An Automatic Coefficient Design Methodology for ... - Semantic Scholar
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 53, NO. 7, JULY 2006
An Automatic Coefficient Design Methodology for High-Order Bandpass Sigma-Delta Modulator With Single-Stage Structure Hwi-Ming Wang, Student Member, IEEE, and Tai-Haur Kuo, Member, IEEE
Abstract—An automatic coefficient design method for synthesis of bandpass sigma-delta modulators (BPSDMs) is presented in this brief. Single-stage BPSDM structures, cascade-of-resonator with distributed feedback, cascade-of-resonator with distributed feedforward, and a new structure with lower coefficient spread, are all used to fit the synthesized coefficients. The automatic coefficient design method is realized in an easy-to-use computer program. Even for inexperienced designers, reliable and high-tolerance BPSDM coefficients for various applications can be automatically and efficiently generated. The methodology covers many design concerns including BPSDM coefficient tolerance for circuit component mismatch, design tradeoffs among in-band noise suppression, oversampling ratio, modulator order and quantizer bit number. Finally, design examples with orders of 6 and 8, and quantizer bit number of 1-bit and 3-bit, respectively, are used for the verification of the proposed automatic coefficient design method. Index Terms—Bandpass sigma-delta modulation (BPSDM), coefficient synthesis, oversampling converter.
I. INTRODUCTION IGMA-DELTA MODULATORs (SDMs) can be classified into low-pass SDMs (LPSDMs) [1] and bandpass SDMs (BPSDMs) [2]. An SDM with high dynamic range (DR) generally results from the use of high-order loop filter, multi-bit quantizer and/or high oversampling ratio (OSR), where OSR is defined as the ratio of sampling frequency to two times of signal bandwidth. To build high-order BPSDMs and achieve high DR, single-stage [3], [4] and cascade or multi-stage noise-shaping (MASH) [5] structures are frequently used. Among these structures, high-order single-stage structures are less sensitive to component mismatch, but design of optimal coefficients for single-stage BPSDMs often suffers from instability problems, long design cycle, and the need for experienced designers. An automatic coefficient synthesis method for high-order LPSDMs was presented by Kuo and Chen [1], making the high-order LPSDM design simpler and more cost-effective. Further, BPSDMs have not been as thoroughly investigated as their LPSDM counterparts. The methods in [3] and [7] were proposed for coefficient design of high-order single-stage BPSDMs. These methods still needed many design iterations between coefficient tolerance and design tradeoffs among
S
Manuscript received June 7, 2005; revised October 22, 2005. This work was supported by the National Science Council of Taiwan under Research Grant NSC 94-2215-E-006-062. This paper was recommended by Associate Editor H. Hashemi. The authors are with the Department of Electrical Engineering, National Cheng Kung University, Tainan City 70101, Taiwan, R.O.C. (e-mail: [email protected]) Digital Object Identifier 10.1109/TCSII.2006.875304
Fig. 1. General single-stage BPSDM structure.
in-band noise suppression, OSR, modulator order, quantizer bit number, and stability. This brief extends the method of Kuo and Chen for high-order single-stage BPSDMs. With the proposed automatic coefficient synthesis methodology, stability and design complexity problems are easily solved. Three general single-stage high-order BPSDM structures, cascade-of-resonator with distributed feedback (CRFB) [3], cascade-of-resonator with distributed feedforward (CRFF) [4], and a new proposed low-spread CRFF (LSCRFF) [2], all are used to fit the synthesized coefficients. The method has been implemented and tested in a high level computer program, making practical BPSDM design convenient. II. DESIGN OF HIGH-ORDER BPSDMS In the single-stage case, the BPSDM has a bandpass loop filter, a quantizer, and a feedback digital-to-analog converter (DAC) as shown in Fig. 1. A multibit quantizer can be embedded in the BPSDM loop to improve the DR for a given OSR, or to reduce OSR for a given DR [6]. The discrete-time linear model [7] is used in the BPSDM analysis. The quantizer in BPSDM is modeled as a unity-gain amplifier with an additive quantization noise source. The overall BPSDM can be characterized by two transfer functions, referred to respectively as signal transfer function and noise transfer function. The two transfer functions can be derived as , where is input signal, is noise, is output signal, is the is the noise transfer funcsignal transfer function, and and are first discussed. tion. The designs of A. Design of
and
For a BPSDM, is required to minimize in-band noise power and thus maximize peak signal-to-noise ratio (PSNR). can be approximated by conventional bandreject The
1057-7130/$20.00 © 2006 IEEE
WANG AND KUO: AUTOMATIC COEFFICIENT DESIGN METHODOLOGY FOR HIGH-ORDER BPSDM
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Fig. 3. BPSDM coefficients design flow.
P N
Fig. 2. NPG-versusplot for: (a) 1-bit and SCR = 4, (b) 1-bit and SCR = 16, (c) 3-bit and SCR = 4, (d) 3-bit and SCR = 16 BPSDMs with = 8, 16, 32, 64 and = 8.
Q
analog filter functions. In this brief, inverse-Chebyshev func. A -domain bandreject functions are used for the tion with an order of is obtained from an -domain low-pass through frequency denormalizafunction with an order of tion, low-pass-to-bandreject transformation and -to- bilinear characteristics are related to retransformation [7]. , modulator order , bandquired stopband attenuation , signal center frequency , and sampling frewidth quency [8]. is also related to , , quality factor , and sampling frequency to signal center frequency ratio and , re(SCR) when and SCR are defined as spectively. The relation between and PSNR is (1) where is the maximum-input magnitude and is the quantization step size of the quantizer in Fig. 1. When PSNR, , SCR, , , , and the quantizer bit number are specwhose satisfies (1) is obified, an initial BPSDM tained. , only zero positions are designed since For and share poles in CRFB, CRFF, and LSCRFF, as will on be shown in the next section. By placing the zeros of the unit circle, the signal is suppressed at the corresponding freare placed quency. For an th-order BPSDM, zeros of zeros at dc, zeros at symmetrically, with , and one at the center of the unit circle [9]. B. Stability Constraints Infinite numbers of can be synthesized from (1), but not all of these can be used to design a stable BPSDM. Stability constraints must be established to exso that the unstable ones can be excluded. amine these For low-pass SDMs, noise power gain (NPG) defined as
can be used to examine stability [1]. In this brief, is applied for the is associated with BPSDMs stability evaluation. Since specifications such as , , and SCR, therefore is also is decreased when increases, SCR related to them. increases or a higher quantizer bit number is applied [1]. To upper bound , the relation between decide the and must be first assured because is associated with . -versusplots for 1-bit and 3-bit A variety of 4, eighth-order BPSDMs are shown in Fig. 2, covering 16 and 8, 16, 32, and 64. For convenient use, BPSDM stability constraints need to be expressed by numerical equations. approximated by a first-order For 1-bit BPSDMs, the equation is close enough to the stability bounds, while a thirdorder equation is close enough for 3-bit BPSDMs. Hence, the stability constraint is expressed as (2) Equation (2) represents an approximated numerical equation for BPSDM stability bound. The values of , , and are chosen such that the approximated curve is close to the instaand are set to zero. bility edge. For first-order equations, For practical BPSDM design, component mismatch usually redue to coeffisults in coefficients variation. The change in where cient variation is denoted as is its nominal value and is its variation. When is positive, the BPSDM may become unstable. To the have a safety margins for the stability bound, the relations bedeviation and coefficient mismatch are tween maximum obtained by Monte Carlo simulation. Maximum deviation of versus different maximum coefficient variations (0.5%, 1% and 2%) and different orders are investigated. It condoes not exceed 10% cludes that the maximum ) for maximum coefficient variations of 1% (0.5% for when the range of is from 8 to 64 and SCR is from 4 to 32. The final stability bound is selected so as to tolerate a maximum of 10%, as shown in Fig. 2. Table I variation of
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 53, NO. 7, JULY 2006
Fig. 4. (a) CRFB, (b) CRFF, and (c) LSCRFF structures for
N th-order m-bit BPSDMs.
lists the and values for 1-bit BPSDMs, and the , , , and values for 3-bit BPSDMs with loop filter orders 6 oband 8, and SCR values 4, 8, 16, and 32. The initial tained from (1) can then be examined by using Table I and thus are obtained. only stable
TABLE I COEFFICIENTS OF STABILITY BOUND [SEE (2)] FOR (A) 1-BIT AND (B) 3-BIT BPSDMS WITH ORDERS OF 6, 8, AND SCR FROM 4 TO 32
III. COEFFICIENT DESIGN FLOW When the of a synthesized is larger than , quantizer bit number and/or and/or SCR may be with a smaller [1]. increased to give a different is corrected by For lower order loop filters, excessive SCR increase. On the other hand, for designs requiring lower excess is corrected by increase. Gencircuit speeds, erally, Fig. 3 shows a coefficient design flow for a single-stage BPSDM. The procedure starts from designating system spec, range of ifications such as PSNR, maximum input
SCR ( and
, and ), range of modulator order ( ), maximum quantizer bit number , ,
,
WANG AND KUO: AUTOMATIC COEFFICIENT DESIGN METHODOLOGY FOR HIGH-ORDER BPSDM
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TABLE II SYNTHESIZED BPSDM COEFFICIENTS AND PERFORMANCE FOR EXAMPLES I AND II
and the specified preference of low-order or low OSR. The is generated from (1). The stability constraint, initial i.e., whether is satisfied, is examined by (2) and Table I. By iterated increase of the quantizer bit number, can be obtained. SCR or , an appropriate Three single-stage structures of CRFB, CRFF, and LSCRFF and . The are used to fit the designed th-order, -bit CRFB, CRFF, and LSCRFF structures are is an inteshown in Fig. 4(a)–(c), respectively, where and , , grator with transfer function of , , , and are modulator coefficients. The transfer functions for CRFB are and
and
For CRFF, they are , where
(6)
(7)
, where (3) (8)
(4)
(5)
By equating the or , and the synthesized , these coefficients can be obtained. For and in (3), are decided by equating the CRFB case, and synthesized ; in (4) numerator of are determined by equating the denominator of and synthesized ; in (5) are determined by equating , and . For CRFF case, the numerator of and the using similar flow such as equating the to find the corresponding coefficients synthesized
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Fig. 5. (a) Pole–zero diagram. (b) Response of
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 53, NO. 7, JULY 2006
NTF(z) and STF(z).
unit circle, two and three zeros at dc and two and three zeros at for modulator orders 6 and 8, respectively. In eighth-order and are example, the pole–zero positions of shown in Fig. 5(a), and the frequency responses in Fig. 5(b), showing the bandreject and bandpass characteristic for and . The synthesized coefficients and achieved PSNR and DR are summarized in Table II. To investigate PSNR degradation due to coefficient variation, Monte Carlo analysis with 100 different samples each is applied to sixth- and eighth-order design examples, each with CRFB, CRFF, and LSCRFF and quantizer bit number of 3. Histograms of PSNR deviation, subject to a 1% maximum mismatch, are shown in Fig. 6. For comparable sixth-order example, the degraded PSNR are, respectively, 10 dB, 8 dB for CRFB and CRFB, and 4 dB for LSCRFF. For the eighth-order example, these respective values are 9 dB, 10 dB, and 5 dB. V. CONCLUSION An automatic coefficient design method for high-order single-stage BPSDMs has been presented. Tradeoffs among quantizer bit number, modulator order and SCR were made to from those stable . The select an appropriate whole design method was realized in a high-level computer program convenient for BPSDM designers. Based on the presented design method, an optimum set of BPSDM coefficients can be automatically designed to fit given specifications for the convenient CRFB and CRFF BPSDM structures, and also the recent single-stage LSCRFF structure which optimized selected coefficients for reduced coefficient spread and less sensitivity to coefficient mismatch.
Fig. 6. Histograms of PSNR deviation for design examples with 1% max. coefficient mismatch and (a) CRFB, (b) CRFF, and (c) LSCRFF among 100 trials.
from (6)-(8). The same flows and derivations can be applied in LSCRFF structure shown in [2]. To avoid internal clipping and to maximize BPSDM DR, coefficients are scaled as is done in switched-capacitor (SC) filters. , , and the The presented method for designing coefficients was realized in a computer program. Hence, one set of coefficients that satisfies the BPSDM specifications is efficiently generated by the program. IV. INVESTIGATION OF PERFORMANCE AND TOLERANCE TO COEFFICIENT MISMATCH Two BPSDM design examples are used to verify the design method. Each example has the maximum input of 6 dB, value of 20 with signal center frequency of 20 MHz and signal bandwidth of 1 MHz, SCR of 4, and modulator order of 6 and 8, for example, I and II, respectively. Each example includes CRFB, CRFF, and LSCRFF subexamples and each of those conzeros tains 1-bit and 3-bit cases. In all examples, the are placed symmetrically, with one zero at the center of the
REFERENCES [1] T.-H. Kuo, K.-D. Chen, and J.-R. Chen, “Automatic coefficients design for high-order sigma-delta modulators,” IEEE Trans. Circuits Syst. II, Analog. Digit. Signal Process., vol. 6, no. 1, pp. 6–15, Jan. 1999. [2] H.-M. Wang and T.-H. Kuo, “The design of high-order bandpass sigma-delta modulators using low-spread single-stage structure,” IEEE Trans. Circuits Syst. II, Expr. Briefs, vol. 51, no. 4, pp. 202–208, Apr. 2004. [3] L. Louis, J. Abcarius, and G. W. Roberts, “An eight-order bandpass sigma-delta modulator for A/D conversion in digital radio,” IEEE J. Solid-State Circuits, vol. 34, no. 4, Apr. 1999. [4] Y. Botteron, B. Nowrouzian, and A. T. G. Fuller, “Design and cascade-of-resonators sigma-delta converter configuration,” in Proc. IEEE Int. Symp. Circuits Syst., 1999, vol. 2, pp. 45–48. [5] D. B. Ribner, “Multistage bandpass delta sigma modulators,” IEEE Trans. Circuits Syst. II, Analog. Digit. Signal Process., vol. 41, no. 6, pp. 402–405, Jun. 1994. [6] T.-H. Kuo, K.-D. Chen, and H.-R. Yeng, “A wideband CMOS sigmadelta modulator with incremental data weighted averaging,” IEEE J. Solid-State Circuits, vol. 37, no. 1, pp. 11–17, Jan. 2002. [7] S. R. Norsworthy, R. Schreier, and G. C. Temes, Delta-Sigma Data Converters: Theory, Design, and Simulation. Piscataway, NJ: IEEE Press, 1997. [8] A. B. Willianms and F. J. Tayor, Electronic Filter Design Handbook. New York: McGraw-Hill, 1995. [9] P. Cusinato, D. Tonietto, F. Stefani, and A. Baschirotto, “A 3.3 V CMOS 10.7 MHz sixth-order bandpass modulator with 74 dB dynamic range,” IEEE J. Solid-State Circuits, vol. 36, no. 4, pp. 629–638, Apr. 2001.
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 53, NO. 7, JULY 2006
An Automatic Coefficient Design Methodology for High-Order Bandpass Sigma-Delta Modulator With Single-Stage Structure Hwi-Ming Wang, Student Member, IEEE, and Tai-Haur Kuo, Member, IEEE
Abstract—An automatic coefficient design method for synthesis of bandpass sigma-delta modulators (BPSDMs) is presented in this brief. Single-stage BPSDM structures, cascade-of-resonator with distributed feedback, cascade-of-resonator with distributed feedforward, and a new structure with lower coefficient spread, are all used to fit the synthesized coefficients. The automatic coefficient design method is realized in an easy-to-use computer program. Even for inexperienced designers, reliable and high-tolerance BPSDM coefficients for various applications can be automatically and efficiently generated. The methodology covers many design concerns including BPSDM coefficient tolerance for circuit component mismatch, design tradeoffs among in-band noise suppression, oversampling ratio, modulator order and quantizer bit number. Finally, design examples with orders of 6 and 8, and quantizer bit number of 1-bit and 3-bit, respectively, are used for the verification of the proposed automatic coefficient design method. Index Terms—Bandpass sigma-delta modulation (BPSDM), coefficient synthesis, oversampling converter.
I. INTRODUCTION IGMA-DELTA MODULATORs (SDMs) can be classified into low-pass SDMs (LPSDMs) [1] and bandpass SDMs (BPSDMs) [2]. An SDM with high dynamic range (DR) generally results from the use of high-order loop filter, multi-bit quantizer and/or high oversampling ratio (OSR), where OSR is defined as the ratio of sampling frequency to two times of signal bandwidth. To build high-order BPSDMs and achieve high DR, single-stage [3], [4] and cascade or multi-stage noise-shaping (MASH) [5] structures are frequently used. Among these structures, high-order single-stage structures are less sensitive to component mismatch, but design of optimal coefficients for single-stage BPSDMs often suffers from instability problems, long design cycle, and the need for experienced designers. An automatic coefficient synthesis method for high-order LPSDMs was presented by Kuo and Chen [1], making the high-order LPSDM design simpler and more cost-effective. Further, BPSDMs have not been as thoroughly investigated as their LPSDM counterparts. The methods in [3] and [7] were proposed for coefficient design of high-order single-stage BPSDMs. These methods still needed many design iterations between coefficient tolerance and design tradeoffs among
S
Manuscript received June 7, 2005; revised October 22, 2005. This work was supported by the National Science Council of Taiwan under Research Grant NSC 94-2215-E-006-062. This paper was recommended by Associate Editor H. Hashemi. The authors are with the Department of Electrical Engineering, National Cheng Kung University, Tainan City 70101, Taiwan, R.O.C. (e-mail: [email protected]) Digital Object Identifier 10.1109/TCSII.2006.875304
Fig. 1. General single-stage BPSDM structure.
in-band noise suppression, OSR, modulator order, quantizer bit number, and stability. This brief extends the method of Kuo and Chen for high-order single-stage BPSDMs. With the proposed automatic coefficient synthesis methodology, stability and design complexity problems are easily solved. Three general single-stage high-order BPSDM structures, cascade-of-resonator with distributed feedback (CRFB) [3], cascade-of-resonator with distributed feedforward (CRFF) [4], and a new proposed low-spread CRFF (LSCRFF) [2], all are used to fit the synthesized coefficients. The method has been implemented and tested in a high level computer program, making practical BPSDM design convenient. II. DESIGN OF HIGH-ORDER BPSDMS In the single-stage case, the BPSDM has a bandpass loop filter, a quantizer, and a feedback digital-to-analog converter (DAC) as shown in Fig. 1. A multibit quantizer can be embedded in the BPSDM loop to improve the DR for a given OSR, or to reduce OSR for a given DR [6]. The discrete-time linear model [7] is used in the BPSDM analysis. The quantizer in BPSDM is modeled as a unity-gain amplifier with an additive quantization noise source. The overall BPSDM can be characterized by two transfer functions, referred to respectively as signal transfer function and noise transfer function. The two transfer functions can be derived as , where is input signal, is noise, is output signal, is the is the noise transfer funcsignal transfer function, and and are first discussed. tion. The designs of A. Design of
and
For a BPSDM, is required to minimize in-band noise power and thus maximize peak signal-to-noise ratio (PSNR). can be approximated by conventional bandreject The
1057-7130/$20.00 © 2006 IEEE
WANG AND KUO: AUTOMATIC COEFFICIENT DESIGN METHODOLOGY FOR HIGH-ORDER BPSDM
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Fig. 3. BPSDM coefficients design flow.
P N
Fig. 2. NPG-versusplot for: (a) 1-bit and SCR = 4, (b) 1-bit and SCR = 16, (c) 3-bit and SCR = 4, (d) 3-bit and SCR = 16 BPSDMs with = 8, 16, 32, 64 and = 8.
Q
analog filter functions. In this brief, inverse-Chebyshev func. A -domain bandreject functions are used for the tion with an order of is obtained from an -domain low-pass through frequency denormalizafunction with an order of tion, low-pass-to-bandreject transformation and -to- bilinear characteristics are related to retransformation [7]. , modulator order , bandquired stopband attenuation , signal center frequency , and sampling frewidth quency [8]. is also related to , , quality factor , and sampling frequency to signal center frequency ratio and , re(SCR) when and SCR are defined as spectively. The relation between and PSNR is (1) where is the maximum-input magnitude and is the quantization step size of the quantizer in Fig. 1. When PSNR, , SCR, , , , and the quantizer bit number are specwhose satisfies (1) is obified, an initial BPSDM tained. , only zero positions are designed since For and share poles in CRFB, CRFF, and LSCRFF, as will on be shown in the next section. By placing the zeros of the unit circle, the signal is suppressed at the corresponding freare placed quency. For an th-order BPSDM, zeros of zeros at dc, zeros at symmetrically, with , and one at the center of the unit circle [9]. B. Stability Constraints Infinite numbers of can be synthesized from (1), but not all of these can be used to design a stable BPSDM. Stability constraints must be established to exso that the unstable ones can be excluded. amine these For low-pass SDMs, noise power gain (NPG) defined as
can be used to examine stability [1]. In this brief, is applied for the is associated with BPSDMs stability evaluation. Since specifications such as , , and SCR, therefore is also is decreased when increases, SCR related to them. increases or a higher quantizer bit number is applied [1]. To upper bound , the relation between decide the and must be first assured because is associated with . -versusplots for 1-bit and 3-bit A variety of 4, eighth-order BPSDMs are shown in Fig. 2, covering 16 and 8, 16, 32, and 64. For convenient use, BPSDM stability constraints need to be expressed by numerical equations. approximated by a first-order For 1-bit BPSDMs, the equation is close enough to the stability bounds, while a thirdorder equation is close enough for 3-bit BPSDMs. Hence, the stability constraint is expressed as (2) Equation (2) represents an approximated numerical equation for BPSDM stability bound. The values of , , and are chosen such that the approximated curve is close to the instaand are set to zero. bility edge. For first-order equations, For practical BPSDM design, component mismatch usually redue to coeffisults in coefficients variation. The change in where cient variation is denoted as is its nominal value and is its variation. When is positive, the BPSDM may become unstable. To the have a safety margins for the stability bound, the relations bedeviation and coefficient mismatch are tween maximum obtained by Monte Carlo simulation. Maximum deviation of versus different maximum coefficient variations (0.5%, 1% and 2%) and different orders are investigated. It condoes not exceed 10% cludes that the maximum ) for maximum coefficient variations of 1% (0.5% for when the range of is from 8 to 64 and SCR is from 4 to 32. The final stability bound is selected so as to tolerate a maximum of 10%, as shown in Fig. 2. Table I variation of
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 53, NO. 7, JULY 2006
Fig. 4. (a) CRFB, (b) CRFF, and (c) LSCRFF structures for
N th-order m-bit BPSDMs.
lists the and values for 1-bit BPSDMs, and the , , , and values for 3-bit BPSDMs with loop filter orders 6 oband 8, and SCR values 4, 8, 16, and 32. The initial tained from (1) can then be examined by using Table I and thus are obtained. only stable
TABLE I COEFFICIENTS OF STABILITY BOUND [SEE (2)] FOR (A) 1-BIT AND (B) 3-BIT BPSDMS WITH ORDERS OF 6, 8, AND SCR FROM 4 TO 32
III. COEFFICIENT DESIGN FLOW When the of a synthesized is larger than , quantizer bit number and/or and/or SCR may be with a smaller [1]. increased to give a different is corrected by For lower order loop filters, excessive SCR increase. On the other hand, for designs requiring lower excess is corrected by increase. Gencircuit speeds, erally, Fig. 3 shows a coefficient design flow for a single-stage BPSDM. The procedure starts from designating system spec, range of ifications such as PSNR, maximum input
SCR ( and
, and ), range of modulator order ( ), maximum quantizer bit number , ,
,
WANG AND KUO: AUTOMATIC COEFFICIENT DESIGN METHODOLOGY FOR HIGH-ORDER BPSDM
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TABLE II SYNTHESIZED BPSDM COEFFICIENTS AND PERFORMANCE FOR EXAMPLES I AND II
and the specified preference of low-order or low OSR. The is generated from (1). The stability constraint, initial i.e., whether is satisfied, is examined by (2) and Table I. By iterated increase of the quantizer bit number, can be obtained. SCR or , an appropriate Three single-stage structures of CRFB, CRFF, and LSCRFF and . The are used to fit the designed th-order, -bit CRFB, CRFF, and LSCRFF structures are is an inteshown in Fig. 4(a)–(c), respectively, where and , , grator with transfer function of , , , and are modulator coefficients. The transfer functions for CRFB are and
and
For CRFF, they are , where
(6)
(7)
, where (3) (8)
(4)
(5)
By equating the or , and the synthesized , these coefficients can be obtained. For and in (3), are decided by equating the CRFB case, and synthesized ; in (4) numerator of are determined by equating the denominator of and synthesized ; in (5) are determined by equating , and . For CRFF case, the numerator of and the using similar flow such as equating the to find the corresponding coefficients synthesized
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Fig. 5. (a) Pole–zero diagram. (b) Response of
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 53, NO. 7, JULY 2006
NTF(z) and STF(z).
unit circle, two and three zeros at dc and two and three zeros at for modulator orders 6 and 8, respectively. In eighth-order and are example, the pole–zero positions of shown in Fig. 5(a), and the frequency responses in Fig. 5(b), showing the bandreject and bandpass characteristic for and . The synthesized coefficients and achieved PSNR and DR are summarized in Table II. To investigate PSNR degradation due to coefficient variation, Monte Carlo analysis with 100 different samples each is applied to sixth- and eighth-order design examples, each with CRFB, CRFF, and LSCRFF and quantizer bit number of 3. Histograms of PSNR deviation, subject to a 1% maximum mismatch, are shown in Fig. 6. For comparable sixth-order example, the degraded PSNR are, respectively, 10 dB, 8 dB for CRFB and CRFB, and 4 dB for LSCRFF. For the eighth-order example, these respective values are 9 dB, 10 dB, and 5 dB. V. CONCLUSION An automatic coefficient design method for high-order single-stage BPSDMs has been presented. Tradeoffs among quantizer bit number, modulator order and SCR were made to from those stable . The select an appropriate whole design method was realized in a high-level computer program convenient for BPSDM designers. Based on the presented design method, an optimum set of BPSDM coefficients can be automatically designed to fit given specifications for the convenient CRFB and CRFF BPSDM structures, and also the recent single-stage LSCRFF structure which optimized selected coefficients for reduced coefficient spread and less sensitivity to coefficient mismatch.
Fig. 6. Histograms of PSNR deviation for design examples with 1% max. coefficient mismatch and (a) CRFB, (b) CRFF, and (c) LSCRFF among 100 trials.
from (6)-(8). The same flows and derivations can be applied in LSCRFF structure shown in [2]. To avoid internal clipping and to maximize BPSDM DR, coefficients are scaled as is done in switched-capacitor (SC) filters. , , and the The presented method for designing coefficients was realized in a computer program. Hence, one set of coefficients that satisfies the BPSDM specifications is efficiently generated by the program. IV. INVESTIGATION OF PERFORMANCE AND TOLERANCE TO COEFFICIENT MISMATCH Two BPSDM design examples are used to verify the design method. Each example has the maximum input of 6 dB, value of 20 with signal center frequency of 20 MHz and signal bandwidth of 1 MHz, SCR of 4, and modulator order of 6 and 8, for example, I and II, respectively. Each example includes CRFB, CRFF, and LSCRFF subexamples and each of those conzeros tains 1-bit and 3-bit cases. In all examples, the are placed symmetrically, with one zero at the center of the
REFERENCES [1] T.-H. Kuo, K.-D. Chen, and J.-R. Chen, “Automatic coefficients design for high-order sigma-delta modulators,” IEEE Trans. Circuits Syst. II, Analog. Digit. Signal Process., vol. 6, no. 1, pp. 6–15, Jan. 1999. [2] H.-M. Wang and T.-H. Kuo, “The design of high-order bandpass sigma-delta modulators using low-spread single-stage structure,” IEEE Trans. Circuits Syst. II, Expr. Briefs, vol. 51, no. 4, pp. 202–208, Apr. 2004. [3] L. Louis, J. Abcarius, and G. W. Roberts, “An eight-order bandpass sigma-delta modulator for A/D conversion in digital radio,” IEEE J. Solid-State Circuits, vol. 34, no. 4, Apr. 1999. [4] Y. Botteron, B. Nowrouzian, and A. T. G. Fuller, “Design and cascade-of-resonators sigma-delta converter configuration,” in Proc. IEEE Int. Symp. Circuits Syst., 1999, vol. 2, pp. 45–48. [5] D. B. Ribner, “Multistage bandpass delta sigma modulators,” IEEE Trans. Circuits Syst. II, Analog. Digit. Signal Process., vol. 41, no. 6, pp. 402–405, Jun. 1994. [6] T.-H. Kuo, K.-D. Chen, and H.-R. Yeng, “A wideband CMOS sigmadelta modulator with incremental data weighted averaging,” IEEE J. Solid-State Circuits, vol. 37, no. 1, pp. 11–17, Jan. 2002. [7] S. R. Norsworthy, R. Schreier, and G. C. Temes, Delta-Sigma Data Converters: Theory, Design, and Simulation. Piscataway, NJ: IEEE Press, 1997. [8] A. B. Willianms and F. J. Tayor, Electronic Filter Design Handbook. New York: McGraw-Hill, 1995. [9] P. Cusinato, D. Tonietto, F. Stefani, and A. Baschirotto, “A 3.3 V CMOS 10.7 MHz sixth-order bandpass modulator with 74 dB dynamic range,” IEEE J. Solid-State Circuits, vol. 36, no. 4, pp. 629–638, Apr. 2001.
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