FUNCTIONALS OF BROWNIAN BRIDGES ARISING IN THE CURRENT MISMATCH IN D/A CONVERTERS

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1 Probability in the Engineering and Informational Sciences, 3, 009, Printed in the U.S.A. doi: /s FUNCTIONALS OF BROWNIAN BRIDGES ARISING IN THE CURRENT MISMATCH IN D/A CONVERTERS MARKUS HEYDENREICH AND REMCO VAN DER HOFSTAD Department of Mathematics and Computer Science Eindhoven University oftechnology 5600 MB Eindhoven, The Netherlands GEORGI RADULOV Department of Electrical Engineering Eindhoven University oftechnology EH 5.15, 5600 MB Eindhoven, The Netherlands Digital-to-analog converters DAC transform signals from the abstract digital domain to the real analog world. In many applications, DACs play a crucial role. Due to variability in the production, various errors arise that influence the performance of the DAC. We focus on the current errors, which describe the fluctuations in the currents of the various unit current elements in the DAC. A key performance measure of the DAC is the Integrated Nonlinearity INL, which we study in this article. There are several DAC architectures. The most widely used architectures are the thermometer and the binary and the segmented architectures. We study the two extreme architectures, namely the thermometer and the binary architectures. We assume that the current errors are independent and identically normally distributed and reformulate the INL as a functional of a Brownian bridge. We then proceed by investigating these functionals. For the thermometer case, the functional is the maximal absolute value of the Brownian bridge, which has been investigated in the literature. For the binary case, we investigate properties of the functional, such as its mean, variance, and density. 009 Cambridge University Press /09 $

2 150 M. Heydenreich, R. van der Hofstad, and G. Radulov 1. CURRENT MISMATCH IN DIGITAL-TO-ANALOG CONVERTERS Digital-to-analog converters DAC transform signals from the abstract digital domain to the real analog world. For many applications, this conversion enables the usage of the computational power of robust digital electronics; for example, digital audio and video, digital control, and telecommunications are fields that require digital-to-analog conversion. The advantageous intelligence of the applications in these fields is implemented with digital logic e.g., microprocessors and used via digital-to-analog D/A conversion in the real analog world. However, the DAC errors, at the end of the application chain, might decrease the performance of the whole system. Therefore, predicting and controlling these errors is crucial. This requirement is further emphasized in the highly integrated mixed-signal systems-on-a-chip SoC, as we will explain now. Currently, the SoC solutions often integrate DAC functionality together with the digital logic for cost-effectiveness. This requires optimal usage of the DAC resources while keeping the errors of the DAC within specified margins, primarily because of the two following reasons. First, the price of the mixed-signal SoC includes the price of the DAC resources even when customers are not interested in the DAC functionality. Second, if the DAC does not comply with its specifications, then the entire SoC chip should be discarded. By a careful design, the DAC errors mainly arise from the uncertainty in the manufacturing process and, hence, they are random. Therefore, statistical rules are used to predict the overall performance for high-volume chip production. Knowledge is required that accurately links the DAC resources with the DAC error margins. An important example of such a relationship is the dependence of the D/A conversion accuracy on the DAC area i.e., the silicon area of the chip that is used for the DAC. Higher conversion accuracy is achieved for larger chip areas [,14]. However, too large areas introduce additional problems that might degrade the conversion accuracy. Thus, precise knowledge and understanding of the relationship between accuracy and chip area is crucial, particularly for high-volume chip systems including DAC. For a general introduction to DAC converters, we refer to [,9,18], whereas Razavi [17] also focuses on technical aspects. The D/A conversion is carried out by switching certain analog quantities, such as voltages, currents, or charges, ON or OFF. For the sake of simplicity, and without loss of generality, this article will assume current as a basic analog quantity. The switching process is controlled by the digital input signal w {0, 1} N, where N is the length of the binary input signal. If all combinations of 0 s and 1 s between the digital bits produce valid input signals [i.e., N = log n + 1], then the coding is called binary and N is called the resolution of the DAC. A DAC that uses binary coding to control its current quantities is called binary DAC. The switched ON current quantities I ui are summed to construct the analog output signal current I out. We will assume that the current quantities are random and that {I uj } n are independent and identically distributed i.i.d. random variables. The sum of all current quantities I outmax = n I uj 1.1

3 FUNCTIONALS OF BROWNIAN BRIDGES 151 is associated to the maximal digital input word w n = 1, 1,...,1 and is called the full-scale current of the DAC. The smallest meaningful difference in the analog output, defined as the output least significant bit LSB, is defined as the full-scale current divided by the number of digital input codes; that is, I lsb := I out max n. 1. For general DAC, errors can be classified as static or dynamic. We will focus on the static errors, which we now introduce. For every digital input word w k {0, 1} N, there is an analog output value I outk. The code words w k {0, 1} N will be assumed to be ordered. The difference between two adjacent output values would ideally be Ī u = E[I uj ], but, in practice, it deviates due to mismatch errors coming from the uncertainty of the manufacturing process; that is, I outk I outk 1 = Ī u + I lsb Ī u + DNLk I lsb, k = 1,..., n. 1.3 Here, I lsb Ī u represents the linear error, that is independent of k, and DNL k = I out k I outk 1 I lsb I lsb 1.4 is the differential nonlinearity in I lsb scale. As the DAC are required to be linear devices, the nonlinear errors are the main concern. The integrated nonlinearity INL measures the nonlinearity of the whole DAC transfer characteristic i.e., the cumulated individual nonlinear errors. This article concentrates on the INL, as defined for a code w k by INL k := k i=1 DNL i = I out k ki lsb I lsb, k = 1,..., n. 1.5 This definition excludes the linear errors by forcing INL 0 = INL n = 0 and normalizes INL k to LSB scale. The maximal absolute INL over all codes is given by INL max := max k=0,...,n INL k. 1.6 The statistic INL max is an important specification, because it indicates how linear the DAC is. Another important figure isyield INL, which indicates how many of the manufactured chips have INL max under a certain specified threshold. This is to say, if a DAC manufacturer guarantees certain linearity, then Yield INL describes what proportion of the produced chips fall within these specifications. Integrated nonlinearity is the most popular DAC specification; see, for example, [9,18]. Its practical importance is very high in all DAC application fields. This general

4 15 M. Heydenreich, R. van der Hofstad, and G. Radulov definition of INL is valid for all DAC architectures and all DAC resolutions. The most commonly used architectures are the binary, segmented, and thermometer architectures. The segmented architecture interpolates between the binary and the thermometer architectures. In this article, we consider the two extreme cases of thermometer and binary DAC architectures. We now describe these DAC architectures. For a thermometer DAC see Fig. 1, I outk = I out T k is given by I T out k = k I uj, k = 0,..., n. 1.7 For a binary DAC see Fig., on the other hand, I outk I B out k = N m 1 B k,m = I B out k is given by j= m 1 I uj, k = 0,..., n, 1.8 FIGURE 1. Thermometer DAC. The codeword w k corresponds to switching T k,i ON for i k and switching T k,i OFF for i > k. FIGURE. Binary DAC. The codeword w k corresponds to switching those i for which B k,i = 1 ON, whereas the i for which B k,i = 0 are switched OFF. The matrix B is given in 1.9.

5 FUNCTIONALS OF BROWNIAN BRIDGES 153 where the switching matrix B is an n N matrix given by B = The matrix B is constructed by writing the first n integers in reversed order binary coding into the rows of B. Note that the columns of the matrix represent the DAC bits i.e., the switches of the grouped current sources. The 0 s represent switched OFF currents, and the 1 s represent switched ON currents. The leftmost column gives the switches for the LSB current I B out 1 switches for the most significant bit MSB current I B out N = I u1, whereas the rightmost column gives the = N 1 j= I N 1 u j. Note that I B out n = N m 1 B n,m j= m 1 I uj = n I uj = ni lsb, 1.10 so that INL n = 0. In practice, the most popular DAC architecture is the segmented one. The segmented DAC architectures implement one part of the input digital bits in a binary way and the other part in a thermometer way. That is how the advantages of both the binary and the thermometer architectures can be combined. Figure 3 shows a segmented DAC architecture. The LSB part is implemented in a binary way and the MSB part is implemented in a thermometer way. The output FIGURE 3. Segmented DAC.

6 154 M. Heydenreich, R. van der Hofstad, and G. Radulov current for a given level of segmentation S is expressed by N S m 1 I out S k = B k,m j= m 1 I uj + k/ N S m+1 N S 1 j=m N S I uj The parameter S {0,..., N} determines the interpolation between the binary and the thermometer parts of the DAC architecture. Indeed, for S = 0, the segmented DAC architecture is transformed to a fully binary architecture. On the other hand, for S = N, the segmented DAC architecture is transformed to a fully thermometer architecture. Note that in both extreme cases, the same unit current sources I uj are used. Thus, the performance difference is only due to the way the I uj are combined to construct the output current I outk. In the binary DAC, the unit currents are first grouped and then switched ON or OFF, whereas in the thermometer DAC, the unit currents are individually switched ON or OFF. More detailed discussions on DAC architectures can be found in the literature [9,18,19]. As discussed, due to manufacturing-related mismatch, the currents I uj always deviate from their designed values i.e., I uj = Ī u + ε j. The mean value Ī u is chosen by the DAC designer, whereas ε j is the random error due to the manufacturing process that we model as an i.i.d. sequence of normal random variable with zero mean and variance σu. Nevertheless, our results remain valid when the {ε j} are i.i.d. with sufficiently many moments. The ratio σ u /Ī u is known as the relative current matching i.e., the unit current matching. The relative matching determines the required transistor area for the particular manufacturing process. The smaller σ u /Ī u, the more accurate I uj, but the larger the required area. For more details on the dependence between relative matching and transistor area, we refer to [14]. Once σ u /Ī u is specified, the required transistor area can be calculated. However, the relationship between σ u /Ī u and DAC nonlinearity, which is crucial to determine the proportion of chips complying to specifications, has never been determined analytically. So far, DAC engineers have used either Monte Carlo simulations or approximations. The Monte Carlo simulations of a DAC model produce empirical results to suggest some design specifications σ u /Ī u. Although not accurate, this approach is very practical; see [5]. A problem arises for the design of a high-resolution DAC, for which N is large. The complexity of the DAC model increases by a factor of for every additional bit in resolution; for example, for N = 14, n = unit elements have to be simulated. Therefore, Monte Carlo simulations are not practical for higher resolutions because they become complex and slow. On the other hand, a number of analytical approximations can be found in the literature. The analytical attempts to describe INL, and in particular INL max, started in The approach in [11] disregards the correlation between the DAC outputs for different input codes. Bastos [] proposed a much simpler formula, which considers only the deviation of the transfer characteristic at the midscale DAC output, which can be a rough, though simple, estimation of the INL max.another approximation was given

7 FUNCTIONALS OF BROWNIAN BRIDGES 155 by van den Bosch, Steyaert, and Sansen [4] by assuming that if the DAC static transfer characteristic at any code INL has error equal to the target value e.g., INL k = 0.5I lsb, then there should be a 50% chance that ultimately INL max is smaller than the target value i.e., INL k 0.5 I lsb for all k. The major approximation inaccuracy is in the probability that both the positive and negative INL limits are reached for the same DAC sample; for example, INL k < 0.5 I lsb is disregarded. Although this approach derives a convenient Normal distribution for INL max, it is inaccurate for higher resolutions, as we show in more detail in this article. In general, approximations lead to transistor overdesign i.e., a transistor area that is too large. In this article, on the other hand, we will present an exact analytical formula, for which no approximation is necessary. Due to the lack of exact analytical formulation of INL and the high complexity of DAC model simulations, the statistics used in industry for a high-volume chip production are hard to predict. Here, we think of the statistics Yield INL, the INL max distribution, the INL max deviation, and the mean. Furthermore, the advantages of some redundancy-based approaches relying on the DAC statistical INL properties cannot be theoretically estimated; see, for example, [16]. Finally, up to now the main DAC architectures i.e., binary, thermometer, and segmented cannot be distinguished with respect to their static linearity properties, so they are wrongly considered identical [,4]. One conclusion from our results is that the INL for binary and thermometer architectures are different. Implications of the results derived in this article for the field of DACs can be found in a second article [15]. A comparison with the results in [4,5,11] is summarized in [15, Table 1].. THERMOMETER CODING: MAXIMUM OF A BROWNIAN BRIDGE For the thermometer coding, we can describe the INL as functional of a Brownian bridge as follows. THEOREM.1 INL max for the Thermometer Coding: As n, I lsb σ u n INL max X,.1 in distribution, in L 1 and L, where the limit X is characterized by for a Brownian bridge {B s } s [0,1], and X = max t [0,1] B t. π E[X] = ln , VarX = π 1 π ln ,.3

8 156 M. Heydenreich, R. van der Hofstad, and G. Radulov and PX x = k e k x, x > 0..4 k=1 Note that in.1, we multiply by I lsb rather than by Ī u. For the convergence in distribution, this makes no difference whatsoever, since I lsb converges to Ī u a.s. by the strong law of large numbers. However, the convergence in L 1 and L fails if we multiply by Ī u, since, in this case, the expected value of INL k = I outk ki lsb /I lsb is not defined, whereas INL k has infinite mean. We recall that a Brownian bridge {B s } s [0,1] is a Markov process on [0, 1] that is obtained from a Wiener process or Brownian motion in either of the following two ways: B1 B s = W s sw 1, where {W s } s [0,1] is a Wiener process. B B s = W s, where {W s } s [0,1] is a Wiener process conditioned on W 1 = 0. For an introduction to Wiener processes and Brownian bridges, we refer to [7]. For the equivalence of B1 and B, cf. [3, pp ]. PROPOSITION.: For the thermometer coding, max I T k=0,...,n out k ki lsb D = σu n max B k/n,.5 k=1,...,n where {B t } t [0,1] is a Brownian Bridge process and D = denotes equality in distribution. PROOF: Since {I uj },...,n is a family of i.i.d. normally distributed random variables with mean Ī u and variance σu, we have that I T out k ki lsb = k Iuj k Ī u.6 is normally distributed with mean 0 and variance kσu. Hence, for a Brownian motion {W t } t 0, I out T D D k ki lsb = σu W k = σu nwk/n,.7 where we used Brownian scaling in the last distributional equality. By substituting k = n and using 1., we obtain ni lsb nī u = D σ u nw1. Combined with.7,

9 FUNCTIONALS OF BROWNIAN BRIDGES 157 this yields I T out k ki lsb D = σu n W k/n k n W 1 D = σ u nbk/n.8 for a Brownian bridge process {B s } s [0,1], where we used B1. After taking absolute values on both sides of.8 and the maximum over k = 1,..., n, we obtain.5. The maximum over the discrete-time points k/n, k = 1,..., n, in.5 can be replaced by the supremum over the whole interval [0, 1] by using the following lemma: LEMMA.3: For C > 4, P max k=1,...,n Bk/n max B log n t C 4n 1 C /8 + n Cn/8..9 t [0,1] n In particular, max k=1,...,n Bk/n converges to maxt [0,1] B t in distribution as n. Moreover, L 1 - and L -convergence holds. The proof of Lemma.3 is deferred to the Appendix. We now use Lemma.3 to complete the proof of Theorem.1. PROOF OF THEOREM.1: The convergence in.1 follows from Proposition. and Lemma.3. For the probability of the upper tail of X [.4], we refer to [3, 11.39]; the computation of mean and variance of X is straightforward integration. For example, we have that E[X] = PX > x dx = 1 k 1 e k x dx = k=1 1 k 1 π 4k k=1 π = ln. The interchange of summation and integral can be justified by looking at X ε = X ε. 3. BINARY CODING: BLOCK INCREMENTS OF A BROWNIAN BRIDGE 3.1. Results and Overview Proof In this section we prove the following theorem characterizing the binary coding statistic.

10 158 M. Heydenreich, R. van der Hofstad, and G. Radulov THEOREM 3.1 INL max for the Binary Coding: As n, I lsb σ u n INL max M, 3.1 in distribution, in L 1 and L, where the limit M is characterized by M = 1 l+1/ l 1 l j Z j Z l 3. for a family {Z l } of i.i.d. standard normal random variables. The expectation of M is given by E[M] =π 1/ l l 1/ , 3.3 and the variance VarM is computed explicitly in 3.3 and can be approximated as VarM Note that in 3.1, we multiply by I lsb rather than by Ī u. As explained for Theorem.1, this makes no difference for the convergence in distribution, although the convergence in L 1 and L fails. The proof of Theorem 3.1 is organized as follows. In Lemma 3.3 we prove the convergence in 3.1, where the limit is characterized in terms of increments of Brownian bridges. After this, Proposition 3.4 shows that the weak limit M can be expressed as the weighted sum of standard normal variables, which proves 3.. Lemma 3.7 states the expression for mean and variance of M. Finally, we give an approximation to the density of M in Section A Brownian Bridge Representation of the Binary INL max Our aim is to derive an expression for INL max for the binary coding. First, we express the nonlinearity of the current steering DAC in terms of a functional of a Brownian bridge. LEMMA 3.: max k=1,...,n I B out k ki lsb D σ u N = n B m 1/n B m 1 1/n, 3.5 where {B s } s [0,1] is a Brownian bridge.

11 FUNCTIONALS OF BROWNIAN BRIDGES 159 Proof: Let {W t } t [0, be a Wiener process; then I uj Ī u D = σu n Wj/n W j 1/n, j = 1,..., n, 3.6 where D = represents equality in distribution. We will further write = rather than D =, because we are interested in the distribution only. Furthermore, I lsb Ī u = 1 n n Iuj Ī u = 1 n σ uw n = 1 n σ u W 1, 3.7 where we recall that the I uj are i.i.d. N Ī u, σ u -distributed. We want to calculate I B out k ki lsb for k being a power of first. The advantage is that, for k = m 1, only the mth block {I uj j = m 1,..., m 1} contributes: I B out m 1 I m 1 lsb = I B out m 1 Ī m 1 u + m 1 Ī u I lsb = m 1 j= m 1 Iuj Ī u + m 1 Ī u I lsb = σ u n W m 1/n m 1 W 1 n W m 1 1/n m 1 1 W 1 n = σ u n B m 1/n B m 1 1/n, where, in the last line, we usedthe representation B1 of Brownian bridges. For calculating max k=1,...,n I B out k ki lsb, we need to take the maximum over every configuration of contributing blocks; hence, max I B k=1,...,n out k ki lsb = σu n max B I {1,...,N} m 1/n B m 1 1/n. 3.8 m I }{{} :=M N We denote by I the subset of {1,..., N} for which the maximum in 3.8 is achieved, and we use the abbreviation ˆB n,m := B m 1/n B m 1 1/n, m = 1,..., N, 3.9 for the increment of the Brownian bridge on the interval [ m 1 1/n, m 1/n ]. Without loss of generality, we may assume that ˆB m I n,m is positive, otherwise the

12 160 M. Heydenreich, R. van der Hofstad, and G. Radulov same argument for B holds. Clearly, Furthermore, 0 = B 1 B 0 = m I ˆB n,m N ˆB n,m = N N ˆB n,m 1l ˆB n,m 0 + ˆB n,m 1l ˆB n,m Using 3.10 in the first line and 3.11 in the second, we obtain N M N = ˆB n,m 1l ˆB n,m 0 which is 3.5. We write as in 3.8 and define = 1 = 1 N ˆB n,m 1l ˆB n,m N ˆB n,m, M := 1 where B denotes a Brownian bridge. N ˆB n,m 1l ˆB n,m 0 M N = 1 max I B σ u n out k=1,...,n k ki lsb 3.1 B l 1 B l, 3.13 LEMMA 3.3: There exists a constant C > 0, such that P M N M >ε CN ε N/ 3.14 for every ε>0. In particular, M N converges to M in distribution as N. Moreover, it converges in L 1 and in L. We show in Proposition 3.4 that M D = M. The proof of Lemma 3.3 is deferred to the Appendix. The combination of Lemmas 3. and 3.3 yields the convergence 1 σ u n max I B k=1,...,n 1 out k ki lsb B l B l in distribution, in L 1 and L,asN and, thus, also n = N 1. This proves 3.1. l=0

13 FUNCTIONALS OF BROWNIAN BRIDGES A Representation of M in Terms of i.i.d. Standard Normals In this section, we prove the following representation formula, which expresses M in terms of independent standard normal random variables. PROPOSITION 3.4 Rewrite of M in Terms of Standard Normals: Let Z 1, Z,... be a sequence of i.i.d. standard normal distributed random variables. Then M can be expressed as M = D 1 l 1 l+1/ l j Z j Z l In other words, M in 3.13 has the same distribution as M in 3.. In order to obtain the limit law of M, we have made use of the representation B1 of the Brownian bridge. Now, we will primarily use B. We will essentially use the following well-known property of Brownian motion. LEMMA 3.5 The Conditional Law of the Middle Point: Let {W s } s=0 be a standard Brownian motion. Then the distribution of W t/ conditional on W t = z is a Normal distribution with mean z/ and variance t/4. LEMMA 3.6 Distribution of {B l} : The distribution of {B l} is equal to B l = l j+1/ l j Z j, 3.17 where {Z j } are i.i.d. standard normal random variables. PROOF: The proof is by induction in l.forl = 1, we use Lemma 3.5 together with the fact that {B s } s [0,1] is a Brownian motion conditioned on B 1 = 0. This implies that the distribution of B 1/ is equal to a normal random variable with mean 0 and variance 1/4. Therefore, B 1/ = 1 Z 1, 3.18 where Z 1 is a standard Normal distribution. This initializes the induction hypothesis. To advance it, assume that l 1 B l 1 = j+1/ l 1 j Z j Then, again using Lemma 3.5, the distribution of B l conditionally on B l 1 is a Normal distribution with mean 1/ B l 1 and variance l+1. Therefore, if we

14 16 M. Heydenreich, R. van der Hofstad, and G. Radulov denote Z l = l+1/ B l 1 B l 1, then Z l is a standard normal random variable independent of B l 1. As a consequence, we have that B l = l+1/ Z l + 1 l B l 1 = j+1/ l j Z j, 3.0 where, in the last step, we have used the induction hypothesis. PROOF OF PROPOSITION 3.4: As a consequence of Lemma 3.6, we obtain the identity l 1 B l 1 B l = j+1/ l j Z j l+1/ Z l. 3.1 Thus, by 3.13, we obtain The Moments of M In this section, we identify the first two moments of M. LEMMA 3.7 Moments of M: The expectation of M is given by 3.3; that is, E[M] = 1 l l 1/ , 3. π and the variance is given by VarM = 1 l l 1/ k k 1/ 3.3 π 1 l<k 1 ρlk 1 + ρ lk arcsinρ lk + 1 1, 6 π with l+k ρ lk = l. 3.4 l k k The variance of M can be approximated by VarM In the proof of Lemma 3.7, we will make use of the following property of the bivariate Normal distribution. LEMMA 3.8 Expected Product of Absolute Values of Normals: Let Y 1 and Y be two standard normal random variables having a bivariate Normal joint distribution with correlation coefficient ρ. Then E[ Y 1 Y ] = 1 ρ + ρ π π arctan ρ = 1 ρ + ρ arcsinρ. 1 ρ π 3.5

15 FUNCTIONALS OF BROWNIAN BRIDGES 163 For a proof of Lemma 3.8, see, for example, [10, Excercise 15.6]. PROOF OF LEMMA 3.7: The expression for the mean is easily derived. We note that l 1 N l := B l 1 B l = j+1/ l j Z j l+1/ Z l, l = 1,,..., 3.6 is normally distributed with mean 0 and variance l v l = j+1 l j = l l, 3.7 and we rewrite M in 3. as M = 1 N l. Since, for a normal random variable Z with mean 0 and variance σ, we have that σ E[ Z ] = π, 3.8 the representation formula 3.16 allows us to identify the mean of the random variable M as E[M] = 1 E [ N l ] = 1 vl = 1 l l 1/ π π 3.9 For the variance of M, we expand VarM = 1 Cov N l, N k + Var N l l<k The variance term is not too hard, as and, therefore, Var N l = Var N l = E[N l ] E[ N l ] = 1 v l π 1 l l = π 3 π Now, fix 1 l < k. Then l 1 CovN l, N k = Cov j+1/ l j Z j l+1/ Z l, l j+1/ l j Z j l 1 = j+1 l j k j l+1 k l 3.33 = l+k.

16 164 M. Heydenreich, R. van der Hofstad, and G. Radulov Therefore, N l, N k is a bivariate Normal distribution with mean 0, 0, variances vl, v k, and correlation coefficient CovN l, N k ρ lk = VarNl VarN k = l+k l l k k Denoting Y l, Y k a bivariate normal distribution with means 0, variances 1, and correlation coefficient ρ lk,wehave Cov N l, N k = v l v k Cov Y l, Y k = v l v k E[ Yl Y k ] E[ Y l ] E[ Y k ] = v l v k E[ Y l Y k ] π By Lemma 3.8, as well as , we obtain VarM = 1 vl v k 1 ρlk π 1 + ρ lk arcsinρ lk π 1 l<k Using 3.7 and 3.34, we can approximate VarM numerically, which yields 3.4. Having completed the proofs of , the proof of Theorem.1 is complete Approximating the Density of M We now derive a formula for the density of M. Without loss of generality, we may assume that Ī u = 0 and σ u = 1; that is, the I u are standard normal distributed. By denoting N = N 1, N,...and Z = Z 1, Z,..., we rewrite 3.6 with the help of the infinite matrix L as N = L Z, where j+1/ l j if j < l L jl = l+1/ if j = l 0 if j > l Note that L is a lower triangular matrix. We will approximate the density of the infinite sum M = 1/ N l by the finite sum M m = 1 m N l, m N Writing N m = N 1,..., N m, Z m = Z 1,..., Z m, and L m for the upper left m m corner of the infinite matrix L, we have that N m = L m Z m. In particular, N m is

17 FUNCTIONALS OF BROWNIAN BRIDGES 165 normally distributed with mean 0,...,0 and covariance matrix m = L m L m T, where we write L m T for the transpose of the matrix L m. Note that m jl = { j+l if j = l l l if j = l Given the mean and covariance matrix of a multivariate Normal distribution, its density is known to be { 1 1 f N mn = π m/ det m exp 1 1/ n m } 1 n T, 3.40 where n = n 1,..., n m R m. We write N m for the pointwise absolute value N 1,..., N m of the m-dimensional vector N m. Its density is given by f N m n = 1 1 π m/ det m 1/ σ { 1,1} m { exp 1 } σ n m 1 σ n T, n [0, m, 3.41 where σ n = σ 1 n 1,..., σ m n m. See, for example, [1, 6.3.0]. The determinant det m and the inverse m 1 of the covariance matrix are easy to compute since L m is a triangular matrix. The results are stated in the following two lemmas. LEMMA 3.9: For all m N, the determinant of m is PROOF: First, we note that det m = mm+3/. 3.4 det m = det L m Since L m is a triangular matrix, its determinant is obtained by multiplying the entries on the diagonal and, hence, m det m = L m jj m = l+1/ = mm+3/ LEMMA 3.10: For all m N, the inverse of m is given by { m 1 j l j+l/ 1 for j = l = jl j + 1 j 1 for j = l. 3.45

18 166 M. Heydenreich, R. van der Hofstad, and G. Radulov PROOF: Since L m is a triangular matrix, it is easy to see that the inverse L m 1 is given by j 1/ if j > l L m 1 jl = j+1/ if j = l if j < l. For j < l, we have that m 1 jl = j l L m 1 jk L m 1 lk = k=1 whereas, for j = l, m 1 jj = j j 1/ l 1/ = j j+l/ 1, 3.47 k=1 j 1 j L m 1 jk = j 1/ + j+1/ = j + 1 j k=1 k=1 In order to calculate the density of M m = 1 N1 + + N m at y 0, we have to integrate 3.40 over the m 1-dimensional surface {n 1,..., n m y = n n m }. This leads to the formula y f M my = 0 y n 1 0 y n 1 n m 0 f N m n 1, n,..., n m 1, y n 1 n m 1 dn m 1 dn dn 1, y [0, Note that there are m 1 integrals. Thus, for m = 1, there are no integrals and f M 1y = f N 1 y = 4 π exp{ 8y }, y [0, Finally, 3.49 gives us a formula for the density of M m. The only numerical problem is the integrals. For larger values of m, an intelligent way of numerical integration seems necessary. However, for sufficiently small m, a mathematical standard package, such as Mathematica, gives good approximations see Fig. 4.

19 FUNCTIONALS OF BROWNIAN BRIDGES m =1 m = m =3 m =4 m =5 m =6 m =7 m =8 density of X for different values of m, in contrast to the den- FIGURE 4. The density of M m sity of X A Disintegration Approach to the Density of M In this section we present a different approach to the density of M. We define the quantity M = 1 W l 1 W l 3.51 for a Wiener process {W s } s [0,1]. It is obvious that M = D M W1 [cf and B]. =0 Let ˆf be the Fourier transform of the joint distribution of M and W 1 ; that is, ˆf k 1, k = E [ exp{ik 1 M + k W 1 } ]. 3.5 Using the independence and stationarity of the increments of the Wiener process in 3.51; we obtain ˆf k 1, k = E [ exp{ik 1 W l 1 W l +k W l 1 W l} ], 3.53 }{{} =ĥ l k 1,k where ĥ l k 1, k = E [ exp{ik 1 Z +k Z} ], 3.54

20 168 M. Heydenreich, R. van der Hofstad, and G. Radulov and Z is a N 0, l -distributed normal random variable. Once we have computed ĥ l, we obtain the joint density of M, W 1 via inverse Fourier transformation as f M,W1 x, y = exp{ ik 1 x ik y} ˆf k 1, k dk 1 dk π The density of M equals the density of M conditioned on W 1 = 0, from which f M x = f M,W 1 x,0 f W1 0 = π It remains to calculate ĥ l k 1, k.ifwelet exp{ ik 1 x} ˆf k 1, k dk 1 dk π ĥ l k = E [ e ikz 1l {Z 0} ], 3.57 then ĥ l k 1, k = ĥ l k 1 + k + ĥ l k 1 k. The dependence of ĥ l on l can easily be eliminated by scaling: ĥ l k = ĥ l/ k, 3.58 where ĥ is as in 3.57 with Z being N 0, 1-distributed. It can be shown that ĥk = 1 e k / +i e k / π k but the inverse Fourier transform in 3.56 seems intractable. k e x / dx, CONCLUSIONS We have derived the distribution of the INL max in terms of Brownian bridges. This distributional identity holds for all architectures, in particular also for the segmented one. In the thermometer case, we have identified the limiting distribution of INL max as the absolute maximum of Brownian bridges, which is well known in the literature. For the binary case, we have identified the limiting distribution of INL max in terms of a Brownian bridge. We have further provided a representation in terms of independent standard normal variables and have computed the mean and variance of INL max. Finally, we have given a procedure that approximates the density. We want to emphasize that the INL max in the thermometer case and in the binary case behave differently. Although the densities look alike e.g., the upper tails are quite close to each other, there are significant changes in the lower tail. The thermometer case has, compared to the binary case, a slightly larger mean but a slightly smaller variance. Even though the distributions in the two cases are close, they are not the same.

21 FUNCTIONALS OF BROWNIAN BRIDGES 169 We still miss the distribution function for the binary case. Random sums of the type in 3.13 have appeared in the literature. In particular, the quantity S = l V l, 4.1 where {V l } are i.i.d. exponential random variables, arises in a variety of applications; see, for example, Ott, Kemperman, and Mathis [13], Guillemin, Robert, and Zwart [8], and Litvak and van Zwet [1]. The density of S can be expressed in terms of an infinite sum; cf. [13, Sect. 5]. When the summands are independent uniform random variables [i.e., {V l } are i.i.d. uniform random variables on 0, 1] the density of S can be computed explicitly [6]. Furthermore, it would be interesting to extend the results to the segmented case. In particular, it would be of interest to investigate which limiting INL max distribution has the smallest mean. This should correspond to the optimal DAC architecture. Practical implications of our results can be found in a companion article [15]. Acknowledgements The work of MH and RvdH was supported by the Netherlands Organisation for Scientific Research NWO, and the work of GR was supported by STW, project ECS We thank Marko Boon for help with the simulations in Figure 4. We thank David Brydges for enlightening discussions on multivariate Normal distributions and Olaf Wittich for pointing our attention to the disintegration approach in Section 3.6. References 1. Bain, L.J. & Engelhardt, M Introduction to probability and mathematical statistics, nd ed. Pacific Grove, CA: Duxbury.. Bastos, J Characterization of MOS transistor mismatch for analog design. Ph.D. thesis, Katholieke Universiteit Leuven. 3. Billingsley, P Convergence of probability measures. New York: Wiley. 4. Bosch, A.v.d., Steyaert, M. & Sansen, W Static and dynamic performance limitations for high-speed D/A converters. Amsterdam: Kluwer Academic Publications. 5. Conroy, C.S.G., Lane, W.A., Moran, M.A., Lakshmikumar, K.R., Copeland, M.A., & Hadaway, R.A Comments, with reply, on Characterization and modeling of mismatch in MOS transistors for precisision analog design. IEEE Journal of Solid-State Circuits 31: Fey-den Boer, A. 006 Personal Communication. 7. Grimmett, G.R. & Stirzaker, D.R Probability and random processes, 3rd ed. NewYork: Oxford University Press. 8. Guillemin, F., Robert, P., & Zwart, B AIMD algorithms and exponential functionals. The Annals of Applied Probability, 141: Jesper, P.G.A Integrated converters. Oxford: Oxford University Press. 10. Kendall, M. & Stuart, A The advanced theory of statistics, Volume 1, 4th ed. London: Griffin. 11. Lakshmikumar, K., Hadaway, R., & Copeland, M Characterization and modeling of mismatch in MOS transistors for precision analog design. IEEE Journal of Solid-State Circuits 16: Litvak, N. & van Zwet, W.R On the minimal travel time needed to collect n items on a circle. The Annals of Applied Probability, 14: Ott, T.J., Kemperman, J.H.B., & Mathis, M The stationary behavior of ideal TCP Congestion Avoidance. Available at

22 170 M. Heydenreich, R. van der Hofstad, and G. Radulov 14. Pelgrom, M.J.M., Duinmaijer, A.C.J., & Welbers, A.P.G Matching properties of MOS transistors. IEEE Journal of Solid-State Circuits 45: Radulov, G.I., Heydenreich, M., Hofstad, R.v.d., Hegt, J.A., & Roermund, A.H.M.v Brownian bridge based statistical analysis of DAC INL caused by current mismatch. IEEE Transactions on Circuits and Systems II: Express Briefs 54: Radulov, G.I., Quinn, P.J., van Beek, P.C.W., Hegt, J.A., & van Roermund, A.H.M A binaryto-thermometer decoder with built-in redundancy for improved DAC yield. In ISCAS, Kos, Greece. 17. Razavi, B Design of analog CMOS integrated circuits. New York: McGraw-Hill. 18. Van de Plassche, R.J Integrated analog-to-digital and digital-to-analog converters. Amsterdam: Kluwer Academic Publishers. 19. Wikner, J.J Studies on CMOS digital-to-analog converters. Ph.D. thesis, Department of Electrical Engineering, Linköping University. APPENDIX A Proof of Lemmas.3 and 3.3 PROOF OF LEMMA.3: We first prove.9. Let {B s } s [0,1] be a Brownian bridge. Then max Bk/n max B t k=1,...,n t [0,1] max k=1,...,n t [ k 1 n, k n ] Bk/n B t. A1 Using representation B1 we can further bound A1 from above by max Wk/n W t 1 + k=1,...,n n W 1 t [ k 1 n, k n ] A for a Wiener process {W s } s [0,1]. Using the Markov property and Brownian scaling, we obtain that for k = 1,..., n, max Wk/n W t D 1 = n max W t, A3 t [ k 1 n, k n ] t [0,1] where D = stands for equality in distribution. Hence, for C > 0, P max Bk/n max B log n t C k=1,...,n t [0,1] n { P max max Wk/n W t C log n k=1,...,n t [ k 1 n, k n ] n n P max W t C log n + P W 1 C t [0,1] } + P 1 n W 1 C log n n n log n. A4 For the first term, we bound for every b 0, P max W t b P max W t b = 4 PW 1 b 4e b /, A5 t [0,1] t [0,1]

23 FUNCTIONALS OF BROWNIAN BRIDGES 171 where we use the reflection principle [7, Them. 6, p. 56] in the second and a standard bound on the tail of standard normals in the third step. Substituting b = C/ log n, we obtain n P max W t C log n n 4n C/ / = 4n 1 C /8. A6 t [0,1] For the second term in A4, we obtain analogously P W 1 C n log n P W 1 C n log n n Cn/8. The bound A4 together with A6 and A7 proves.9. For the convergence in L 1, we use that, with X n = max Bk/n, k=1,...,n X = max t [0,1] B t, A7 A8 we have that E X n X = P X n X t dt. A9 0 We split between t 4 log n/n and t > 4 log n/n. Fort 4 log n/n, we bound P X n X t 1, whereas for t > 4 log n/n, weusea6 and A7 to bound P X n X t 6ne nt /. A10 Substitution of these bounds yields log n E X n X = P X n X t dt 4 + 6n e nt / log n dt = O. 0 n log n 4 n n A11 The convergence in L follows similarly, now using E[X n X ]= t P X n X t dt. 0 We leave the details to the reader. A1 PROOF OF LEMMA 3.3: We observe that, by 3.5, 3.1 and 3.13, and replacing m by N m, MN M N 1 B m B N m 1/n + B m B m 1. A13 m=n Therefore, for an arbitrary chosen constant ε>0, we have that P MN M N 1 >ε P B m B ε N m 1/n > + P B m B m 1 > ε. m=n }{{}}{{} i ii A14

24 17 M. Heydenreich, R. van der Hofstad, and G. Radulov Using representation B1, we see that for a Brownian bridge {B s } s [0,1], and any 0 s < t 1, the following holds for every constant C > 0: P B t B s > C P W t W s t sw 1 > C P W t W s > C/ + P t sw 1 > C/. A15 Using the Markov inequality and Gaussian scaling, we obtain P B t B s > C C t s E[ W1 ] + C t s E[ W 1 ] 4 t s. C π A16 We use A16 to bound i in A14 from above by N 1 B P m B N m 1/n > N 1 ε N 1 8N 1 1 ε m N m 1 N 1 8N ε N/, A17 which converges to 0 as N. The second term ii in A14 can be bounded using the Markov inequality by [ ii ] ε E B m B m 1. A18 m=n Using 3.9, this expectation can be computed as [ ] 1 E B m B m 1 = 1 m m 1/ m=n π m=n and ii ε π m=n m m 1/ 4 ε π 1 N/, A19 A0 which converges to 0 as N. The combination of A14, A17, and A0 shows that for C = π π 1, P MN M >ε CN ε N/ A1 for every ε>0, that is, M N converges to M in distribution as N. The convergence in L 1 and in L follows as in the proof of Lemma.3.