MULTI-CORRELATION ESTIMATION (MCE)

TECHNIQUE

6.1 Introduction

In the following sections, we propose a novel calibration scheme that can accurately estimate and correct errors arising from the residue amplifiers and continuously track and update the correction parameters against environmental variations. Because us-ing the open-loop amplifier rather than closed-loop one in the pipeline stages, the amplifier may substantially changes its transfer function due to the absence of feed-back. This condition dictates the need of fast updating the correction parameters.

Under this condition, the proposed scheme enables fast and continuous estimation for the varying amplifier in short time intervals as compared to [35]. Meanwhile, the scheme operates during the normal ADC conversion with no scheduled calibra-tion cycles or use of redundant hardware [15] or slots queues [4, 40] to enable the background feature.

To estimate the error information about the MDAC or the amplifier, many cal-ibration techniques have been proposed. Some techniques need additional stages [3, 18, 37] to reduced the backend ADC quantization noise, enabling precise estima-tion of the transiestima-tion heights that relate to amplifier’s gain. Fig. 6.1 indicates that the transition height of the residue is proportional to the amplifier’s gain [41–43].

However, some of such techniques need to stop the input then a calibration signal can be applied as in [18], which is not allowed in many applications. Other ap-proaches acquire the nonlinear information by using a parallel ADC to compare the difference of an ideal output (from the parallel ADC) and a real one (from the main ADC) [1, 44]. Fig. 6.2 demonstrates this idea.

For instance, the split-ADC architecture extracts the nonlinear information using separated ADC channels [5, 7]. If these channels were non-perfect, the unbalanced channels generate different outputs that can be used for nonlinearity correction as shown by Fig. 6.3 and 6.4. While such techniques have the advantages of de-terministic error extraction therefore reducing the calibration time, they increases

0

0 H

V

V

V

−V

1/4V

3/4V

−3/4V

−1/4V

Figure 6.1: Transition height of digitized residue.

S/H Pipelined

ADC

Slow-but-accurate ADC

f (D)

e

V

D

D

Figure 6.2: Error correction of pipelined ADC [1].

analog circuits’ complexity, imposing penalties of larger die area, power, and

Figure 6.3: “Split” ADC architecture [5].

Some calibration techniques, called the correlation-based technique, use statisti-cal functions to estimate errors therefore being able to correct gain errors have been presented in literature [17,35,44]. In these techniques, the analog error is modulated using a pseudo-random noise sequence and then the digital output is processed in order to extract the error information useful to calibrate the ADC.

Having described the features of above techniques, we propose a technique that has the following superiorities:

• No limitation on the input amplitude:

The benefit of the proposed technique relative to that presented in [3] is that it works for any input signal, and the benefits relative to that presented in [17]

are that it does not have restrictions on dc input and it is not sensitive to amplifier offsets.

• Reduced circuit complexity:

The proposed calibration scheme cooperated with statistics-based estimation enables the use of a low resolution backend ADC. Unlike the work in [3], the resolution of the backend ADC is no longer limited by the target resolution minus one. Therefore, employing simple circuits yields the potential toward high speed and/or low power. To facilitate the estimation, the required pseu-dorandom noise sequences (RNGs) only needs negligible modifications on the sub-DAC.

ADC DAC

Figure 6.4: Two channel ADC architecture [7].

• Digital background calibration:

All the calibration circuits are built using digital circuitry. Given that the input acts as a stimulus and is modulated with the RNGs, the scheme performs error estimation and calibration during the normal operation of the ADC.

• Unbiased gain error information extraction:

With the simple statistical function (mean function) and the estimation pro-cedure, the linear and nonlinear gain error information of the amplifier can be extracted independently. Notably, error information of high order nonlineari-ties, e.g., 5th order, is possible if more RNGs are merged.

In the following sections, we will describe the technique with the associated functions in detail.

6.2 Modulation Approach

The proposed scheme makes use of the fact that the offsets in the sub-ADC does not affect the ADC conversion results based on the digital redundancy [21]. As a result, the scaled random noise sequences whose values no more than the tolerable offsets can be applied using either the sub-ADC or the sub-DAC. Because of the added random sequence, the residue moves up/down. Fig. 6.5 shows one possible residue plot when the RNG is added. It shows that one input signal may have two different residues. However, their conversion results agree provided that the gain errors are perfectly corrected. Similar approaches can be found in [3, 7, 17, 35, 44].

V _in V res

V

−V

V

−V

RNG=1 RNG = -1

Figure 6.5: Residue plot when adding RNGs.

In order to unbiasedly estimate the correction parameters, the RNG is designed as a uniformly distributed pseudorandom binary number sequence, i.e., (RN G ∈ {1, −1}) and is uncorrelated with the input. As a result, the RNG is continuously applied to the stage being calibrated to continuously estimate and update the cor-rection parameters.

In the remainder of this chapter, we propose a multi-correlation estimation (MCE) technique using the modulation approach, allowing continuous background estimation of the correction parameters p₁ and p₃ described in the previous chapter.

6.3 Multi-Correlation Estimation (MCE) Technique

In this section we will describe a technique based on statistics that can estimate the correction parameters. Using two different modulated sequences, this approach results in the residue having different distributions. Then the statistical results associated with the residues are used to find the nonlinearities, i.e., the error infor-mation. With the help of this information, we can approach the optimum values of p₁ and p₃ using Least Mean Square (LMS) algorithm [46].

Calibration

Figure 6.6: Reduced model with proposed calibration scheme.

Considering Fig. 6.6, the digitized residue D_b when the random sequences are applied is

D_bi = a₁(V_x) + a₃(V_x)³ + ε_b

= a₁(−ε_a+ R_i· V_di) + (−ε_a+ R_i· V_di)³+ ε_b, i ∈ {1, 2} (6.1) where R are the pseudo-random number sequences that are uniformly distributed

Therefore, R_i times V_di, i.e., R_iV_di add offsets of ±V_d1 or ±V_d2 LSB (of local sub-ADC) to the sub-DAC. They are

R₁V_d1 ∈ {+V_d1, −V_d1}, R₂V_d2 ∈ {+V_d2, −V_d2}.

Taking the correlations of D_bi and R_i, we have

E[R_iD_bi] = Ea₁(−R_iε_a− V_di) + a₃(−R_iε³_a− 3ε²_aV_di− 3R_iε_aV_di² − V_di³) + R_iε_b . (6.2) Because R_i are uncorrelated with the input, correlations of R_i and the quantization errors ε_a, ε_b will be zero. Under this circumstance, (6.2) is further reduced to

E[R_iD_bi] = a₁(−V_di) + a₃(−3ε²_aV_di− V_di³). (6.3) This finding reveals the quantization noise of backend ADC has no effect on the estimation accuracy as compared with [3].

Considering the terms a1(−Vdi) and a3(−3ε²_aVdi) in (6.3), if they can be elimi-nated, the result is proportional to a₃. For such a reason, we propose a technique called “multi-correlation estimation technique” that can accurately estimate the er-ror information.

Using (6.3) and V_d2= V_d1/2 = LSB/4 gives

ε₃ = E[R₁D_b1] − 2E[R₂D_b2] = −3

4a₃V_d1³. (6.4) In this equation, ε₃ represents the sum of correlations E[R₁V_d1], E[R₂V_d2] and is directly proportional to a₃, leading to an unbiased estimation. If the correction function (5.11) is applied, we obtain

ε₃ = −3

4a₃V_d1³ = −3

4a³₁V_d1³ · (p_3,opt− p₃). (6.5) This result indicates the deviation of parameter p₃ from its ideal value is directly proportional to ε₃. According to this result, we can use iterative functions, e.g., LMS algorithm, trying to minimize the deviation so as to obtain the ideal value of p₃.

As can be seen from the derivation of ε₃, it only represents the degree of nonlinear term a₃. Hence, we need another error information related to the linear gain a₁. Indicated by (6.3), the resulting correlation is proportional to a₁ when a₃ = 0. That is, when the nonlinear gain error has been corrected, we define (6.3) as

ε₁ = E[R₁D_b1] = a₁(−V_d1), (6.6) where ε₁ represents the linear gain error information. In addition, correlation of R₂ and D_b2 can be used as well. Since p_1,opt = a₁ as indicated in (5.11), it is straightforward that dividing ε₁ by V_d1is p_1,opt. However, this procedure takes large number of samples for reducing the variance of p₁. For an N-bit ADC, roughly 2^2N

samples are required to obtain sufficiently accurate estimate of a₁ [5]. Above results suggest that fast updates of p₁ and p₃ is desirable. As a result, we employ LMS algorithm to achieve this goal. Although making use of LMS still needs an amount of time to converge the corrections parameters to a sufficient accuracy, once they have converged, each update is fast enough to track the environment variations. In order to be merged in the LMS loop, ε₁ is modified as

ε⁰₁ = ε₁

p₁ + V_d1 = −a₁V_d1

p₁ + V_d1. (6.7)

In this modification, p₁ will approaches p_1,opt when ε⁰₁ = 0 by using LMS.

6.4 Adaptive Signal Processing

G(z) d(n)

y(n)

e(n) x(n)

W(z)