1396
`
`IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 59, NO. 7, JULY 2012
`
`Statistical Analysis of ENOB and Yield in Binary
`Weighted ADCs and DACS With Random
`Element Mismatch
`
`Jeffrey A. Fredenburg and Michael P. Flynn
`
`the design deci-
`Abstract—Mismatch motivates many of
`sions for binary weighted, ratiometric converters, such as
`successive approximation (SAR) analog-to-digital converters
`(ADC), but the statistical relationship between mismatch and
`signal-to-noise-plus-distortion ratio (SNDR) has not been pre-
`cisely quanti(cid:191)ed. In this paper, we analyze the effects of capacitor
`mismatch in a binary weighted, charge redistribution SAR ADC
`and derive a new analytic expression relating capacitor mismatch
`and the effective-number-of-bits (ENOB). We then explore the
`statistics of this new expression and develop a model that accu-
`rately predicts yield in terms of ENOB. Finally, the major results
`of this paper are generalized into a simple and compact design
`equation that relates resolution, mismatch, and ENOB to yield
`for all binary weighted, ratiometric converters. The expressions
`derived in this paper offer practical insight into the relationship
`between mismatch and performance for all binary, weighted ratio-
`metric converters with these results validated through numerical
`simulations.
`Index Terms—Analog-digital conversion, analog integrated cir-
`cuits, mismatch, successive approximation registers, yield.
`
`I. INTRODUCTION
`
`S AR ADCs OFFER an attractive solution in low power
`
`applications. Due to the inherent energy ef(cid:191)ciency of
`charge redistribution DACs and the leveraged bene(cid:191)ts of
`scaling [1], SAR ADCs can provide power ef(cid:191)cient analog to
`digital conversion in systems that require moderate resolution
`and speed. However, speci(cid:191)c applications have speci(cid:191)c needs,
`and to ensure those needs are met, it is important for designers
`to have complete understanding of the design tradeoffs in the
`key building blocks of SAR ADCs, such as the capacitor DAC,
`the comparator, and the successive approximation registers.
`It is well established that mismatch degrades the overall per-
`formance of ADCs, and various techniques have been proposed
`to overcome this degradation [2]–[8]. However, a precise for-
`mulation of the relationship between mismatch, the effective
`number of bits (ENOB), and yield is still lacking. In practice,
`an ADC designer may need to target a particular ENOB spec-
`i(cid:191)cation, but when estimating the yield, only indirect metrics
`
`Manuscript received March 08, 2011; revised July 19, 2011 and July 19, 2011;
`accepted October 24, 2011. Date of publication January 16, 2012; date of current
`version June 22, 2012. This paper was recommended by Associate Editor R.
`Lot(cid:191).
`The authors are with the Department of Electrical Engineering and Com-
`puter Science, University of Michigan, Ann Arbor, MI 48109 USA (e-mail:
`fredenbu@umich.edu).
`Digital Object Identi(cid:191)er 10.1109/TCSI.2011.2177006
`
`such as integral nonlinearity (INL) or differential nonlinearity
`(DNL) are available. Although ENOB, INL, and DNL are im-
`portant indicators of ADC performance, ENOB is a better in-
`dicator of the overall system level performance, and with the
`yield expressions derived in this paper, ADC designers can more
`easily target system level performance objectives.
`The use of INL as a yield metric for data converters is preva-
`lent in literature, but has limited utility in system design. Al-
`though the bulk of the analytic work has focused on developing
`INL yield models for current-steering DACs [9]–[13] in the
`presence of transistor drain current mismatch [14], the major
`results of these works are also generally applicable to ADCs.
`According to [13], the analytical development of INL as a yield
`metric begins with [9], where the maximum deviation of the
`INL is introduced as a measure for distinguishing between good
`and bad current-steering DACs. Later in [10]–[13], we see a
`progression of re(cid:191)nements aimed towards improving the sta-
`tistical accuracy of INL based yield estimates. However, none
`of these works [9]–[13] offer a detailed comparison between
`INL yield measurements and other performance metrics such
`as signal-to-noise-plus-distortion ratio (SNDR), and it is unclear
`how to precisely interpret INL based yield estimates when tar-
`geting a speci(cid:191)c ENOB yield for ADCs and DACs.
`Examples of analysis relating INL/DNL to ENOB can be
`found in [16]–[18], [24], but these works do not contain a de-
`tailed statistical treatment relating ENOB and yield. In [16],
`DNL is related to signal-to-noise ratio (SNR) by considering
`DNL errors as an additive noise in (cid:192)ash ADCs. In [17], SNDR
`is related to INL errors as a function of the input signal proba-
`bility density function (PDF). In [18], ENOB is related to INL
`through a harmonic analysis for thermometer-coded structures,
`and in [24] an approximate relationship between ENOB and
`INL is given for resistor strings based on analysis in [15]. Al-
`though these works provide a convenient sketch relating ENOB
`and DNL/INL, it is unclear how to extract accurate ENOB yield
`information.
`In this paper, we develop an alternative statistical model using
`ENOB as a yield metric. First, we examine the effects of mis-
`match in a binary weighted, charge redistribution SAR ADC.
`We then derive an exact algebraic formulation relating capac-
`itor mismatch to the average noise power of the ADC output,
`and from this algebraic formulation, we derive ENOB as a func-
`tion of capacitor mismatch. Next, we explore the statistics of
`this ENOB expression and develop a statistical expression that
`predicts yield in terms of ENOB and mismatch. Finally, we gen-
`eralize the results of this work by presenting a compact design
`
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`
`1549-8328/$31.00 © 2012 IEEE
`
`IPR2021-00294
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`
`

`

`FREDENBURG AND FLYNN: STATISTICAL ANALYSIS OF ENOB AND YIELD IN BINARY WEIGHTED ADCS AND DACS
`
`1397
`
`squared value of the output noise voltage [18]. Assuming the
`DAC output codes are uniformly distributed, we can calculate
`this quantity as shown in (1) and (2)—where
`is the DAC
`resolution in bits,
`is the LSB,
`is the output noise voltage,
`and
`is the mean output noise voltage.
`
`(1)
`
`(2)
`
`We incorporate mismatch into this expression by modifying
`the limits of integration in (1) to include the INL errors of the
`DAC. Since the -th code transition voltage of a mismatched
`DAC is offset from the ideal transition voltage by the INL error
`of that code, we offset the integration limits in (1) by the INL
`error as shown in (3)—where
`is the INL error of the -th code
`expressed in LSB.
`
`(3)
`
`Evaluating the integral in (3) and simplifying the resulting
`expression, we obtain an expression for the noise power in terms
`of the INL, which is given by (4).2 A more intuitive formulation
`of (4) is also presented in (5).
`
`(4)
`
`(5)
`
`, the contribution from the mismatch
`In the limit of a large
`induced noise power can be approximated as the variance of the
`INL as shown in (6).3
`
`(6)
`
`Expressions (4)–(6) describe the average noise power of a
`single-ended DAC as the sum of the ideal quantization noise and
`the mean square of the INL errors. These results are generally
`applicable to all ADCs and DACs with both (cid:191)xed quantization
`levels and uniformly distributed DAC outputs and indicate that
`nonlinearities in the quantization levels manifest as an additive
`noise. This conclusion is also suggested in [16] for DNL errors.
`
`B. Analytic Formulation of DNL and INL
`We continue by formulating an expression for the noise power
`contributed by the INL errors as a function of capacitor mis-
`match. To this end, we (cid:191)rst introduce a capacitor mismatch
`model and then derive expressions for DAC DNL errors in terms
`of this model. Finally, we convert these DNL expressions into
`INL expressions and solve for the mean squared INL in terms
`of the capacitor mismatch parameters.
`
`Fig. 1. Transfer function and residual noise voltage of a capacitor DAC with
`mismatch (solid) and without mismatch (dashed). Without mismatch, the code
`transitions and DAC outputs occur in regular LSB intervals.
`
`equation, which accurately relates resolution, mismatch, and
`ENOB to yield for all binary weighted, ratiometric converters.
`The design equation offered is accurate to within
`0.17 bits for
`yield values between 0.5% and 99.5% and is consistent with
`standard test methodology.
`Section II analyzes the effects of mismatch and derives an
`expression for ENOB as a function of capacitor mismatch.
`Section III explores the statistics of this ENOB expression,
`and Section IV formulates an expression for yield. Section V
`develops a compact design equation for yield, ENOB and
`mismatch, which generalizes the results of this work.
`
`II. ANALYTICAL ENOB DERIVATION
`In this section, we derive an analytic expression for the ENOB
`of a binary weighted SAR ADC in terms of capacitor mismatch.
`Although we derive this expression from the perspective of a ca-
`pacitor DAC, our results are equally valid form the perspective
`of the ADC. We begin this derivation by relating the INL errors
`of a DAC to its average noise power. Next, we formulate an
`expression for the INL in terms of capacitor mismatch parame-
`ters, and use this relationship to express the mismatch induced
`noise power as a function of the capacitor mismatch. Finally, we
`translate the mismatch induced noise expression into an analyt-
`ical expression for ENOB which supports differential sinusoidal
`signals and is consistent with standard ADC test methodology.
`
`A. Mismatch Induced Noise Power
`The relationship between the mismatch induced noise power
`and INL can be derived in a manner similar to the calculation of
`ideal quantization noise power. By including INL errors into this
`calculation, we can capture the noise power contributed from
`INL errors.1 Fig. 1 shows the transfer function of a DAC and its
`corresponding noise voltage with and without INL errors. For
`an ideal single-ended DAC without mismatch, the established
`quantization noise expression represents the mean-
`
`1A related result in [16] expresses the average noise power of a (cid:192)ash ADC to
`its DNL errors.
`
`2The DC power contributed by the INL errors is not removed from (4).
`3A similar result to (6) is derived in [17] using a probabilistic approach.
`
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`
`

`

`1398
`
`IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 59, NO. 7, JULY 2012
`
`We model the mismatch of capacitors within the array as an
`additive random error—i.e.,
`, where
`is the nominal design capacitance and
`is a normally dis-
`tributed random error with zero mean with
`variance. Fur-
`thermore, we de(cid:191)ne a mismatch parameter
`to describe the
`fractional error of each binary weighted capacitor group from
`its ideal value. Assuming that the capacitors are carefully ar-
`ranged, we neglect pathological errors and effects from spatial
`gradients.4
`is the total
`The mismatch model is provided in (7)—where
`number of capacitors in the array,
`is the average unit ca-
`pacitance of the array,
`is the capacitance of the -th binary
`weighted capacitor group, and
`is the associated fractional
`mismatch of the -th group. In addition, we let
`represent
`the MSB,
`the LSB, and
`the termination capacitor.
`Note that the effective unit capacitance,
`, is distinct from the
`nominal design capacitance,
`.
`
`(7)
`
`Since the sum of the binary weighted capacitors, de(cid:191)ned in
`(7) as
`, must equal the total capacitance of the array, the
`weighted sum of the fractional mismatch parameters
`must
`sum to zero. This condition is enforced by (8).5
`
`(8)
`
`Using the capacitor mismatch model de(cid:191)ned in (7) and (8),
`we now relate the DNL errors of the DAC to the fractional
`mismatch parameter
`. The DNL error of a DAC can be ex-
`pressed by (9)—where
`is the difference between suc-
`cessive DAC output voltages [19].
`
`Using the DAC output voltage expression in (11) and the def-
`inition for DNL given in (9), we calculate the DNL errors for
`each of the
`DAC codes. For an
`bit, single-ended, binary
`weighted capacitor DAC, however, the DNL errors are uniquely
`determined by
`distinct DNL values, and these
`values rep-
`resent the DNL error at the major code transitions—speci(cid:191)cally,
`codes
`where
`.
`Intuitively, we can understand why the DAC has only
`unique DNL by examining the odd numbered codes. Since all
`the odd numbered codes have a binary representation ending in
`one, the difference in the DAC output voltage between these
`codes and one code less is determined solely by the DAC LSB
`capacitor. Therefore, the DNL error for every odd code is the
`same and is equal to the DNL error for code 2 , which is an odd
`code. Using similar examples, we can show through induction
`that only
`unique values are needed to describe the entire DNL
`of the DAC and these unique values are equal to the DNL at the
`major code transitions.
`We now calculate the DNL errors at the major code transi-
`tions by substituting (11) into (9)—where
`from (9) is
`the difference in the DAC output voltages between codes
`and
`. An expression for the
`unique DNL values is
`provided in (12)—where
`represents the DNL error at code
`, and
`
`.
`
`(12)
`
`The distribution of the DNL values given in (12) across each
`of the DAC codes can be described by the recursively ordered
`set shown in (13)—where
`is ordered set of DNL values,
`and
`, as described by (12), represents the DNL at the most
`signi(cid:191)cant code in the level of hierarchy. The arrangement of
`DNL values given by (13) describes a sequence in which the
`unique DNL values are distributed across the DAC codes in an
`“ modulo
`” manner.
`
`(9)
`
`(13)
`
`Furthermore, we can express the DAC output voltages
`in terms of the binary weighted capacitors as shown in
`(10)—where
`is the resolution in bits,
`is the LSB,
`is the -th binary weighted capacitor group, and
`represents the digital bits in the DAC code.
`
`(10)
`
`from (7) into (10), we relate
`Substituting the expression for
`the DAC output voltage to the fractional mismatch parameter
`as in (11).
`
`(11)
`
`As an example of how (13) describes the DNL distribution,
`, the arrangement of the
`we consider a 3 bit DAC. For
`DNL errors for this DAC is shown in (14)—where
`is again
`described by (12).
`
`(14)
`
`With both the DNL values and their arrangement calculated,
`we now relate the INL to the DNL and work towards expressing
`the mean-squared INL in terms of the mismatch parameter
`. The relationship between INL and DNL is shown in (15)
`[19]—where
`is the INL error at code , and
`is the DNL
`error at code
`. Furthermore,
`assumes one of the values
`described by (12) in an order determined by (13).
`
`(15)
`
`4An analysis relating INL errors to spatial gradients is found in [20].
`5Although based on the physical construction of the capacitor array, the con-
`given by (8) also ensures that gain errors in the transfer curve are
`straint on
`not counted as distortion since (8) imposes a unity gain for the DAC transfer
`curve.
`
`Substituting the DNL expression from (12) into (15) and
`simplifying the resulting summation by exploiting the inherent
`folding symmetry of (13), we derive the mean-squared INL in
`
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`

`

`FREDENBURG AND FLYNN: STATISTICAL ANALYSIS OF ENOB AND YIELD IN BINARY WEIGHTED ADCS AND DACS
`
`1399
`
`terms of the mismatch parameter
`shown in (16).
`
`. The simpli(cid:191)ed result is
`
`(16)
`
`Substituting this expression for the INL noise power from
`(16) into the noise power expression from (5), we obtain an
`explicit expression for the average noise power of an
`bit
`single-ended DAC with capacitor mismatch, which is shown in
`(17)—where
`is the LSB, and
`is the fractional mismatch of
`the -th capacitor group as de(cid:191)ned in (7). Furthermore, we let
`represent the MSB,
`the LSB, and
`the termi-
`nation capacitor.
`
`(17)
`
`The expression given in (17) describes the average noise
`power of a binary weighted DAC as the sum of the ideal
`quantization noise and a linear combination of the mismatch
`parameters squared. Similar to INL and DNL, the mismatch
`parameter manifests as additive noise.
`
`C. Differential Conversion
`Since most high-performance SAR ADCs process differen-
`tial signals, we now convert the noise power expression given
`by (17) from a single-ended result into a differential result. If
`we imagine constructing an
`bit, differential DAC using two
`bit, single-ended DACs, each with identical mismatch
`and opposite polarity,6 the average noise power of this com-
`posite differential DAC is the average of the two single-ended
`DAC noise powers. Using the results from (17) to describe the
`noise powers of the two
`bit single-ended DACs and av-
`eraging, we obtain the noise power of an
`bit, differential, bi-
`nary weighted DAC as given in (18)—where
`now describes
`the differential LSB,
`is the composite fractional mismatch of
`the -th capacitor groups, and
`and
`are the individual
`mismatch parameters from the positive and negative arrays.
`
`(18)
`
`Equation (18) presents an exact algebraic solution for the
`average noise power of a binary weighted
`bit, differential
`DAC with uniformly distributed INL errors. Furthermore, since
`the differential DAC output voltages are perfectly symmetrical
`about the origin, the noise power given by (18) is zero mean.
`Additionally, the constraint on
`given by (8) properly accounts
`for gain errors throughout the development of (18).
`
`D. Analytic Formulation of ENOB
`We now formulate an expression for ENOB in terms of the
`mismatch parameter
`. For a perfectly matched DAC, only the
`
`6This DAC structure represents a generic sign/magnitude encoded structure
`utilizing a (cid:191)xed common-mode output.
`
`quantization errors contribute noise and the average noise power
`is
`, as shown in (1). If we de(cid:191)ne an effective LSB size,
`which generates an average noise power equivalent to the noise
`power of a mismatched DAC, we can explicitly relate ENOB to
`the average noise power of the mismatched DAC as is done in
`(19)—where
`is the differential full scale range of the DAC
`output voltage, and
`is the effective LSB size.
`
`(19)
`
`Substituting the differential noise expression from (18) into
`(19), we can relate the ENOB of the DAC to the mismatch pa-
`rameters
`. Solving this resulting expression for ENOB, we ob-
`tain (20).
`
`(20)
`
`Equation (20) offers an exact analytic expression relating
`bit, differential, binary weighted capacitor
`the ENOB of an
`DAC to capacitor mismatch for uniformly distributed signals.
`Although we derived (20) from the perspective of a SAR ADC,
`the result provided in (20) is applicable to all binary weighted
`ratiometric converters.7
`
`E. Correction for Sinusoidal Distributions
`the
`The ENOB expression given in (20) assumes that
`DAC codes are uniformly distributed. In practice, however, the
`ENOB of an ADC is typically measured using a sinusoidal input
`signal, not a uniformly distributed signal. With a full-scale, uni-
`formly distributed signal, all of the INL errors across the entire
`code range each contribute equally to noise. On the other hand,
`since sinusoidal signals tend to dwell more near their peaks
`than their mean, the INL errors at the outer codes contribute a
`larger fraction of the noise than the INL errors near the center
`codes. Therefore, the noise power contributed by INL errors
`depends on the probability distribution of the signal.8
`To reconcile the ENOB expression in (20) with this preferred
`sinusoidal testing method, we introduce the scalar correction
`factor,
`, to convert the ENOB expression given by (20) into an
`equivalent expression describing the ENOB of a sinusoidally
`distributed input signal. The modi(cid:191)ed ENOB expression is
`given in (21)—where
`represents the composite fractional
`mismatch parameter of the binary weighted capacitor groups
`as de(cid:191)ned in (18) and (7), and
`is approximated as the ratio
`
`7For binary weighted ratiometric converter without an explicit termination
`is still de(cid:191)ned as in (8), but should instead be interpreted as either
`element,
`the mean of the single-ended INL errors or a description of the INL induced
`gain error of the converter transfer function.
`8If the INL error were constant across the code range, the INL induced noise
`power would be independent of the signal distribution. This is why the quanti-
`zation noise does not need to be scaled. By de(cid:191)nition, however, the INL errors
`across an extended code range must sum to zero and therefore cannot remain
`constant across the codes.
`
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`

`1400
`
`IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 59, NO. 7, JULY 2012
`
`between the INL noise contributions from a sinusoidal distribu-
`tion and a uniform distribution.9 A derivation for the estimated
`value of
`used in (21) is offered in Appendix I.
`
`termination capacitor,
`LSB capacitor,
`.
`
`, follows the same distribution as the
`
`(21)
`
`Equation (21) provides an accurate estimate for the ENOB
`of
`bit, differential, binary weighted ratiometric converters,
`which is consistent with the standard sinusoidal testing of the
`ADCs and DACs. Had we not introduced the correction factor
`, the ENOB expression would overestimate the mismatch in-
`duced noise power by 18%.10 Using (21), we can now accu-
`rately estimate the ENOB of a sinusoidal distribution over a
`wide range of
`values and compare results with standard ADC
`and DAC test measurements.
`
`III. STATISTICAL ENOB DERIVATION
`Section II provides an analytic expression relating ENOB and
`mismatch (21), and in this section, we examine the statistics of
`this ENOB expression. First, we derive the probability density
`functions (PDF) for the single-ended mismatch parameters
`and
`. Next, we use these PDFs for
`and
`to derive
`the PDF for the differential, composite parameter
`, and subse-
`quently, the PDF for the square of
`. Finally, we combine these
`results with the ENOB expression given by (21) and obtain a
`statistical expression for ENOB. Finally, we compare this ex-
`pression to results from numerical ADC simulations.
`
`A. PDF for the Single-Ended Mismatch Parameter
`In the capacitor mismatch model presented in (7), each ca-
`pacitor is modeled as
`, where
`is the
`nominal design capacitance and
`is a normally distributed
`error with zero mean with
`variance. The PDF for
`is shown
`in (22)—where the PDF is expressed using the notation
`.
`
`(22)
`
`Furthermore, both the binary weighted capacitor groups
`and the total array capacitance can be represented as sums
`of the individual capacitors. Since the sum of independent
`normal random variables is itself normal with a mean and
`variance equal to the sum of the constituent means and vari-
`ances, we obtain the marginal PDFs for the binary weighted
`capacitors directly from (22) as given in (23)—where
`is
`the single-ended resolution, and
`is the capacitance of the
`-th binary weighted capacitor group in one of the single-ended
`arrays. To avoid parametric equations, we will omit the PDF of
`the termination capacitor and note that the distribution for the
`
`9Alternatively, the ENOB of sinusoidally distributed DAC codes can be de-
`rived by replacing the “averaging” in (3) with the probability mass function
`(PMF) of a sinusoidal distribution, but it is unclear whether a tractable ENOB
`expression can be obtained due to the complexity of the sinusoidal PMF.
`10Since
`linearly scales only the mismatch induced noise power, the 18%
`.
`overestimation can be approximated by
`
`(23)
`
`Similarly, we derive the PDF for the total single-ended array
`capacitance from (22) as shown in (24)—where
`is the single-
`ended resolution, and
`is the total capacitance for one of the
`single-ended arrays.
`
`(24)
`
`Using the de(cid:191)nition of
`from the mismatch model given in
`(7), we next reformulate
`in terms of the new variables
`and
`as shown in (25). For convenience, we will denote the
`single-ended fractional mismatch parameter with
`. When we
`derive the composite mismatch parameter, we will clarify the
`notation with
`and
`.
`
`(25)
`
`is determined by ratio of
`As shown in (25), the PDF for
`two dependent normal variables,
`and
`, which results in a
`prohibitively complicated expression for the PDF.11 In order to
`simplify this PDF into a form amenable to further analysis, we
`will therefore expand (25) and approximate the capacitance of
`the array,
`, as a constant in the denominator. The expansion
`of (25) is given by (26) with
`approximated as
`in the
`denominator12—where
`is the mean capacitance of the array
`as de(cid:191)ned in (7).
`
`(26)
`
`Using the PDFs for
`from (23) and (24), we now
`and
`derive an approximation of the marginal PDF for
`through the
`expansion given by (26). The simpli(cid:191)ed PDF for
`is provided
`in (27)—note that the correlation between
`and
`in the nu-
`merator has not been neglected.
`
`(27)
`
`We now calculate the PDF for the composite mismatch factor.
`Using (27) to describe the distributions for
`and
`, we ob-
`tain the PDF for composite mismatch factor using the relation-
`ship for the mismatch factors given by (20), which states that the
`
`11An exact formulation of this PDF is derived in [15] to analyze nonlinearities
`in resistor strings.
`12Approximating
`follows from the weak law of large numbers
`as
`is well approximated by
`and is equivalent to assuming that
`when the number of capacitors is large.
`
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`

`

`FREDENBURG AND FLYNN: STATISTICAL ANALYSIS OF ENOB AND YIELD IN BINARY WEIGHTED ADCS AND DACS
`
`1401
`
`composite mismatch factor is the average of the single-ended
`mismatch factors. The PDF for the composite mismatch factor
`is provided in (28)—where
`is the differential resolution and
`is related to single-ended resolution,13
`, by
`.
`
`(28)
`
`Equation (28) provides an analytic expression for the PDF of
`the composite mismatch parameter
`of an
`bit differential
`DAC—where
`follows the same distribution as
`.
`
`B. Statistical ENOB Expression
`Since the ENOB expression in (21) depends on a linear com-
`, we now derive the PDF for the square of the
`bination of
`composite mismatch factor from the PDF of the composite mis-
`match factor. Letting
`—where
`represents the scalar
`coef(cid:191)cients from the ENOB expression given by (21), the PDF
`of
`follows a Chi-Squared distribution [21]. Using the PDF
`described from (28) and replacing the scalars
`with the ap-
`propriate values from (21), we calculate the distribution for
`,
`which is shown in (29)14—where the distributions for
`is de-
`scribed by the distribution for
`.
`
`(29)
`
`Substituting
`into the ENOB expression (21), we express
`the ENOB in terms of
`as shown in (30)—where the distribu-
`tions for
`are described in (29).
`
`(30)
`
`Equation (30) provides an analytic model describing the sta-
`tistics for the ENOB of an
`bit, binary weighted, differential
`SAR ADC with a normally distributed capacitor mismatch. Fur-
`thermore, this model includes a sinusoidal correction factor, so
`this statistical model is valid for sinusoidally distributed signals
`and is thus compatible with standard ADC test methods.
`
`C. Expected Value and Variance
`We verify the validity of (30) by comparing analytical expres-
`sions for the expected value and variance of ENOB to numer-
`ical simulations of randomly generated SAR ADCs. Because the
`ENOB expression in (30) contains a logarithmic term, we will
`estimate the expected value and variance using a Taylor series
`expansion.
`
`13This DAC structure represents the generic sign/magnitude encoded struc-
`ture described in II-C which utilizes a (cid:191)xed common-mode output.
`14This PDF is an approximation for the marginal PDF for
`.
`
`Fig. 2. Comparison between simulated and calculated expected values (32) for
`ENOB across various resolutions. The numerical simulation results are obtained
`using a 1024 point FFT of 300,000 randomly mismatched ADCs at each reso-
`lution and each standard deviation of capacitor mismatch.
`
`represent the sum of
`Letting
`expansion for the ENOB centered at
`
`in (30), the Taylor series
`is shown in (31).
`
`(31)
`Taking the expected value of (31) and dropping higher order
`terms, we obtain the approximation for the expected ENOB
`shown in (32).
`
`(32)
`
`Due to the complexity of including correlations between each
`in later analysis, we will neglect all correlations.15 Therefore,
`treating the
`from (30) as independent variables, we can ap-
`proximate the expected value and variance of
`as the sum of
`the expected values and variances of
`. Figs. 2 and 3 offer a
`comparison between the calculated and simulated values for the
`expected ENOB of a SAR ADC. As shown in Fig. 2, the calcu-
`lated ENOB values track the simulated values reasonably well,
`and in Fig. 3, we see that the analytic expected value is within
`1.0% of the simulated value over a wide range of resolution and
`mismatch.16
`Next, we obtain an expression for the ENOB variance. Taking
`the variance of (31) and dropping higher order terms, we derive
`(33).
`
`(33)
`
`15A comparison between the (cid:191)rst four moments of the ENOB expression
`given in (30) and the moments calculated from simulation data showed rea-
`sonable similarity, which included correlations, and the moments derived from
`treated as independent random variables.
`(30) with
`16In Fig. 3, however, the error in the expected ENOB is non-monotonic with
`respect to resolution, we attribute this to the (cid:191)xed 1024 point FFTs used to gen-
`erate the simulation data. With a (cid:191)xed 1024 FFT, only a subset of the output
`codes is measured for resolutions beyond 11 bits.
`
`Authorized licensed use limited to: OREGON STATE UNIV. Downloaded on March 10,2021 at 04:23:05 UTC from IEEE Xplore. Restrictions apply.
`
`

`

`1402
`
`IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 59, NO. 7, JULY 2012
`
`IV. YIELD ANALYSIS
`We complete the statistical analysis of ENOB with an exami-
`nation of the ENOB yield for an
`bit, binary weighted, differ-
`ential SAR ADC. Using the ENOB expression given in (30), we
`can express the probability of achieving some minimal ENOB
`in terms of the probabilities for
`as shown in (34)—where
`is the minimal desired ENOB, and
`is the ADC
`resolution in bits.
`
`(34)
`
`A. Full Yield Approximation
`We next derive an approximate ENOB yield expression in
`terms of the cumulative distribution function (CDF) for
`from
`(34). The details of this derivation are provided in Appendix II.
`When N is even number of bits, the CDF of
`can be approx-
`imated as in (35)—where
`denotes the CDF of
`,
`is the -th even
`from (29) including
`,
`is -th odd
`from (29), and
`denotes the value in the sequence.
`
`Fig. 3. Comparison between simulated and calculated expected values (32) for
`ENOB across various resolutions expressed as percent error. The analytic ex-
`1.0% of simulated values.
`pected ENOB values are within
`
`Fig. 4. Comparison between the simulated and calculated ENOB variances
`(33) across various resolutions. The numerical simulation results are obtained
`using a 1024 point FFT of 300,000 randomly mismatched ADCs at each reso-
`lution and each standard deviation of capacitor mismatch.
`
`When N is odd number of bits, the CDF of X can be approxi-
`mated as in (36)—where, again, where
`denotes the CDF
`of
`,
`is the -th even
`from (29) including
`,
`is
`-th odd
`from (29), and
`denotes the last value from (29).
`
`(35)
`
`from
`Similar to the expected value calculation, we treat the
`(30) as independent variables and approximate the variance of
`as the sum of the
`variances. Fig. 4 compares the calculated
`and simulated values for the ENOB variance.
`As shown in Fig. 4, the calculated variances compress at
`higher resolutions. This compression indicates a nonlinear rela-
`tionship between the calculated and simulated variances. Since
`the inclusion of higher order terms up to the fourth moment of
`in the Taylor series expansion did not reduce this error, we
`attribute the causes of this discrepancy to the scalar correction
`factor,
`, and the assumption that the
`are independent. While
`the correction factor
`correctly scales the expected ENOB to
`approximate a sinusoidal distribution,
`does not properly scale
`the higher moments. Furthermore, the
`are not independent
`since the mismatch parameters are correlated, which is evi-
`denced by (8). Nevertheless, the magnitude of the error between
`the calculated and simulated variances is small compared to the
`resolution of the ADC.
`
`(36)
`
`Using the ENOB relationship given by (34) along with