C IRCUITS S YSTEMS S IGNAL P ROCESSING VOL . 23, N O . 6, 2004, P P. 439–462
c Birkh¨auser Boston (2004) DOI: 10.1007/s00034-004-0328-8
A N OVEL S TATISTICAL T ECHNIQUE FOR THE E STIMATION OF DC S TABILITY IN H IGHER -O RDER - A/D C ONVERTERS * Neil A. Fraser1 and Behrouz Nowrouzian2 Abstract. The existing techniques available for the statistical estimation of the dc input signal stability in general-order - analog-to-digital (A/D) converters are based on the assumption that the constituent quantizer input signal has a Gaussian distribution. However, empirical investigations reveal that this assumption holds adequately true only for the special case of conventional first-order - A/D converters. This paper presents an alternative technique for the accurate estimation of the dc input signal stability for higher-order - A/D converters. This estimation technique is based on the practical assumption that the constituent quantizer operates in its overload-free region, permitting the characterization of the quantizer output signal digit-pattern for the determination of the statistical moments of the corresponding quantizer input signal. The resulting statistical moments are subsequently incorporated in a Gram-Charlier series for an accurate quasilinear modeling of the quantizer. A typical application example is given to demonstrate the accuracy of the proposed statistical technique for predicting the existence of multiple regions of instability and stability in the - A/D converter operation, and particularly for predicting the point where the A/D converter operation becomes unstable. Key words: Oversampled A/D converters, sigma-delta (-) modulators, statistical estimation, stability analysis.
1. Introduction In recent years, the process of converting an analog signal to a corresponding digital signal (and vice versa) has been gaining steady prominence in signal processing applications. In general, one can distinguish two classes of analog-to∗ Received March 24, 2003; revised July 15, 2004; Work supported in part by the Natural Sciences
and Engineering research Council of Canada (NSERC) through Operating Grant A6715, by Micronet and by Nortel Networks and Gennum Corporation. 1 MacDonald, Dettwiler and Associates Ltd., Suite 60, 1000 Windmill Road, Dartmouth, Nova Scotia, Canada B3B-1L7. E-mail:
[email protected] 2 Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Alberta, Canada T6G-2V4. E-mail:
[email protected]
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F RASER AND N OWROUZIAN q(n)
x(n) N
y(n) Q
y(n) Figure 1. General configuration of an oversampled - A/D converter.
digital (A/D) converters [5]: the Nyquist rate A/D converters and oversampled A/D converters. Nyquist rate A/D converters offer high-speed and high-resolution performance but require high-tolerance analog components. Oversampled A/D converters, on the other hand, offer high resolution for lower-speed applications and require low-tolerance analog components [6]. These converters trade off the sampling rate to achieve a high resolution, making them relatively inexpensive compared to their Nyquist rate counterparts [3]. Typical examples include the oversampled - A/D converters. An oversampled - A/D converter consists of a two-input one-output linear time-invariant network N and a single-bit quantizer Q embedded in a feedback loop from the output port to internal input port of the network N as shown in Figure 1. In this configuration, x(n) represents the input signal, y(n) represents the corresponding output signal, q(n) represents the quantizer input signal, and n represents the discrete-time time index. The two-input one-output network N in the - A/D converter in Figure 1 is characterized by an input-to-output relationship of the form x(n) . . , (1) q(n) = t1 (n) . t2 (n) ∗ y(n) where ∗ represents the convolution operator, and where t1 (n) and t2 (n) represent unit-impulse responses given by t1 (n) = q(n)x(n)=δ(n), y(n)=0,
(2)
t2 (n) = q(n) y(n)=δ(n), x(n)=0,
(3)
and
with δ(n) representing a unit-impulse function. Moreover, the single-bit quantizer Q is characterized by its (nonlinear) input-output relationship +1 if q(n) ≥ 0, y(n) = (4) −1 if q(n) < 0. In this paper, the quantizer Q in the - A/D converter in Figure 1 is represented by a corresponding quasi-linear model consisting of an additive white
DC S TABILITY IN S IGMA -D ELTA A/D C ONVERTERS q(n)
441
y(n) Q |||
e(n) k q(n)
+
+ Σ
y(n)
Figure 2. A quasi-linear model for the quantizer Q.
noise source e(n) [4] and a variable noise gain element k, as depicted in Figure 2. Consequently, y(n) = kq(n) + e(n).
(5)
In [11], an investigation was undertaken of the stability of - A/D converters in the presence of a dc input signal x(n) of amplitude µx . This investigation led to a statistical estimation of the maximum dc input signal amplitude µxmax for a stable A/D converter operation. The resulting technique was based on several assumptions, the most stringent of which is that the quantizer input signal q(n) possesses a white Gaussian distribution. However, empirical investigations reveal that in higher-order - A/D converters, q(n) is indeed a weighted sum of signals with Gaussian-like distributions, leading to an overall quantizer input signal with nonGaussian distribution [8]. Consequently, the statistical technique in [11] may fail to accurately predict the maximum dc input signal amplitude µxmax , particularly when multiple regions of instability and stability are present. This paper presents a novel statistical technique for an accurate estimation of the maximum dc input signal amplitude µxmax for a stable operation in higherorder - A/D converters. This technique is based on the following practical assumptions, without making any recourse to the assumption of a Gaussian quantizer input signal q(n). Assumption 1. The quantizer Q operates in its overload-free region. Assumption 2. (a) The quantization error e(n) is a white noise process with zero mean. (b) The zero-mean component of the quantizer input signal q(n) is uncorrelated with e(n). The proposed technique exploits the concept of the noise power gain (NPG) associated with the - A/D converter on the one hand, and the Gram-Charlier series [9] characterization of the quantizer input signal q(n) probability density
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F RASER AND N OWROUZIAN
function pd f (q) =
1 √
σq 2π
2
e
− ζ2
N
C j H j (ζ ) ,
(6)
j=0
on the other. Here, ζ =
q − µq , σq
(7)
where µq and σq represent the mean and standard deviation of the quantizer input signal q(n), respectively, and where N represents the number of terms in the series expansion (controlling the accuracy of the estimation). Moreover, H j (ζ ) represents the jth Hermite polynomial defined recursively as H j (ζ ) = ζ H j−1 (ζ ) − ( j − 1)H j−2 (ζ ),
(8)
for j ≥ 1, with H0 (ζ ) = 1. Finally, C j represents the jth Hermite coefficient in accordance with
2 1 i v j−2i Cj = − , 2 i!( j − 2i)! i=0 j
where
vj = E ζ j
(9)
(10)
represents the jth normalized central moment of the quantizer input signal q(n), and where E {·} represents the expectation operator. The proposed estimation of the maximum dc input signal amplitude µxmax is facilitated by making use of the following (empirical) properties. Property 1. In the region where the - A/D converter in Figure 1 incorporating the quasi-linear quantizer model in Figure 2 is bounded-input, boundedoutput (BIBO) stable, N P G is a convex function of the quantizer noise gain k. Property 2. In the neighborhood of the dc input signal instability point, N P G is a monotonically decreasing function of the dc input signal amplitude µx . From Property 1, the N P G will possess a well-defined minimum N P G = N P G min . Then, from Property 2, the maximum dc input signal amplitude can be determined as the point where N P G as a function of µx attains the value N P G min . Therefore, the problem of the dc input signal stability estimation reduces to that of determining the functional relationship between N P G and µx . This paper is organized as follows. Section 2 is concerned with the characterization of the - A/D converter in terms of its noise power gain N P G via the constituent generalized signal and noise transfer functions. In Section 3, by
DC S TABILITY IN S IGMA -D ELTA A/D C ONVERTERS x(n)
T1 (z)
q1 (n) + Σ
y(n)
443
T2 (z)
q(n) +
q2 (n) Figure 3. Equivalent model for the two-input one-output network N .
employing the Gram-Charlier series, a closed-form solution is obtained for the functional relationship between N P G and the dc input signal amplitude µx . The resulting relationship requires the knowledge of the normalized higher-order central moments of q(n). In Section 4, the required higher-order central moments are determined by recognizing the fact that in the course of an overload-free quantizer operation, its input signal q(n) can be estimated in terms of the corresponding A/D converter output signal y(n) digit-pattern. In Section 5, a practical example is given to demonstrate the application of the resulting statistical technique to the investigation of the dc input signal stability of a sixth-order lowpass - A/D converter and to demonstrate the accuracy of the estimation technique for predicting the maximum dc input signal amplitude µxmax for a stable A/D converter operation.
2. Theoretical background This section is concerned with the characterization of the - A/D converter in Figure 1 in terms of the corresponding N P G by employing the quasi-linear quantizer model in Figure 2. In order to facilitate the developments, equation (1) is used to decompose the two-input one-output network N in Figure 1 into an equivalent network, as shown in Figure 3. In Figure 3, T1 (z) and T2 (z) represent the z-transforms of t1 (n) and t2 (n), respectively, and z represents the discrete-time complex frequency variable. (In this paper, upper case letters are used to represent z-transformed signals, and lower case letters are used to represent the corresponding time domain signals.) Moreover, q1 (n) = t1 (n) ∗ x(n)
(11)
q2 (n) = t2 (n) ∗ y(n),
(12)
q(n) = q1 (n) + q2 (n).
(13)
and
where
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F RASER AND N OWROUZIAN
2.1. Generalized signal and noise transfer functions By invoking equation (5) in equation (1), and by applying the z-transform to both sides of the resulting equation, one can obtain the relationship Y (z) = ST F(z)X (z) + N T F(z)Q(z),
(14)
where kT1 (z) 1 − kT2 (z) represents the signal transfer function, and where ST F(z) =
(15)
1 (16) 1 − kT2 (z) represents the noise transfer function associated with the - A/D converter. The design of higher-order - A/D converters usually begins with the approximation of the corresponding unity-quantizer-noise-gain signal and noise transfer functions ST F(z) = ST F(z)|k=1 and N T F(z) = N T F(z)|k=1 from a given set of high-level system design specifications. This approximation is followed by the derivation of the transfer functions N T F(z) =
Author: always cap delta right?
T1 (z) =
ST F(z) N T F(z)
(17)
and 1 (18) N T F(z) and by the synthesis of the two-input one-output network N . Finally, the corresponding generalized signal and noise transfer functions are obtained as T2 (z) = 1 −
ST F(z) =
k ST F(z) (1 − k)N T F(z) + k
(19)
and N T F(z) (20) (1 − k)N T F(z) + k for the investigation of the characteristics of the resulting - A/D converter. This investigation usually includes a statistical estimation of the maximum dc input signal amplitude µxmax for a stable converter operation. N T F(z) =
2.2. Noise power gain The noise power gain N P G is formally defined as π 1 jwT 2 N PG = N T F e dw, 2π −π
(21)
DC S TABILITY IN S IGMA -D ELTA A/D C ONVERTERS
445
where ω represents the discrete-time (normalized) real-frequency variable. By using equation (20), N P G can be expressed in terms of the unity-quantizer-noisegain noise transfer function N T F(z) as 2 π N T F e jω 1 (22) N PG = dw. 2π −π (1 − k)N T F e jω + k In accordance with equation (22), one can derive an empirical relationship for N P G as a function of the quantizer noise gain k. The resulting relationship reveals that N P G is a convex function of the quantizer noise gain k, implying that it possesses a well-defined minimum value N P G = N P G min in the region where the quasi-linear - A/D converter is BIBO stable. Through a statistical analysis of the - A/D converter, one can derive another empirical relationship for N P G as a function of the dc input signal amplitude µx . Then, the maximum dc input signal amplitude µxmax can be determined as the point where N P G as a function of µx attains the value N P G min . 3. Statistical relationship between noise power gain and dc input signal amplitude The investigation of the dc input signal stability in lower-order - A/D converters is usually based on the assumption that the quantizer input signal q(n) has a Gaussian distribution [2]. This is justified by the fact that in the conventional first-order - A/D converters, the quantizer input signal q(n) is formed as the discrete-time integration of the quantization error e(n) (via the noise-shaping characteristics of the converter), where e(n) itself possesses a uniform distribution. However, for higher-order - A/D converters, the quantizer input signal q(n) usually possesses a non-Gaussian distribution. This section is concerned with a statistical analysis - A/D converter for the derivation of the functional relationship between the noise power gain N P G and the dc input signal amplitude µx . From Assumption 1, the quantizer output signal y(n) will have the same longterm average µ y as the input signal x(n), so that µ y = µx .
(23)
Consequently, N PG =
1 − µ2x , σe2
(24)
where σe2 represents the variance of the quantization error e(n). Moreover, from Assumption 2a, the quantization error variance σe2 can be obtained in accordance with 2 σe2 = 1 − µ2x + kσq . (25)
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F RASER AND N OWROUZIAN
By substituting for σe2 from equation (25) into equation (24), one obtains N PG =
1 − µ2x 2 . 1 − µ2x + kσq
(26)
Therefore, one can obtain the desired relationship between N P G and µx through the determination of the term kσq in terms of the dc input signal amplitude µx . 3.1. Determination of the term kσq This subsection is concerned with the derivation of a general relationship between the term kσq in equation (26) for higher-order - A/D converters in terms of the normalized mean and the normalized higher-order central moments v j of the quantizer input signal q(n). In accordance with equations (9) and (10), N j=0
C j H j (ζ ) = 1 +
N
C j H j (ζ ),
(27)
j=2
simplifying the probability density function pd f (q) to N 2 1 − ζ2 pd f (q) = √ e 1+ C j H j (ζ ) . σq 2π j=2
(28)
From Assumption 2b [11], kσq = E {y(n)ζ } . By using equation (28), equation (29) can be written explicitly as ∞ N 2 1 − ζ2 kσq = 1+ y(n) √ ζ e C j H j (ζ ) dq. σq 2π −∞ j=2 Then, by invoking equation (4) in equation (30), one can write 0 N 2 1 − ζ2 1+ (−1) × √ ζ e C j H j (ζ ) dq kσq = σq 2π −∞ j=2 ∞ N 2 1 − ζ2 1+ (+1) × √ ζ e C j H j (ζ ) dq. + σq 2π 0 j=2
(29)
(30)
(31)
By separating terms, one gets 0 0 N 2 ζ2 1 1 − ζ2 ζe dq − √ Cj ζ e− 2 H j (ζ ) dq kσq = − √ σq 2π −∞ σq 2π j=2 −∞ ∞ N ∞ ζ2 ζ2 1 1 ζ e− 2 dq + √ Cj ζ e− 2 H j (ζ ) dq. (32) + √ σq 2π 0 σq 2π j=2 0
DC S TABILITY IN S IGMA -D ELTA A/D C ONVERTERS
447
By using equation (7) to change the integration variables, equation (32) can be rewritten as − µq − µq N 2 σq σq ζ2 1 1 − ζ2 kσq = − √ ζe dζ − √ Cj ζ e− 2 H j (ζ ) dζ 2π −∞ 2π j=2 −∞ ∞ ∞ N 2 ζ2 1 1 − ζ2 ζ e dζ + C ζ e− 2 H j (ζ ) dζ . (33) +√ √ j µq µq 2π − σq 2π j=2 − σq Let us invoke the following two identities: ζ2 ζ2 ζ e− 2 dζ = −e− 2 and
ζ e−
ζ2 2
H j (ζ ) dζ = −e−
ζ2 2
ζ H j−1 (ζ ) + H j−2 (ζ )
(34)
(35)
in equation (33), and manipulate the result to arrive at the desired relationship 2
N µq µq µq 2 − 12 µσqq kσq = 1+ + H j−2 − . e C j − H j−1 − π σq σq σq j=2 (36) A close inspection of equations (26) and (36) reveals that the problem of the determination of N P G in terms of the dc input signal amplitude µx amounts to obtaining the normalized mean µq /σq of the quantizer input signal q(n) on the one hand, and obtaining the Hermite coefficients C j on the other, all in terms of µx . Moreover, a close inspection of equations (9) and (10) reveals that the problem of obtaining the Hermite coefficients C j in terms of µx amounts to determining the normalized higher-order central moments v j of q(n) in terms of µx . In conclusion, the problem of the determination of N P G in terms of µx reduces to analyzing the - A/D converter to obtain the quantizer input signal q(n) in terms of µx . In accordance with these discussions, one can obtain an empirical relationship for N P G as a function of the dc input signal amplitude µx . Then, the maximum dc input signal amplitude µxmax for a stable - A/D converter operation can be determined as the point where N P G as a function of µx attains the minimum value N P G min . From equations (26) and (36), the resulting µxmax is given by the solution of the implicit equation 2 N P G min π N P G min − 1 2
N µ µq µq µq − σqq ×e 1+ + H j−2 − . (37) C j − H j−1 − σq σq σq j=2
µxmax = 1 −
In practical situations, the desired solution µxmax is best obtained through an
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F RASER AND N OWROUZIAN
iterative optimization procedure. The required normalized mean µq /σq may be obtained directly from the quantizer input signal q(n), or, more appropriately, indirectly by using the statistical estimation technique discussed in the following subsection. 3.2. Statistical estimation of the term µq /σq An approach similar to that in the previous subsection can be used to derive a general relationship between the normalized mean µq /σq of the quantizer input signal q(n) and the dc input signal amplitude µx in higher-order - A/D converters. In accordance with equation (4), the mean µ y of the quantizer output signal y(n) can be determined as the probability that the quantizer input signal q(n) is positive (or zero) (leading to a +1 output), minus the probability that q(n) is negative (leading to a −1 output), in accordance with ∞ 0 µ y = Pr q(n) ≥ 0 − Pr q(n) < 0 = pd f (q) dq − pd f (q) dq. 0
−∞
(38) By substituting equation (6) into equation (38), one obtains N ∞ 2 1 − ζ2 1+ C j H j (ζ ) dq µy = √ e σq 2π 0 j=2 0 N ζ2 1 C j H j (ζ ) dq . − √ e− 2 1 + −∞ σq 2π j=2 By separating terms, one gets ∞ ∞ N 2 ζ2 1 1 − ζ2 µy = e dq + √ Cj e− 2 H j (ζ ) dq √ σq 2π 0 σq 2π j=2 0 0 N 0 ζ2 ζ2 1 1 e− 2 dq + √ Cj e− 2 H j (ζ ) dq . − √ σq 2π −∞ σq 2π j=2 −∞
(39)
(40)
By using equation (7) to change the integration variables, equation (40) can be rewritten as ∞ ∞ N 2 ζ2 1 1 − ζ2 e dζ + C e− 2 H j (ζ ) dζ µy = √ √ j µq µq σq 2π − σq σq 2π j=2 − σq µ µ − q − q N 2 2 σq σq 1 1 − ζ2 − ζ2 − √ e dζ + √ Cj e H j (ζ ) dζ . (41) σq 2π −∞ σq 2π j=2 −∞
DC S TABILITY IN S IGMA -D ELTA A/D C ONVERTERS
449
By substituting equation (23) into equation (41), invoking the following three identities:
2 π ζ − ζ2 dζ = er f √ , (42) e 2 2 er f (−ζ ) = −er f (ζ ), and
e−
ζ2 2
H j (ζ )dζ = −e−
ζ2 2
H j−1 (ζ )
in the result, and by manipulating, one can arrive at the relationship 2 N
µq µq 2 − 12 µσqq . e C j H j−1 − µx = er f + √ π σq σq 2 j=2 Then, one can recast equation (45) in the implicit equation 2 N
√ µq µq 2 − 12 µσqq −1 = 2er f , µx − C j H j−1 − e σq π σq j=2
(43)
(44)
(45)
(46)
to solve for the normalized mean µq /σq of the quantizer input signal q(n) in terms of the dc input signal amplitude µx . Once again, a practical approach to obtain the solution is through an iterative optimization, e.g., by using the interval-halving technique [10].
4. Estimation of higher-order moments of quantizer input signal The determination of the maximum input signal amplitude µxmax for a stable A/D converter operation developed in the preceding section relies on a prior knowledge of the normalized mean µq /σq as well as the normalized higher-order central moments v j of the quantizer input signal q(n). This section presents a novel technique for a statistical estimation of the normalized central moments of q(n). 4.1. The case of conventional first-order - A/D converters Let us consider the case of the conventional first-order - A/D converters, where the transfer functions T1 (z) and T2 (z) are given by T1 (z) =
z −1 1 − z −1
(47)
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F RASER AND N OWROUZIAN
and z −1 . (48) 1 − z −1 Moreover, let the dc input signal amplitude µx have a rational value in accordance with µx = rs , where r and s are relatively prime integers (with r < s). In this way, the quantizer output signal y(n) will consist of a stream of “+1” and “−1” digits having an average value that is approximately equal to the dc input signal amplitude rs . Let α represent the percentage of “+1” digits, and let (1 − α) represent the percentage of “−1” digits in the output signal y(n). Then, the dc input signal amplitude rs must satisfy a relationship of the form r α × (+1) + (1 − α) × (−1) = . (49) s From equation (49), one can obtain T2 (z) = −
2αs = s + r,
(50)
2(1 − α)s = s − r .
(51)
and By recalling that r and s are integers, both 2αs and 2(1 − α)s will also become integers. In this way, out of every 2s quantizer output signal y(n) digits, there are (r + s) “+1” digits and (r − s) “−1” digits, rendering the output signal y(n) digitpattern periodic with a period of 2s. Then, if one assumes that the “+1” digits are interleaved among the “−1” digits as regularly as possible, the average length of 2s . Using this result, one can conclude that the spectrum these smaller cycles is s−r of y(n) will contain a strong tone of frequency [12] s −r f tone = (52) fs , 2s where f s represents the sampling frequency. This tone dominates all the other tones in the quantizer output signal y(n) spectrum, representing the dominant period of oscillations. This tone is responsible for the generation of the correct mean value for the output signal y(n). In this way, in order for the mean of the output signal y(n) to be equal to rs , the ith sample of y(i) for i ≥ 2 will be a “−1” if the mean of y(0) to y(i − 1) is greater than rs , and a “+1” otherwise. This leads to Procedure 1 as follows for the estimation of the quantizer output signal y(n). Procedure 1 Set y(1) = 1 for i = 2 to 2s { if E y(n)|1≤n
rs then y(i) = −1 otherwise y(i) = 1 }.
DC S TABILITY IN S IGMA -D ELTA A/D C ONVERTERS
451
Number of Occurrences (Normalized)
0.05 0.04 0.03 0.02
0.01 Actual Estimated 0.0 – 1.0
– 0.5
0.0
0.5
1.0
Quantizer Input Signal q(n)
Figure 4. Histograms of the actual and estimated quantizer input signal q(n) for the conventional first-order − A/D converter.
Having determined the quantizer output signal y(n), the corresponding quantizer input signal q(n) can be estimated in accordance with equation (13) in terms of its components q1 (n) and q2 (n) by invoking the inverse z-transforms of the transfer functions T1 (z) and T2 (z) of equations (47), (48) in equations (11), (12) to get r q1 (n) = n (53) s and n q2 (n) = − y(i). (54) i=1
In order to confirm the validity of Procedure 1, let us choose the dc input signal 23 amplitude as µx = 100 . Moreover, let us determine the actual quantizer input signal q(n) through a simulation of the nonlinear - A/D converter, and the estimated q(n) by employing Procedure 1. In this way, the histogram of the actual quantizer input signal q(n) and that of the estimated q(n) are obtained, as shown in Figure 4. Consequently, the normalized central moments v j of the actual quantizer input signal q(n) and those of the estimated q(n) can be determined, as given in Table 1. 4.2. The case of higher-order - A/D converters In the case of higher-order - A/D converters, it is difficult to determine the quantizer output signal y(n) based on the knowledge of the dc input signal amplitude µx alone. In this case, despite the fact that the output signal y(n) tracks the mean µx of the dc input signal x(n), its digit-pattern is not nearly as uniform
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F RASER AND N OWROUZIAN
Table 1. Actual and estimated central moments v j of the quantizer input signal q(n) for the conventional first-order - A/D converter
i
Actual ith moment v j
Estimated ith moment v j
2 3 4 5 6 7 8 9 10
0.575897885 0.000000000 0.345518575 0.000000000 0.246776359 0.000000000 0.191913177 0.000000000 0.156995012
0.575897885 0.000000000 0.345518575 0.000000000 0.246776359 0.000000000 0.191913177 0.000000000 0.156995012
and as predictable as that of the conventional first-order - A/D converter. Moreover, despite the fact that the tone at the frequency given in equation (52) is still present in the signal y(n), it is surrounded by other tones affecting the digit-pattern of y(n). This subsection introduces a novel technique for an accurate approximation of the digit-pattern of the quantizer output signal y(n) for higherorder - A/D converters based on other considerations. For higher-order - A/D converters, equation (13) can still be applied to the determination of the quantizer input signal q(n). In this way, as the dc input signal amplitude µx is already known, it is a simple matter to obtain the first component q1 (n) of q(n) from equation (11). In order to obtain the second component q2 (n), on the other hand, let us recall that if the - A/D converter operates in its overload-free region, then the digit-pattern of the quantizer output signal y(n) is generated in such a manner as to minimize the difference between the components q1 (n) and q2 (n) of q(n). In this way, the ith output signal digit can be determined by considering which value, either a +1 or a −1, produces a minimum difference between q1 (i) and q2 (i). These considerations lead to Procedure 2 for the determination of the output signal y(n). Procedure 2 Set x(1) = µx and y(1) = 1 for i = 2 to 2q { x(i) if 1 ≤ n ≤ i − 1 Set x(n) = µx if n = i 0 if n = i + 1 Find q1 (n) = t1 (n) ∗ x(n) y(i) if 1 ≤ n ≤ i − 1 Set y ± (n) = ±1 if n = i 0 if n = i + 1
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453
Table 2. High-level system design specifications for the lowpass COI - A/D converter
Order
Bandwidth
Oversampling ratio
SQNR
6
10 kHz
64
85 dB
Table 3. Noise transfer function coefficients for the COI lowpass - A/D converter
i
Coefficient n i
Coefficient di
1 2 3 4 5 6
−5.996560946 14.986246667 −19.979371441 14.986246667 −5.996560946 1.000000000
−5.347278883 11.940946337 −14.251651912 9.586544165 −3.445261347 0.516707845
Determine q2± (n) = t2 (n) ∗ y ± (n) ± Obtain q ± (i + 1) = q2 (i + 1) − q1 (i + 1) −1 if q + (i + 1) ≥ q + (i + 1) Set y(i + 1) = +1 if q + (i + 1) < q + (i + 1) Find q2 (n) = t2 (n) ∗ y(n) Obtain q(n) = q1 (n) + q2 (n)
} Having estimated the quantizer input signal q(n) by employing Procedure 2, the normalized central moments v j of q(n) can be determined in a straightforward fashion. However, computational investigations reveal that the required normalized mean µq /σq of the estimated quantizer input signal (q) is quite inaccurate. Fortunately, this problem can be circumvented by resorting to a statistical estimation of µq /σq as outlined in Section 3.2. Let us consider the application of Procedure 2 to the estimation of the quantizer input signal q(n) for a cascade-of-integrators (COI) lowpass - A/D converter [7] satisfying the high-level system specifications given in Table 2. Through the application of genetic algorithms, the corresponding unityquantizer-noise-gain noise transfer function N T F(z) is obtained as [1] N T F(z) =
1 + n 1 z −1 + n 2 z −2 + · · · + n 6 z −6 , 1 + d1 z −1 + d2 z −2 + · · · + d6 z −6
(55)
where the numerator coefficients n i and the denominator coefficients di are as given in Table 3. An important characteristic of the COI - A/D converters is that the unityquantizer-noise-gain signal and noise transfer functions ST F(z) and N T F(z) are
454
F RASER AND N OWROUZIAN Number of Occurrences (Normalized)
0.10 Actual Estimated
0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00 – 1.0
– 0.5
0.0
0.5
1.0
Quantizer Input Signal q(n)
Figure 5. Histograms of the actual and estimated quantizer input signal q(n) for the sixth-order lowpass COI - A/D converter for µx = 0.23.
Table 4. Actual and estimated central moments v j of the quantizer input signal q(n) for the COI - A/D converter for µx = 0.23
j
Actual moment v j
Estimated moment v j
2 3 4 5 6 7 8 9 10
0.349886819 0.007682884 0.085127539 0.002996222 0.025650188 0.001096448 0.008580233 0.000409438 0.003054415
0.345566746 0.007781357 0.080960438 0.002862518 0.023424201 0.001015171 0.007518243 0.000364804 0.002572717
complementary by the - A/D converter proper so that N T F(z) − ST F(z) = 1.
(56)
Therefore, the unity-quantizer-noise-gain signal transfer function ST F(z), as required for the realization of the - A/D converter, can be determined in terms of the corresponding noise transfer function N T F(z) by using equation (56). In order to demonstrate the usefulness of Procedure 2, let us consider two different values for the dc input signal amplitude, namely µx = 0.23 and µx = 0.46. For µx = 0.23, the histogram of the actual quantizer input signal q(n) and that of the estimated q(n) for the COI lowpass - A/D converter are obtained, as shown in Figure 5. The corresponding normalized central moments v j of the actual quantizer input signal q(n) and those of the estimated q(n) are obtained as given in Table 4.
DC S TABILITY IN S IGMA -D ELTA A/D C ONVERTERS
455
Number of Occurrences (Normalized)
0.10 Actual Estimated
0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00 – 1.0
– 0.5
0.0
0.5
1.0
Quantizer Input Signal q(n)
Figure 6. Histograms of the actual and estimated quantizer input signal q(n) for the sixth-order lowpass COI - A/D converter for µx = 0.46.
Table 5. Actual and estimated central moments v j of the quantizer input signal q(n) for the COI - A/D converter for µx = 0.46
j
Actual moment v j
Estimated moment v j
2 3 4 5 6 7 8 9 10
0.347263420 0.024273961 0.083810656 0.012518203 0.025422161 0.005662767 0.008680813 0.002502991 0.003193869
0.347703130 0.023192860 0.084833877 0.011483993 0.026140627 0.004831345 0.009163370 0.001864531 0.003553280
Similarly, for µx = 0.46, the histogram of the actual quantizer input signal q(n) and that of the estimated q(n) are obtained as shown in Figure 6. Finally, the corresponding central moments v j of the actual quantizer input signal q(n) and those of the estimated q(n) are obtained as given in Table 5. By comparing the actual and estimated histograms in Figure 5 and the corresponding central moments in Table 4, and comparing the actual and estimated histograms in Figure 6 and the corresponding moments in Table 5, one can confirm the accuracy of Procedure 2 for the estimation of the higher-order normalized central moments v j of the quantizer input signal q(n).
456
F RASER AND N OWROUZIAN 20 0
Magnitude, dB
-20 -40 -60 -80 -100
| ST F (e jω ) |
.......... .... .................................... .. ... .......................... ................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ..................................... . .................................................... .. ...................................................................................... ... ............................................................................................................................................................. . .. .... . .... . .... . ... .. ... .. jω ... .. .... . ... .. .... . ... .. ... .. ... .. .... ... .. ... .. ........ ....... ..................... ............. ............ ... ..... ... ... ... ... ... .. ... . ..
| N T F (e )|
-120 0.00
0.25
0.50
0.75
1.00
Normalized Frequency
Figure 7. Magnitude-frequency responses associated with signal and noise transfer functions ST F(z) and N T F(z) for the COI lowpass - A/D converter.
5. Application example This section is concerned with an application of the proposed statistical technique to the estimation of the dc input signal stability for the COI lowpass - A/D converter obtained in the preceding section to satisfy the high-level design specifications given in Table 2. In accordance with equations (55) and (56) and the entries in Table 3, the magnitude-frequency responses associated with the signal and noise transfer functions N T F(z) and ST F(z) of the COI - A/D converter are obtained as shown in Figure 7. The power spectral density of the quantizer output signal y(n) in the preceding - A/D converter for a sinusoidal input signal x(n) with an amplitude of −6 dB (referred to the quantizer full-scale range) and with a frequency of 5 kHz can be obtained as shown in Figure 8. By performing further computational investigations, it becomes evident that the converter gives rise to an achievable signal-to-quantization noise ratio (SQNR) of 114 dB and a dynamic range of 120 dB. It can be shown that the COI lowpass - A/D converter employing the quasilinear quantizer model in Figure 2 is BIBO stable for a quantizer noise gain in the interval k ≥ 0.54. In this interval, the noise power gain N P G is a convex function of k, attaining a minimum value of N P G min = 1.7459 as shown in Figure 9. The fact that the quantizer noise gain k changes relative to the dc input signal amplitude µx plays a major role in understanding the mechanism of stability (or lack thereof) in the operation of higher-order - A/D converters. To illustrate this point, let us consider the root-locus of the poles of the unity-noise-gain noise transfer function N T F(z) as shown in Figure 10 and magnified in Figure 11. It can readily be verified that a gradual increase in the dc input signal amplitude
457
PSD y
DC S TABILITY IN S IGMA -D ELTA A/D C ONVERTERS
Figure 8. Power spectral density P S D(y) of the quantizer output signal y(n) for the COI lowpass - A/D converter. 8.0 7.0
N PG
6.0 5.0 4.0 3.0 2.0
... ... ... ... . ... .. ... ... .. ... ... ... ... ... . . ... .. ... ... ... .. ... ... ... ... ... ... ... ... . . ... . ... ... ... ... .... ... .... ... .... ... ... ... .... . . ... . . ... ..... ... ..... ... ..... ... ..... ...... ... ...... ... ...... . . ... . . . ... ..... ....... ... ....... ... ........ ... ........ ... ......... ... .......... ........... ... . . . . . . . . . . ... .... ............. ... ............. ... ............... ... ................ ... .................. ..... min .................... ........................ ........ ............................................................................................................
1.0 0.5
N PG
1.0
1.5
2.0
2.5
3 .0
Quantizer Gain k
Figure 9. Noise power gain N P G as a function of the quantizer noise gain k for the quasi-linear COI - A/D converter.
µx gives rise to a gradual decrease in the quantizer noise gain k in such a manner as to push the poles outside the unit-circle in the z-plane (causing instability). Of course, the preceding root-locus presentation cannot adequately be applied to an accurate estimation of the region(s) of nonlinear instability and stability for the A/D converter. Alternatively, by using the proposed Gram-Charlier estimation technique for N = 40, and by using the actual - A/D converter quantizer input signal q(n), one can obtain the quantizer noise gain k as a function of the dc input signal amplitude µx as shown in Figure 12. The erratic fluctuations in the quantizer noise gain k for dc input signal amplitudes past µx ≈ 0.66 in Figure 12 clearly predict the existence of multiple regions of nonlinear instability and stability in the course of - A/D converter operation.
458
F RASER AND N OWROUZIAN
Imaginary Axis
1.0
0.5
.
. . . . . . . . . . . . .
0
-0.5 -1.0
.
.
.
.
.
.
.
. . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
. .
........ ....... ......... .. ...... . .. ....... ............... .............................................................................................................................................................................................................. .......................... . .... ........ ......... ....... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.0
0.5
0.0
0.5
1.0
Real Axis
Figure 10. Root-locus of pole locations for the unity-noise-gain noise transfer function N P G(z).
0.2
. .................................................... ................... .............. . .......... .......... ....... ........ . .... ...... ...... ..... . ..... ... . .... ..... ........ ... . .......... ... . . ... . ... . .. . ... ... . .. . . . . ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .................. ........... .... ............. ........... ....... . ....... ....... ...... ............................................. ...... ..... .... . .... ................. .......................................... . ... ............ ...... ....................... .................................................................................................. . .... ........... ... .......................................... . ... ................. .... . ..... ..... ...... ...... ................................................ ......... ..... ....... . ............ .............. . ..... . . . . . . . ..................... . . . . . . . . . . .... .............................................................. . ... ... . ... ... . .. . . ... . . . ... . . . . . . . . . . . . .... ... ....... . . . . . . .. ..... . . . ..... . . .... ... . ....... ..... . ......... ........ ........... ............. .......................................................................... .
-0.1
0 ≤ k < 0.54
0.0
.xx x x xx.
Imaginary Axis
0.1
-0.2
0.80
0.85
0.90
0.95
1.00
1.05
1.10
Real Axis
Figure 11. A selective magnification of the root-locus in Figure 10.
To investigate the stability of the - A/D converter further, let us proceed to determine N P G as a function of the input signal amplitude µx as shown in Figures 13–16 with the following particulars: • Figure 13: Obtained by using the existing statistical estimation technique [11]. • Figures 14, 15: Obtained by employing actual simulation of the nonlinear - A/D converter to determine the quantizer input signal q(n), and by using equation (36) to determine the term kσq via the Gram-Charlier representation of q(n) with N = 6 and N = 20, respectively. • Figure 16: Obtained by employing actual simulation of the (nonlinear) - A/D converter, and by using equation (29) for the determination of the term kσq .
DC S TABILITY IN S IGMA -D ELTA A/D C ONVERTERS
Quantizer Noise Gain k
2.5 2.0
1.5 1.0
0.5
459
. . . ................................................................. . . ........................ . . . .... ......................................................................... ... ..................... ........................ . ........................ . .............. ............ ..................... ..... ...... ...................... .. ..... ......................... ................. ... ................ ................. ................ ................ ............ . .............. .......... ............. .......... .. ............ .......... .......... ................. ............... ............... ......................... ........................ ................................. . .................................... ................................... ............................. ........................................ ............................. ...................................... .............................. ... .......................... .... ................................. . .... ..................................... .. ........................ ... .................... ... .......................... ... ........................... .... .................................... .. .......................... ... .......................... .. .................... ... ................... .. ........... ...... ... .............. .... .. .. ..... ................................ ... ..................... ... .................... ... ..................... .. ................... ... ................... . .... ................................ .
0.0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
DC Input Signal Amplitude µ x
Figure 12. Quantizer noise gain k as a function of the dc input signal amplitude µx using the actual quantizer input signal q(n) with N = 40.
Noise Power Gain NPG
4.0
3.5 3.0
2.5 2.0
............................................................................... ................................. ........................ ..................... ................... .................. ................. ................ ............... ............... .............. .............. .............. ............. ............. ............. ............. ............. ............ ............ ............ ............ ............ ............ ............ min ... ...........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
N PG
1.5 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0 .7
DC Input Signal Amplitude µ x
Figure 13. Noise power gain N P G as a function of the dc input signal amplitude µx obtained by employing the existing estimation technique [11].
In this way, the maximum µxmax of the dc input signal amplitude µx at the verge of instability can be obtained, as shown in Table 6. For completeness, Table 7 shows the corresponding maximum dc input signal amplitudes µxmax obtained with the following particulars: • Obtained by employing the estimated quantizer input signal q(n), and by using the Gram-Charlier representation of q(n) with N = 6 and N = 20, respectively, via equation (36). • Obtained by employing the estimated q(n) and by using equation (29). From the results in Tables 6 and 7, it can be observed that the existing technique
460
F RASER AND N OWROUZIAN
Noise Power Gain NPG
4.0
3.5 3.0
2.5 2.0
. ......................................................... . .... . .................................................. . . ... .......... ............................................................ ...................................... .. .... ....................................... . ... .............................. ... . .. ....................... ................. ................ ...... ... ....................... ................ . .................... . . .................... ............................... . ................. . ...... .. ................................. . ................................... ........ ...................... ... . ... ...... ............................... . . ... ........................ . ............................. ... ..... . ............................ .. ......................................... .... ................................... .. .... ................. .......................... ... ........................... . ......................... ........... .... .......................... .. ........................ .. . .................... ..... ... ................. . .... . . ... . ............................. . .............. ..................... .......... .............................. ............... ...... .................. .......................................... . .......................... ............................. .................................. min .. ........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ............................... ....................................... ............................... ............................ ........................
N PG
1.5 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0 .7
DC Input Signal Amplitude µ x
Figure 14. N P G as a function of the dc input signal amplitude µx , obtained employing the actual q(n) together with Gram-Charlier series with N = 6 via equation (29).
Noise Power Gain NPG
4.0
3.5 3.0
2.5 2.0
.. ......... ... .......... .. .... ............................................................................................................ .... ......................... .. . . ..... ........................................................... .. . . ............................... .. . ..... .... ................................... ... .................................. . . ... .............. ..... .. ........................ ... ........... . ... ......................... ........... ....... .. ....... ............... ....... . ........ .................. ...................................... . ... . ... ..................................................... . .... ... .. .................................................... .. .. .... . .................................................... ....... .... ... ..... ....... ..... ........ .............................. ..... .. . ... ... ... . ... .................................................... ..... ...................................................... ...... ............................................................ . .. .. .... ............................... ....... ........................................... .... ................................. . .............................. ................... ..... ... ............................... . ....................... .............. ....
N P G min
1.5 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
DC Input Signal Amplitude µ x
F gure 15 N P G as a unc on o he dc npu s gna amp ude µx ob a ned by emp oy ng he ac ua q n oge he w h G am Cha e se es w h N = 20 v a equa on 29
fa s shor of produc ng an accura e es ma on of he po n where he - A D conver er becomes uns ab e However no e ha he proposed echn que produces a subs an a y more accura e es ma on of he po n of ns ab y no on y for N = 20 bu a so for N = 6 In prac ca s ua ons a cho ce of 15 < N < 30 s usua y suffic en for an accura e s a s ca es ma on of he max mum dc npu s gna amp ude µxm for s ab e opera on n h gher-order - A D conver ers
DC S TABILITY IN S IGMA -D ELTA A/D C ONVERTERS
461
Noise Power Gain NPG
4.0
3.5 3.0
2.5 2.0
... ... . .... ......... . . .................... ............................... ......... ..... . ..................................................................................................................................... .......................... ... .. . ........ . .......................................................... .. . . . .. .............................. .. ....................... . ................................ ...................... ...................... ....... .. ...................... .... ........... . ................ ........... ...... ................... ...... ...... ...... .................. ....... .......... ............ ............... . ..... ...................... . ................... . ..... ......................... . ......... .... .................. . ........... . ............... ............ . ............ . ................ . ......... . ............. ........... ............. .......... ......... .... .. .......... ...... ............. ............... ........ .......... ......................... ......... ...... .............. ....................... ............................ .............................. .............................. ... ................................... ...................................................... ............................... ........................................ min .. .............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. ............................................ .. ......................... ... .......................... ... ......................... . .
N PG
1.5 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
DC Input Signal Amplitude µ x
Figure 16. Actual N P G as a function of the dc input signal amplitude µx , obtained by employing the actual q(n) and equation (26).
Table 6. Maximum dc input signal amplitude µxmax at the verge of instability using the actual quantizer input signal q(n)
Existing technique 0.6994
Proposed technique N =6 N = 20 0.6627
Using equation (29)
0.6642
0.6641
Table 7. Maximum dc input signal amplitude µxmax at the verge of instability using the estimated quantizer input signal q(n)
Proposed technique N =6 N = 20 0.6664
0.6665
Using equation (29) 0.6665
6. Conclusions A novel technique has been presented for the estimation of dc input signal stability in higher-order - A/D converters. The resulting estimation technique is based on an accurate modeling of the probability distribution of the constituent quantizer input signal in terms of Gram-Charlier series. The Gram-Charlier series itself requires the knowledge of the normalized higher-order central moments of the quantizer input signal. These higher-order central moments have been determined by recognizing the fact that in the course of an overload-free quantizer operation, the probability distribution of the quantizer input signal can be determined in terms of the corresponding quantizer output signal digit-pattern. The proposed estimation technique has been illustrated through its application to the investiga-
462
F RASER AND N OWROUZIAN
tion of the dc input signal stability for a practical sixth-order lowpass - A/D converter.
Acknowledgments This was supported, in part, by NSERC under Operating Grant #A6715, by Micronet, and by Nortel Networks and Gennum Corporation. Thanks are due to A. Alavi for compiling an initial draft of the manuscript of this paper.
References [1] A. Alavi and B. Nowrouzian, A novel genetic algorithm with applications to the design of higher-order sigma-delta D/A converters, Proceedings of 47th IEEE Midwest Symposium on Circuits and Systems, Hiroshima, Japan, vol. I, pp. 349–352, July 2004. [2] S. Ardalan and J. J. Papulos, An analysis of nonlinear behavior in delta-sigma modulators, IEEE Trans. Circuits and Systems, vol. CAS-34, pp. 593–603, June 1987. [3] H. S. P. M. Aziz and J. V. der Spiegel, An overview of - converters, IEEE Signal Process., vol. 13, pp. 61–84, Jan. 1996. [4] J. Candy and O. Benjamin, The structure of quantization noise from sigma-delta modulation, IEEE Trans. Comm., vol. COM-29, pp. 1316–1323, Sept. 1981. [5] J. Candy and E. G. C. Temes, Oversampling Delta-Sigma Data Converters: Theory, Design and Simulation, IEEE Press, Washington, D.C., 1992. [6] J. C. Candy, A use of double integration in sigma delta modulation, IEEE Trans. Comm., vol. COM-33, pp. 249–258, March 1985. [7] N. Fraser and B. Nowrouzian, Design and Monte-Carlo analysis of multiple-feedback oversampled - A/D converter configurations, Proceedings of 2000 World Automation Conference, Maui, HI, June 2000. [8] N. Fraser and B. Nowrouzian, Stability analysis of feedforward and multiple-feedback - converter configurations, Proceedings of IEEE Midwest Symposium on Circuits and Systems, Lansing, MI, Aug. 2000. [9] R. Freedman, On Gram-Charlier series, IEEE Trans. Comm., vol. COM-29, pp. 122–125, Feb. 1981. [10] G. R. A. Ravindran and K. Ragsdell, Engineering Optimization, Methods and Applications, John Wiley & Sons, New York, 1983. [11] L. Risbo, Stability predictions for high-order - modulators based on quasilinear modeling, Proceedings of IEEE International Symposium on Circuits and Systems, London, England, vol. 5, pp. 361–364, May 1994. [12] M. Vogels and G. Gielen, Efficiency analysis of the stability of sigma-delta modulators using wavelets, Proceedings of IEEE International Symposium on Circuits and Systems, Geneva, Switzerland, pp. 758–761, May 2000.
