Phase Uncertainty of a Sampled Quantizer J.A.E.P. van Engelen and B.E. Sarroukh
Eindhoven University of Technology, Department of Electrical Engineering Electronic Signal Processing Systems Group (SES), P.O. Box 513, 5600 MB Eindhoven, The Netherlands tel. +31 (0)40 247 3393 fax. +31 (0)40 245 5674
[email protected]
[email protected]
Abstract | In this paper the phase uncertainty or +- π/4 phase error of a sampled quantizer is examined. Phase uncertainty results from a limited accuracy of detection of the level crossings, as the input signal generally crosses the quantization levels in between sample moments. In the case that the input signal frequency is a rational fraction of the sample Ts frequency, the input signal may be shifted in phase without a ecting the output samples. Closed-form expressions for the phase uncertainties have been derived in the cases of a one bit and two bit quantizer. Also a piece-wise linear approxima−3π/2 −π −π/2 0 π/2 π 3π/2 phase tion scheme for the phase uncertainty of a one bit quantizer is presented. Fig. 1. Phase uncertainty of a single-bit quantized and sampled sine-wave with freq. f4s Keywords | Sampled Quantizer, Phase Uncertainty. I. Introduction
Sampled quantizers are used as analog-to-digital converters (ADC) in which time- and amplitudecontinuous signals are converted into time- and amplitude-discrete signals. Usually this is done with a direct path from the analog input to the digital output. In some cases e.g. a sigma-delta modulator, the sampled quantizer resides within a feedback loop, and the phase transfer of the quantizer a ects the stability of the feedback loop. An often used method to analyze the stability of a non-linear feedback system is the describing function method, in which non-linear elements are modeled using a linear signal dependent transfer function. Because the stability of the feedback loop is governed by both the amplitude and phase transfer of the elements inside the loop, a model of a sampled quantizer should consist of a variable gain to incorporate the e ects of amplitude quantization, and a phase transfer. We will show that the phase transfer is in fact a phase uncertainty, whose range depends on the frequency and amplitude of the input signal.
Sampling of the input signal causes a quantizationlevel crossing to be detected by the sample moment following this crossing. As the crossing could have occured anywhere within the previous sample period, an uncertainty in the phase of the signal is introduced. Figure 1 shows an example for the case of a single bit quantizer. The input signal (solid line) can be shifted in phase without changing the output of the quantizer (impulses) [1]. In the following sections we will analyze the phase uncertainty for a sampled quantizer. In the cases of a one bit and two bit quantizer closed-form expressions for the phase uncertainty are derived. In section IV an approximation scheme for the phase uncertainty of a single bit quantizer is presented. II. Phase Uncertainty
The phase uncertainty follows from the accuracy with which the sampled quantizer detects the quantizationlevel crossings of the input signal. Let the input signal be a sine wave: A sin(2ft + ) with amplitude A > 0, frequency f and phase .
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Let the quantization levels qp be equal to qp = q p, with q > 0 the quantization step size and p 2 Z (uniform quantization). Let the sample moments tk be equal to tk = k Ts , in which Ts = 1=fs represents the sample period and k 2 Z (uniform sampling). The phase error of the sampled quantizer is zero in the case that the input sine wave is equal to one of the quantization levels qp = q p at any of the sample moments tk = k Ts. We have
B. Input frequencies equal to rational fraction of the sample frequency If the input frequency f is rational fraction of the sample frequency fs , this fraction can be written as
f =m (4) fs M with m; M 2 N and gcd 1 (m; M )=1. In the case that the Nyquist criterion ( ffs 21 ) is taken into account, m A sin(2fkTs + ) = p q: (1) and M also satisfy M 2 : Set of phases of the input signal for which the (5) m b M2 c error is zero, can be obtained by solving from (1) for k; p 2 Z. Note that equation (1) has solutions for Substituting (4) into (3) and using Bezouts theo for which p 2 f?P; : : : ; P g, with P = b Aq c the index rem [3]:
of the highest quantization level reached by the input sine wave. The total number of quantization levels crossed by the input signal equals 2P + 1. Any phase 62 of the input sine wave will result in a phase error of ? in which 2 which is nearest to . As a result, the maximum absolute phase uncertainty equals half the maximum distance between two adjacent solutions in : max = 12 max( 2 ? 1 ) 2 > 1 2 ( 1 ; 2 ) \ = ; (2)
8m;M 2Z j gcd(m; M ) = 1 9 1; 2 2Z 1 m + 2 M = 1 equation (3) can be simpli ed to
arcsin( pqA ) + 2M k M even (6) : = arcsin( pq ) + k M odd A M The phase uncertainty for input frequencies satisfying (4) approaches zero for an in nite number of quantization-level crossings. This is shown by regardset in (6) as a subset 0 which is repeatedly A. Input frequencies not equal to a rational fraction ing transposed over 2=M (M even) or =M (M odd). of the sample frequency 0 is de ned by The phase uncertainty is zero for input frequencies not equal to rational fraction of the sample frequency 0 : =arcsin pq A p 2f?P; : : : ; P g: (7) i.e. f=fs 2 R n Q . This can be shown by solving set of solutions from (1): In order to nd an upper bound for the phase uncertainty, the largest transposition equal to is consid pq 2 f ered. : = arcsin A + l ? k f De nition: The enclosure " of a set is de ned as s k; l 2 Z; p 2 f?P; : : : ; P g (3) the smallest closed interval encompassing all 2 , and is given by According to [2], any set n + m with m; n arbitrary "fg = [min (); max ()]: (8) integers and an irrational number will be dense in R. As we assumed f=fs to be an irrational number, set of solutions of (3) will also be dense in R. As a result, As the enclosure of the subset 0 is encompassed the di erence between any two adjacent solutions and by [?=2; =2] and the transposition of the subsets the maximum phase uncertainty max is zero. equals , the enclosures of all transposed subsets are In the case that the input frequency is not a rational disjunct. As a result, the largest interval between two fraction of the sample frequency, any change in the adjacent 2 occurs between the highest element phase of the input sine wave will always result in a 1 gcd(m; M ) represents the greatest common divisor of m change of at least one of the output samples. and M .
Phase Uncertainty of a Sampled Quantizer
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in 0 and the lowest element in 0 + , for the largest sine wave amplitude A resulting in P quantizationlevel crossings: max(A)jb Aq c=P = q(P + 1)? :
π/8
(9)
As a result, an upper bound for the maximum phase uncertainty is given by = ? arcsin
π/4
0
−π/8
P (10) P +1 : 2 If the number of quantization-level crossings goes to in nity (P ! 1), the upper bound approaches zero. Fig. 2. Phase uncertainty of a single bit quantizer. Because the actual maximum phase uncertainty lies between zero and this upper bound, the phase uncer- In the case that the input frequency equals half the tainty also decreases to zero. This follows with sample frequency (M = 2), the maximum phase uncertainty for a single bit quantizer equals =2. It 0 Alim max should be noted that for a sampled sine wave with b q c!1 frequency f = fs =2 a phase shift is indistinguish ? arcsin P = 0: (11) able from a change in amplitude, regardless of quanlim P +1 P !1 2 tization. As a result, the phase uncertainty is undeNote that this does not imply that the maximum tectable in the case that only the output samples of phase uncertainty decreases with every increase of the the quantizer are considered. number of quantization levels. B. Two bit quantizer III. Closed Form Expressions A two bit quantizer has three quantization levels Here we will derive closed form expressions for the at p 2 f?1; 0; 1g. De ning A as the phase di ermaximum phase uncertainty max for the cases of ence between a zero crossing and an adjacent p = 1 a single bit and a two bit quantizer, where input level crossing frequencies are a rational fraction of the sample freq (14) A = arcsin( A ) quency and satisfy (4) and (5). the solutions for zero phase uncertainty can be written A. Single bit quantizer as a repeatedly transposed subset f? A ; 0; A g: 2 A single bit quantizer has one quantization level even : = f? A ; 0; A g + M kk M at p = 0. As a result, equation (6) is reduced to M M odd (15) 2 k M even M (12) with k 2 Z. Because the actual adjacent solutions : = k M odd k 2 Z: M depend on the values A and M , the maximum phase The maximum phase uncertainty can be derived di- uncertainty has to be determined in four cases. For M rectly using (2). This gives even the cases are: 1. The enclosures of repeatedly transposed subsets M even are disjunct. The distribution of the solutions M max = M odd : (13) in (15) is depicted in Fig. 3a. Each separate sub2M set is marked by a di erent symbol (+; 3; or ). From (13) it follows that the maximum phase uncerThe enclosures are disjunct for 2=M > 2 A . As tainty does not depend on the sine wave amplitude, a result, the set of solutions satis es as could be expected from the presence of a single quantization level. In Fig. 2 the phase uncertainty is : f:::; ? 2M + A ; ? A ; 0; A ; 2 ? ; 2 ; 2 + ; :::g: shown for M = 3; : : : ; 64. A M M A M
u:b:
−π/4
0
1/8
1/4 f/fs
3/8
1/2
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The resulting maximum phase uncertainty, equal to half the maximum distance between consecutive solutions, equals max =
1 2
A 2 A < 2M 3 A ? A 3 A < 2M (16)
M
(a)
(b)
(c)
2. The enclosures of two adjacent subsets intersect ( A < 2=M 2 A ). The distribution of Fig. 3. 4 possible distributions of the zero phase error solution set of a two bit quantizer. solution subsets is shown in Fig. 3b. The set of solutions written in ascending order equals (d)
: f:::; ? A ; ? 2M + A ; 0; 2 ? ; ; 2 ; 2 + ; :::g: A A M M A M
π/4 A=1.2q
1 3 2 M ? 2 A 2 A < 2M 23 A A ? M A < M 2 A
(17)
−π/4 0
1/4 f/fs
3/8
1/2
1/8
1/4 f/fs
3/8
1/2
1/8
1/4 f/fs
3/8
1/2
A=1.8q π/8
−π/4 0
π/4
4. Four or more enclosures of subsets intersect (0 < 2=M A =2). Here the size of the transposition is smaller than half the interval between two solutions in the same subset, and any interval will be smaller or equal to 2=M . Figure 3d shows the distribution of the solutions of four subsets. The maximum phase uncertainty is equal to the half of the transposition. (19) max = M 2M 21 A Similar expressions can be obtained for odd values of M . In this case every occurance of 2=M in (16) through (19) should to be replaced by =M . Equations (16) through (19), together with the expressions for odd values of M constitute the closed-form expression for the maximum phase uncertainty of a two bit sampled quantizer. In Fig. 4 the maximum phase un-
worst-case π/8
(c)
(18)
0
−π/8
The maximum phase uncertainty is described by 2 1 ? A 2 A < 2 A 1M A ?2 1 3 A < 2M 2 A 2 M 2 M 3
1/8
π/4
3. The enclosures of three subsets intersect ( A =2 < 2=M A ). Figure 3c shows the solutions of three subsets. The set of solutions satis es : f:::; ? A ; ? 2M ; 2M ? A ; 0; ? 2M + A; 2M ; A ; :::g: max =
0
−π/8
(b)
= max
(a)
Here the maximum phase uncertainty equals
π/8
0
−π/8
−π/4 0
Fig. 4. Phase uncertainty of a two bit quantizer: (a) for amplitude A = 1:4q, (b) for amplitude A = 1:99q and (c) worst-case values.
certainty of a two bit sampled quantizer is shown for a sine wave amplitude A = 1:2q and A = 1:8q, together with the worst-case maximum phase uncertainty for each frequency. As is clear from equation (19) and Fig. 4c, the worstcase value for the phase uncertainty is equal to that of a single bit quantizer. The two bit quantizer exhibits the worst-case phase uncertainty in the case that the input phase is such that the range of the output samples is within two levels. In particular, this situation occurs for amplitudes of the input sine wave just exceeding the additional quantization levels.
Phase Uncertainty of a Sampled Quantizer
IV. Approximation
Because of the complexity and discrete character of the maximum phase uncertainty of a sampled quantizer, application to standard methods for stability analysis such as root-locus and Nyquist methods is complicated. The maximum phase uncertainty relates to an in nite time interval (k 2 Z). The phase uncertainty over a short time interval may be considerably larger. This is undesirable, as it may a ect e.g. the stability of a feedback loop. Limiting the time interval for which the maximum phase uncertainty is determined gives an approximation with reduced accuracy and complexity. This method will be applied to a single-bit sampled quantizer. For a single bit quantizer, the set for which the phase error is zero, is described by (3) with p = 0. Limiting the number of sample periods to k 2 f?K; :::; K g gives K
: = (l ? 2k ff ) l 2 Z s p 2f?P; : : : ; P g k 2f?K; : : : ; K g (20)
157 π/4 K=1
π/8 0 −π/8 −π/4 0
1/8
1/4 f/fs
3/8
1/2
1/8
1/4 f/fs
3/8
1/2
1/8
1/4 f/fs
3/8
1/2
π/4 K=2
π/8 0 −π/8 −π/4 0
π/4 K=3
π/8 0 −π/8 −π/4 0
π/4 K=4
π/8 0 −π/8 −π/4
The approximation for the maximum phase uncertainty is governed by functions linear in f=fs, and the 5. Piece-wise linear approximation of the maximum maximum phase uncertainty max (f ) is a piece-wise Fig.phase uncertainty of a single bit sampled quantizer. linear function. It should be noted, that this function is not necessarily a strictly continuous function. As an example, the approximation for K = 1 is given by It is shown that the phase uncertainty is zero for input ( f frequencies that are no rational fraction of the samf 1 0 fs fs 4 1max = (21) ple frequency. The phase uncertainty approaches zero ( 21 ? ffs ) 41 < ffs 21 for any frequency if the number of quantization levels goes to in nity. The approximation K = 1 of the maximum phase uncertainty for a single bit sampled quantizer is shown Closed-form expressions for the maximum phase unin Fig. 5, together with approximations using other certainty for a single bit and two bit quantizer were presented. Despite the additional quantization levels, values for K . The approximation K = 1 of the maximum phase the worst-case maximum phase uncertainty of a two uncertainty has been applied in stability analysis of bit quantizer is equal to that of a single bit quantizer. low-pass Sigma-Delta modulators using the root-locus The worst-case values depend on the denominator of the relative prime ratio between the input frequency method [4]. and the sample frequency. V. Conclusions 0
In this paper the phase uncertainty of a sampled quantizer was analyzed. The phase uncertainty arises from the limited accuracy in time with which a quantizer can detect the quantization-level crossings of the input signal.
1/8
1/4 f/fs
3/8
1/2
An approximation for the maximum phase uncertainty of a single bit quantizer has been presented. The piece-wise linear nature of the approximation allows application to stability analysis using for example the root locus method.
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Acknowledgments
This work was supported by Philips Research Labs, Eindhoven, the Netherlands under project number RWC-061-ps-940028-ps. References [1] M.H.H. Hofelt, \On the stability of a 1-bit-quantized feedback system," in Proc. ICASSP, Washington, 1979, pp. 844{848. [2] Paul R. Halmos, Measure Theory, p. 69, Graduate Text in Mathematics (GTM). Springer Verlag, 1974. [3] Harold M. Stark, An Introduction to Number Theory, pp. 21{22, Markham Publishing Company, Chicago, 1970. [4] J.A.E.P. van Engelen and R.J. van de Plassche, \New stability criteria for the design of low-pass sigma-delta modulators," in Proc. of the Int. Symp. on Low Power Electronics and Design (ISLPED), Monterey, 1997, pp. 114{118.
