High-Order Mismatch-Shaping in Multibit DACs - IEEE Xplore
346
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 58, NO. 6, JUNE 2011
High-Order Mismatch-Shaping in Multibit DACs Nan Sun, Member, IEEE
Abstract—This brief reports a high-order mismatch-shaping technique for multibit ΔΣ digital-to-analog converters (DACs). It builds upon the vector-based method but with two key improvements. First, a new vector quantization (VQ) scheme exploits the information in all vector variables. This VQ scheme allows the second-order mismatch shaping for large input signals and makes possible the stable third- and fourth-order mismatch shaping. Second, a hardware-efficient method maintains the average of the vector variables bounded. Simulations under various conditions prove its validity. Index Terms—Data converter, mismatch shaping, ΔΣ DAC.
II. R EVIEW OF THE M ISMATCH -S HAPED M ULTIBIT ΔΣ DAC The structure of a mismatch-shaped multibit ΔΣ DAC is in Fig. 1. By supposing that the quantizer in the ΔΣ modulator has M steps with each step size of 1/M , then, v[n] takes a value in {0, 1/M, 2/M, . . . , (M − 1)/M, 1}. The element selection logic (ESL) block decodes v[n] into vector sv[n] as follows: sv[n] = [sv 1 [n], sv 2 [n], . . . , sv M [n]]
where each sv i [n] takes a value of either 1 or 0 and drives one element in the unit-element DAC. The ESL block ensures that
I. I NTRODUCTION MULTIBIT ΔΣ DAC is more favorable than a binary one for its better stability and SNR. The relative drawback of the multibit DAC is that it does not guarantee linearity due to element mismatches. To address this issue, researchers have developed various mismatch-shaping techniques [1]–[14], among which the techniques in [1]–[7] provide the first-order mismatch shaping and the techniques in [8]–[14] provide the second-order mismatch shaping. Nevertheless, the progress on developing higher order (> 2) mismatch-shaping techniques has been limited, for it is difficult to keep the existing mismatchshaping algorithms stable at higher orders. This brief reports a stable high-order mismatch-shaping technique. By building upon the conventional vector-based scheme in [12]–[14], we developed a new architecture with a new vector quantization (VQ) scheme that extends the stable region of the second-order mismatch shaping and makes possible the thirdand fourth-order mismatch shaping. Unlike the conventional VQ scheme that uses only one vector variable, our VQ scheme exploits the information in all vector variables. In addition to the new architecture and the new VQ scheme, we also developed a hardware-efficient technique that keeps the average of the vector variables constant. This brief is organized as follows: Section II reviews the operation of a mismatch-shaped multibit ΔΣ DAC. Section III presents our technique for the second-order mismatch shaping. Sections IV and V generalize our technique to the thirdand fourth-order mismatch shaping. We finally conclude in Section VI.
A
(1)
≡ v[n] = sv[n]
M 1 sv i [n] M i=1
(2)
where the overline stands for taking the average. Equation (2) is equivalent to the number conservation rule in [10]. The block diagram of a typical unit-element DAC is shown in Fig. 2. The output of the entire DAC w(t) is given by w(t) =
M i=1
wi (t) =
M ∞
wi, n (t)
(3)
i=1 n=0
where wi (t) is the output of the ith 1-bit DAC and wi, n (t) corresponds to wi (t) during the nth sample period and is nonzero only when t ∈ [nT, nT + T ). The relationship between wi, n (t) and sv i [n] is given by 1 + 2M p(t−nT )+eh, i (t−nT ), if sv i [n] = 1 wi, n (t) = (4) 1 p(t−nT )+el, i (t −nT ), if sv i [n] = 0 − 2M where p(t) is the output of an ideal 1-bit DAC and eh, i (t) and el, i (t) are the mismatch error pulses that assume different values for different 1-bit DACs. p(t), eh, i (t), and el, i (t) are nonzero only when t ∈ [0, T ). The two cases in (4) can be combined into one expression given as wi, n (t) = sv i [n] · αi (t − nT ) + βi (t − nT )
(5)
where 1 p(t) + eh, i (t) − el, i (t) M 1 βi (t) = − p(t) + el, i (t). 2M
αi (t) =
(6) (7)
We assume that the ESL block ensures that each sv i [n] can be decomposed into Manuscript received December 10, 2010; revised February 7, 2011; accepted March 26, 2011. Date of current version July 1, 2011. This paper was recommended by Associate Editor P. Mak. The author is with the Department of Electrical and Computer Engineering, University of Texas at Austin, Austin, TX 78712 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCSII.2011.2158163
sv i [n] = v[n] + ei [n]
(8)
where ei [n] is a noise term uncorrelated with v[n]. By plugging (5) and (8) into (3), we obtain w(t) =
1549-7747/$26.00 © 2011 IEEE
∞ n=0
α(t − nT )v[n] + β(t) + e(t)
(9)
SUN: HIGH-ORDER MISMATCH-SHAPING IN MULTIBIT DACs
347
III. O UR S ECOND -O RDER M ISMATCH -S HAPING T ECHNIQUE A. New ESL Architecture
Fig. 1.
Mismatch-shaped multibit ΔΣ DAC.
Fig. 2.
Unit-element DAC model.
where α(t) =
M
αi (t)
(10)
i=1
β(t) =
M ∞
βi (t − nT )
(11)
n=0 i=1
e(t) =
M ∞
αi (t − nT )ei [n].
(12)
n=0 i=1
Among the three terms on the right-hand side of (9), the first term is the signal that is linear with v[n], the second term is the offset that is independent of v[n] and sv[n], and the third term is the error caused by element mismatches. The goal of the mismatch shaping is to make e(t) noise shaped regardless of ai (t − nT ) [see (12)], which is equivalent to making every ei [n] noise shaped; in other words, we want the power spectral density (PSD) of ei [n] to have the following format: ∗ PSDi (z) = HMNTF (z)HMNTF (z)
(13)
where HMNTF (z) is the mismatch-noise transfer function (e.g., 1 − z −1 for the first-order mismatch shaping and (1 − z −1 )2 for the second-order mismatch shaping). From the foregoing derivations, we see that the mismatch shaping is equivalent to the validity of (8) and (13), which are both guaranteed by the ESL block. Researchers have developed various mismatch-shaping techniques that satisfy (8) and (13). One widely used second-order mismatch-shaping technique is the vector-based method [12]–[14]. Our technique builds upon the vector-based method with key improvements that enhance the stability and make possible the higher order (> 2) mismatch shaping.
Fig. 3 shows our new ESL architecture that is built upon the vector-based method in [12]–[14]. The main difference is that the conventional vector-based method uses an error-feedback structure, whereas ours uses a standard feedback structure. Our new architecture looks similar to a standard second-order ΔΣ modulator but with three key differences: 1) The state variables (i.e., sx[n], sy[n], and sv[n]) are vectors with the length of M . 2) The quantizer is a vector quantizer. Following the same definition of the conventional vector-based method, the vector quantizer sets those components of sv[n] to 1 that correspond to the M · v[n] largest components of sy[n] and set the rest of the components of sv[n] to 0. 3) The modified 1/(1 − z −1 ) block, whose internal block diagram is shown in Fig. 4, contains extra blocks that are used to ensure the minimum of the output vector to be 0. This modification is inherited from the conventional vector-based method. It is necessary because the ESL architecture (see Fig. 3) has no feedback mechanism on sx[n] and sy[n], which would go unbounded without this modification. The mathematical expressions for sx[n] and sy[n] are ∗ [n] − min sx ∗ [n] sx[n] = sx (14) ∗ [n] − min sy ∗ [n] sy[n] = sy (15) where sx∗ [n] and sy ∗ [n] are given by ∗ [n] = sx[n − 1] − sv[n − 1] sx ∗ − 1] + sx[n] − sv[n − 1]. sy [n] = sy[n
(16) (17)
Despite the structural differences, our new ESL architecture and the conventional vector-based method are essentially equivalent because they produce the same sv[n] and share the same mismatch-noise transfer function of HMNTF (z) = (1 − z −1 )2 . We develop this new architecture because it facilitates the implementation of the two key improvements, which will be described in Sections III-B and III-C. B. Simple Method to Hold the Vector Average Constant As discussed earlier, the conventional way to keep the vector average (sx[n] and sy[n]) bounded is to add extra blocks that subtract the minimum value of a vector from the vector (see Fig. 4). However, this method is hardware intensive, for it requires searching for the minimum element of a vector. In addition, it does not keep the vector average constant; sx[n] and sy[n] may experience large variations. This subsection presents a simple method that can hold the vector average constant. Fig. 5 shows the modified ESL architecture equipped with our method.1 Different from the one in Fig. 3, the modified ESL architecture uses standard 1/(1 − z −1 ) blocks. To prove that 1 This modification does not change sv[n] and HMNTF (z), for the vector quantizer considers only the relative magnitudes among SYi [n] (i ∈ [1, M ]).
348
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 58, NO. 6, JUNE 2011
Fig. 3. Our new ESL architecture for the second-order mismatch shaping.
Fig. 6. Example sequences of sv[n] for A = 0.85 with (a) the conventional VQ scheme and (b) our new VQ scheme.
Fig. 4. Modified 1/(1 − z −1 ) block.
Fig. 7. Output spectra for A = 0.85 with the conventional VQ scheme and our new VQ scheme for 218 points. Fig. 5. Modified ESL architecture that holds the vector average constant.
our method can keep sx[n] and sy[n] constant, let us first write down the new expressions for sx[n] and sy[n] as follows: sx[n] = sx[n − 1] − sv[n − 1] + v[n − 1] (18) sy[n] = sy[n − 1] + sx[n] − sv[n − 1] + v[n − 1]. (19) By taking the average of (18) and (19) and by using (2), we obtain sx[n] = sx[n − 1] sy[n] = sy[n − 1] + sx[n]
(20) (21)
Thus, by induction, we have the following: sx[n] = sx[0] sy[n] = sy[0] + n · sx[0].
(22) (23)
= sy[0] = 0, we have Setting the initial conditions of sx[0] sx[n] = sy[n] = 0.
(24)
Thus, sx[n] and sy[n] are anchored at 0, which is achieved essentially by exploiting the number conservation rule of (2). This method is simple and hardware efficient since it only requires adding v[n − 1] [see Fig. 5, (18) and (19)].
C. Stable VQ Scheme The vector quantizer plays the key role in keeping the difference between vector elements bounded. The conventional VQ scheme, which is described in Section III-A, works for small inputs but is unstable for large inputs. Let us consider an example of a 4-bit (M = 15) third-order ΔΣ modulator with an oversampling ratio (OSR) of 32, an element mismatch of 1%, and an input of u[n] = A · sin(nπ/32), where A = 1 corresponds to the full amplitude. When A = 0.85, sv[n] [see Fig. 6(a)2 ] shows repetitive or thermometer-like patterns, and the mismatch noise is not shaped (see Fig. 7), for both sx[n] and sy[n] become unstable/unbounded. This instability results from the fact that the conventional VQ scheme only uses sy[n] and does not take into account of sx[n] (see Fig. 5). Consider the case where sy j [n] = sy i [n] + k1 > 0, sxi [n] = sxj [n] + k2 > 0, and k2 k1 > 0. The conventional VQ scheme would set sv j [n] ≥ sv i [n]. However, sy i has a higher chance to become unbounded than sy j because sy i is the integration of sxi and the much larger sxi [n] leads to a much larger sy i value in the subsequent samples. Thus, the vector quantizer should have set sv j [n] ≤ sv i [n] to maintain stability. To understand this issue more thoroughly, we can make an analogy between the ESL architecture (see Fig. 5) and a system of cross-coupled pendulums: We may consider sx[n] and sy[n] 2 In
Fig. 6, the solid box represents 1, and the empty box represents 0.
SUN: HIGH-ORDER MISMATCH-SHAPING IN MULTIBIT DACs
349
Fig. 8. max{sy} versus the input amplitude A for different K. Fig. 10. max{sz} versus Kx and Ky for A = 0.8.
IV. O UR T HIRD -O RDER M ISMATCH -S HAPING T ECHNIQUE Our mismatch-shaping technique can be generalized readily to the third order (see Fig. 9). It is easy to prove that if we set sx[0] = sy[0] = sz[0] =0
(26)
sx[n] = sy[n] = sz[n] = 0.
(27)
we have
Fig. 9.
Our stable third-order mismatch-shaping architecture.
as the speed and the displacement of the pendulums and sv[n] as the restoring force. The conventional VQ scheme applies the stronger restoring force sv i [n] to a pendulum with the larger displacement sy i [n] without considering its speed sxi [n]. A naturally better strategy would be to distribute the restoring force sv[n] according to the tendency of the pendulums going unstable, which is related to both sx[n] and sy[n]. Our new VQ scheme considers a linear combination of sx[n] and sy[n], which is given by S[n] = sy[n] + K · sx[n].
(25)
We set those components of sv[n] to 1 that corresponds to the M · v[n] largest components of S[n]. To study the effect of K on the bound of sy, we plot max{sy} versus the input amplitude A for different K in Fig. 8. As K increases, max{sy} decreases, which demonstrates the stability improvements. For K ≥ 2, sy is bounded for A ≤ 0.98. Further increasing K leads to limited reduction in max{sy}. We choose K = 4 for our new VQ scheme. For clear comparisons with the conventional VQ scheme, we plot sv[n] and the output spectrum for A = 0.85 with the new VQ scheme in Figs. 6(b) and 7. We see that sv[n] is randomized and the output spectrum shows the 40-dB/dec behavior, which validates that our new VQ scheme is stable for A = 0.85. This stability improvement can also be explained from the view of the mismatch-noise transfer function HMNTF (z) = (1 − z −1 )2 /(1 + Kz −1 − Kz −2 ), which is derived by assuming a simple linear model for the vector quantizer. The increase in K reduces max{|HMNTF (z)|}, which leads to better stability according to the analysis for standard ΔΣ modulators [15]. This HMNTF (z) also shows that the second-order mismatch shaping is maintained by the new VQ scheme of (25).
Our new VQ scheme for the third-order mismatch shaping directly extends from the one for the second-order shaping (see Section III-C): We set those components of sv[n] to 1 that corresponds to the M · v[n] largest components of T [n], where T [n] = sz[n] + Kx · sx[n] + Ky · sy[n].
(28)
To study the effect of Kx and Ky on the bound of sz[n], let us consider an example of a 4-bit (M = 15) fourth-order ΔΣ modulator with an OSR of 64, an element mismatch of 1%, and an input of u[n] = A · sin(nπ/64). Fig. 10 shows 1/ max{sz} versus Kx and Ky for A = 0.8. When Kx and Ky are small, 1/ max{sz} = 0, which indicates that sz is unstable/unbounded. As Kx and Ky increase, max{sz} decreases, showing the stability improvement. The reduction in max{sz} saturates for Kx > 50 and Ky > 10. We choose Kx = 64 and Ky = 16 for our new VQ scheme. Kx and Ky are set intentionally to be at a power of 2 so that the multiplication in (28) can be implemented simply through bit aligning. Simulation shows that sz is bounded for A ≤ 0.95. By contrast, if we use the conventional VQ scheme with Kx = Ky = 0, sz is unbounded for any A. Fig. 11 shows the output spectra for A = 0.5. The mismatch noise is not shaped with the conventional VQ scheme. By contrast, with our new VQ scheme, the spectrum shows the 60-dB/dec behavior, which demonstrates clearly the third-order mismatch shaping. V. O UR F OURTH -O RDER M ISMATCH -S HAPING T ECHNIQUE Our fourth-order mismatch-shaping technique is in Fig. 12. The vector quantizer sets those components of sv[n] to 1 that corresponds to the M · v[n] largest components of R[n], where R[n] = sw[n] + 512 · sx[n] + 64 · sy[n] + 16 · sz[n]. (29)
350
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 58, NO. 6, JUNE 2011
TABLE I SNR S AND B EHAVIORS
Fig. 11. Output spectra for A = 0.5 and 218 points with the conventional VQ scheme and our new VQ scheme for the third-order mismatch shaping.
linearity limit of the current ΔΣ DACs can be extended. In addition, the matching requirement of the unit-element DAC can be further relaxed, thus reducing the layout efforts and allowing the use of smaller unit elements for power saving. ACKNOWLEDGMENT The author would like to thank O. Yildirim of Harvard University, Dr. B. Zhang of Broadcom Inc. for the discussions, and Dr. R. Schreier of Analog Devices Inc. for the discussions and sharing the ΔΣ toolbox. R EFERENCES
Fig. 12. Our stable fourth-order mismatch-shaping architecture.
Fig. 13. Output spectra of various mismatch-shaping orders for A = 0.5 and 218 points.
We apply our fourth-order mismatch-shaping technique to a 4-bit (M = 15) fifth-order ΔΣ modulator with an OSR of 64, an element mismatch of 1%, and an input of u[n] = A · sin(nπ/64). Simulations show that our fourth-order mismatchshaping technique is stable for A ≤ 0.76. Fig. 13 shows the output spectra for the thermometer coding, the second-order, the third-order, and the fourth-order mismatch shaping at A = 0.5. Their SNRs and behaviors are summarized in Table I. Our technique can be generalized to even higher order (≥ 5) mismatch shaping, but we omit it to keep this brief concise. VI. C ONCLUSION This brief has described a stable high-order mismatchshaping technique for multibit ΔΣ DACs. With the high-order mismatch shaping now being available, we envision that the
[1] B. H. Leung and S. Sutarja, “Multibit sigma-delta A/D converter incorporating a novel class of dynamic element matching techniques,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 39, no. 1, pp. 35–51, Jan. 1992. [2] H. S. Jackson, “Circuit and method for cancelling nonlinearity error associated with component value mismatches in a data converter,” U.S. Patent 5 221 926, Jun. 22, 1993. [3] R. W. Adams and T. W. Kwan, “Data-directed scrambler for multi-bit noise shaping D/A converters,” U.S. Patent 5 404 142, Apr. 4, 1995. [4] R. T. Baird and T. S. Fiez, “Linearity enhancement of multi-bit ΣΔ A/D and D/A converters using data weighted averaging,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 42, no. 12, pp. 753–762, Dec. 1995. [5] K. D. Chen and T. H. Kuo, “An improved technique for reducing baseband tones in sigma-delta modulators employing data weighted averaging algorithm without adding dither,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 46, no. 1, pp. 63–68, Jan. 1999. [6] A. A. Hamoui and K. W. Martin, “High-order multibit modulators and pseudo data-weighted-averaging in low-oversampling ΔΣ ADCs for broadband applications,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 51, no. 1, pp. 72–85, Jan. 2004. [7] H.-Y. Hsieh and L. Lin, “A first-order tree-structured DAC with reduced signal-band noise,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 54, no. 5, pp. 392–396, May 2007. [8] R. K. Henderson and O. J. A. P. Nys, “Dynamic element matching techniques with arbitrary noise shaping functions,” in Proc. IEEE Int. Symp. Circuits Syst., May 1996, pp. 293–296. [9] A. Keady and C. Lyden, “Tree structure for mismatch noise-shaping multibit DAC,” Electron. Lett., vol. 33, no. 17, pp. 1431–1432, Aug. 1997. [10] I. Galton, “Spectral shaping of circuit errors in digital-to-analog converters,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 44, no. 10, pp. 808–817, Oct. 1997. [11] E. Fogleman, J. Welz, and I. Galton, “An audio ADC delta-sigma modulator with 100-dB peak SINAD and 102-dB DR using a second-order mismatch-shaping DAC,” IEEE J. Solid-State Circuits, vol. 36, no. 3, pp. 339–348, Mar. 2001. [12] R. Schreier and B. Zhang, “Noise-shaped multibit D/A convertor employing unit elements,” Electron. Lett., vol. 31, no. 20, pp. 1712–1713, Sep. 1995. [13] R. Schreier, “Mismatch-shaping digital-to-analog conversion,” in Proc. 103rd Conv. Audio Eng. Soc., preprint no. 4529, Sep. 1997. [14] A. Yasuda and H. Tanimoto, “Noise shaping dynamic element matching method using tree structure,” Electron. Lett., vol. 33, no. 2, pp. 130–131, Jan. 1997. [15] R. Schreier, “An empirical study of high-order single-bit delta-sigma modulators,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 40, no. 8, pp. 461–466, Aug. 1993.
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 58, NO. 6, JUNE 2011
High-Order Mismatch-Shaping in Multibit DACs Nan Sun, Member, IEEE
Abstract—This brief reports a high-order mismatch-shaping technique for multibit ΔΣ digital-to-analog converters (DACs). It builds upon the vector-based method but with two key improvements. First, a new vector quantization (VQ) scheme exploits the information in all vector variables. This VQ scheme allows the second-order mismatch shaping for large input signals and makes possible the stable third- and fourth-order mismatch shaping. Second, a hardware-efficient method maintains the average of the vector variables bounded. Simulations under various conditions prove its validity. Index Terms—Data converter, mismatch shaping, ΔΣ DAC.
II. R EVIEW OF THE M ISMATCH -S HAPED M ULTIBIT ΔΣ DAC The structure of a mismatch-shaped multibit ΔΣ DAC is in Fig. 1. By supposing that the quantizer in the ΔΣ modulator has M steps with each step size of 1/M , then, v[n] takes a value in {0, 1/M, 2/M, . . . , (M − 1)/M, 1}. The element selection logic (ESL) block decodes v[n] into vector sv[n] as follows: sv[n] = [sv 1 [n], sv 2 [n], . . . , sv M [n]]
where each sv i [n] takes a value of either 1 or 0 and drives one element in the unit-element DAC. The ESL block ensures that
I. I NTRODUCTION MULTIBIT ΔΣ DAC is more favorable than a binary one for its better stability and SNR. The relative drawback of the multibit DAC is that it does not guarantee linearity due to element mismatches. To address this issue, researchers have developed various mismatch-shaping techniques [1]–[14], among which the techniques in [1]–[7] provide the first-order mismatch shaping and the techniques in [8]–[14] provide the second-order mismatch shaping. Nevertheless, the progress on developing higher order (> 2) mismatch-shaping techniques has been limited, for it is difficult to keep the existing mismatchshaping algorithms stable at higher orders. This brief reports a stable high-order mismatch-shaping technique. By building upon the conventional vector-based scheme in [12]–[14], we developed a new architecture with a new vector quantization (VQ) scheme that extends the stable region of the second-order mismatch shaping and makes possible the thirdand fourth-order mismatch shaping. Unlike the conventional VQ scheme that uses only one vector variable, our VQ scheme exploits the information in all vector variables. In addition to the new architecture and the new VQ scheme, we also developed a hardware-efficient technique that keeps the average of the vector variables constant. This brief is organized as follows: Section II reviews the operation of a mismatch-shaped multibit ΔΣ DAC. Section III presents our technique for the second-order mismatch shaping. Sections IV and V generalize our technique to the thirdand fourth-order mismatch shaping. We finally conclude in Section VI.
A
(1)
≡ v[n] = sv[n]
M 1 sv i [n] M i=1
(2)
where the overline stands for taking the average. Equation (2) is equivalent to the number conservation rule in [10]. The block diagram of a typical unit-element DAC is shown in Fig. 2. The output of the entire DAC w(t) is given by w(t) =
M i=1
wi (t) =
M ∞
wi, n (t)
(3)
i=1 n=0
where wi (t) is the output of the ith 1-bit DAC and wi, n (t) corresponds to wi (t) during the nth sample period and is nonzero only when t ∈ [nT, nT + T ). The relationship between wi, n (t) and sv i [n] is given by 1 + 2M p(t−nT )+eh, i (t−nT ), if sv i [n] = 1 wi, n (t) = (4) 1 p(t−nT )+el, i (t −nT ), if sv i [n] = 0 − 2M where p(t) is the output of an ideal 1-bit DAC and eh, i (t) and el, i (t) are the mismatch error pulses that assume different values for different 1-bit DACs. p(t), eh, i (t), and el, i (t) are nonzero only when t ∈ [0, T ). The two cases in (4) can be combined into one expression given as wi, n (t) = sv i [n] · αi (t − nT ) + βi (t − nT )
(5)
where 1 p(t) + eh, i (t) − el, i (t) M 1 βi (t) = − p(t) + el, i (t). 2M
αi (t) =
(6) (7)
We assume that the ESL block ensures that each sv i [n] can be decomposed into Manuscript received December 10, 2010; revised February 7, 2011; accepted March 26, 2011. Date of current version July 1, 2011. This paper was recommended by Associate Editor P. Mak. The author is with the Department of Electrical and Computer Engineering, University of Texas at Austin, Austin, TX 78712 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCSII.2011.2158163
sv i [n] = v[n] + ei [n]
(8)
where ei [n] is a noise term uncorrelated with v[n]. By plugging (5) and (8) into (3), we obtain w(t) =
1549-7747/$26.00 © 2011 IEEE
∞ n=0
α(t − nT )v[n] + β(t) + e(t)
(9)
SUN: HIGH-ORDER MISMATCH-SHAPING IN MULTIBIT DACs
347
III. O UR S ECOND -O RDER M ISMATCH -S HAPING T ECHNIQUE A. New ESL Architecture
Fig. 1.
Mismatch-shaped multibit ΔΣ DAC.
Fig. 2.
Unit-element DAC model.
where α(t) =
M
αi (t)
(10)
i=1
β(t) =
M ∞
βi (t − nT )
(11)
n=0 i=1
e(t) =
M ∞
αi (t − nT )ei [n].
(12)
n=0 i=1
Among the three terms on the right-hand side of (9), the first term is the signal that is linear with v[n], the second term is the offset that is independent of v[n] and sv[n], and the third term is the error caused by element mismatches. The goal of the mismatch shaping is to make e(t) noise shaped regardless of ai (t − nT ) [see (12)], which is equivalent to making every ei [n] noise shaped; in other words, we want the power spectral density (PSD) of ei [n] to have the following format: ∗ PSDi (z) = HMNTF (z)HMNTF (z)
(13)
where HMNTF (z) is the mismatch-noise transfer function (e.g., 1 − z −1 for the first-order mismatch shaping and (1 − z −1 )2 for the second-order mismatch shaping). From the foregoing derivations, we see that the mismatch shaping is equivalent to the validity of (8) and (13), which are both guaranteed by the ESL block. Researchers have developed various mismatch-shaping techniques that satisfy (8) and (13). One widely used second-order mismatch-shaping technique is the vector-based method [12]–[14]. Our technique builds upon the vector-based method with key improvements that enhance the stability and make possible the higher order (> 2) mismatch shaping.
Fig. 3 shows our new ESL architecture that is built upon the vector-based method in [12]–[14]. The main difference is that the conventional vector-based method uses an error-feedback structure, whereas ours uses a standard feedback structure. Our new architecture looks similar to a standard second-order ΔΣ modulator but with three key differences: 1) The state variables (i.e., sx[n], sy[n], and sv[n]) are vectors with the length of M . 2) The quantizer is a vector quantizer. Following the same definition of the conventional vector-based method, the vector quantizer sets those components of sv[n] to 1 that correspond to the M · v[n] largest components of sy[n] and set the rest of the components of sv[n] to 0. 3) The modified 1/(1 − z −1 ) block, whose internal block diagram is shown in Fig. 4, contains extra blocks that are used to ensure the minimum of the output vector to be 0. This modification is inherited from the conventional vector-based method. It is necessary because the ESL architecture (see Fig. 3) has no feedback mechanism on sx[n] and sy[n], which would go unbounded without this modification. The mathematical expressions for sx[n] and sy[n] are ∗ [n] − min sx ∗ [n] sx[n] = sx (14) ∗ [n] − min sy ∗ [n] sy[n] = sy (15) where sx∗ [n] and sy ∗ [n] are given by ∗ [n] = sx[n − 1] − sv[n − 1] sx ∗ − 1] + sx[n] − sv[n − 1]. sy [n] = sy[n
(16) (17)
Despite the structural differences, our new ESL architecture and the conventional vector-based method are essentially equivalent because they produce the same sv[n] and share the same mismatch-noise transfer function of HMNTF (z) = (1 − z −1 )2 . We develop this new architecture because it facilitates the implementation of the two key improvements, which will be described in Sections III-B and III-C. B. Simple Method to Hold the Vector Average Constant As discussed earlier, the conventional way to keep the vector average (sx[n] and sy[n]) bounded is to add extra blocks that subtract the minimum value of a vector from the vector (see Fig. 4). However, this method is hardware intensive, for it requires searching for the minimum element of a vector. In addition, it does not keep the vector average constant; sx[n] and sy[n] may experience large variations. This subsection presents a simple method that can hold the vector average constant. Fig. 5 shows the modified ESL architecture equipped with our method.1 Different from the one in Fig. 3, the modified ESL architecture uses standard 1/(1 − z −1 ) blocks. To prove that 1 This modification does not change sv[n] and HMNTF (z), for the vector quantizer considers only the relative magnitudes among SYi [n] (i ∈ [1, M ]).
348
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 58, NO. 6, JUNE 2011
Fig. 3. Our new ESL architecture for the second-order mismatch shaping.
Fig. 6. Example sequences of sv[n] for A = 0.85 with (a) the conventional VQ scheme and (b) our new VQ scheme.
Fig. 4. Modified 1/(1 − z −1 ) block.
Fig. 7. Output spectra for A = 0.85 with the conventional VQ scheme and our new VQ scheme for 218 points. Fig. 5. Modified ESL architecture that holds the vector average constant.
our method can keep sx[n] and sy[n] constant, let us first write down the new expressions for sx[n] and sy[n] as follows: sx[n] = sx[n − 1] − sv[n − 1] + v[n − 1] (18) sy[n] = sy[n − 1] + sx[n] − sv[n − 1] + v[n − 1]. (19) By taking the average of (18) and (19) and by using (2), we obtain sx[n] = sx[n − 1] sy[n] = sy[n − 1] + sx[n]
(20) (21)
Thus, by induction, we have the following: sx[n] = sx[0] sy[n] = sy[0] + n · sx[0].
(22) (23)
= sy[0] = 0, we have Setting the initial conditions of sx[0] sx[n] = sy[n] = 0.
(24)
Thus, sx[n] and sy[n] are anchored at 0, which is achieved essentially by exploiting the number conservation rule of (2). This method is simple and hardware efficient since it only requires adding v[n − 1] [see Fig. 5, (18) and (19)].
C. Stable VQ Scheme The vector quantizer plays the key role in keeping the difference between vector elements bounded. The conventional VQ scheme, which is described in Section III-A, works for small inputs but is unstable for large inputs. Let us consider an example of a 4-bit (M = 15) third-order ΔΣ modulator with an oversampling ratio (OSR) of 32, an element mismatch of 1%, and an input of u[n] = A · sin(nπ/32), where A = 1 corresponds to the full amplitude. When A = 0.85, sv[n] [see Fig. 6(a)2 ] shows repetitive or thermometer-like patterns, and the mismatch noise is not shaped (see Fig. 7), for both sx[n] and sy[n] become unstable/unbounded. This instability results from the fact that the conventional VQ scheme only uses sy[n] and does not take into account of sx[n] (see Fig. 5). Consider the case where sy j [n] = sy i [n] + k1 > 0, sxi [n] = sxj [n] + k2 > 0, and k2 k1 > 0. The conventional VQ scheme would set sv j [n] ≥ sv i [n]. However, sy i has a higher chance to become unbounded than sy j because sy i is the integration of sxi and the much larger sxi [n] leads to a much larger sy i value in the subsequent samples. Thus, the vector quantizer should have set sv j [n] ≤ sv i [n] to maintain stability. To understand this issue more thoroughly, we can make an analogy between the ESL architecture (see Fig. 5) and a system of cross-coupled pendulums: We may consider sx[n] and sy[n] 2 In
Fig. 6, the solid box represents 1, and the empty box represents 0.
SUN: HIGH-ORDER MISMATCH-SHAPING IN MULTIBIT DACs
349
Fig. 8. max{sy} versus the input amplitude A for different K. Fig. 10. max{sz} versus Kx and Ky for A = 0.8.
IV. O UR T HIRD -O RDER M ISMATCH -S HAPING T ECHNIQUE Our mismatch-shaping technique can be generalized readily to the third order (see Fig. 9). It is easy to prove that if we set sx[0] = sy[0] = sz[0] =0
(26)
sx[n] = sy[n] = sz[n] = 0.
(27)
we have
Fig. 9.
Our stable third-order mismatch-shaping architecture.
as the speed and the displacement of the pendulums and sv[n] as the restoring force. The conventional VQ scheme applies the stronger restoring force sv i [n] to a pendulum with the larger displacement sy i [n] without considering its speed sxi [n]. A naturally better strategy would be to distribute the restoring force sv[n] according to the tendency of the pendulums going unstable, which is related to both sx[n] and sy[n]. Our new VQ scheme considers a linear combination of sx[n] and sy[n], which is given by S[n] = sy[n] + K · sx[n].
(25)
We set those components of sv[n] to 1 that corresponds to the M · v[n] largest components of S[n]. To study the effect of K on the bound of sy, we plot max{sy} versus the input amplitude A for different K in Fig. 8. As K increases, max{sy} decreases, which demonstrates the stability improvements. For K ≥ 2, sy is bounded for A ≤ 0.98. Further increasing K leads to limited reduction in max{sy}. We choose K = 4 for our new VQ scheme. For clear comparisons with the conventional VQ scheme, we plot sv[n] and the output spectrum for A = 0.85 with the new VQ scheme in Figs. 6(b) and 7. We see that sv[n] is randomized and the output spectrum shows the 40-dB/dec behavior, which validates that our new VQ scheme is stable for A = 0.85. This stability improvement can also be explained from the view of the mismatch-noise transfer function HMNTF (z) = (1 − z −1 )2 /(1 + Kz −1 − Kz −2 ), which is derived by assuming a simple linear model for the vector quantizer. The increase in K reduces max{|HMNTF (z)|}, which leads to better stability according to the analysis for standard ΔΣ modulators [15]. This HMNTF (z) also shows that the second-order mismatch shaping is maintained by the new VQ scheme of (25).
Our new VQ scheme for the third-order mismatch shaping directly extends from the one for the second-order shaping (see Section III-C): We set those components of sv[n] to 1 that corresponds to the M · v[n] largest components of T [n], where T [n] = sz[n] + Kx · sx[n] + Ky · sy[n].
(28)
To study the effect of Kx and Ky on the bound of sz[n], let us consider an example of a 4-bit (M = 15) fourth-order ΔΣ modulator with an OSR of 64, an element mismatch of 1%, and an input of u[n] = A · sin(nπ/64). Fig. 10 shows 1/ max{sz} versus Kx and Ky for A = 0.8. When Kx and Ky are small, 1/ max{sz} = 0, which indicates that sz is unstable/unbounded. As Kx and Ky increase, max{sz} decreases, showing the stability improvement. The reduction in max{sz} saturates for Kx > 50 and Ky > 10. We choose Kx = 64 and Ky = 16 for our new VQ scheme. Kx and Ky are set intentionally to be at a power of 2 so that the multiplication in (28) can be implemented simply through bit aligning. Simulation shows that sz is bounded for A ≤ 0.95. By contrast, if we use the conventional VQ scheme with Kx = Ky = 0, sz is unbounded for any A. Fig. 11 shows the output spectra for A = 0.5. The mismatch noise is not shaped with the conventional VQ scheme. By contrast, with our new VQ scheme, the spectrum shows the 60-dB/dec behavior, which demonstrates clearly the third-order mismatch shaping. V. O UR F OURTH -O RDER M ISMATCH -S HAPING T ECHNIQUE Our fourth-order mismatch-shaping technique is in Fig. 12. The vector quantizer sets those components of sv[n] to 1 that corresponds to the M · v[n] largest components of R[n], where R[n] = sw[n] + 512 · sx[n] + 64 · sy[n] + 16 · sz[n]. (29)
350
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 58, NO. 6, JUNE 2011
TABLE I SNR S AND B EHAVIORS
Fig. 11. Output spectra for A = 0.5 and 218 points with the conventional VQ scheme and our new VQ scheme for the third-order mismatch shaping.
linearity limit of the current ΔΣ DACs can be extended. In addition, the matching requirement of the unit-element DAC can be further relaxed, thus reducing the layout efforts and allowing the use of smaller unit elements for power saving. ACKNOWLEDGMENT The author would like to thank O. Yildirim of Harvard University, Dr. B. Zhang of Broadcom Inc. for the discussions, and Dr. R. Schreier of Analog Devices Inc. for the discussions and sharing the ΔΣ toolbox. R EFERENCES
Fig. 12. Our stable fourth-order mismatch-shaping architecture.
Fig. 13. Output spectra of various mismatch-shaping orders for A = 0.5 and 218 points.
We apply our fourth-order mismatch-shaping technique to a 4-bit (M = 15) fifth-order ΔΣ modulator with an OSR of 64, an element mismatch of 1%, and an input of u[n] = A · sin(nπ/64). Simulations show that our fourth-order mismatchshaping technique is stable for A ≤ 0.76. Fig. 13 shows the output spectra for the thermometer coding, the second-order, the third-order, and the fourth-order mismatch shaping at A = 0.5. Their SNRs and behaviors are summarized in Table I. Our technique can be generalized to even higher order (≥ 5) mismatch shaping, but we omit it to keep this brief concise. VI. C ONCLUSION This brief has described a stable high-order mismatchshaping technique for multibit ΔΣ DACs. With the high-order mismatch shaping now being available, we envision that the
[1] B. H. Leung and S. Sutarja, “Multibit sigma-delta A/D converter incorporating a novel class of dynamic element matching techniques,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 39, no. 1, pp. 35–51, Jan. 1992. [2] H. S. Jackson, “Circuit and method for cancelling nonlinearity error associated with component value mismatches in a data converter,” U.S. Patent 5 221 926, Jun. 22, 1993. [3] R. W. Adams and T. W. Kwan, “Data-directed scrambler for multi-bit noise shaping D/A converters,” U.S. Patent 5 404 142, Apr. 4, 1995. [4] R. T. Baird and T. S. Fiez, “Linearity enhancement of multi-bit ΣΔ A/D and D/A converters using data weighted averaging,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 42, no. 12, pp. 753–762, Dec. 1995. [5] K. D. Chen and T. H. Kuo, “An improved technique for reducing baseband tones in sigma-delta modulators employing data weighted averaging algorithm without adding dither,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 46, no. 1, pp. 63–68, Jan. 1999. [6] A. A. Hamoui and K. W. Martin, “High-order multibit modulators and pseudo data-weighted-averaging in low-oversampling ΔΣ ADCs for broadband applications,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 51, no. 1, pp. 72–85, Jan. 2004. [7] H.-Y. Hsieh and L. Lin, “A first-order tree-structured DAC with reduced signal-band noise,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 54, no. 5, pp. 392–396, May 2007. [8] R. K. Henderson and O. J. A. P. Nys, “Dynamic element matching techniques with arbitrary noise shaping functions,” in Proc. IEEE Int. Symp. Circuits Syst., May 1996, pp. 293–296. [9] A. Keady and C. Lyden, “Tree structure for mismatch noise-shaping multibit DAC,” Electron. Lett., vol. 33, no. 17, pp. 1431–1432, Aug. 1997. [10] I. Galton, “Spectral shaping of circuit errors in digital-to-analog converters,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 44, no. 10, pp. 808–817, Oct. 1997. [11] E. Fogleman, J. Welz, and I. Galton, “An audio ADC delta-sigma modulator with 100-dB peak SINAD and 102-dB DR using a second-order mismatch-shaping DAC,” IEEE J. Solid-State Circuits, vol. 36, no. 3, pp. 339–348, Mar. 2001. [12] R. Schreier and B. Zhang, “Noise-shaped multibit D/A convertor employing unit elements,” Electron. Lett., vol. 31, no. 20, pp. 1712–1713, Sep. 1995. [13] R. Schreier, “Mismatch-shaping digital-to-analog conversion,” in Proc. 103rd Conv. Audio Eng. Soc., preprint no. 4529, Sep. 1997. [14] A. Yasuda and H. Tanimoto, “Noise shaping dynamic element matching method using tree structure,” Electron. Lett., vol. 33, no. 2, pp. 130–131, Jan. 1997. [15] R. Schreier, “An empirical study of high-order single-bit delta-sigma modulators,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 40, no. 8, pp. 461–466, Aug. 1993.
