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D converters
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Measurement 41 (2008) 186–191 www.elsevier.com/locate/measurement
Integral non-linearity in memoryless A/D converters A. Moschitta a
a,*
, D. Petri
b,1
DIEI – Department of Information and Electronic Engineering, University of Perugia, Perugia, Italy b DIT – Department of Informatics and Telecommunications, University of Trento, Trento, Italy Received 23 January 2006; received in revised form 19 July 2006; accepted 4 September 2006 Available online 16 September 2006
Abstract This paper investigates the statistical properties of quantization noise. In particular, a theoretical model is discussed, which evaluates the power of quantization noise introduced by a memoryless analog to digital converter (ADC) as a function of both the converted signal distribution and the ADC thresholds positioning. Expressions have also been derived to express the integral non-linearity (INL) contribution to quantization noise power as an additive term, and to evaluate such a term with a simple formula. Simulation results that validate the proposed expression are provided. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Quantization noise; Integral non-linearity
1. Introduction Analog to digital (A/D) and digital to analog (D/ A) converters are widely used in many modern fields of application, allowing to replace analog systems with digital high performance implementations. Modeling the behavior of A/D and D/A converters is important both for characterizing the performance of the converters themselves and for embedding such devices in real systems. The matter has been subject of several investigations, but the effects of ADC and DAC non-idealities upon the properties of quantization noise have not been deeply *
Corresponding author. Tel.: +39 075 5853933; fax: +39 075
E-mail addresses: [email protected] Moschitta), [email protected] (D. Petri). 1 Tel.: +39 0461 883902; fax: +39 0461 882093.
(A.
investigated yet [1]. In particular, an additive, white, and uniformly distributed quantization noise is usually considered, whose power does not depend on the statistical properties of the input signal. However, such assumptions are not verified when the converter is affected by INL, which may both change the noise power and introduce harmonics in the noise spectrum. Thus, modeling the effects of INL on the quantization error is of interest, especially when highly accurate A/D conversion is required, as for example, in digital measurement intrumentation, in digital receivers for telecommunications, or when the A/D converter is used to characterize a device and its nonlinearities. This paper is focused on the effects of INL on the noise power of a memoryless A/D converter, fed with various kinds of stochastic signals. An exact model is discussed, which describes the quantization noise power as a function of both the input signal
0263-2241/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.measurement.2006.09.002
A. Moschitta, D. Petri / Measurement 41 (2008) 186–191
probability density function (pdf) and the transition levels of the quantizer [2,3]. The model is then extended to evaluate the effects of INL and input signal statistical properties. In particular, the INL effect is modeled as an additive contribution, extending the results presented in [4]. It is worth of notice that the noise power of a quantizer affected by INL may noticeably depend on the amplitude of the quantizer stimulus. As an exact model may lead to very complicated expressions, a simplified formula has been derived, which accurately describes the effects of INL on the noise power when the stimulus covers enough quantizer levels. This model has been applied to various stochastic stimuli, showing a very good agreement with simulation results. In particular, uniform, Gaussian and noisy sinewave inputs have been considered.
fe ðeÞ ¼
ð1Þ
Ak ¼ ½y k1 sk ; y k1 sk1 ; where sk is the kth quantizer decision threshold, yk is the kth output level, fx(Æ) is the input signal pdf, and i(Æ) is the indicator function [2]. Thus, quantization noise power can be evaluated according to: Z 1 2 re ¼ e2 fe ðeÞ de; ð2Þ 1
which, using (1), leads to the following: Z y k1 sk1 1 X e2 fx ðy k1 eÞ de: r2e ¼ k¼1
ð3Þ
y k1 sk
Fig. 1(a,b), obtained for an ideal uniform quantizer, report quantization noise power as a function of signal standard deviation rIN, normalized to the quantization step D, for uniform and Gaussian input signals respectively. The error power is normalized to the error power r20 of an uniform ADC fed with an uniformly distributed stimulus. As it is well known, with very high accuracy we can assume r20 ffi D2 /12 [1]. In all the considered situations the model (3) shows a good agreement with simulation results.
2.1. Quantizer model The theoretical model, whose detailed derivation is shown in Appendix A, has been obtained by assuming that the quantizer thresholds define a partition of the real axis, and by evaluating the conditional error pdf in each of the partition subsets. In particular, quantizers with infinite decision thresholds and quantization levels are considered. Notice that such an approach does not need any particular hypothesis about the threshold positioning, except for their monotonicity, and can be applied indifferently to uniform or non-uniform converters. In particular, it can be shown that quantization noise pdf can be expressed as
2.2. Uniform quantizers affected by integral non-linearity The presented model can easily keep into account INL, by replacing the ideal decision threshold values in (3) with the ones affected by INL. In particular, the quantizer thresholds may be expressed as follows:
2.5
2.5
2
2
2 2 σ e/(Δ /12)
σ e/(Δ /12)
1.5
1.5
2
2
fx ðy k1 eÞ iðAk Þ;
k¼1
2. Analysis results
1
0.5
1 X
187
1
0
2
4
6
8
10
σ in/Δ
12
14
16
18
20
0.5
0
2
4
6
8
10
12
14
16
18
20
σ in/Δ
Fig. 1. Quantization noise power, normalized to D2/12, obtained in absence of INL for an uniformly distributed input signal (a) and for a Gaussian distributed one (b). Dots are simulation results, while the continuous line is obtained by means of the theoretical model (3).
188
A. Moschitta, D. Petri / Measurement 41 (2008) 186–191
sk ¼ s0k þ INLk ;
ð4Þ
where erf(Æ) is the error function [5]. For a stimulus uniformly distributed in [A,A], we have
where s0k is the kth ideal threshold and INLk is the offset caused by INL. Notice that this INL definition slightly differs from the one reported in [7,8], where INL is expressed as a percentage of the ADC full scale range. By using (4), Eq. (3) can be further refined, obtaining an equivalent form in which the ideal quantizer and the INL contributions to the noise power appear as two distinct additive terms. In fact, by assuming that jINLkj < D/2 in order to ensure the monotonicity, the properties of the integral operator allow to obtain the following (see Appendix B): r2e ¼ r20 þ dINL ; 1 X dINL ¼ 2D ek ; ek ¼
Z
1 ið½A; AÞ; 2A 1 b2k a2k ek ¼ s0k ðbk ak Þ ; 2A 2
fx ðxÞ ¼
½ak ; bk ¼ ½s0k ; s0k þ INLk \ ½A; A
where \ is the intersection operator. Finally, for a sinusoidal stimulus of amplitude A, it results 1 fx ðxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ið A; A½Þ; p A2 x 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 1 k 2 2 ðA ak Þ þ s0k arcsin ek ¼ p A qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 b 2 2 ðA bk Þ þ s0k arcsin k ; p A
ð5Þ
k¼1 s0k þINLk
ð5cÞ
½ak ; bk ¼ ½s0k ; s0k þ INLk \ A; A½
ðx s0k Þfx ðxÞ dx
Fig. 2(a,b) report the noise power curves obtained in presence of INL for the considered input signals, normalized to D2/12, as a function of rIN/D. In particular, both stimuli have been applied to an ADC affected by deterministic INL, where each INLk has been taken from a set of values uniformly distributed between D/2 and D/2. Again, the model shows a good agreement with simulation results. While the theoretical model provide very accurate results, it’s derivation and application may lead to complicated closed form expressions, depending on the expression of the input signal pdf. This may easily happen in practical applications, where the quantizer stimulus is usually distorted or corrupted by noise. A typical case is the usage of dither
s0k
r20
where N is the number of quantizer levels, is the noise power generated in absence of INL, and dINL is the INL contribution to quantization noise power. This term has been evaluated for uniform, sinusoidal and Gaussian input signals. In particular, by assuming a Gaussian stimulus of variance r2, after some algebraic manipulations it results: 2 1 x fx ðxÞ ¼ pffiffiffiffiffiffi e 2r2 ; r 2p s2 ðs þINL Þ2 r 0k 0k 2 k 2r ek ¼ pffiffiffiffiffiffi e 2r2 e 2p s s0k þ INLk 0k s0k erf erf ; r r
ð5aÞ
2.5
2
2
2 2 σ e/(Δ /12)
2 2 σ e/(Δ /12)
2.5
1.5
1.5
1
1
0.5
ð5bÞ
0.5 0
2
4
6
8
10
σ in/Δ
12
14
16
18
20
0
2
4
6
8
10
12
14
16
18
20
σ in Δ
Fig. 2. Quantization noise power, normalized to D2/12, obtained in presence of INL for an uniformly distributed input signal (a) and for a Gaussian distributed one (b). Dots are simulation results, while the continuous line is obtained by means of the theoretical model (5).
A. Moschitta, D. Petri / Measurement 41 (2008) 186–191 2.5
2
2
2 2 σ e/(Δ /12)
2 2 σ e/(Δ /12)
2.5
1.5
1.5
1
0.5
189
1
0
2
4
6
8
10
12
14
16
18
σ in/Δ
20
0.5
0
2
4
6
8
10
12
14
16
18
20
σ in/Δ
Fig. 3. Quantization noise power, normalized to D2/12, obtained in presence of INL for an uniformly distributed input signal (a) and for a Gaussian distributed one (b). Dots are simulation results, while the continuous line is obtained by means of the simplified theoretical model (7).
[4,6]. To this aim, by assuming that the input signal pdf fx(Æ) is almost constant in [s0k, s0k + INLk], a simplified model results, expressed as follows: 1 X
dINL ffi D
fx ðs0k Þ INL2k :
ð6Þ
k¼1
Then, by substituting Eq. (6) in Eq. (5), the noise power of a quantizer affected by INL may be expressed as r2e ffi
1 X D2 þD fx ðs0k Þ INL2k : 12 k¼1
ð7Þ
It should be noticed that for an uniformly distributed input signal whose dynamic range equals the ADC one, (5) and (6) reduce to the INL contribution reported in Eq. (A.8) of [4]. Fig. 3(a,b) show the results obtained by using (7) for both uniform 2.5
2 2 σ e/(Δ /12)
2
1.5
1
0.5
0
2
4
6
8
10
12
14
16
18
20
σ in/Δ
Fig. 4. Quantization noise power, normalized to D2/12, obtained in presence of INL for an input signal consisting in a sinewave affected by an additive dither, uniformly distributed in [D/2, D/2]. Dots are simulation results, while the continuous line is obtained by means of the simplified theoretical model (7).
and Gaussian input signals. It can be seen that as far as the input signal excites a few quantizer levels, Eq. (7) shows a good agreement with simulation results. In order to analyze another situation of practical p interest, a sinewave signal of amplitude A = rIN 2, affected by an additive dither uniformly distributed in [D/2, D/2], has been considered. Fig. 4 reports both simulation results and the values returned by Eq. (7), in which the signal pdf derived in [6] has been used. It can be seen that also in this case the simplified model (7) provide a very good accuracy. 3. Conclusions A theoretical exact model has been presented, which describes the quantization noise distribution and power of memoryless A/D converters as a function of both threshold spacing and input signal pdf. The model has been extended to keep into account the INL contribution to quantization noise power, which has been expressed as an additive term, and may be used for characterizing the performance of highly accurate A/D converters. A simplified model has been also proposed, which provides good results as far as the input signal covers a few quantizer steps. The model has been applied to various input signals, including a noisy sinewave. Appendix A Quantizer error model Let us assume that sk is the kth quantizer decision threshold, and yk is the kth output level, such that sk < skþ1 ;
y k < y kþ1 ;
sk < y k < skþ1
190
A. Moschitta, D. Petri / Measurement 41 (2008) 186–191
The quantizer decision thresholds define a partition of the real axis, given by [ X k; X k ¼sk1 ; sk : ðA:1Þ R¼ k
The quantizer error can be obtained as the difference between the quantizer input x, which may be modeled as a random variable with pdf fx(x), and the quantizer output y. By keeping in mind that the value of y depends on which decision interval Xk the input x belongs to, and by defining xjXk as the input x conditioned to Xk, that is, x such that x 2 Xk, it results: ejX k ¼ xjX k y k1 ;
ðA:2Þ
where ejXk, is the quantizer error conditioned to Xk. Hence, the error pdf, conditioned to Xk, can be obtained as ( fx ðy k1 ejX k Þ ; ejX k 2 Ak P ðX k Þ fejX k ðejX k Þ ¼ ðA:3Þ 0; elsewhere Ak ¼ ½y k1 sk ; y k1 sk1 ½ where P(Xk) is the probability that x 2 Xk. The unconditioned error pdf fe(e) can be obtained as [5] fe ðeÞ ¼
þ1 X
fejX k ðeÞP ðX k ÞiðAk Þ;
ðA:4Þ
k¼1
where i(Æ) is the indicator function, which equals 1 if e 2 Ak and equals 0 if e 62 Ak. By inserting (A.3) in (A.4), (1) results. In particular, if the considered quantizer is uniform, it results that Ak = [D/2, D/2[, regardless of k. Hence, (1) reduces to 8 1 < P f ðy eÞ; jej < D=2 x k1 fe ðeÞ ¼ k¼1 ðA:5Þ : 0; elsewhere Appendix B
eðxÞ ¼ e0 ðxÞ þ eINL ðxÞ;
Let us express the quantizer thresholds affected by INL as ðB:1Þ
where s0k is the k-th ideal threshold and INLk is the offset caused by INL. Thus, the effect of INL is to change the decision intervals Xk, which may cause the quantizer to produce incorrect output levels.
ðB:2Þ
where, 1 X
eINL ðxÞ ¼
DsignðINLk ÞiðI k Þ;
ðB:3Þ
ðx s0k D=2ÞiðX kþ1 Þ
ðB:4Þ
k¼1
e0 ðxÞ ¼
1 X k¼1
where x is the input signal, e0(Æ) is the quantization error of a quantizer not affected by INL, eINL(Æ) is the INL contribution to the error, sign(Æ) is the sign function and Ik = [min(sk, s0k), max(sk, s0k)] is the input interval where INL causes the quantizer to produce an incorrect output level. In fact, when INLk is positive, if x belongs to [s0k, sk], the quantizer output equals yk1, rather than the correct value yk. Conversely, when INLk is negative, the quantizer output equals yk, rather than the correct value yk1, only if x belongs to [sk, s0k]. Eq. (B.3) shows that the sign of eINL(Æ) is always opposite to the sign of e0(Æ), and its magnitude is always D. The error power can be evaluated according to (2), which, by applying a change of variable, can be expressed in terms of the input x, obtaining Z skþ1 1 X 2 2 re ¼ ðe0 ðxÞ þ eINL ðxÞÞ fx ðxÞ dx ðB:5Þ k¼1
sk
Eq. (B.5) can be expanded into r2e ¼ r2e0 þ r2INL þ C INL;
ðB:6Þ
where r2e0 is the error power of an ideal quantizer, given by (3), r2INL is the power of the INL error, given by Z 1 X 2 eINL ðxÞ fx ðxÞ dx r2INL ¼ k¼1
Effects of integral non-linearity on quantizer error power
sk ¼ s0k þ INLk ;
By assuming that jINLkj < D/2, it can be shown that when INL is introduced, the quantization error can be expressed as
¼ D2
Ik
1 X
P ðI k Þ;
ðB:7Þ
k¼1
and CINL is a cross product term between the ideal error and the INL error, expressed as Z 1 X e0 ðxÞeINL ðxÞfx ðxÞ dx ðB:8Þ C INL ¼ 2 k¼1
Ik
This term can be further developed by substituting the expressions for e0(Æ) and eINL(Æ), thus obtaining
A. Moschitta, D. Petri / Measurement 41 (2008) 186–191
C INL ¼ 2D
Z 1 X k¼1
ðx s0k D=2Þ
Ik
signðINLk Þfx ðxÞ dx
ðB:9Þ
which can be expanded into 1 X C INL ¼ D2 P ðI k Þ k¼1 1 X
þ 2D
k¼1
Z
s0k þINLk
ðx s0k Þfx ðxÞ dx
ðB:10Þ
s0k
where the properties of Ik and the sign function have been used to simplify the integral in (B.9). Finally, by inserting (B.7)and (B.10) in (B.6), the first addendum of the right part of (B.10) cancels r2INL , and Eq. (5) is obtained. References [1] Gray, Quantization noise spectra, IEEE Transactions on Information Theory 36 (6) (1990).
191
[2] N.S. Jayant, P. Noll, Digital Coding of Waveforms, Prentice Hall, 1984. [3] F.H. Irons, K.J. Riley, D.M. Hummels, G.A. Friel, The Noise Power Ratio, IEEE Transactions on Instrumentation and Measurement 49 (3) (1999). [4] P. Carbone, D. Petri, Noise sensitivity of the ADC histogram test, IEEE Transactions on Instrumentation and Measurement 47 (4) (1998). [5] A. Papoulis, ‘‘Probability, Random Variables, and Stochastic Processes’’, McGraw Hill International Editions, 1991. [6] M.F. Wagdy, N.G. Wai-Man, Validity of uniform quantization error model for sinusoidal signals without and with dither, IEEE Transactions on Instrumentation and Measurement 38 (3) (1989). [7] IEEE standard 1241-2000, ‘‘IEEE Standard for Terminology and Test Methods for Analog-to-Digital Converters’’. [8] IEEE standard 1057-1994, ‘‘IEEE Standard for Digitizing Waveform Recorders’’.
Measurement 41 (2008) 186–191 www.elsevier.com/locate/measurement
Integral non-linearity in memoryless A/D converters A. Moschitta a
a,*
, D. Petri
b,1
DIEI – Department of Information and Electronic Engineering, University of Perugia, Perugia, Italy b DIT – Department of Informatics and Telecommunications, University of Trento, Trento, Italy Received 23 January 2006; received in revised form 19 July 2006; accepted 4 September 2006 Available online 16 September 2006
Abstract This paper investigates the statistical properties of quantization noise. In particular, a theoretical model is discussed, which evaluates the power of quantization noise introduced by a memoryless analog to digital converter (ADC) as a function of both the converted signal distribution and the ADC thresholds positioning. Expressions have also been derived to express the integral non-linearity (INL) contribution to quantization noise power as an additive term, and to evaluate such a term with a simple formula. Simulation results that validate the proposed expression are provided. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Quantization noise; Integral non-linearity
1. Introduction Analog to digital (A/D) and digital to analog (D/ A) converters are widely used in many modern fields of application, allowing to replace analog systems with digital high performance implementations. Modeling the behavior of A/D and D/A converters is important both for characterizing the performance of the converters themselves and for embedding such devices in real systems. The matter has been subject of several investigations, but the effects of ADC and DAC non-idealities upon the properties of quantization noise have not been deeply *
Corresponding author. Tel.: +39 075 5853933; fax: +39 075
E-mail addresses: [email protected] Moschitta), [email protected] (D. Petri). 1 Tel.: +39 0461 883902; fax: +39 0461 882093.
(A.
investigated yet [1]. In particular, an additive, white, and uniformly distributed quantization noise is usually considered, whose power does not depend on the statistical properties of the input signal. However, such assumptions are not verified when the converter is affected by INL, which may both change the noise power and introduce harmonics in the noise spectrum. Thus, modeling the effects of INL on the quantization error is of interest, especially when highly accurate A/D conversion is required, as for example, in digital measurement intrumentation, in digital receivers for telecommunications, or when the A/D converter is used to characterize a device and its nonlinearities. This paper is focused on the effects of INL on the noise power of a memoryless A/D converter, fed with various kinds of stochastic signals. An exact model is discussed, which describes the quantization noise power as a function of both the input signal
0263-2241/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.measurement.2006.09.002
A. Moschitta, D. Petri / Measurement 41 (2008) 186–191
probability density function (pdf) and the transition levels of the quantizer [2,3]. The model is then extended to evaluate the effects of INL and input signal statistical properties. In particular, the INL effect is modeled as an additive contribution, extending the results presented in [4]. It is worth of notice that the noise power of a quantizer affected by INL may noticeably depend on the amplitude of the quantizer stimulus. As an exact model may lead to very complicated expressions, a simplified formula has been derived, which accurately describes the effects of INL on the noise power when the stimulus covers enough quantizer levels. This model has been applied to various stochastic stimuli, showing a very good agreement with simulation results. In particular, uniform, Gaussian and noisy sinewave inputs have been considered.
fe ðeÞ ¼
ð1Þ
Ak ¼ ½y k1 sk ; y k1 sk1 ; where sk is the kth quantizer decision threshold, yk is the kth output level, fx(Æ) is the input signal pdf, and i(Æ) is the indicator function [2]. Thus, quantization noise power can be evaluated according to: Z 1 2 re ¼ e2 fe ðeÞ de; ð2Þ 1
which, using (1), leads to the following: Z y k1 sk1 1 X e2 fx ðy k1 eÞ de: r2e ¼ k¼1
ð3Þ
y k1 sk
Fig. 1(a,b), obtained for an ideal uniform quantizer, report quantization noise power as a function of signal standard deviation rIN, normalized to the quantization step D, for uniform and Gaussian input signals respectively. The error power is normalized to the error power r20 of an uniform ADC fed with an uniformly distributed stimulus. As it is well known, with very high accuracy we can assume r20 ffi D2 /12 [1]. In all the considered situations the model (3) shows a good agreement with simulation results.
2.1. Quantizer model The theoretical model, whose detailed derivation is shown in Appendix A, has been obtained by assuming that the quantizer thresholds define a partition of the real axis, and by evaluating the conditional error pdf in each of the partition subsets. In particular, quantizers with infinite decision thresholds and quantization levels are considered. Notice that such an approach does not need any particular hypothesis about the threshold positioning, except for their monotonicity, and can be applied indifferently to uniform or non-uniform converters. In particular, it can be shown that quantization noise pdf can be expressed as
2.2. Uniform quantizers affected by integral non-linearity The presented model can easily keep into account INL, by replacing the ideal decision threshold values in (3) with the ones affected by INL. In particular, the quantizer thresholds may be expressed as follows:
2.5
2.5
2
2
2 2 σ e/(Δ /12)
σ e/(Δ /12)
1.5
1.5
2
2
fx ðy k1 eÞ iðAk Þ;
k¼1
2. Analysis results
1
0.5
1 X
187
1
0
2
4
6
8
10
σ in/Δ
12
14
16
18
20
0.5
0
2
4
6
8
10
12
14
16
18
20
σ in/Δ
Fig. 1. Quantization noise power, normalized to D2/12, obtained in absence of INL for an uniformly distributed input signal (a) and for a Gaussian distributed one (b). Dots are simulation results, while the continuous line is obtained by means of the theoretical model (3).
188
A. Moschitta, D. Petri / Measurement 41 (2008) 186–191
sk ¼ s0k þ INLk ;
ð4Þ
where erf(Æ) is the error function [5]. For a stimulus uniformly distributed in [A,A], we have
where s0k is the kth ideal threshold and INLk is the offset caused by INL. Notice that this INL definition slightly differs from the one reported in [7,8], where INL is expressed as a percentage of the ADC full scale range. By using (4), Eq. (3) can be further refined, obtaining an equivalent form in which the ideal quantizer and the INL contributions to the noise power appear as two distinct additive terms. In fact, by assuming that jINLkj < D/2 in order to ensure the monotonicity, the properties of the integral operator allow to obtain the following (see Appendix B): r2e ¼ r20 þ dINL ; 1 X dINL ¼ 2D ek ; ek ¼
Z
1 ið½A; AÞ; 2A 1 b2k a2k ek ¼ s0k ðbk ak Þ ; 2A 2
fx ðxÞ ¼
½ak ; bk ¼ ½s0k ; s0k þ INLk \ ½A; A
where \ is the intersection operator. Finally, for a sinusoidal stimulus of amplitude A, it results 1 fx ðxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ið A; A½Þ; p A2 x 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 1 k 2 2 ðA ak Þ þ s0k arcsin ek ¼ p A qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 b 2 2 ðA bk Þ þ s0k arcsin k ; p A
ð5Þ
k¼1 s0k þINLk
ð5cÞ
½ak ; bk ¼ ½s0k ; s0k þ INLk \ A; A½
ðx s0k Þfx ðxÞ dx
Fig. 2(a,b) report the noise power curves obtained in presence of INL for the considered input signals, normalized to D2/12, as a function of rIN/D. In particular, both stimuli have been applied to an ADC affected by deterministic INL, where each INLk has been taken from a set of values uniformly distributed between D/2 and D/2. Again, the model shows a good agreement with simulation results. While the theoretical model provide very accurate results, it’s derivation and application may lead to complicated closed form expressions, depending on the expression of the input signal pdf. This may easily happen in practical applications, where the quantizer stimulus is usually distorted or corrupted by noise. A typical case is the usage of dither
s0k
r20
where N is the number of quantizer levels, is the noise power generated in absence of INL, and dINL is the INL contribution to quantization noise power. This term has been evaluated for uniform, sinusoidal and Gaussian input signals. In particular, by assuming a Gaussian stimulus of variance r2, after some algebraic manipulations it results: 2 1 x fx ðxÞ ¼ pffiffiffiffiffiffi e 2r2 ; r 2p s2 ðs þINL Þ2 r 0k 0k 2 k 2r ek ¼ pffiffiffiffiffiffi e 2r2 e 2p s s0k þ INLk 0k s0k erf erf ; r r
ð5aÞ
2.5
2
2
2 2 σ e/(Δ /12)
2 2 σ e/(Δ /12)
2.5
1.5
1.5
1
1
0.5
ð5bÞ
0.5 0
2
4
6
8
10
σ in/Δ
12
14
16
18
20
0
2
4
6
8
10
12
14
16
18
20
σ in Δ
Fig. 2. Quantization noise power, normalized to D2/12, obtained in presence of INL for an uniformly distributed input signal (a) and for a Gaussian distributed one (b). Dots are simulation results, while the continuous line is obtained by means of the theoretical model (5).
A. Moschitta, D. Petri / Measurement 41 (2008) 186–191 2.5
2
2
2 2 σ e/(Δ /12)
2 2 σ e/(Δ /12)
2.5
1.5
1.5
1
0.5
189
1
0
2
4
6
8
10
12
14
16
18
σ in/Δ
20
0.5
0
2
4
6
8
10
12
14
16
18
20
σ in/Δ
Fig. 3. Quantization noise power, normalized to D2/12, obtained in presence of INL for an uniformly distributed input signal (a) and for a Gaussian distributed one (b). Dots are simulation results, while the continuous line is obtained by means of the simplified theoretical model (7).
[4,6]. To this aim, by assuming that the input signal pdf fx(Æ) is almost constant in [s0k, s0k + INLk], a simplified model results, expressed as follows: 1 X
dINL ffi D
fx ðs0k Þ INL2k :
ð6Þ
k¼1
Then, by substituting Eq. (6) in Eq. (5), the noise power of a quantizer affected by INL may be expressed as r2e ffi
1 X D2 þD fx ðs0k Þ INL2k : 12 k¼1
ð7Þ
It should be noticed that for an uniformly distributed input signal whose dynamic range equals the ADC one, (5) and (6) reduce to the INL contribution reported in Eq. (A.8) of [4]. Fig. 3(a,b) show the results obtained by using (7) for both uniform 2.5
2 2 σ e/(Δ /12)
2
1.5
1
0.5
0
2
4
6
8
10
12
14
16
18
20
σ in/Δ
Fig. 4. Quantization noise power, normalized to D2/12, obtained in presence of INL for an input signal consisting in a sinewave affected by an additive dither, uniformly distributed in [D/2, D/2]. Dots are simulation results, while the continuous line is obtained by means of the simplified theoretical model (7).
and Gaussian input signals. It can be seen that as far as the input signal excites a few quantizer levels, Eq. (7) shows a good agreement with simulation results. In order to analyze another situation of practical p interest, a sinewave signal of amplitude A = rIN 2, affected by an additive dither uniformly distributed in [D/2, D/2], has been considered. Fig. 4 reports both simulation results and the values returned by Eq. (7), in which the signal pdf derived in [6] has been used. It can be seen that also in this case the simplified model (7) provide a very good accuracy. 3. Conclusions A theoretical exact model has been presented, which describes the quantization noise distribution and power of memoryless A/D converters as a function of both threshold spacing and input signal pdf. The model has been extended to keep into account the INL contribution to quantization noise power, which has been expressed as an additive term, and may be used for characterizing the performance of highly accurate A/D converters. A simplified model has been also proposed, which provides good results as far as the input signal covers a few quantizer steps. The model has been applied to various input signals, including a noisy sinewave. Appendix A Quantizer error model Let us assume that sk is the kth quantizer decision threshold, and yk is the kth output level, such that sk < skþ1 ;
y k < y kþ1 ;
sk < y k < skþ1
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A. Moschitta, D. Petri / Measurement 41 (2008) 186–191
The quantizer decision thresholds define a partition of the real axis, given by [ X k; X k ¼sk1 ; sk : ðA:1Þ R¼ k
The quantizer error can be obtained as the difference between the quantizer input x, which may be modeled as a random variable with pdf fx(x), and the quantizer output y. By keeping in mind that the value of y depends on which decision interval Xk the input x belongs to, and by defining xjXk as the input x conditioned to Xk, that is, x such that x 2 Xk, it results: ejX k ¼ xjX k y k1 ;
ðA:2Þ
where ejXk, is the quantizer error conditioned to Xk. Hence, the error pdf, conditioned to Xk, can be obtained as ( fx ðy k1 ejX k Þ ; ejX k 2 Ak P ðX k Þ fejX k ðejX k Þ ¼ ðA:3Þ 0; elsewhere Ak ¼ ½y k1 sk ; y k1 sk1 ½ where P(Xk) is the probability that x 2 Xk. The unconditioned error pdf fe(e) can be obtained as [5] fe ðeÞ ¼
þ1 X
fejX k ðeÞP ðX k ÞiðAk Þ;
ðA:4Þ
k¼1
where i(Æ) is the indicator function, which equals 1 if e 2 Ak and equals 0 if e 62 Ak. By inserting (A.3) in (A.4), (1) results. In particular, if the considered quantizer is uniform, it results that Ak = [D/2, D/2[, regardless of k. Hence, (1) reduces to 8 1 < P f ðy eÞ; jej < D=2 x k1 fe ðeÞ ¼ k¼1 ðA:5Þ : 0; elsewhere Appendix B
eðxÞ ¼ e0 ðxÞ þ eINL ðxÞ;
Let us express the quantizer thresholds affected by INL as ðB:1Þ
where s0k is the k-th ideal threshold and INLk is the offset caused by INL. Thus, the effect of INL is to change the decision intervals Xk, which may cause the quantizer to produce incorrect output levels.
ðB:2Þ
where, 1 X
eINL ðxÞ ¼
DsignðINLk ÞiðI k Þ;
ðB:3Þ
ðx s0k D=2ÞiðX kþ1 Þ
ðB:4Þ
k¼1
e0 ðxÞ ¼
1 X k¼1
where x is the input signal, e0(Æ) is the quantization error of a quantizer not affected by INL, eINL(Æ) is the INL contribution to the error, sign(Æ) is the sign function and Ik = [min(sk, s0k), max(sk, s0k)] is the input interval where INL causes the quantizer to produce an incorrect output level. In fact, when INLk is positive, if x belongs to [s0k, sk], the quantizer output equals yk1, rather than the correct value yk. Conversely, when INLk is negative, the quantizer output equals yk, rather than the correct value yk1, only if x belongs to [sk, s0k]. Eq. (B.3) shows that the sign of eINL(Æ) is always opposite to the sign of e0(Æ), and its magnitude is always D. The error power can be evaluated according to (2), which, by applying a change of variable, can be expressed in terms of the input x, obtaining Z skþ1 1 X 2 2 re ¼ ðe0 ðxÞ þ eINL ðxÞÞ fx ðxÞ dx ðB:5Þ k¼1
sk
Eq. (B.5) can be expanded into r2e ¼ r2e0 þ r2INL þ C INL;
ðB:6Þ
where r2e0 is the error power of an ideal quantizer, given by (3), r2INL is the power of the INL error, given by Z 1 X 2 eINL ðxÞ fx ðxÞ dx r2INL ¼ k¼1
Effects of integral non-linearity on quantizer error power
sk ¼ s0k þ INLk ;
By assuming that jINLkj < D/2, it can be shown that when INL is introduced, the quantization error can be expressed as
¼ D2
Ik
1 X
P ðI k Þ;
ðB:7Þ
k¼1
and CINL is a cross product term between the ideal error and the INL error, expressed as Z 1 X e0 ðxÞeINL ðxÞfx ðxÞ dx ðB:8Þ C INL ¼ 2 k¼1
Ik
This term can be further developed by substituting the expressions for e0(Æ) and eINL(Æ), thus obtaining
A. Moschitta, D. Petri / Measurement 41 (2008) 186–191
C INL ¼ 2D
Z 1 X k¼1
ðx s0k D=2Þ
Ik
signðINLk Þfx ðxÞ dx
ðB:9Þ
which can be expanded into 1 X C INL ¼ D2 P ðI k Þ k¼1 1 X
þ 2D
k¼1
Z
s0k þINLk
ðx s0k Þfx ðxÞ dx
ðB:10Þ
s0k
where the properties of Ik and the sign function have been used to simplify the integral in (B.9). Finally, by inserting (B.7)and (B.10) in (B.6), the first addendum of the right part of (B.10) cancels r2INL , and Eq. (5) is obtained. References [1] Gray, Quantization noise spectra, IEEE Transactions on Information Theory 36 (6) (1990).
191
[2] N.S. Jayant, P. Noll, Digital Coding of Waveforms, Prentice Hall, 1984. [3] F.H. Irons, K.J. Riley, D.M. Hummels, G.A. Friel, The Noise Power Ratio, IEEE Transactions on Instrumentation and Measurement 49 (3) (1999). [4] P. Carbone, D. Petri, Noise sensitivity of the ADC histogram test, IEEE Transactions on Instrumentation and Measurement 47 (4) (1998). [5] A. Papoulis, ‘‘Probability, Random Variables, and Stochastic Processes’’, McGraw Hill International Editions, 1991. [6] M.F. Wagdy, N.G. Wai-Man, Validity of uniform quantization error model for sinusoidal signals without and with dither, IEEE Transactions on Instrumentation and Measurement 38 (3) (1989). [7] IEEE standard 1241-2000, ‘‘IEEE Standard for Terminology and Test Methods for Analog-to-Digital Converters’’. [8] IEEE standard 1057-1994, ‘‘IEEE Standard for Digitizing Waveform Recorders’’.