Conclusions and Recommendations

Abstract

With the fast advancement of CMOS fabrication technology, more and more signal-processing functions are implemented in the digital domain for a lower cost, lower power consumption, higher yield, and higher re-configurability. The trend of increasing integration level for integrated circuits has forced the A/D converter interface to reside on the same silicon in complex mixed-signal ICs containing mostly digital blocks for DSP and control. However, specifications of the converters in various applications emphasize high dynamic range and low spurious spectral performance. It is nontrivial to achieve this level of linearity in a monolithic environment where post-fabrication component trimming or calibration is cumbersome to implement for certain applications or/and for cost and manufacturability reasons. Additionally, as CMOS integrated circuits are accomplishing unprecedented integration levels, potential problems associated with device scaling – the short-channel effects – are also looming large as technology strides into the deep-submicron regime. The A/D conversion process involves sampling the applied analog input signal and quantizing it to its digital representation by comparing it to reference voltages before further signal processing in subsequent digital systems. Depending on how these functions are combined, different A/D converter architectures can be implemented with different requirements on each function. As discussed in Chapter 2, practical realizations show the trend that to a first order, converter power is directly proportional to sampling rate. However, power dissipation required becomes nonlinear as the speed capabilities of a process technology are pushed to the limit. Pipeline and two-step/multi-step converters tend to be the most efficient at achieving a given resolution and sampling rate specification.

Appendix

6.1.1 A.1 Time Mismatch

Let the original sampled data sequence S = [x(t ₀ ), x(t ₁ ), x(t ₂ ), …, x(t _m ), …, x(t _N ), x(t _N+1 ),…] be divided into N subsequences S ₀ , S ₁ , S ₂ , …, S _N−1 as follows [133]:

$$ \begin{array}{lll} {S_0} = [x({t_0}),x({t_N}),x({t_{2N}}),...] \hfill \\\vdots \hfill \\{S_n} = [x({t_n}),x({t_{N + n}}),x({t_{2N + n}}),...] \hfill \\\vdots \hfill \\{S_N} = [x({t_{N - 1}}),x({t_{2N - 1}}),x({t_{3N - 1}}),... \hfill \\ \end{array} $$

(A.1)

The S _n is obtained by uniformly sampling the signal x(t + t _n ) at the rate 1/NT. Assume

$$\overline{{{S_n}}} = [x({t_n}),0,0,...(N - 1{ }zeros),x({t_{N + n}}),0,0,...] $$

(A.2)

the original sequence S, can be represented as

$$ S = \sum\limits_{n = 0}^{N - 1} {\overline {{S_n}} {z^{ - n}}} $$

(A.3)

Then, the digital spectrum, X(ω), of S can be represented as

$$ X(\omega ) = \frac{1}{{NT}}\sum\limits_{n = 0}^{N - 1} {\left[ {\sum\limits_{k = - \infty }^\infty {{X^a}\left( {\omega - \frac{{2\pi k}}{{NT}}} \right){e^{j[\omega - (2\pi k/NT)] \times {t_n}}}} } \right]} \times {e^{ - jn\omega T}} $$

(A.4)

Let r _n T, n = 0,1,N − 1 be the sampling time offset encountered at the nth sample-hold unit (positive r _n means that the nth sample is delayed), and t _n real sampling time for nth sample-hold unit,

$$ {r_n}T = nT - {t_n} $$

(A.5)

then (A.4) can be rewritten as

$$ X(\omega ) = \frac{1}{T}\sum\limits_{k = - \infty }^\infty {\left[ {\sum\limits_{n = 0}^{N - 1} {\frac{1}{N}{e^{ - j\left( {\omega - k\frac{{2\pi }}{{NT}}} \right){r_n}T}}{e^{ - jkn\frac{{2\pi }}{N}}}} } \right]} \times {X^a}\left( {\omega - \frac{{2\pi k}}{{NT}}} \right) $$

(A.6)

For a given sine wave x(t) = sin(ω _in t), the Fourier transform X ^a (ω) is given by

$$ {X^a}(\omega ) = j\pi \left[ {\delta (\omega + {\omega_{in}}) - \delta (\omega - {\omega_{in}})} \right] $$

(A.7)

where δ is unit sample sequence δ[x] = 1 at x = 0, and 0 elsewhere, (A.6) becomes [133]

$$ X(\omega ) = \frac{1}{T}\sum\limits_{k = - \infty }^\infty {\left[ {A(k)j\pi \left( {\delta \left( {\omega + {\omega_{in}} - k\frac{{2\pi }}{{NT}}} \right) - \delta \left( {\omega - {\omega_{in}} - k\frac{{2\pi }}{{NT}}} \right)} \right)} \right]} $$

(A.8)

where

$$ A(k) = \sum\limits_{n = 0}^{N - 1} {\left( {\frac{1}{N}{e^{ - j{r_n}2\pi {f_{in}}/{f_s}}}} \right){e^{ - jkn(2\pi /N)}}} $$

(A.9)

The digital spectrum given by (A.8) has N pairs of line spectra, each pairs centered at the fractional of the sampling frequency, such as f _S /N,…,(N−1) f _S /N.

Fundamental corresponds to k = 0 while k = 1, …, N − 1 corresponds to the distortion. The signal amplitude is determined by A(0) while the distortion amplitudes are determined by A(n), n = 1, …, N − 1. A(k) is a DFT of the sequence of $[(1/N){e^{ - j{r_n}2\pi {f_0}/{f_s}}},n = 0,1,2,...,N - 1]$.

Assuming r _n 2πf _in /f _S «1, the magnitude of the sidebands components can be expressed as:

$$ \left| {A(k) = \sum\limits_{n = 0}^{N - 1} {\left( {{e^{ - j{r_n}2\pi {f_{in}}/{f_S}}}} \right){e^{ - jkn(2\pi /N)}}} } \right| \approx \left| {\frac{1}{N}\sum\limits_{n = 0}^{N - 1} {\left( {1 - j2\pi {r_n}{f_{in}}/{f_s}} \right){e^{ - jkn(2\pi /N)}}} } \right| $$

$$ = \left\{ \begin{array}{ll} {\frac{{2\pi {f_{in}}}}{{N{f_s}}}\left| {\sum\limits_{n = 0}^{N - 1} {{r_n}{e^{ - jkn(2\pi /N)}}} } \right|{\hbox{ for k}} \ne {0, }\pm {\hbox{N}}} \\{\left| {1 - (j2\pi {f_{in}}/{f_s})\left( {\frac{1}{N}\sum\limits_{n = 0}^{N - 1} {{r_n}} } \right)} \right| = 1{\hbox{ for k}} = {0, }\pm {\hbox{N}}}\end{array}\right. $$

(A.10)

The signal power is A ² /2 and power density of a spurious due to the time mismatch is expressed as

$$ {\hbox{P}}_{\rm{r}}^{\rm{spur}}(k) = \frac{{{A^2}}}{{2{N^2}}}{\left| {\sum\limits_{n = 0}^{N - 1} {(1 - j2\pi {r_n}{f_{in}}/{f_s}){e^{ - jkn(2\pi /N)}}} } \right|^2} = \frac{{{A^2}4{\pi^2}f_{in}^2}}{{2{N^2}f_s^2}}\sigma_r^2 $$

(A.11)

6.1.2 A.2 Offset Mismatch

The offset mismatch can be modeled by adding a dc level with the input signal that is unique for each S/H unit. For a input signal A sin(ω _in t) + d _n with n = 0, …, N − 1, the Fourier transform is given by

$$ {X^a}(\omega ) = j\pi A\left[ {\delta (\omega + {\omega_{in}}) - \delta (\omega - {\omega_{in}})} \right] + 2\pi {d_n}\delta (\omega ) $$

(A.12)

If no timing error is assumed r_n is zero. Substituting previous equation in (A.8),

$$ X(\omega ) = \frac{1}{T}\sum\limits_{k = - \infty }^\infty {Aj\pi \delta \left[ {\left( {\omega + {\omega_{in}} - \frac{{2\pi k}}{{NT}}} \right) - \left( {\omega - {\omega_{in}} - \frac{{2\pi k}}{{NT}}} \right)} \right]} + \frac{1}{T}\sum\limits_{k = - \infty }^\infty {A(k)2\pi \delta \left( {\omega - \frac{{2\pi k}}{{NT}}} \right)} $$

(A.13)

where

$$ A(k) = \frac{1}{N}\sum\limits_{n = 0}^{N - 1} {{d_n}{e^{ - jkn(2\pi /N)}}} $$

(A.14)

The first term of (A.12) corresponds to the input signal, while the second term corresponds to the distortion caused by channel offset. The distortion is not signal dependent and appears at nf _S /N, where n = 0,1, …, N − 1. From previous equation can be seen that the distortion consists of a sum of impulses. Each impulse corresponds to a complex exponential signal in the time domain e ^jω. The power of the exponential signal is 1. The factors A(k) can be seen as the DFT of the sequence d _n /N, n = 0, 1, …, N − 1. Power density of a spurious due to the offset mismatch is expressed as

$$ {\hbox{P}}_{\rm{r}}^{\rm{spur}}(k) = \frac{1}{{{N^2}}}{\left| {\sum\limits_{n = 0}^{N - 1} {{d_n}{e^{ - jkn(2\pi /N)}}} } \right|^2} = \frac{1}{{{N^2}}}\sigma_d^2 $$

(A.15)

6.1.3 A.3 Gain Mismatch

To model gain mismatch, magnitude of one of the S/H unit input signals is different. The largest difference occurs at the peaks of the sine wave. The signal is a _n sin(ω _in t) with n = 0, …, N − 1. The Fourier transform is given by

$$ {X^a}(\omega ) = j\pi {a_n}\left[ {\delta (\omega + {\omega_{in}}) - \delta (\omega - {\omega_{in}})} \right] $$

(A.16)

If no timing error is assumed r _n is zero. Substituting previous equation in (A.8),

$$ X(\omega ) = \frac{1}{T}\sum\limits_{k = - \infty }^\infty {\left[ {A(k)j\pi \left( {\delta \left( {\omega + {\omega_{in}} - k\frac{{2\pi }}{{NT}}} \right) - \delta \left( {\omega - {\omega_{in}} - k\frac{{2\pi }}{{NT}}} \right)} \right)} \right]} $$

(A.17)

where

$$ A(k) = \frac{1}{N}\sum\limits_{n = 0}^{N - 1} {{a_n}{e^{ - jkn(2\pi /N)}}} $$

(A.18)

for the gain mismatches in N S/H units with spurious tones at f _S /N ± f _in , 2f _S /N ± f _in , …, (N − 1) f _S /N ± f _in . Power density of a spurious due to the gain mismatch is expressed as

$$ {\hbox{P}}_{\rm{r}}^{\rm{spur}}(k) = \frac{{{A^2}}}{{2{N^2}}}{\left| {\sum\limits_{n = 0}^{N - 1} {{a_n}{e^{ - jkn(2\pi /N)}}} } \right|^2} = \frac{{{A^2}}}{{2{N^2}}}\sigma_a^2 $$

(A.19)

6.1.4 A.4 Bandwidth Mismatch

To model frequency dependent bandwidth mismatch, the S/H Amplifiers are approximated as the ideal one-pole amplifiers. For a one-pole system, A(s) = A ₀ /(1 + s/ω ₀), the closed loop transfer function is

$$ \frac{{{V_{out}}}}{{{V_{in}}}}(s) = \frac{{A(s)}}{{1 + A(s)\beta }} = \frac{{{A_0}}}{{1 + {A_0}\beta + \frac{s}{{{\omega_0}}}}} = \frac{{\frac{{{A_0}}}{{1 + {A_0}\beta }}}}{{1 + \frac{s}{{(1 + {A_0}\beta ){\omega_0}}}}} $$

(A.20)

Recognizing that A ₀ β » 1, the previous equation becomes

$$ \frac{{{V_{out}}}}{{{V_{in}}}}(s) = \frac{{1/\beta }}{{1 + \frac{s}{{\beta {A_0}{\omega_0}}}}} = \frac{{1/\beta }}{{1 + j\frac{{{f_{in}}}}{{\beta {A_0}{f_0}}}}} = \frac{1}{{1 + j\frac{{{f_{in}}}}{{{f_1}}}}} \approx 1 - j\frac{{{f_{in}}}}{{{f_1}}} = {b_n} $$

(A.21)

for β = 1. For a given sine wave x(t) = sin(ω _in t), the Fourier transform X ^a (ω) is given by

$$ {X^a}(\omega ) = j\pi \left[ {\delta (\omega + {\omega_{in}}) - \delta (\omega - {\omega_{in}})} \right] $$

(A.22)

and Eq. (A.8) with r _n = 0 becomes

$$ X(\omega ) = \frac{1}{T}\sum\limits_{k = - \infty }^\infty {\left[ {A(k)j\pi \left( {\delta \left( {\omega + {\omega_{in}} - k\frac{{2\pi }}{{NT}}} \right) - \delta \left( {\omega - {\omega_{in}} - k\frac{{2\pi }}{{NT}}} \right)} \right)} \right]} $$

(A.23)

where

$$ A(k) = \frac{1}{N}\sum\limits_{n = 0}^{N - 1} {{b_n}{e^{ - jkn(2\pi /N)}}} $$

(A.24)

with f ₁ unity-gain frequency and b _n bandwidth offset experienced by the nth S/H amplifier with spurious tones at f _S /N ± f _in , 2f _S /N ± f _in , …, (N − 1) f _S /N ± f _in . The signal power is A ² /2 and power density of a spurious due to the gain-bandwidth mismatch is expressed as

$$ {\hbox{P}}_{\rm{r}}^{\rm{spur}}(k) = \frac{{{A^2}}}{{2{N^2}}}{\left| {\sum\limits_{n = 0}^{N - 1} {{b_n}{e^{ - jkn(2\pi /N)}}} } \right|^2} = \frac{{{A^2}f_{in}^2}}{{2{N^2}f_1^2}}\sigma_b^2 $$

(A.25)

6.1.5 A.5 General Expression

The general expression for spurious-free dynamic range (SFDR) is given by

$$ SFDR = 10{\log_{10}}\left( {{P_s}/{P_{spur}}} \right) = 10{\log_{10}}\left( {{N^2}{\lambda_x}} \right)\begin{array}{ll} {{\lambda_{off}} = {A^2}/2\sigma_d^2} & {{\lambda_{time}} = {{\left( {{f_s}/(2{f_{in}})} \right)}^2}/({\pi^2}\sigma_r^2)} \\{{\lambda_{gain}} = 1/\sigma_a^2} & {{\lambda_{bandwidth}} = {{\left( {{f_1}/{f_{in}}} \right)}^2}/(\sigma_b^2)} \end{array}$$

(A.26)

where the input signal power is defined as P _s = A ² /2.

6.1.6 B.1 Histogram Measurement of ADC Nonlinearities Using Sine Waves

The histogram or output code density is the number of times every individual code has occurred. For an ideal A/D converter with a full scale ramp input and random sampling, an equal number of codes is expected in each bin. The number of counts in the ith bin H(i) divided by the total number of samples N _t , is the width of the bin as a fraction of full scale. By compiling a cumulative histogram, the cumulative bin widths are the transition levels.

The use of sine wave histogram tests for the determination of the nonlinearities of analog-to-digital converters (ADCs) has become quite common and is described in [336, 414]. When a ramp or triangle wave is used for histogram tests (as in [433]), additive noise has no effect on the results; however, due to the distortion or nonlinearity in the ramp, it is difficult to guarantee the accuracy.

For a differential nonlinearity test, a one percent change in the slope of the ramp would change the expected number of, codes by one percent. Since these errors would quickly accumulate, the integral nonlinearity test would become unfeasible. From brief consideration it is clear that the input source should have better precision than the converter being tested. When a sine wave is used, an error is produced, which becomes larger near the peaks. However, this error can be made as small and desired by sufficiently overdriving the A/D converter.

The probability density p(V) for a function of the form A sin ωt is

$$ p(V) = \frac{1}{{\pi \sqrt{{{A^2} - {V^2}}}}} $$

(B.27)

Integrating this density with respect to voltage gives the distribution function P(V _a , V _b )

$$ P({V_a},{V_b}) = \frac{1}{\pi }\left\{ {{{\sin }^{ - 1}}\left[ {\frac{{{V_b}}}{A}} \right] - {{\sin }^{ - 1}}\left[ {\frac{{{V_a}}}{A}} \right]} \right\} $$

(B.28)

which is in essence, the probability of a sample being in the range V _a to V _b . If the input has a dc offset, it has the form V _o + A sin ωt with density

$$ p(V) = \frac{1}{{\pi \sqrt {{{A^2} - {{(V - {V_o})}^2}}} }} $$

(B.29)

The new distribution is shifted by V _o as expected

$$ P({V_a},{V_b}) = \frac{1}{\pi }\left\{ {{{\sin }^{ - 1}}\left[ {\frac{{{V_b} - {V_o}}}{A}} \right] - {{\sin }^{ - 1}}\left[ {\frac{{{V_a} - {V_o}}}{A}} \right]} \right\} $$

(B.30)

The statistically correct method to measure the nonlinearities is to estimate the transitions from the data. The ratio of bin width to the ideal bin width P(i) is the differential linearity and should be unity.

Subtracting on LSB gives the differential nonlinearity in LSBs

$$ DNL(i) = \frac{{H(i)/{N_t}}}{{P(i)}} - 1 $$

(B.31)

Replacing the function P(V _a , V _b ) by the measured frequency of occurrence H/N _t , taking the cosine of both sides of (A.67) and solving for ${\hat{V}_b}$, which is an estimate of V _b , and using the following identities

$$ \cos (\alpha - \beta ) = \cos (\alpha )\cos (\beta ) + \sin (\alpha )\sin (\beta ) $$

(B.32)

$$ \cos \left( {{{\sin }^{ - 1}}\frac{V}{A}} \right) = \frac{{\sqrt {{{A^2} - {V^2}}} }}{A} $$

(B.33)

yields to

$$ \hat{V}_b^2 - \left( {2{V_a}\cos \left( {\frac{{\pi H}}{{{N_t}}}} \right)} \right){\hat{V}_b} - {A^2}\left( {1 - {{\cos }^2}\left( {\frac{{\pi H}}{{{N_t}}}} \right)} \right) + V_a^2 = 0 $$

(B.34)

In this consideration, the offset V _o is eliminated, since it does not effect the integral or differential nonlinearity. Solving for ${\hat{V}_b}$and using the positive square root term as a solution so that ${\hat{V}_b}$ is greater than V _a

$$ {\hat{V}_b} = {V_a}\cos \left( {\frac{{\pi H}}{{{N_t}}}} \right) + \sin \left( {\frac{{\pi H}}{{{N_t}}}} \right)\sqrt {{{A^2} - V_a^2}} $$

(B.35)

This gives ${\hat{V}_b}$in terms of V _a . ${\hat{V}_k}$ can be computed directly by using the boundary condition V _o = −A and using

$$ CH(k) = \sum\limits_{i = 0}^k {H(i)} $$

(B.36)

the estimate of the transition level ${\hat{V}_b}$denoted as a T _k can be expressed as

$$ \begin{array}{lll} {{T_k} = - A\cos \left( {\pi \frac{{C{H_{k - 1}}}}{{{N_t}}}} \right),} & {k = 1, \ldots, N - 1}\end{array}$$

(B.37)

A is not known, but being a linear factor, all transitions can be normalized to A so that the full range of transitions is ±1.

6.1.7 B.2 Mean Square Error

As the probability density function associated with the input stimulus is known, the estimators of the actual transition level T _k and of the corresponding INL _k value expressed in least significant bits (LSBs) are represented as random variables defined, respectively, for a coherently sampled sinewave

$$ s[m] = d + A\sin \left(2\pi \frac{D}{M} m + \theta_0\right) \quad m = 0,1, \ldots, M - 1 $$

(B.38)

$$ \begin{aligned}{ll}{T_k} &= d - A\cos \left( \pi \frac{{C{H_k}}}{M} \right), {k = 1, \ldots, N - 1} { {IN{L_k}}} \\&= \left( {{T_k} - T_k^i} \right)/\Delta\quad {k = 1, \ldots, N - 1}\end{aligned} $$

(B.39)

where A, d, θ ₀ are the signal amplitude, offset and initial phase, respectively, M is the number of collected data, D/M represents the ratio of the sinewave over the sampling frequencies. T _k ⁱ is the ideal kth transition voltage, and Δ = FSR/2^B is the ideal code-bin width of the ADC under test, which has a full-scale range equal to FSR. A common model employed for the analysis of an analog-to digital converter affected by integral nonlinearities describes the quantization error ε as the sum of the quantization error of a uniform quantizer ε _q and the nonlinear behavior of the considered converter ε _n . For simplicity assuming that |INL _k | < Δ/2, we have:

$$ {\varepsilon_n} = \sum\limits_{k = 1}^{N - 1} {\Delta {\rm sgn} (} IN{L_k})i(s \in {I_k}) $$

(B.40)

where sgn(.) and i(.) represent the sign and the indicator functions, respectively, s denotes converter stimulus signal and the non-overlapping intervals I _k are defined as

$$ {I_k}\hat{ = }\left\{ {\begin{array} {lll}{\begin{array} {lll}{(T_k^i - IN{L_k},T_k^i),} & {IN{L_k} >0} \\\end{array} } \\{\begin{array} {lll}{(T_k^i,T_k^i - IN{L_k}),} & {IN{L_k} < 0} \\\end{array} } \\\end{array} } \right. $$

(B.41)

The nonlinear quantizer mean-square-error, evaluated under the assumption of uniform stimulation of all converter output codes, is given by

$$ mse = \int\limits_{ - \infty }^\infty {{{[{\varepsilon_q}(s) + {\varepsilon_n}(s)]}^2}{f_s}(s)} ds $$

(B.42)

where f _s represent PDF of converter stimulus. Stimulating all device output codes with equal probability requires that

$$ {f_s}(s) = \frac{1}{{{V_M} - {V_m}}} \cdot i({V_m} \leq s < {V_M}) $$

(B.43)

Thus, mse becomes

$$ mse = \frac{1}{{{V_M} - {V_m}}}\int\limits_{{V_m}}^{{V_M}} {[\varepsilon_q^2(s) + 2{\varepsilon_q}(s){\varepsilon_n}(s) + \varepsilon_n^2(s)]} ds $$

(B.44)

Assuming Δ = (V _M -V _m )/N, and exploiting the fact the mse associated with the uniform quantization error sequence is Δ ² /12

$$ mse = \frac{{{\Delta^2}}}{{12}} + \frac{1}{{N\Delta }}\sum\limits_{k = 1}^{N - 1} {\int\limits_{{I_k}} {[2\Delta {\rm sgn} (IN{L_k}){\varepsilon_q}(s) + {\Delta^2}]} ds} $$

(B.45)

Since, for a rounding quantizer, ε _q (s) = Δ/2−Δ(s/Δ−1/2), it can be verified that sgn(INL _k ) · ε _q (s) < 0, so that

$$ mse = \frac{{{\Delta^2}}}{{12}} + \frac{1}{N}\sum\limits_{k = 1}^{N - 1} {INL_k^2} $$

(B.46)

When characterizing A/D converters the SINAD is more frequently used than the mse. The SINAD is defined as

$$ SINAD = 20{\log_{10}}\frac{{rm{s_{(signal)}}}}{{rm{s_{(noise)}}}}[dB] $$

(B.47)

Let the amplitude of the input signal be A _dBFS , expressed in dB relative full scale. Hence, the rms value is then

$$ rm{s_{(signal)}} = \frac{{\Delta {{10}^{\frac{{{A_{dBFS}}}}{{20}}}}{2^{b - 1}}}}{{\sqrt {2} }} $$

(B.48)

The rms _(noise) amplitude is obtained from the mse expression above so that

$$ \begin{aligned}{ll}{rms_{(noise)}} &= \sqrt {{mse}} \,SINA{D_{INL}}\\&= 20b{\log_{10}}2 + 10{\log_{10}}\frac{3}{2} + {A_{dBFS}} - 10{\log_{10}}\left( {\frac{{mse}}{{{\Delta^2}/12}}} \right)[dB]\end{aligned} $$

(B.49)

To calculate the effective number of bits ENOB, firstly express the SINAD for an ideal uniform ADC and than solve for b

$$ SINA{D_{(ideal)}} = 20{\log_{10}}\left( {\frac{{\sqrt {6} A{2^b}}}{{FSR}}} \right) $$

$$ ENOB = \frac{{{{\log }_2}10}}{{20}}SINAD + {\log_2}\frac{{FSR}}{{\sqrt {6} A}} $$

(B.50)

Letting the amplitude A = 10^A(dBFS)/20 FSR/2, and incorporating above equation, the ENOB can be expressed as

$$ ENO{B_{INL}} = b - \frac{1}{2}{\log_2}\left( {\frac{{mse}}{{{\Delta^2}/12}}} \right)[dB] $$

(B.51)

6.1.8 B.3 Measurement Uncertainty

To estimate the uncertainty on the DNL and INL it is necessary to know the probability distribution of the cumulative probability Q _i to realize a measurement V < UB _i , with UB _i the uperbound of the ith level

$$ {Q_i} = P(V < U{B_i}) = \int_{{V_o} - V}^{U{B_i}} {p(V)dV} $$

(B.52)

and using linear transformation

$$ U{B_i} = - \cos \pi {Q_i} $$

(B.53)

The variance and cross-correlation of UB _i is derived using linear approximations. To realize the value Q _i , it is necessary to have N _i measurements with a value <UB _i , and (N − N _i ) measurements with a value >UB _i . The distribution of Q _i is a binomial distribution, which can be very well approximated by a normal distribution [414]

$$ \begin{array}{ll}P(Q_i^{\prime}) &= C_N^{{N_i}}P{{(V < U{B_i})}^{{N_i}}}(1 - P{{(V >U{B_i})}^{N - {N_i}}} \\&\qquad = C_N^{{N_i}}Q_i^{{N_i}}{{(1 - {Q_i})}^{N - {N_i}}}\end{array} $$

(B.54)

with Q _i ^’ the estimated value of Q _i . The mean and the standard deviation is given by

$$ \begin{array} {lll}{{\mu_{Q_i^{\prime}}} = {Q_i}} & {{ }{\sigma_{Q_i^{\prime}}}} \\\end{array} = \sqrt {{{Q_i}(1 - {Q_i})/N}} $$

(B.55)

which states that Q _i ^’ is an unbiased estimate of Q _i . To calculate the covariance between Q _i and Q _j , firstly, let’s define

$$ \begin{array}{lll} {{Q_0} = P(V > U{B_j}){ }} \\{{Q_{ij}} = P(U{B_i} < V < U{B_j}) = 1 - {Q_i} - {Q_j}} \\\end{array} $$

(B.56)

and the relation

$$ \begin{array} {lll}{{N_j} = {N_i} + {N_{ij}}{ }} \\{{N_i} + {N_{ij}} + {N_0} = N} \\\end{array} \,\begin{array} {lll}{\sigma_{{N_i}{N_j}}^2 = \sigma_{{N_i}}^2 + \sigma_{{N_i}{N_{ij}}}^2{ }} \\{\sigma_{{N_0}}^2 = \sigma_{{N_i}}^2 + \sigma_{{N_{ij}}}^2 + 2\sigma_{{N_i}{N_{ij}}}^2} \\\end{array} $$

(B.57)

which leads to

$$ \sigma_{{N_i}{N_j}}^2 = [\sigma_{{N_i}}^2 + \sigma_{{N_0}}^2 - \sigma_{{N_{ij}}}^2{]/2} $$

(B.58)

with

$$ \begin{array} {lll}{\sigma_{{N_i}}^2 = N{Q_i}(1 - {Q_i})} \\{\sigma_{{N_0}}^2 = N{Q_0}(1 - {Q_0})} \\{\sigma_{{N_{ij}}}^2 = N{Q_{ij}}(1 - {Q_{ij}})} \\\end{array} $$

(B.59)

or

$$ \begin{array} {lll}{\sigma_{{N_i}{N_j}}^2 = N{Q_i}{Q_0} = N{Q_i}(1 - {Q_j})} \\{\sigma_{{Q_i}{Q_j}}^2 = {Q_i}(1 - {Q_j})/N{ }} \\\end{array} $$

(B.60)

To calculate the variance σ _UB ²

$$ \sigma_{U{B_i}}^2 = E[dU{B_i}dU{B_i}] = {\pi^2}{\sin^2}\pi {Q_i}\sigma_{{Q_i}}^2 = {\pi^2}{\sin^2}\pi {Q_i}{Q_i}(1 - {Q_i})/N $$

(B.61)

Similarly,

$$ \sigma_{U{B_i}U{B_j}}^2 = E[dU{B_i}dU{B_j}] = {\pi^2}\sin \pi {Q_i}\sin \pi {Q_j}{Q_i}(1 - {Q_j})/N $$

(B.62)

Since the differential nonlinearity of the ith level is defined as the ratio

$$ DN{L_i} = \frac{{U{B_i} - U{B_{i - 1}}}}{{{L_R}}} - 1 $$

(B.63)

where L _R is the length of the record, the uncertainty in DNL _i and INL _i measurements can be expressed as

$$ \begin{array} {lll}{\sigma_{DN{L_i}}^2 = \sqrt {{[\sigma_{U{B_i}}^2 + \sigma_{U{B_{i - 1}}}^2 - 2\sigma_{U{B_i}U{B_j}}^2]}} /{L_R}} \\{\sigma_{IN{L_i}}^2 = {\sigma_{U{B_i}}}/{L_R}{ }} \\\end{array} $$

(B.64)

The maximal uncertainty occurs for Q _i = 0.5, thus the previous equation can be approximated with

$$ \begin{array} {lll}{\sigma_{DN{L_i}}^2 \approx \sqrt {{\pi /{L_R}}} \times 1/\sqrt {N} } \\{\sigma_{IN{L_i}}^2 = \pi /2{L_R} \times 1/\sqrt {N} { }} \\\end{array} $$

(B.65)