is nonlinear and does not meet the linear observation requirements of standard Kalman filtering.

[0129] Step 3, Linearization of the original observation equation (core innovation of this invention): To solve the nonlinear problem, this invention introduces the concept of working response of the iterative weighted least squares method, and performs local linearization of the original observation equation at the k-th iteration point.

[0130] Step 4: Kalman filter recursive estimation;

[0131] Substitute the state equation constructed in step 2 and the linear pseudo-observation equation constructed in step 3 into the standard Kalman filter framework to perform recursive optimal estimation; where the observations in the standard Kalman filter are pseudo-observations under the current conditions. Instead of the original comparator output observations The observation noise is the pseudo-observation noise of the pseudo-observation equation. .

[0132] The algorithm execution flow is shown in the appendix. Figure 2 As shown:

[0133] 1. Input known parameters (initialization): comparator noise standard deviation The planned number of repeated comparisons N, and the binary observation sequence generated by repeated LSB comparisons. Let Boltzmann constant K, total CDAC capacitance C, and absolute temperature T. Assume the process noise variance Q = KT / C. The initial residual voltage estimate is a typical value. The reason is that the residual voltage itself follows a Gaussian distribution independent of the input, with a statistical mean of 0. Initial estimation of state error covariance. This value is included in the empirical range. middle.

[0134] 2. Iterative processing: For k=1,2,...,N, execute the loop:

[0135] a. Prediction: Based on time k-1, predict the state at time k: ; Calculate the covariance of the predicted state error at time k: .

[0136] b. Linearization: computation , , , , .

[0137] c. Update: Combine with current observed codewords Calculate Kalman gain And update to obtain the optimal estimate. Covariance .

[0138] 3. Output and Calibration: After iteration, the final residual voltage estimate is output. .

[0139] Finally, the raw output code value of the SAR ADC is... minus This means completing high-precision digital calibration.

[0140] It should be noted that the process noise variance Q and the initial estimated state error covariance are related. It does not have a unique value; its value can be fine-tuned under different application conditions (such as different comparator noise levels, LSB repetition counts, etc.) to achieve better algorithm optimization results.

[0141] The verification workflow is as follows:

[0142] 1. System Modeling: Establish a behavioral-level model of the SAR ADC in MATLAB. Set the CDAC weights (e.g., high-order capacitor weights [174, 90, 48, ...]), using a bridging capacitor structure; see appendix for details. Figure 3 Gaussian white noise is injected as a noise source for the sampling network and comparator.

[0143] 2. LSB Repeated Comparison Simulation: This involves performing normal successive approximation quantization on the input signal. After quantization, the residual voltage under ideal conditions is calculated based on the determined capacitor switching states. The comparator in the control model corresponds to the ideal residual voltage described above. The comparison is repeated N times. In each comparison, the comparator generates a series of binary output codes based on whether the sum of the residual voltage and noise is greater than 0. This process generates an observation sequence of length N. As an observation of Kalman filtering.

[0144] 3. Algorithm Execution and Calibration: Input this observation sequence into the independently implemented Kalman filter estimation algorithm module described in this invention. This algorithm module follows the aforementioned process (as shown in the appendix). Figure 2 (As shown) Recursive calculations are performed to finally output the estimated value of the residual voltage. .

[0145] 4. Performance Evaluation: The improvement effect of the method of the present invention on the overall dynamic performance of the ADC is quantitatively evaluated by calculating the signal-to-noise ratio (SNDR) of the ADC output signal before and after calibration.

[0146] Modeling validation results and analysis:

[0147] Based on the above system-level modeling and simulation, the following key conclusions are obtained, as shown in the appendix. Figure 4 To be continued Figure 8 As shown:

[0148] Validity verification (attached) Figure 4 ): at different comparator noise levels ( Under these conditions, the SNDR improvement of the ADC monotonically increases with the increase of the number of repeated comparisons N. This proves the effectiveness of the invention at the algorithm level. The curve also shows that when When the value is too small, the upper limit of the SNDR increase will decrease, which is consistent with the theoretical expectation that "calibration is intended to compensate for noise". The minimum value of the comparator noise standard deviation is set here to 0.4 LSB, rather than a smaller value, because: when When the noise level is below this threshold, the degrading effect of comparator noise on ADC performance (such as SNDR and valid ENOB) is negligible. At this point, the gain (performance improvement) of noise compensation using this technique is lower than the implementation cost (such as hardware overhead and algorithm complexity), and it is not worth implementing from an engineering practicality perspective.

[0149] Quantification of speed advantage (with appendix) Figure 5-7 ): at comparator noise standard deviation Under typical operating conditions of 0.8 LSB, 1 LSB, and 1.2 LSB, respectively, the residual voltage estimation algorithm based on Kalman filtering in this invention significantly outperforms Bayesian estimation (BE) and maximum likelihood estimation (MLE) methods in core performance indicators such as signal-to-noise ratio / distortion ratio (SNDR) improvement and convergence speed. Under typical configuration conditions, to achieve a 5dB improvement in SNDR, the method of this invention only requires 10 repeated comparisons in the model to converge. In the same modeling environment, traditional Bayesian estimation and maximum likelihood estimation methods require 20 and 30 comparisons respectively, representing a reduction of 50%–67% compared to Bayesian estimation and maximum likelihood estimation. This result conclusively demonstrates at the algorithmic level that the present invention has a significant advantage of 2 to 3 times in convergence speed.

[0150] Robustness analysis (with appendix) Figure 8 To test the algorithm's sensitivity to parameter mismatch, the algorithm module was tested using a slightly biased parameter during modeling. (e.g., 1.0 LSB), while the actual noise source in the model uses (e.g., 0.9 LSB or 1.1 LSB). The results show only a slight change in the SNDR improvement curve, indicating that the present invention has good robustness to the estimation error of key noise parameters. In practical chips, this can be updated periodically using foreground noise estimation techniques. This is to further ensure calibration accuracy.

[0151] The above embodiments, through detailed system-level algorithm modeling and mathematical simulation, demonstrate that the residual voltage estimation method based on Kalman filtering proposed in this invention can achieve high-precision noise suppression with a small number of repeated comparisons, providing an efficient and feasible digital calibration scheme for resolving the speed-accuracy contradiction in high-precision SAR ADCs. This modeling result lays a solid theoretical foundation and performance expectations for the subsequent integration and application of this method in specific circuit designs.

Claims

1. A method for suppressing noise in a SAR ADC comparator based on Kalman filtering, characterized in that, Includes the following steps: Step 1: Control the comparator to adjust the residual voltage. The codeword sequence obtained by performing N LSB repeated comparisons. The signal is latched by SAR digital logic and output to an external device. The residual voltage This refers to the SAR ADC completing successive approximation quantization, and the capacitor-to-analog converter (CDAC) array outputting the codewords. The voltage presented at the comparator input after all switches have been switched; Step 2: In the off-chip processing unit, process the codeword sequence of the N comparison results output in Step 1. The Kalman filter method is used for recursive optimal estimation to finally obtain the residual voltage estimate. ; First, the codeword sequence output by the comparator The corresponding representations are as follows: Codeword sequence Given a binary observation sequence, where each element The values ​​of belong to the set {0,1}, and k∈[1,N]; Then, the state equation and the original observation equation are constructed. The original observation equation introduces the concept of working response from the iterative weighted least squares method. The original observation equation is locally linearized at the k-th iteration point. Based on the local linear relationship at the k-th prediction point, it is generalized to a general linear model about the true state, resulting in a linear pseudo-observation equation applicable to the Kalman filter update step. Finally, using binary observation sequences Based on the constructed state equations and linear pseudo-observation equations, the Kalman filter method is used to recursively estimate the corresponding residual voltage state sequence. The optimal estimate sequence of residual voltage is obtained. This allows us to obtain the optimal estimate of the final residual voltage. , This is the estimated residual voltage value. ; Among them, binary observations The residual voltage at the same time k in the state sequence to be estimated A unique correspondence exists, together forming a time-aligned observation-state pair; and the state variables are Kalman filtered. Corresponding to the residual voltage to be estimated Observation Corresponding binary observations ; This is the optimal state estimate at the k-th iteration point of the Kalman filter; Step 3: Take the residual voltage estimate obtained in Step 2. Used for digital calibration and compensation of the raw output code value of SAR ADC.

2. The SAR ADC comparator noise suppression method based on Kalman filtering as described in claim 1, characterized in that, Step two specifically involves: Step 1: Parameter relationship mapping; Kalman filter state variables Corresponding to the residual voltage to be estimated Observation The codeword sequence output by the comparator during N LSB repeated comparisons ; The codeword sequence output by the comparator The corresponding representations are as follows: Codeword sequence Given a binary observation sequence, The values ​​of belong to the set {0,1}, and k∈[1,N]; Using binary observations For the observations of the Kalman filter, the true value of the residual voltage to be estimated. For the state variables of the Kalman filter, It is the state prediction value at the k-th iteration point of the Kalman filter. It is the optimal state estimate at the k-th iteration point of the Kalman filter; Step 2: System modeling, constructing state equations and original observation equations; 1. Construction of state equations; State quantity is defined as residual voltage Its state equation is: （1） in The process noise is represented by the variance Q, which is on the same order of magnitude as the sampling noise power in the SAR ADC. The sampling noise introduced by the sampling network in the SAR ADC has a noise power of KT / C, where K is the Boltzmann constant, T is the absolute temperature, and C is the sampling capacitance.

2. Construction of the original observation equations; Ignoring the comparator offset voltage, the comparator input This is the superposition of the residual voltage and the comparator noise, i.e.: （2） in, The comparator noise has a mean of 0 and a variance of . Gaussian distribution; The comparator's output is a binary codeword, determined by the comparison result between its input voltage and 0: （3） When the residual voltage is When, the comparator output The probability can be derived using the cumulative distribution function (CDF) of a Gaussian distribution: （4） in The cumulative distribution function (CDF) of the standard normal distribution is defined as: Therefore, the observed values In a given state The time follows a Bernoulli distribution: （5） This equation is nonlinear and does not meet the linear observation requirements of the standard Kalman filter; Step 3, Linearization of the original observation equation: Introduce the concept of working response of the iterative weighted least squares method, and perform local linearization of the original observation equation at the k-th iteration point; 3-1. Define normalized state variables The observation probability is: ; 3-2. Constructing pseudo-observations At the k-th iteration point, based on the state prediction value Calculate normalized state predictions and the corresponding probability estimates and probability density function Resulting in pseudo-observations The expression is: （6） in, It is the probability density function PDF of the standard normal distribution; The predicted value at time k; 3-3. Determining Observation Weights: The reliability of the linearized approximation of the original observation equation is determined by the weights. measure; （7） 3-4. The linear pseudo-observation equation is obtained: Constructing pseudo-observations And calculate its weights Then, based on the single state prediction value The local linear relationship can be generalized to the relationship with respect to the true value of the state. The general linear model; For the normalized state prediction value at time k Move to the unknown normalized state true value According to the theory of generalized linear models, the linear relationship equation (6) can be generalized as: （8） in and , Represents higher-order linearization error; Substitution yields: （9） The term is a random variable whose statistical properties depend on binary observations. In order to shape the equations into the standard form of Kalman filtering; ,Will The term and its linearization error are combined and defined as pseudo-observation noise. : （10） right Statistical analysis can prove the existence of spurious observation noise. The expected value is zero, and its variance can be expressed as: ; Although the pseudo-observation equation is theoretically about the true state value Yes, but in the k-th step of the Kalman filter method, the true state value It is unknown; in statistical estimation theory, the standard method for dealing with unknown parameters is to use a first-order approximation: the best available estimate—the state prediction value. At this point, the theoretical variance is calculated locally; Therefore, at the algorithm implementation level, the variance of false observation noise is... The value is assigned to the state prediction value. First-order approximation at: （11） Therefore, it is assumed Follows a pattern with a mean of zero and a variance of . If the Gaussian distribution is true, then we have ; The simultaneous equations (9)-(11) and the spurious observation noise satisfy the following conditions: Based on the assumptions made, we obtain the linear pseudo-observation equation applicable to the Kalman filter update step: （12） Where the observation matrix Thus, the nonlinear Bernoulli observation problem is transformed into a linear observation problem with Gaussian noise. Step 4: Kalman filter recursive estimation; Substitute the state equation constructed in step 2 and the linear pseudo-observation equation constructed in step 3 into the standard Kalman filter framework to perform recursive optimal estimation; where the observations in the standard Kalman filter are pseudo-observations under the current conditions. Instead of the original comparator output observations The observation noise is the pseudo-observation noise of the pseudo-observation equation. .

3. The SAR ADC comparator noise suppression method based on Kalman filtering as described in claim 2, characterized in that, Step 4 specifically involves:

1. Input known parameters: comparator noise standard deviation The planned number of repeated comparisons N, and the binary observation sequence generated by repeated LSB comparisons. Boltzmann constant K, total capacitance of CDAC C, absolute temperature T; Assume the process noise variance Q = KT / C, and the initial residual voltage estimate is a typical value. Initial estimated state error covariance The empirical range is ; 2. Iterative processing: For k=1,2,...,N, execute the loop: a. Based on time k-1, predict the state at time k: ; Calculate the covariance of the predicted state error at time k: ; b. Linearization: computation , , , , ; c. Combine with the currently observed codewords Calculate Kalman gain And update to obtain the optimal estimate. and estimated state error covariance ; Calculate the Kalman gain: Update state estimate: ; Update the estimated state error covariance: ; 3. Output and Calibration: After iteration, the final residual voltage estimate is output. .

4. The SAR ADC comparator noise suppression method based on Kalman filtering as described in claim 1, characterized in that, The digital calibration in step three refers to: adjusting the original output code value of the SAR ADC. minus .

5. The SAR ADC comparator noise suppression method based on Kalman filtering as described in claim 2, characterized in that, Q = KT / C.

6. The SAR ADC comparator noise suppression method based on Kalman filtering as described in claim 3, characterized in that, The .

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