Quasi-Equivalent Feedforward System in Asymptotic ... - IEEE Xplore
482
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 51, NO. 9, SEPTEMBER 2004
Quasi-Equivalent Feedforward System in Modulation Asymptotic
61
Nguyen T. Thao
61
Abstract—It was previously shown that sigma–delta ( ) modulators of “asymptotic” type theoretically yield an equivalent feedforward system where the recursive nonlinear mechanisms are extracted from the feedback loop and reduced to a memoryless function. With time-varying inputs, we show in this paper, partially by mathematical derivations and partially by experiment, that this system is quasi-equivalent to the original modulator in a sense that we explain. This reduction of the nonlinear mechanisms should permit more refined modeling of the errors in future research, with a better account of the original nonlinearities of asymptotic modulation.
61
61
Index Terms—Circuit equivalence, feedback, feedforward, modeling, nonlinear systems, quantization error, sigma–delta ( ) modulation.
61
I. INTRODUCTION
T
HE fundamental difficulty in the signal analysis of ) modulators lies in the presence of a sigma–delta ( nonlinear operation in a feedback loop. This basically prevents the derivation of the explicit and deterministic expression of the modulator is currently error signal. The performance of a predicted either by simulations or by a simplified model that basically linearizes the quantizer error signal. The only exact error analysis available until now applies to the particular case of ideal nonoverloaded single-loop and multiloop modulators [1], [2] where the explicit output expression appears to be accessible. With constant inputs, it was recently found in [3] that such an explicit expression is theoretically possible on a large modulators called the “asymptotic” class of overloaded modulators [4]. This class basically includes the modulators whose noise transfer functions have all their zeros at the zero frequency. The explicit output expression can be given through an equivalent feedforward system where all the nonlinear recursive mechanisms are reduced to a single -dimensional nonlinear but memoryless function of modulo type. In the general case of a time-varying input, we show in this paper that this system can be extended into a virtual feedforward encoder that is quasi-equivalent to the original modulator in a sense that we explain. After reviewing the past results on asymptotic modulators in Section II, we give in Section III the general theoretical properties of the virtual encoder. At this stage, the exact parameters of the virtual system are not in general easy to
Manuscript received October 14, 2003; revised March 20, 2004. This work has been supported in part by the National Science Foundation under Grant CCR-0209431 and Grant DMS-0219053. This paper was recommended by Associate Editor P. Carbone. The author is with the Department of Electrical Engineering, City College, CUNY, New York, NY 10031 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TCSII.2004.834533
Fig. 1. Generic diagram of
61 modulation.
extract. We numerically evaluate the equivalence of the virtual modulafeedforward system at least on one prototype of tion where the explicit extraction of the nonlinear memoryless function has been possible [5]. The general motivation of this error modeling by a study is to start a new framework of more faithful account of the nonlinear mechanisms. II. REVIEW ON ASYMPTOTIC
MODULATORS
In this section, we review the results previously obtained in [4], [3] on asymptotic modulators. As defined in [4], these modulators are described by the block diagram of Fig. 1 with the following characteristics. 1) with and . 2) The transfer function of the quantizer is any increasing with equally spaced output levels staircase function (we normalize the step size to 1, im). plying that This class of modulators includes as a particular case the standard single-loop and multiloop configurations [1], [2]. In the general case, the asymptotic modulators correspond to a generic modulators whose design was presented in class of practical [6], [7]. Condition 2) automatically includes all single-bit quanand ). tizers (particular case where Asymptotic modulators were observed to yield a remarkable property in [8], [9], [4]. Let us define the -dimensional vector1 (1) is the th-order integration of the modulator . In the -domain, is defined by where as specified in condition 1). Note that the above relation implies that
where error
(2) 1The transposition symbol vector
1057-7130/04$20.00 © 2004 IEEE
is used in (1) to denote that U
[k] is a column
THAO: QUASI-EQUIVALENT FEEDFORWARD SYSTEM
483
in all dimensions we mean that the integer translations of do not intersect with each other, while their union covers the whole space. We show an example of this phenomenon in Fig. 2 with the single-bit second-order configuration such that . Mathematically, the tiling property of is expressed as follows: (3) It was derived in [3] that the vector recursive system of equations:
satisfies the following
(4) (5) where
is the identity matrix, , and the coefficients ’s result from the expansion . Note that is an matrix that contains only integer coefficients. Equations (4) and (5) and provide the complete description of the the relation dynamical system of the modulator and lead to the equivalent block diagram of Fig. 3(a). Although not obvious from the orig, Fig. 3(a) shows better that is a state inal definition of vector of the system. , it was then derived that In the case where satisfies the recursive relation
where (6)
[] ( )= = (1 15)
Fig. 2. Representation in black of 1 000 consecutive state points U k starting , for single-bit second-order modulator with A z from U = z and constant inputs x k x: (a) x = ; (b) x = ; = . The shaded areas are the translated region of the polygon (b) x containing the state points by (1, 0) and (1, 1), respectively.
[0] = [0 0] 10(1 2) = (1 6)
[ ]=
61 = (1 500)
With any constant input and under stability conditions, it was observed in [8], [9], [4] at least in the second-order remain in a tile . By this, case that the state points
and is the transfer function of the quantizer. When the system is stable, it is explained in [10] that there exists by necessity that is positively invariant a bounded and nonempty region , i.e., such that . Any sequence of state by points starting from the stability region eventually converges , where is recursively defined by to the set . It was proved in [11] that is by necessity , i.e., , and in fact, is the maximal invariant by invariant set of [12]. Furthermore, it was proved at least for of (6) belongs to a irrational values of that the mapping class of mappings whose maximal invariant sets are by necessity the union of a finite number of disjoint tiles. Experiments presented in [4], [5] (from which Fig. 2 is a sample) show that this union is reduced to a single tile at least with all single-bit second-order asymptotic configurations. The single-tile property was moreover proved and the tile explicitly derived in [5] on the standard multibit multiloop configurations and on one
484
Fig. 3.
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 51, NO. 9, SEPTEMBER 2004
(a) Equivalent dynamical system of nth order asymptotic
61 modulator. (b) Virtual feedforward encoder with initial condition [0] = [0].
Fig. 4. Illustration of the modulo mechanism of the transformation U and V k U k ; .
[ + 1] 0 [ + 1] = (0 2)
V
U
[ ] = mod ( [ ]). In the figure, [ 0 1] 0 [ 0 1] = (1 01) [ ] 0 [ ] = (2 1) k
particular single-bit second-order modulator, i.e., characterized . We have actually shown by a solid by line in Fig. 2 the boundaries of the invariant tile analytically obtained on this single-bit configuration. The figure shows that the analytical derivations and the numerical results match. We base the rest of the paper on the conjecture that the invariant set is always reduced to a single tile at all orders.
V k
V k
U k
;
;V k
U k
;
show to be quasi-equivalent to that of Fig. 3(a) in a certain sense. introduced This system uses a special nonlinear function is in [5], [3] that we review here. From the property that , there exists a a tile, (3) implies that for any vector such that . We denote unique vector . We illustrate the function this unique vector by in Fig. 4. One can easily prove that this function satisfies the following two properties:
III. QUASI-EQUIVALENT FEEDFORWARD SYSTEM of is a tile, we Using the property that the invariant set build a feedforward system as shown in Fig. 3(b), which we will
(7) (8)
THAO: QUASI-EQUIVALENT FEEDFORWARD SYSTEM
485
61 ( ) = 1 0 (1 2)
Fig. 5. Comparison between the outputs of modulator and quasi-equivalent system. (a) Constant input case. (b) Time-varying input case. (i) Input. (ii) Output = z . (iii) Output of corresponding quasi-equivalent system. (iv) Difference between the outputs of of single-bit second-order modulator with A z modulator and quasi-equivalent system.
61
In other words, is 1-periodic in each dimension and pointwise invariant within 2. The system of Fig. 3(b) starts by computing the sequence of recursively defined by vector (9) with the initial condition . Note that (9) is similar to has been replaced by the constant minimal (4), except that quantized level . This similarity implies the following property. Theorem III.1: The difference vector belongs to . since Proof: This is already true at . Suppose that for some . By subtracting (9) to (4), we find that . Because is only composed of , obviously . integer coefficients and The system of Fig. 3(b) proceeds by calculating the vector . By property of , we have by . Then, Theorem III.1 implies necessity that the following property. 2It can be actually shown that there exists a unique function that satisfies (7) and (8).
Theorem III.2: The difference vector belongs . is the following. Now, the outstanding property of of the Theorem III.3: If at a certain instant , the vector original modulator happens to belong to (the invariant set at ), then, by necessity, . Proof: Because of the above assumption, (8) implies . Now, from Theorem III.1, that where . Then, (7) im. plies that Corollary III.4: If is constant and equal to and , then for all . Proof: Because is invariant by for all . The proof is completed by applying Theorem III.3 at all . This theorem implies that under a constant input , the state vector of the original modulator can be obtained and the explicitly from the knowledge of , via the vector the first component block diagram of Fig. 3(b). Let us call and let be the -transform of . of the vector We implement in Fig. 3(b) the computation of the signal defined in the -domain by to
(10)
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 51, NO. 9, SEPTEMBER 2004
61
Fig. 6. Experimental measure of error between modulator and quasi-equivalent system with sinusoidal inputs for the case of single-bit second-order modulator = z . (a) Frequency of the event u k u k per Nyquist period. (b) Comparison of in-band MSE of outputs. with A z
( ) = 1 0 (1 2)
[ ] 6= [ ]
Under the conditions of Corollary III.4, (2) and (10) imply that . In other words, when is constant, the two systems of Fig. 3 are rigorously equivalent under matching initial conditions [see example of Fig. 5(a)]. We retrieve here the previous result of [3]. Now, when the input is time-varying, we can no longer guarfor all . This implies that antee that may happen. When this is the case, we get which implies through (2) and (10). Now, because of Theorem III.2, we obviously have the following constraint. Theorem III.5: and Because , we know that is by necesplus an integer. Fig. 5(iii)(b) gives an illustrasity equal to . Thus, as a first consequence of tion of this, with this theorem, the system of Fig. 3(b) works like a virtual A/D converter, although it does not explicitly include a quantizer. For this reason, we call it “virtual encoder.” Now, note that it has the same range of quantized is not implied here that values as , as can actually be seen in Fig. 5(b). However, and are equal to each one observes from this test that and are derived from other most of the time. Since and from the relations (2) and (10) respectively, let us measure how often and differ from each other. from Theorem III.5, note that the event Since
” can be detected without ambiguity. According to “ the results shown in Fig. 6(a), we find that this event happens no more than once per Nyquist period, regardless of the oversampling ratio. Now, the measure of most significance is the differand in terms of in-band mean-square ence between error (MSE). The results of Fig. 6(b) show that this difference is negligible, and approximately 20 dB or more below the in-band . This demonstrates the quasi-equivalence between MSE of the two systems. With this quasi-equivalence, the particular motivation of this signal analysis by paper is to initiate new directions of studying the virtual encoder instead of the original encoder. The virtual system has the advantage that all recursive nonlinear mechanisms are reduced to the single instantaneous nonlinear . While the description of the set is diffunction ficult to derive in general, the virtual encoder at least offers a nonrecursive structure of the nonlinear mechanisms from which new types of models can be investigated. Meanwhile, current efand the forts are made to find the explicit relation between noise-transfer function . ACKNOWLEDGMENT The author would like to thank S. Güntürk and I. Daubechies for their great influence and helpful discussions on this research topic.
THAO: QUASI-EQUIVALENT FEEDFORWARD SYSTEM
487
REFERENCES [1] R. M. Gray, W. Chou, and P.-W. Wong, “Quantization noise in single-loop sigma-delta modulation with sinusoidal input,” IEEE Trans. Commun., vol. 37, pp. 956–968, Sept. 1989. [2] N. He, F. Kuhlmann, and A. Buzo, “Multiloop sigma-delta quantization,” IEEE Trans. Inform. Theory, vol. 38, pp. 1015–1028, May 1992. [3] N. T. Thao, “Breaking the feedback loop of modulators,” in Proc. IEEE Int. Conf. ASSP, vol. 6, Apr. 2003, pp. 677–680. [4] , “MSE behavior and centroid function of th order asymptotic modulators,” IEEE Trans. Circuits Syst. II, vol. 49, pp. 86–100, Feb. 2002. [5] S. Güntürk and N. T. Thao, “Refined error analysis in second-order modulation with constant inputs,” IEEE Trans. Inform. Theory, vol. 50, pp. 839–860, May 2004, submitted for publication. [6] S. R. Norsworthy, R. Schreier, and G. C. Temes, Eds., Delta-Sigma Data Converters: Theory, Design, and Simulation. Piscataway, NJ: IEEE Press, 1996.
61
61
m
61
[7] R. Schreier, “An empirical study of high-order single-bit delta-sigma modulators,” IEEE Trans. Circuits Syst. II, vol. 40, pp. 461–466, Aug. 1993. [8] I. Daubechies and R. DeVore, “Reconstructing a bandlimited function from very coarsely quantized data: A family of stable sigma-delta modulators of arbitrary order,” Ann. Math., vol. 158, pp. 643–674, Sept. 2003. [9] S. Güntürk, “Harmonic Analysis of Two Problems in Signal Quantization and Compression,” Ph.D. dissertation, Dept. Appl. Comput. Math., Princeton Univ., Princeton, NJ, Oct. 2000. [10] R. Schreier, M. V. Goodson, and B. Zhang, “An algorithm for computing convex positively invariant sets for delta-sigma modulators,” IEEE Trans. Circuits Syst. I, vol. 44, pp. 38–44, Jan. 1997. [11] S. Güntürk and N. T. Thao, “Ergodic dynamics in quantization: Tiling invariant sets and spectral analysis of error,” Adv. Appl. Math., to be published. [12] P. Ashwin, X.-C. Fu, T. Nishikawa, and K. Zyczkowski, “Invariant sets for discontinuous parabolic area-preserving torus maps,” Nonlinearity, pp. 819–835, 2000.
61
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 51, NO. 9, SEPTEMBER 2004
Quasi-Equivalent Feedforward System in Modulation Asymptotic
61
Nguyen T. Thao
61
Abstract—It was previously shown that sigma–delta ( ) modulators of “asymptotic” type theoretically yield an equivalent feedforward system where the recursive nonlinear mechanisms are extracted from the feedback loop and reduced to a memoryless function. With time-varying inputs, we show in this paper, partially by mathematical derivations and partially by experiment, that this system is quasi-equivalent to the original modulator in a sense that we explain. This reduction of the nonlinear mechanisms should permit more refined modeling of the errors in future research, with a better account of the original nonlinearities of asymptotic modulation.
61
61
Index Terms—Circuit equivalence, feedback, feedforward, modeling, nonlinear systems, quantization error, sigma–delta ( ) modulation.
61
I. INTRODUCTION
T
HE fundamental difficulty in the signal analysis of ) modulators lies in the presence of a sigma–delta ( nonlinear operation in a feedback loop. This basically prevents the derivation of the explicit and deterministic expression of the modulator is currently error signal. The performance of a predicted either by simulations or by a simplified model that basically linearizes the quantizer error signal. The only exact error analysis available until now applies to the particular case of ideal nonoverloaded single-loop and multiloop modulators [1], [2] where the explicit output expression appears to be accessible. With constant inputs, it was recently found in [3] that such an explicit expression is theoretically possible on a large modulators called the “asymptotic” class of overloaded modulators [4]. This class basically includes the modulators whose noise transfer functions have all their zeros at the zero frequency. The explicit output expression can be given through an equivalent feedforward system where all the nonlinear recursive mechanisms are reduced to a single -dimensional nonlinear but memoryless function of modulo type. In the general case of a time-varying input, we show in this paper that this system can be extended into a virtual feedforward encoder that is quasi-equivalent to the original modulator in a sense that we explain. After reviewing the past results on asymptotic modulators in Section II, we give in Section III the general theoretical properties of the virtual encoder. At this stage, the exact parameters of the virtual system are not in general easy to
Manuscript received October 14, 2003; revised March 20, 2004. This work has been supported in part by the National Science Foundation under Grant CCR-0209431 and Grant DMS-0219053. This paper was recommended by Associate Editor P. Carbone. The author is with the Department of Electrical Engineering, City College, CUNY, New York, NY 10031 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TCSII.2004.834533
Fig. 1. Generic diagram of
61 modulation.
extract. We numerically evaluate the equivalence of the virtual modulafeedforward system at least on one prototype of tion where the explicit extraction of the nonlinear memoryless function has been possible [5]. The general motivation of this error modeling by a study is to start a new framework of more faithful account of the nonlinear mechanisms. II. REVIEW ON ASYMPTOTIC
MODULATORS
In this section, we review the results previously obtained in [4], [3] on asymptotic modulators. As defined in [4], these modulators are described by the block diagram of Fig. 1 with the following characteristics. 1) with and . 2) The transfer function of the quantizer is any increasing with equally spaced output levels staircase function (we normalize the step size to 1, im). plying that This class of modulators includes as a particular case the standard single-loop and multiloop configurations [1], [2]. In the general case, the asymptotic modulators correspond to a generic modulators whose design was presented in class of practical [6], [7]. Condition 2) automatically includes all single-bit quanand ). tizers (particular case where Asymptotic modulators were observed to yield a remarkable property in [8], [9], [4]. Let us define the -dimensional vector1 (1) is the th-order integration of the modulator . In the -domain, is defined by where as specified in condition 1). Note that the above relation implies that
where error
(2) 1The transposition symbol vector
1057-7130/04$20.00 © 2004 IEEE
is used in (1) to denote that U
[k] is a column
THAO: QUASI-EQUIVALENT FEEDFORWARD SYSTEM
483
in all dimensions we mean that the integer translations of do not intersect with each other, while their union covers the whole space. We show an example of this phenomenon in Fig. 2 with the single-bit second-order configuration such that . Mathematically, the tiling property of is expressed as follows: (3) It was derived in [3] that the vector recursive system of equations:
satisfies the following
(4) (5) where
is the identity matrix, , and the coefficients ’s result from the expansion . Note that is an matrix that contains only integer coefficients. Equations (4) and (5) and provide the complete description of the the relation dynamical system of the modulator and lead to the equivalent block diagram of Fig. 3(a). Although not obvious from the orig, Fig. 3(a) shows better that is a state inal definition of vector of the system. , it was then derived that In the case where satisfies the recursive relation
where (6)
[] ( )= = (1 15)
Fig. 2. Representation in black of 1 000 consecutive state points U k starting , for single-bit second-order modulator with A z from U = z and constant inputs x k x: (a) x = ; (b) x = ; = . The shaded areas are the translated region of the polygon (b) x containing the state points by (1, 0) and (1, 1), respectively.
[0] = [0 0] 10(1 2) = (1 6)
[ ]=
61 = (1 500)
With any constant input and under stability conditions, it was observed in [8], [9], [4] at least in the second-order remain in a tile . By this, case that the state points
and is the transfer function of the quantizer. When the system is stable, it is explained in [10] that there exists by necessity that is positively invariant a bounded and nonempty region , i.e., such that . Any sequence of state by points starting from the stability region eventually converges , where is recursively defined by to the set . It was proved in [11] that is by necessity , i.e., , and in fact, is the maximal invariant by invariant set of [12]. Furthermore, it was proved at least for of (6) belongs to a irrational values of that the mapping class of mappings whose maximal invariant sets are by necessity the union of a finite number of disjoint tiles. Experiments presented in [4], [5] (from which Fig. 2 is a sample) show that this union is reduced to a single tile at least with all single-bit second-order asymptotic configurations. The single-tile property was moreover proved and the tile explicitly derived in [5] on the standard multibit multiloop configurations and on one
484
Fig. 3.
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 51, NO. 9, SEPTEMBER 2004
(a) Equivalent dynamical system of nth order asymptotic
61 modulator. (b) Virtual feedforward encoder with initial condition [0] = [0].
Fig. 4. Illustration of the modulo mechanism of the transformation U and V k U k ; .
[ + 1] 0 [ + 1] = (0 2)
V
U
[ ] = mod ( [ ]). In the figure, [ 0 1] 0 [ 0 1] = (1 01) [ ] 0 [ ] = (2 1) k
particular single-bit second-order modulator, i.e., characterized . We have actually shown by a solid by line in Fig. 2 the boundaries of the invariant tile analytically obtained on this single-bit configuration. The figure shows that the analytical derivations and the numerical results match. We base the rest of the paper on the conjecture that the invariant set is always reduced to a single tile at all orders.
V k
V k
U k
;
;V k
U k
;
show to be quasi-equivalent to that of Fig. 3(a) in a certain sense. introduced This system uses a special nonlinear function is in [5], [3] that we review here. From the property that , there exists a a tile, (3) implies that for any vector such that . We denote unique vector . We illustrate the function this unique vector by in Fig. 4. One can easily prove that this function satisfies the following two properties:
III. QUASI-EQUIVALENT FEEDFORWARD SYSTEM of is a tile, we Using the property that the invariant set build a feedforward system as shown in Fig. 3(b), which we will
(7) (8)
THAO: QUASI-EQUIVALENT FEEDFORWARD SYSTEM
485
61 ( ) = 1 0 (1 2)
Fig. 5. Comparison between the outputs of modulator and quasi-equivalent system. (a) Constant input case. (b) Time-varying input case. (i) Input. (ii) Output = z . (iii) Output of corresponding quasi-equivalent system. (iv) Difference between the outputs of of single-bit second-order modulator with A z modulator and quasi-equivalent system.
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In other words, is 1-periodic in each dimension and pointwise invariant within 2. The system of Fig. 3(b) starts by computing the sequence of recursively defined by vector (9) with the initial condition . Note that (9) is similar to has been replaced by the constant minimal (4), except that quantized level . This similarity implies the following property. Theorem III.1: The difference vector belongs to . since Proof: This is already true at . Suppose that for some . By subtracting (9) to (4), we find that . Because is only composed of , obviously . integer coefficients and The system of Fig. 3(b) proceeds by calculating the vector . By property of , we have by . Then, Theorem III.1 implies necessity that the following property. 2It can be actually shown that there exists a unique function that satisfies (7) and (8).
Theorem III.2: The difference vector belongs . is the following. Now, the outstanding property of of the Theorem III.3: If at a certain instant , the vector original modulator happens to belong to (the invariant set at ), then, by necessity, . Proof: Because of the above assumption, (8) implies . Now, from Theorem III.1, that where . Then, (7) im. plies that Corollary III.4: If is constant and equal to and , then for all . Proof: Because is invariant by for all . The proof is completed by applying Theorem III.3 at all . This theorem implies that under a constant input , the state vector of the original modulator can be obtained and the explicitly from the knowledge of , via the vector the first component block diagram of Fig. 3(b). Let us call and let be the -transform of . of the vector We implement in Fig. 3(b) the computation of the signal defined in the -domain by to
(10)
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 51, NO. 9, SEPTEMBER 2004
61
Fig. 6. Experimental measure of error between modulator and quasi-equivalent system with sinusoidal inputs for the case of single-bit second-order modulator = z . (a) Frequency of the event u k u k per Nyquist period. (b) Comparison of in-band MSE of outputs. with A z
( ) = 1 0 (1 2)
[ ] 6= [ ]
Under the conditions of Corollary III.4, (2) and (10) imply that . In other words, when is constant, the two systems of Fig. 3 are rigorously equivalent under matching initial conditions [see example of Fig. 5(a)]. We retrieve here the previous result of [3]. Now, when the input is time-varying, we can no longer guarfor all . This implies that antee that may happen. When this is the case, we get which implies through (2) and (10). Now, because of Theorem III.2, we obviously have the following constraint. Theorem III.5: and Because , we know that is by necesplus an integer. Fig. 5(iii)(b) gives an illustrasity equal to . Thus, as a first consequence of tion of this, with this theorem, the system of Fig. 3(b) works like a virtual A/D converter, although it does not explicitly include a quantizer. For this reason, we call it “virtual encoder.” Now, note that it has the same range of quantized is not implied here that values as , as can actually be seen in Fig. 5(b). However, and are equal to each one observes from this test that and are derived from other most of the time. Since and from the relations (2) and (10) respectively, let us measure how often and differ from each other. from Theorem III.5, note that the event Since
” can be detected without ambiguity. According to “ the results shown in Fig. 6(a), we find that this event happens no more than once per Nyquist period, regardless of the oversampling ratio. Now, the measure of most significance is the differand in terms of in-band mean-square ence between error (MSE). The results of Fig. 6(b) show that this difference is negligible, and approximately 20 dB or more below the in-band . This demonstrates the quasi-equivalence between MSE of the two systems. With this quasi-equivalence, the particular motivation of this signal analysis by paper is to initiate new directions of studying the virtual encoder instead of the original encoder. The virtual system has the advantage that all recursive nonlinear mechanisms are reduced to the single instantaneous nonlinear . While the description of the set is diffunction ficult to derive in general, the virtual encoder at least offers a nonrecursive structure of the nonlinear mechanisms from which new types of models can be investigated. Meanwhile, current efand the forts are made to find the explicit relation between noise-transfer function . ACKNOWLEDGMENT The author would like to thank S. Güntürk and I. Daubechies for their great influence and helpful discussions on this research topic.
THAO: QUASI-EQUIVALENT FEEDFORWARD SYSTEM
487
REFERENCES [1] R. M. Gray, W. Chou, and P.-W. Wong, “Quantization noise in single-loop sigma-delta modulation with sinusoidal input,” IEEE Trans. Commun., vol. 37, pp. 956–968, Sept. 1989. [2] N. He, F. Kuhlmann, and A. Buzo, “Multiloop sigma-delta quantization,” IEEE Trans. Inform. Theory, vol. 38, pp. 1015–1028, May 1992. [3] N. T. Thao, “Breaking the feedback loop of modulators,” in Proc. IEEE Int. Conf. ASSP, vol. 6, Apr. 2003, pp. 677–680. [4] , “MSE behavior and centroid function of th order asymptotic modulators,” IEEE Trans. Circuits Syst. II, vol. 49, pp. 86–100, Feb. 2002. [5] S. Güntürk and N. T. Thao, “Refined error analysis in second-order modulation with constant inputs,” IEEE Trans. Inform. Theory, vol. 50, pp. 839–860, May 2004, submitted for publication. [6] S. R. Norsworthy, R. Schreier, and G. C. Temes, Eds., Delta-Sigma Data Converters: Theory, Design, and Simulation. Piscataway, NJ: IEEE Press, 1996.
61
61
m
61
[7] R. Schreier, “An empirical study of high-order single-bit delta-sigma modulators,” IEEE Trans. Circuits Syst. II, vol. 40, pp. 461–466, Aug. 1993. [8] I. Daubechies and R. DeVore, “Reconstructing a bandlimited function from very coarsely quantized data: A family of stable sigma-delta modulators of arbitrary order,” Ann. Math., vol. 158, pp. 643–674, Sept. 2003. [9] S. Güntürk, “Harmonic Analysis of Two Problems in Signal Quantization and Compression,” Ph.D. dissertation, Dept. Appl. Comput. Math., Princeton Univ., Princeton, NJ, Oct. 2000. [10] R. Schreier, M. V. Goodson, and B. Zhang, “An algorithm for computing convex positively invariant sets for delta-sigma modulators,” IEEE Trans. Circuits Syst. I, vol. 44, pp. 38–44, Jan. 1997. [11] S. Güntürk and N. T. Thao, “Ergodic dynamics in quantization: Tiling invariant sets and spectral analysis of error,” Adv. Appl. Math., to be published. [12] P. Ashwin, X.-C. Fu, T. Nishikawa, and K. Zyczkowski, “Invariant sets for discontinuous parabolic area-preserving torus maps,” Nonlinearity, pp. 819–835, 2000.
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