FUNDAMENTAL LIMITATIONS OF CONTINUOUS ... - Semantic Scholar
FUNDAMENTAL LIMITATIONS OF CONTINUOUS-TIME DELTA-SIGMA MODULATORS DUE TO CLOCK JITTER Karthikeyan Reddy & Shanthi Pavan Department of Electrical Engineering Indian Institute of Technology, Madras where σ2∆Ts is the variance of the clock jitter. Clearly, σ2e j is dependent on the input signal through y n and y n 1. When the input is nulled, the modulator will exhibit idle-channel noise, which is dependent on clock jitter and quantization noise [4]. Then, σ2dy , the variance of y n y n 1 is given by
ABSTRACT We examine noise due to clock jitter in single-loop low pass continuous-time delta-sigma modulators employing NRZ feedback DACs. Using the discrete-time version of the Bode sensitivity integral, we derive a lower bound on jitter noise and its relationship to the noise transfer function of the modulator. We give intuition to a recent observation (arrived through numerical optimization) that NTFs with peaking in their pass bands have lower jitter noise than maximally flat NTFs.
σ2dy
1. INTRODUCTION
yn
1
∆Ts n T
J
σ2∆Ts T2
σ2∆Ts σ2lsb T 2 πOSR
AJ
jω
e
NT F
e jω 2 dω
(3)
π 0
1
e
jω
NT F
e jω 2 dω
(4)
π 0
1
e
jω
NT F
e jω 2 dω
(5)
The quantization noise Q, on the other hand is given by Q
σ2lsb πOSR
π OSR
0
NT F
e jω 2 dω
(6)
From equation (4), it is clear that the jitter noise is predominantly determined by the behavior of the NTF outside the signal band, since NT F e jω 0 at low frequencies. Quantization noise, on the other hand, only depends on the NTF within the signal bandwidth. It might thus appear that, for a given Q, J could be π . However, the inreduced by making NT F small for ω OSR band and out of band properties of the NTF are closely related. In order to better understand this, consider the set of maximally flat NTFs. For this family, once the order is specified, the NTF is fully defined by the the out-of-band gain (OBG). It is well known [5] that increasing the OBG results in a lower in-band quantization noise Q. However, increasing the OBG results in a large J. Figure 1 shows the peak SNR for modulators as the OBG is varied, for different values of clock jitter. For small values of OBG, Q is large, leading to a low SNR. Increasing OBG increases the SNR, as Q decreases with increasing OBG. However, beyond a certain point, increasing the OBG has a detrimental effect on the SNR, as the noise is swamped by the jitter noise J. A large OBG also reduces the maximum stable amplitude of the modulator. It is thus seen that given the order and the amount of clock jitter, there is an optimum OBG at which the SNR is maximum. Understandably, this optimum OBG decreases with increasing clock jitter. For a given Q and NTF order, the OBG for the NTF with optimally spread passband zeros can be smaller than that for the
(1)
(2)
The first author’s work was supported by Texas Instruments India, Bangalore.
0-7803-9390-2/06/$20.00 ©2006 IEEE
1
It is thus seen [4] that the in band jitter noise J depends only on the area AJ under the 1 e jω NT F e jω 2 curve. (equation 5).
where y n is the nth output sample of the modulator, T is the sampling time and ∆Ts n (assumed white) is the clocking uncertainty of the nth edge of the DAC. The spectral density of e j is white and has a variance σ2e j given by σ2e j σ2dy
0
where, σ2lsb is the variance of the quantization noise of the internal quantizer used in the modulator. Using (3) in (2), we see that the in-band noise due to jitter is
Continuous-time Σ∆ modulators are lower power alternatives to their discrete-time counterparts. Several advantages accrue by implementing the modulator loop filter with continuous-time circuitry. An explicit anti-alias filter, which would be required in a switched-capacitor implementation is not necessary. The bandwidth requirements for the active elements in the loop are relaxed, which results in a lower power consumption. However, clock jitter degrades the SNR of the modulator by influencing the sampling instant of the flash quantizer, as well as the width of the feedback DAC pulse. To the authors’ best knowledge, the effect of clock-jitter in CTDSMs was first studied in [1] and [2]. Both these papers conclude that noise due to the modulation of the width of the feedback DAC pulse is dominant cause of jitter noise. This is intuitively satisfying due to the following - the error due to the variation of the sampling instant of the quantizer is noise shaped due to the high loop gain. Hence, the in-band noise power should not be dominated by this noise. However, the error in the DAC feedback pulse width adds directly at the input of the modulator and is not noise shaped. A similar conclusion was reached in [3]. The effect of jitter can therefore be modeled as an additive sequence at the input of a jitterfree modulator. For the case of NRZ feedback DACs considered in this paper, the error sequence is given by e j n y n
π
σ2lsb π
2021
ISCAS 2006
E(z)
110 σ =0 Ts σ = 10−4 Ts Ts σ = 10−3 Ts Ts σ = 10−2 Ts Ts
100
Y(z)
Xin(z) + -
L(z)
+
Peak SNR (dB)
90
80
Fig. 2. Linearized diagram of a Σ∆ modulator.
70
the following :
60
∞ 50
40 0
log
0
2
4
6
OBG
8
10
12
1 L1 jω
dω 0
(7)
Physically, this means that the rejection of noise cannot be high at all frequencies, low sensitivity in one frequency band must be achieved through high sensitivity outside that band. The same idea applies to discrete-time feedback systems [7], where it can be shown that if the loop filter has no poles and zeros outside the unit circle the sensitivity function is constrained by the following integral.
14
Fig. 1. Peak SNR versus OBG for various clock jitter, for 4th order NTF ,OSR=16 and 4-bit quantizer
∞
log
0
NTF with all zeros at the origin. This means that J will be lower for the modulator with optimally spread zeros. If the (artificial) constraint that the NTF be maximally flat is removed, many NTFs with the same in-band characteristics, but different out of band behavior can be conceived. This begs the following question : for a given quantization noise, is there a specific choice of the out-of-band behavior of the NTF that results in the lowest jitter sensitivity ? In [4], the authors numerically optimized the pole and zero locations of the NTF to minimize J while keeping Q a constant. The optimization process resulted in an NTF with a peak of transmission, as well as zeros spread in the signal band. In this paper, we attempt to analytically justify the shape of the NTF proposed in [4]. For this, we use the Bode sensitivity integral, which relates the out of band behavior of the NTF to the in-band response. We show that there exists a trade-off between quantization noise and jitter noise, one can be decreased only at the expense of another. The rest of the paper is organized as follows. In Section 2, we review the Bode sensitivity integral for discrete-time systems and extend its application to the evaluation of the integral in equation (4). For an NTF with a quantization noise Q, we show there is a lower bound on the jitter noise of the modulator. In Section 3, we derive approximate versions of the lower bound that are valid for a family of commonly used NTFs. Section 3.2 discusses NTF design implications resulting from the jitter bound. Conclusions are given in Section 4.
1 L1 e jω
dω 0
(8)
In the context of a delta-sigma modulator, the sensitivity function is the same as the noise transfer function, assuming the quantizer can be modeled as an additive noise source. Hence, equation (8) can be written as ∞ log NT F e jω dω 0 (9) 0 ∞ Using the relation 0 log 1 e jω dω 0, in conjunction with equation (9), we see that ∞ 0
log NT F e jω 1
e
jω
dω 0
(10)
This means that when log NT F e jω 1 e jω is plotted as a function of ω, the area above the 0 dB line is equal to the area below the line. Consider now the evaluation of the jitter noise J. Let ω1 denote the frequency at which log NT F e jω 1 e jω 0. From equation (5), it is seen that π
AJ
ω1
1
e
jω
e jω 2 dω
NT F
(11)
In order to arrive at a lower bound on J, we use the following inequality, shown in Appendix 1. b a
f x2 dx
b
b
2
a exp
b
a
a
log f x dx
From equations (12) and (11), we have
2. BODE’S SENSITIVITY INTEGRAL FOR DISCRETE-TIME SYSTEMS AND A LOWER BOUND ON JITTER NOISE
AJ
π
ω1 exp
2 π ω1
π ω1
log 1
e
Denoting the area of log NT F e jω 1 line by C, we see that J Jmin where
Consider the discrete-time feedback system shown in Figure 2. In the control systems literature, the quantity 1 1Lz is referred to as the sensitivity function of the loop. It quantifies the ability of the loop to reject disturbances (like E z) in the forward path as a function of frequency. For a continuous-time system, z is replaced by the Laplace variable s. Bode [6] showed that for a minimum-phase loop filter L s (with order 2), the continuous-time sensitivity function satisfies
Jmin
σ2∆Ts σ2lsb π T 2 πOSR
ω1 exp
jω
(12)
NT F
e jω dω (13)
e
above the 0 dB
jω
2C π ω1
(14)
Jmin is therefore a lower bound on the in-band jitter noise for a given NTF and clock jitter. The above expression shows that the jitter
2022
Jitter noise and Jitter bound(Normalised to quantiser range, V2)
noise is exponentially related to the area of the log NT F e jω 1 e jω curve above (below) the 0 dB line. Consider the plots (Figure 3) of log NT F e jω 1 e jω for modulators with 2 different NTFs with identical performance inband. However, the NTF with peaking exhibits a sharper transition when compared to the maximally flat design. This results in smaller values of C and AJ when compared to the maximally flat NTF. It is clear, therefore, that in order to minimize J for the same quantization noise Q, the NTF must be designed to have a sharp transition band, so that lower values of ω1 and C can be obtained. We now list some observations and implications of the jitter bound derived above.
¯ Jmin is a tight bound, since the area under the NT F e jω 1 e jω 2 curve from 0 ω ω1 is negligible, so that (13) is almost an equality. Due to the tightness of this bound, it is useful in design as is shown in the next section. ¯ (14) relates jitter noise to two parameters of the NTF - C and ω1 . For NTFs normally used in practice, it turns out that a simple relation between C and ω1 can be found. This in turn makes Jmin simply a function of ω1 .
−7
2.5
x 10
2
1.5
1
Jitter Bound Jitter Noise
0.5
0 0.55
0.6
0.65
0.7
0.75 ω 0.8 1
0.85
0.9
0.95
1
Fig. 4. Jitter bound and jitter noise as a function of ω1 for a NTFs having same in-band characteristics
20
close to being achievable (notice how close the bound and the actual noise are at around ω1 057.) It turns out that this corresponds to a gentle peak in the NTF. It also turns out that a maximally flat NTF with apppropriately chosen OBG, though not optimal for minimizing J, is sufficiently close to the optimum that the jitter bound may actually be used as an estimate of J. The consequences of this observation are explored further in the next section.
f 0
2
f1 C = 3.9166, f = 0.131 1 1 C = 3.5713, f = 0.103 2
−40
2
C1, C2 = Area below 0dB for solid and dashed curves respectively
−60
jω
log|NTF(e )(1−e
−jω
)|
−20
−80
3. APPLICATION TO COMMONLY USED NTFS A useful approximation for the jitter bound of equation (14) can be derived for the family of maximally flat NTFs. The magnitude of an N th order NTF may be written as
−100 −120 −140 −160 0
NT F 0.1
0.2 0.3 Normalised frequency (f/f )
0.4
0.5
e jω
N l ωl ∏zi 1 ω ωzi Q e jω
(15)
where the ωzi denote the zeros of transmission of the NTF within the signal band. Note that ωzi ω1 . If the NTF does not have excessive peaking, Q e jω is approximately a constant (whose reciprocal is denoted by k) in the range 0 ω ω1 , so that
s
Fig. 3. NTFs with the same in-band performance, but different out of band behavior
NT F
The problem of NTF design for reduced jitter sensitivity can be looked at in a different light as follows. Intuitively, it is seen that the area of the log NT F e jω 1 e jω under the 0 dB line (i.e C) is dependent on the quantization noise Q. By the Bode Sensitivity Integral the area above the 0 dB line must also be C. Varying the locations of the NTF poles will change the way C is distributed in the range ω1 ω π - so for least jitter sensitivity, one “distribute” this C in such a manner as to minimize π should jω e jω 2 . From the discussion in Appendix ω1 NT F e 1 A, this is achieved when NT F e jω 1 e jω is a constant for ω1 ω π. Since 1 e jω is monotonically increasing with ω, it follows that J is minimized when NT F e jω exhibits peaking. This explains the observation in [4]. Figure 4 shows the jitter noise and the jitter bound (computed from (14) for different fourth order NTFs having the same Q. The NTFs were generated by randomly perturbing the poles of the modulator in a manner as to keep the NTF the same at low frequencies. The figure shows that the jitter bound is tight and
e jω k ωl
N l
∏
zi1
ω
ωzi
(16)
From the above equation, it is seen that ω1 k1 1N 1 and C1 N 1 ω1 Using the above in equation (14) results in the approximate relation J
σ2∆Ts σ2q π ω1 2 N 1ω1 exp π OSR π ω1 T2
(17)
Using equation (15), the quantization noise can be approximated as Q
σ2q π2 N π 2 N 1ω12 N
1 2
OSR2 N
1
Cz
(18)
where Cz 1 depends on the position of the NTF zeros ωi within the signal band. Cz 1 when all the zeros are at ω 0. From equations (17) and (18), it is seen that the jitter noise and quantization noise
2023
In−band noise(normalised to Quantiser Range)
4. CONCLUSIONS In this paper, the Bode Sensitivity Integral was used to derive a lower bound on the jitter noise in continuous-time delta-sigma modulators. A consequence of the study is that NTFs with moderate peaking in the transfer function have lesser jitter noise (for the same quantization noise) when compared to their maximally flat counterparts. Intuition was give for the results of [4], where a similar conclusion was reached, but the results were arrived at using numerical optimization. It was shown that jitter noise and quantization noise cannot be reduced simultaneously, and there is an optimum OBG which minimizes the total noise.
−7
4
x 10
Order = 4 = 10−2 Ts σ ∆ Ts
Quantiser = 4 bits OSR = 16
3
2
Q
1
APPENDIX A Consider a real function f x in the interval [a,b]. Using the notation fk f a k b L a and observing that the arithmetic mean is greater than the geometric mean, we have
J+Q J
0 0.2
0.3
0.4
0.5
0.6
ω1
0.7
0.8
0.9
1
a kL
b
∑ fk 2
k0
L
Fig. 5. J, Q and total noise as a function of ω1 , for a fourth order modulator with a 4-bit quantizer and optimized NTF zeros
1
kL 1 1 L a ∏ fk 2
b
(20)
k 0
Simplifying the RHS of the above equation results in a kL
b can be expressed as a function of a single parameter ω1 . As ω1 increases, Q decreases while J increases. Recall that a higher ω1 implies a higher out of band gain. The total in-band noise can be now optimized with respect to frequency ω1 . Q, J and total noise for an example fourth order modulator with a 4-bit quantizer are shown in Figure 5. The OSR is 16, and J is calculated for a 1% RMS clock jitter. From this plot, it seen that an ω1 of about 0.42 results in the least amount of total noise. The knowledge of ω1 , alongwith the constraint that the NTF is maximally flat can be used to determine the OBG that results in the least total in-band noise for the modulator. Alternatively, once ω1 is known, the Bode Sensitivity integral can be used to arrive at an approximate value of the OBG, as discussed below.
L
1
∑ fk 2
k 0
b
k L 1 2 a exp log fk L ∑ k 0
As L ∞ the summation tends to an integral, and we get b a
f x2 dx
b
a exp
b
2 b
a
a
(21)
log f x dx
(22)
Note that the equality is valid only when f x is a constant. Equivalently, we see that if ab log f x dx is constrained to be b a constant, then a f x2 dx is minimized when f x is constant for a x b. 5. REFERENCES [1] J. A. Cherry and W. M. Snelgrove, “Clock jitter and quantizer metastability in continuous-time delta-sigma modulators,” IEEE Trans. Circuits Syst. II, vol. 46, no. 6, pp. 661–676, June 1999.
3.1. Finding the optimum OBG Let ω2 be denote the frequency at which log NT F 0. If the NTF can be approximated by equation (15), it is easily seen that ω2 can
[2] O. Oliaei, “Clock jitter noise spectra in continuous-time deltasigma modulators,” in Proc. IEEE Int. Sym. Circuits and Systems, vol. 2, 1999, pp. 192–195.
be related to ω1 as ω2 ω1 . The area of log NT F below zero can be shown to be N ω2 . Approximating the area above zero as π ω2 log OBG and using Bode’s sensitivity integral, we see that Nω1NN1 (19) OBG exp N 1 π ω1 N N 1 N
[3] H. Tao, L. Toth, and J. M. Khoury, “Analysis of timing jitter in bandpass sigma-delta modulators,” IEEE Trans. Circuits Syst. II, vol. 46, no. 8, pp. 991–1001, August 1999. [4] L. Hernandez, A. Wiesbauer, S. Paton, and A. D. Giandomencio, “Modelling and optimization of low pass continuous-time sigma delta modulators for clock jitter noise reduction,” in Proc. IEEE Int. Sym. Circuits and Systems, vol. 1, May 2004, pp. 1072–1075.
3.2. Design Implications Excess loop delay due to the latency of the quantizer is known to cause peaking in continuous-time delta sigma modulators, if nothing is done to compensate for the excess delay. Hence, a modulator with a nominal NTF that is maximally flat will exhibit peaking in the presence of excess loop delay. The magnitude of the peak depends on the amount of excess delay. From the discussion in Section 2, it is apparent that a small amount of excess delay is beneficial as it reduces the jitter noise J. Lack of space prevents more details here. Simulations for a 4th order modulator with a 4-bit quantizer (maximally flat NTF with OBG=2.5) show that with an RMS clock jitter of 1%, the jitter noise in the presence of excess loop delay of 01 Ts and without delay and 98 10 8 and 12 10 7 respectively.
[5] S. R. Norsworthy, R. Schreier, and G. C. Temes, Eds., DeltaSigma Data Converters:Theory, Design, and Simulation. New York: IEEE Press., 1997. [6] H. W. Bode, Network Analysis and Feedback Amplifier Design, 2nd ed. Princeton: Van Nostrand Reinhold Publishing Company, 1945. [7] C. Mohtadi, “Bode’s integral theorem for discrete-time systems,” IEE Proc. Control Theory and Applications, vol. 137, no. 2, pp. 57–66, March 1990.
2024
ABSTRACT We examine noise due to clock jitter in single-loop low pass continuous-time delta-sigma modulators employing NRZ feedback DACs. Using the discrete-time version of the Bode sensitivity integral, we derive a lower bound on jitter noise and its relationship to the noise transfer function of the modulator. We give intuition to a recent observation (arrived through numerical optimization) that NTFs with peaking in their pass bands have lower jitter noise than maximally flat NTFs.
σ2dy
1. INTRODUCTION
yn
1
∆Ts n T
J
σ2∆Ts T2
σ2∆Ts σ2lsb T 2 πOSR
AJ
jω
e
NT F
e jω 2 dω
(3)
π 0
1
e
jω
NT F
e jω 2 dω
(4)
π 0
1
e
jω
NT F
e jω 2 dω
(5)
The quantization noise Q, on the other hand is given by Q
σ2lsb πOSR
π OSR
0
NT F
e jω 2 dω
(6)
From equation (4), it is clear that the jitter noise is predominantly determined by the behavior of the NTF outside the signal band, since NT F e jω 0 at low frequencies. Quantization noise, on the other hand, only depends on the NTF within the signal bandwidth. It might thus appear that, for a given Q, J could be π . However, the inreduced by making NT F small for ω OSR band and out of band properties of the NTF are closely related. In order to better understand this, consider the set of maximally flat NTFs. For this family, once the order is specified, the NTF is fully defined by the the out-of-band gain (OBG). It is well known [5] that increasing the OBG results in a lower in-band quantization noise Q. However, increasing the OBG results in a large J. Figure 1 shows the peak SNR for modulators as the OBG is varied, for different values of clock jitter. For small values of OBG, Q is large, leading to a low SNR. Increasing OBG increases the SNR, as Q decreases with increasing OBG. However, beyond a certain point, increasing the OBG has a detrimental effect on the SNR, as the noise is swamped by the jitter noise J. A large OBG also reduces the maximum stable amplitude of the modulator. It is thus seen that given the order and the amount of clock jitter, there is an optimum OBG at which the SNR is maximum. Understandably, this optimum OBG decreases with increasing clock jitter. For a given Q and NTF order, the OBG for the NTF with optimally spread passband zeros can be smaller than that for the
(1)
(2)
The first author’s work was supported by Texas Instruments India, Bangalore.
0-7803-9390-2/06/$20.00 ©2006 IEEE
1
It is thus seen [4] that the in band jitter noise J depends only on the area AJ under the 1 e jω NT F e jω 2 curve. (equation 5).
where y n is the nth output sample of the modulator, T is the sampling time and ∆Ts n (assumed white) is the clocking uncertainty of the nth edge of the DAC. The spectral density of e j is white and has a variance σ2e j given by σ2e j σ2dy
0
where, σ2lsb is the variance of the quantization noise of the internal quantizer used in the modulator. Using (3) in (2), we see that the in-band noise due to jitter is
Continuous-time Σ∆ modulators are lower power alternatives to their discrete-time counterparts. Several advantages accrue by implementing the modulator loop filter with continuous-time circuitry. An explicit anti-alias filter, which would be required in a switched-capacitor implementation is not necessary. The bandwidth requirements for the active elements in the loop are relaxed, which results in a lower power consumption. However, clock jitter degrades the SNR of the modulator by influencing the sampling instant of the flash quantizer, as well as the width of the feedback DAC pulse. To the authors’ best knowledge, the effect of clock-jitter in CTDSMs was first studied in [1] and [2]. Both these papers conclude that noise due to the modulation of the width of the feedback DAC pulse is dominant cause of jitter noise. This is intuitively satisfying due to the following - the error due to the variation of the sampling instant of the quantizer is noise shaped due to the high loop gain. Hence, the in-band noise power should not be dominated by this noise. However, the error in the DAC feedback pulse width adds directly at the input of the modulator and is not noise shaped. A similar conclusion was reached in [3]. The effect of jitter can therefore be modeled as an additive sequence at the input of a jitterfree modulator. For the case of NRZ feedback DACs considered in this paper, the error sequence is given by e j n y n
π
σ2lsb π
2021
ISCAS 2006
E(z)
110 σ =0 Ts σ = 10−4 Ts Ts σ = 10−3 Ts Ts σ = 10−2 Ts Ts
100
Y(z)
Xin(z) + -
L(z)
+
Peak SNR (dB)
90
80
Fig. 2. Linearized diagram of a Σ∆ modulator.
70
the following :
60
∞ 50
40 0
log
0
2
4
6
OBG
8
10
12
1 L1 jω
dω 0
(7)
Physically, this means that the rejection of noise cannot be high at all frequencies, low sensitivity in one frequency band must be achieved through high sensitivity outside that band. The same idea applies to discrete-time feedback systems [7], where it can be shown that if the loop filter has no poles and zeros outside the unit circle the sensitivity function is constrained by the following integral.
14
Fig. 1. Peak SNR versus OBG for various clock jitter, for 4th order NTF ,OSR=16 and 4-bit quantizer
∞
log
0
NTF with all zeros at the origin. This means that J will be lower for the modulator with optimally spread zeros. If the (artificial) constraint that the NTF be maximally flat is removed, many NTFs with the same in-band characteristics, but different out of band behavior can be conceived. This begs the following question : for a given quantization noise, is there a specific choice of the out-of-band behavior of the NTF that results in the lowest jitter sensitivity ? In [4], the authors numerically optimized the pole and zero locations of the NTF to minimize J while keeping Q a constant. The optimization process resulted in an NTF with a peak of transmission, as well as zeros spread in the signal band. In this paper, we attempt to analytically justify the shape of the NTF proposed in [4]. For this, we use the Bode sensitivity integral, which relates the out of band behavior of the NTF to the in-band response. We show that there exists a trade-off between quantization noise and jitter noise, one can be decreased only at the expense of another. The rest of the paper is organized as follows. In Section 2, we review the Bode sensitivity integral for discrete-time systems and extend its application to the evaluation of the integral in equation (4). For an NTF with a quantization noise Q, we show there is a lower bound on the jitter noise of the modulator. In Section 3, we derive approximate versions of the lower bound that are valid for a family of commonly used NTFs. Section 3.2 discusses NTF design implications resulting from the jitter bound. Conclusions are given in Section 4.
1 L1 e jω
dω 0
(8)
In the context of a delta-sigma modulator, the sensitivity function is the same as the noise transfer function, assuming the quantizer can be modeled as an additive noise source. Hence, equation (8) can be written as ∞ log NT F e jω dω 0 (9) 0 ∞ Using the relation 0 log 1 e jω dω 0, in conjunction with equation (9), we see that ∞ 0
log NT F e jω 1
e
jω
dω 0
(10)
This means that when log NT F e jω 1 e jω is plotted as a function of ω, the area above the 0 dB line is equal to the area below the line. Consider now the evaluation of the jitter noise J. Let ω1 denote the frequency at which log NT F e jω 1 e jω 0. From equation (5), it is seen that π
AJ
ω1
1
e
jω
e jω 2 dω
NT F
(11)
In order to arrive at a lower bound on J, we use the following inequality, shown in Appendix 1. b a
f x2 dx
b
b
2
a exp
b
a
a
log f x dx
From equations (12) and (11), we have
2. BODE’S SENSITIVITY INTEGRAL FOR DISCRETE-TIME SYSTEMS AND A LOWER BOUND ON JITTER NOISE
AJ
π
ω1 exp
2 π ω1
π ω1
log 1
e
Denoting the area of log NT F e jω 1 line by C, we see that J Jmin where
Consider the discrete-time feedback system shown in Figure 2. In the control systems literature, the quantity 1 1Lz is referred to as the sensitivity function of the loop. It quantifies the ability of the loop to reject disturbances (like E z) in the forward path as a function of frequency. For a continuous-time system, z is replaced by the Laplace variable s. Bode [6] showed that for a minimum-phase loop filter L s (with order 2), the continuous-time sensitivity function satisfies
Jmin
σ2∆Ts σ2lsb π T 2 πOSR
ω1 exp
jω
(12)
NT F
e jω dω (13)
e
above the 0 dB
jω
2C π ω1
(14)
Jmin is therefore a lower bound on the in-band jitter noise for a given NTF and clock jitter. The above expression shows that the jitter
2022
Jitter noise and Jitter bound(Normalised to quantiser range, V2)
noise is exponentially related to the area of the log NT F e jω 1 e jω curve above (below) the 0 dB line. Consider the plots (Figure 3) of log NT F e jω 1 e jω for modulators with 2 different NTFs with identical performance inband. However, the NTF with peaking exhibits a sharper transition when compared to the maximally flat design. This results in smaller values of C and AJ when compared to the maximally flat NTF. It is clear, therefore, that in order to minimize J for the same quantization noise Q, the NTF must be designed to have a sharp transition band, so that lower values of ω1 and C can be obtained. We now list some observations and implications of the jitter bound derived above.
¯ Jmin is a tight bound, since the area under the NT F e jω 1 e jω 2 curve from 0 ω ω1 is negligible, so that (13) is almost an equality. Due to the tightness of this bound, it is useful in design as is shown in the next section. ¯ (14) relates jitter noise to two parameters of the NTF - C and ω1 . For NTFs normally used in practice, it turns out that a simple relation between C and ω1 can be found. This in turn makes Jmin simply a function of ω1 .
−7
2.5
x 10
2
1.5
1
Jitter Bound Jitter Noise
0.5
0 0.55
0.6
0.65
0.7
0.75 ω 0.8 1
0.85
0.9
0.95
1
Fig. 4. Jitter bound and jitter noise as a function of ω1 for a NTFs having same in-band characteristics
20
close to being achievable (notice how close the bound and the actual noise are at around ω1 057.) It turns out that this corresponds to a gentle peak in the NTF. It also turns out that a maximally flat NTF with apppropriately chosen OBG, though not optimal for minimizing J, is sufficiently close to the optimum that the jitter bound may actually be used as an estimate of J. The consequences of this observation are explored further in the next section.
f 0
2
f1 C = 3.9166, f = 0.131 1 1 C = 3.5713, f = 0.103 2
−40
2
C1, C2 = Area below 0dB for solid and dashed curves respectively
−60
jω
log|NTF(e )(1−e
−jω
)|
−20
−80
3. APPLICATION TO COMMONLY USED NTFS A useful approximation for the jitter bound of equation (14) can be derived for the family of maximally flat NTFs. The magnitude of an N th order NTF may be written as
−100 −120 −140 −160 0
NT F 0.1
0.2 0.3 Normalised frequency (f/f )
0.4
0.5
e jω
N l ωl ∏zi 1 ω ωzi Q e jω
(15)
where the ωzi denote the zeros of transmission of the NTF within the signal band. Note that ωzi ω1 . If the NTF does not have excessive peaking, Q e jω is approximately a constant (whose reciprocal is denoted by k) in the range 0 ω ω1 , so that
s
Fig. 3. NTFs with the same in-band performance, but different out of band behavior
NT F
The problem of NTF design for reduced jitter sensitivity can be looked at in a different light as follows. Intuitively, it is seen that the area of the log NT F e jω 1 e jω under the 0 dB line (i.e C) is dependent on the quantization noise Q. By the Bode Sensitivity Integral the area above the 0 dB line must also be C. Varying the locations of the NTF poles will change the way C is distributed in the range ω1 ω π - so for least jitter sensitivity, one “distribute” this C in such a manner as to minimize π should jω e jω 2 . From the discussion in Appendix ω1 NT F e 1 A, this is achieved when NT F e jω 1 e jω is a constant for ω1 ω π. Since 1 e jω is monotonically increasing with ω, it follows that J is minimized when NT F e jω exhibits peaking. This explains the observation in [4]. Figure 4 shows the jitter noise and the jitter bound (computed from (14) for different fourth order NTFs having the same Q. The NTFs were generated by randomly perturbing the poles of the modulator in a manner as to keep the NTF the same at low frequencies. The figure shows that the jitter bound is tight and
e jω k ωl
N l
∏
zi1
ω
ωzi
(16)
From the above equation, it is seen that ω1 k1 1N 1 and C1 N 1 ω1 Using the above in equation (14) results in the approximate relation J
σ2∆Ts σ2q π ω1 2 N 1ω1 exp π OSR π ω1 T2
(17)
Using equation (15), the quantization noise can be approximated as Q
σ2q π2 N π 2 N 1ω12 N
1 2
OSR2 N
1
Cz
(18)
where Cz 1 depends on the position of the NTF zeros ωi within the signal band. Cz 1 when all the zeros are at ω 0. From equations (17) and (18), it is seen that the jitter noise and quantization noise
2023
In−band noise(normalised to Quantiser Range)
4. CONCLUSIONS In this paper, the Bode Sensitivity Integral was used to derive a lower bound on the jitter noise in continuous-time delta-sigma modulators. A consequence of the study is that NTFs with moderate peaking in the transfer function have lesser jitter noise (for the same quantization noise) when compared to their maximally flat counterparts. Intuition was give for the results of [4], where a similar conclusion was reached, but the results were arrived at using numerical optimization. It was shown that jitter noise and quantization noise cannot be reduced simultaneously, and there is an optimum OBG which minimizes the total noise.
−7
4
x 10
Order = 4 = 10−2 Ts σ ∆ Ts
Quantiser = 4 bits OSR = 16
3
2
Q
1
APPENDIX A Consider a real function f x in the interval [a,b]. Using the notation fk f a k b L a and observing that the arithmetic mean is greater than the geometric mean, we have
J+Q J
0 0.2
0.3
0.4
0.5
0.6
ω1
0.7
0.8
0.9
1
a kL
b
∑ fk 2
k0
L
Fig. 5. J, Q and total noise as a function of ω1 , for a fourth order modulator with a 4-bit quantizer and optimized NTF zeros
1
kL 1 1 L a ∏ fk 2
b
(20)
k 0
Simplifying the RHS of the above equation results in a kL
b can be expressed as a function of a single parameter ω1 . As ω1 increases, Q decreases while J increases. Recall that a higher ω1 implies a higher out of band gain. The total in-band noise can be now optimized with respect to frequency ω1 . Q, J and total noise for an example fourth order modulator with a 4-bit quantizer are shown in Figure 5. The OSR is 16, and J is calculated for a 1% RMS clock jitter. From this plot, it seen that an ω1 of about 0.42 results in the least amount of total noise. The knowledge of ω1 , alongwith the constraint that the NTF is maximally flat can be used to determine the OBG that results in the least total in-band noise for the modulator. Alternatively, once ω1 is known, the Bode Sensitivity integral can be used to arrive at an approximate value of the OBG, as discussed below.
L
1
∑ fk 2
k 0
b
k L 1 2 a exp log fk L ∑ k 0
As L ∞ the summation tends to an integral, and we get b a
f x2 dx
b
a exp
b
2 b
a
a
(21)
log f x dx
(22)
Note that the equality is valid only when f x is a constant. Equivalently, we see that if ab log f x dx is constrained to be b a constant, then a f x2 dx is minimized when f x is constant for a x b. 5. REFERENCES [1] J. A. Cherry and W. M. Snelgrove, “Clock jitter and quantizer metastability in continuous-time delta-sigma modulators,” IEEE Trans. Circuits Syst. II, vol. 46, no. 6, pp. 661–676, June 1999.
3.1. Finding the optimum OBG Let ω2 be denote the frequency at which log NT F 0. If the NTF can be approximated by equation (15), it is easily seen that ω2 can
[2] O. Oliaei, “Clock jitter noise spectra in continuous-time deltasigma modulators,” in Proc. IEEE Int. Sym. Circuits and Systems, vol. 2, 1999, pp. 192–195.
be related to ω1 as ω2 ω1 . The area of log NT F below zero can be shown to be N ω2 . Approximating the area above zero as π ω2 log OBG and using Bode’s sensitivity integral, we see that Nω1NN1 (19) OBG exp N 1 π ω1 N N 1 N
[3] H. Tao, L. Toth, and J. M. Khoury, “Analysis of timing jitter in bandpass sigma-delta modulators,” IEEE Trans. Circuits Syst. II, vol. 46, no. 8, pp. 991–1001, August 1999. [4] L. Hernandez, A. Wiesbauer, S. Paton, and A. D. Giandomencio, “Modelling and optimization of low pass continuous-time sigma delta modulators for clock jitter noise reduction,” in Proc. IEEE Int. Sym. Circuits and Systems, vol. 1, May 2004, pp. 1072–1075.
3.2. Design Implications Excess loop delay due to the latency of the quantizer is known to cause peaking in continuous-time delta sigma modulators, if nothing is done to compensate for the excess delay. Hence, a modulator with a nominal NTF that is maximally flat will exhibit peaking in the presence of excess loop delay. The magnitude of the peak depends on the amount of excess delay. From the discussion in Section 2, it is apparent that a small amount of excess delay is beneficial as it reduces the jitter noise J. Lack of space prevents more details here. Simulations for a 4th order modulator with a 4-bit quantizer (maximally flat NTF with OBG=2.5) show that with an RMS clock jitter of 1%, the jitter noise in the presence of excess loop delay of 01 Ts and without delay and 98 10 8 and 12 10 7 respectively.
[5] S. R. Norsworthy, R. Schreier, and G. C. Temes, Eds., DeltaSigma Data Converters:Theory, Design, and Simulation. New York: IEEE Press., 1997. [6] H. W. Bode, Network Analysis and Feedback Amplifier Design, 2nd ed. Princeton: Van Nostrand Reinhold Publishing Company, 1945. [7] C. Mohtadi, “Bode’s integral theorem for discrete-time systems,” IEE Proc. Control Theory and Applications, vol. 137, no. 2, pp. 57–66, March 1990.
2024
