H-ADC device

Measurement 25 (1999) 265–283 www.elsevier.com / locate / measurement

Finite memory non-linear model of a S / H-ADC device a, a b c Domenico Mirri *, Gaetano Pasini , Fabio Filicori , Gaetano Iuculano , Gabriella Pellegrini d a

Department of Electrical Engineering, Viale Risorgimento 2, 40136 Bologna, Italy b Department of Electronics, Via S. Marta 2, 50125 Firenze, Italy c Department of Electronics, Facolta` di Ingegneria, Viale Risorgimento 2, 40136 Bologna, Italy d Department of Mathematics, Via S. Marta 2, 50125 Firenze, Italy Received 19 August 1998; received in revised form 15 January 1999; accepted 25 January 1999

Abstract A new model has been proposed for the characterisation of the non-linear dynamic effects of a S / H-ADC device and a new mathematical approach has been introduced to describe its non-ideal dynamic behaviour. The measurement procedure to deduce the parameters of the proposed model is given together with the experimental results on a commercial device.  1999 Elsevier Science Ltd. All rights reserved. Keywords: S / H-ADC device; Non-linear dynamic effects; Mathematical approach

1. Introduction Any digital instrument for continuous-time varying signals must convert the input signal into a sequence of numbers that are processed digitally; this is accomplished through two successive basic steps: the signal is sampled through a sample and hold circuit (S / H) and the amplitude of each sampled value is quantized with an analog to digital converter (ADC). Therefore, the characterisation of the S / HADC device is a relevant problem that must be faced in order to evaluate the performance of the digital instruments; in particular this paper investigates the non-linear dynamic effects in a S / H-ADC device. According to previous studies of the authors [1,2], a *Corresponding author. Tel.: 139-51-644-3472; fax: 139-51644-3470. E-mail address: [email protected] (D. Mirri)

suitable black-box non-linear dynamic model of the S / H-ADC is outlined in Section 2. In this model the non-linear dynamic effects are taken into account through a continuous-time system which can be mathematically described through a modified Volterra integral series previously introduced by the authors [3,4]. This approach, like the one proposed in Refs. [5,6], has the main goal of modelling the non-linear dynamic effects in the S / H-ADC devices. However, while in Refs. [5,6] an empirical characterisation procedure is proposed which directly leads to a real time error correction algorithm, our main goal is to define a general-purpose mathematical model, based on the rigorous Volterra approach. This model provides a complete characterisation of the non-linear dynamic behaviour of the S / H-ADC devices and can be used for error prediction purposes or, possibly, for off-line error correction algorithms. The model identification procedure to evaluate the

0263-2241 / 99 / $ – see front matter  1999 Elsevier Science Ltd. All rights reserved. PII: S0263-2241( 99 )00011-1

266

D. Mirri et al. / Measurement 25 (1999) 265 – 283

parameters of the proposed model is described in Section 3 and the experimental results on a commercial S / H-ADC device are given in Section 4.

2. Finite-memory non-linear integral model of a S / H-ADC device Fig. 1 shows the block diagram of a data acquisition system, i.e. a S / H circuit cascaded by an m-bit ADC; sstd is the input signal, vCstd the clock signal which controls the two operation modes of the S / H and u q f k g the discrete-time discrete-amplitude sequence at the output of the ADC, sequence which is memorised in the latch. An ideal ADC device is characterised by a non-linear operation due to quantization which can be considered an intrinsic, ‘a priori’ known, error source. A practical ADC exhibits however additional error sources; therefore it can be represented by an ideal m-bit ADC cascaded to a separate non-linear memoryless device which char-

acterises its non-idealities (Fig. 2). The S / H circuit, when described as a system with two inputs, to which the input signal sstd and the clock signal vCstd are applied (Fig. 1), can be assumed to be timeinvariant. Several errors are associated to this circuit. A sampling command in the instant t k9 becomes effective at the instant t k 5 t k9 1 d due to the aperture delay d. This quantity is also affected by the uncertainty due to time-jitter and, in the block diagram in Fig. 2, is described by an equivalent delay block. Moreover, the S / H is characterised both by nonlinearity and memory effects which influence the output also after the current sampling instant t k , so that the system is virtually anticipative. More precisely, since the clock signal vCstd is completely characterised 1 by the sampling frequency fs , the generic kth output value u f k g of the S / H-ADC 1

In the following we will assume that signal sampling is carried out at a constant sampling rate fs and that the behaviour of the clock generator is practically ideal (e.g. jitter is neglected).

Fig. 1. Block diagram of a data acquisition system.

Fig. 2. Block diagram of a S / H-ADC device in which are separately considered the non-idealities of the ADC and the delay in the sampling instants.

D. Mirri et al. / Measurement 25 (1999) 265 – 283

267

Fig. 3. Functional model used to characterise the non-linear dynamic effects of a S / H-ADC device.

system non-linearly depends on both fs and all the values of the input signal sst k 2 td over a timeinterval defined by 2 t 1 # t # t 2 . This functional dependence on sst k 2 td, by adopting the original symbolism 2 introduced by Volterra [7], can be expressed in the form: t2

u[k] 5 lim F2 u[s(t k 2t ); fs ]u t 1 →`

t1

(1)

t 2 →`

where the lim has been introduced since, in real t 1 →` t 2 →`

systems, the ‘memory’ effects are practically ‘vanishing’ for large values of t but not strictly limited in time. This allows for a simpler description of nonideal effects in the S / H-ADC device. In fact, Eq. (1) can also be interpreted as the ‘ideal’ sampling (i.e. carried out by an ideal S / H device) of the output of an equivalent continuous time system whose output is given by: t2

u(t) 5 lim F2 u[s(t 2t ); fs ]u t 1 →`

t1

(2)

t 2 →`

as shown in the block diagram in Fig. 3. The output sequence u f k g can be therefore considered the result of an ideal sampling of a continuous-time signal ustd 2

According to the original Volterra symbolism, Fu[x(t TTAB )]u represents a non-linear ‘line’ function (i.e. a functional) since the value of F depends on all the values of x in the interval TA # t # TB . It should not be confused with an algebraic function of the absolute value of its argument.

at the output of a non-linear system with memory. It takes into account the non-idealities both of the S / H and the ADC [1,2], apart from the delay in the sampling instant. When the sampling rate fs is not too large in relation to the system dynamics, it can be assumed that the memory effects associated with any sampling instant do not influence the remaining ones (when the sampling frequency fs is not too large, or a random sampling strategy is used), the functionals of Eqs. (1) and (2) become independent from fs . Thus in the following, for simplicity of presentation, we will neglect the dependence on fs . However, all the results presented can be easily extended also to operating conditions where dependence on fs is not negligible. Mathematical modelling and identification of a non-linear dynamic system is very complex and therefore presents a number of challenges [8]. The proposed approach uses a modified Volterra series still based on an integral representation [3,4]; in this case, however, the multifold integral terms are expressed in terms of the dynamic deviations of the input signal with respect to its value in the instant in which the output is evaluated. It can be shown that this new series can be truncated to the single-fold integral provided that the ‘memory’ effects are ‘short’ enough with respect to the signal period [4]. In order to make this hypothesis acceptable, it is convenient to partition the non-linear dynamic block of Fig. 3 into the cascade of a linear-block with memory and a non-linear one with ‘shorter’, residual memory (Fig. 4). To this end we can impose, through a suitable choice of the transfer function

D. Mirri et al. / Measurement 25 (1999) 265 – 283

268

Fig. 4. Final functional model to characterise the non-linear dynamic effects of a S / H-ADC device.

Hs fd of the linear block, that the behaviour of the non-linear block is memoryless in small signal operation at a given bias (e.g. zero bias). Without loss of generality we can also impose Hs0d 5 1. In fact, the small-signal low-frequency gain of the S / H-ADC device can be indifferently accounted for either in the linear system with memory or in the cascaded non-linear system with short memory. The choice Hs0d 5 1 simply means that the small-signal lowfrequency gain will be accounted for in the static characteristic of the non-linear block with short memory. Because the linear block takes into account a large part of the linear dynamic effects, it is reasonable to assume that the memory of the nonlinear block is practically finite; in such conditions, in order to reduce the model complexity, we can truncate the memory duration to a finite interval 2 T 1 # t # T 2 , where T 1 ,T 2 are suitable chosen memory time limits in order to make the memory truncation error ´MT small enough. Thus the output ustd of the continuous time non-linear system can be expressed as:

E

2T 1

2T 1

OE ? ? ?E g hs `

5 z 0 hs INTstd j 1

n51

P fs

2 s INTstd g dt 1 ´MT 1 ´ST 5 u˜ std 1 ´MT 1 ´ST

1T 2 2T 1

n

INT

std, t1 , . . . ,tn j

(4)

n

i 51

st 2 tid 2 s INTstd g dti 1 ´MT

INT

1T 2

ustd 5 z 0 hs INTstd j 1 g1 hs INTstd, t jf s INTst 2 td

1T 2

ustd 5Fu[s(t 2 t )]u

3

are identically null, i.e. it is the response of the non-linear block with memory to a DC input equal to s INTstd; the second one represents the sum of the convolution integrals of the dynamic deviations of the signal weighted by the correspondent order kernel gn h ? j of this series, kernel which is a nonlinear function of the reference input signal s INTstd, multiplied by n dynamic deviations of the input signal with respect to its value in the instant in which the output is evaluated. Because we have imposed Hs0d 5 1, the static characteristic does not depend on the linear block and the low-frequency small-signal gain with null offset of the entire system coincides with that of the non-linear block. Moreover, if the extension T 2 1 T 1 of the memory time-interval is relatively short (i.e. the maximum frequency fmax of the input signal satisfies the constraint fmax < 1 /sT 1 1 T 2d), it can also be shown [4] that the multifold integral series in Eq. (3) can be truncated to the first single fold term with a negligible series truncation error ´ST :

(3)

where the second right-hand side term is obtained by applying the modified Volterra series expansion proposed in Refs. [3,4]. The term z 0 hs INTstd j, where s INTstd is the signal at the input of the non-linear block with short memory (Fig. 4), represents the response of the system when the dynamic deviations

Eq. (4) represents the non-linear dynamic model which completely characterises the non-ideal behaviour of the S / H-ADC system. The convolution integral in Eq. (4) can also be expressed in an equivalent, more convenient, way by considering the Fourier transform of the input signal s(t). To this end, by introducing a suitable finite window function wst, T 1 , T 2d non null only in the prefixed memory interval T 1 , T 2 (as a particular case the window is a

D. Mirri et al. / Measurement 25 (1999) 265 – 283

unit-amplitude rectangular one), Eq. (4) can also be expressed as follows:

269

OS 1N

u˜ std 5 z 0 hs INTstd j 1

INT, q

e j 2 p fq t

q 52N 1`

ustd 5 z 0 hs INTstd j 3

1`

E w st, T , T d g hs

1

1

2

1

1

INT

3E

2`

std, t jf sINTst 2 td

1`

2`

2 s INTstd g dt 1 ´FM 1 ´TS

w 1st ; T 1 , T 2d g1 hs INTstd, t je 2j 2 p fqt

(5)

3 dt 2

E w st ; T , T d g hs 1

1

2

1

std, t jdt

INT

2`

4

5 z 0 hs INTstd j By indicating with G1 hs INTstd, f j and W1s f; T 1 , T 2d the Fourier transform respectively of the first-order non-linear kernel g1 h ? j of Eq. (5) and of the window w 1s ?d, by recalling the convolution property of the Fourier transform, the frequency domain kernel G˜ 1 hs INTstd, f; T 1 , T 2 j, that is the Fourier transform of the product w 1st ; T 1 , T 2d g1 hs INTstd, t j, can be evaluated at a generic frequency f as follows:

OS 1N

1

INT, q

q52N

e j 2 p fq t f G˜ 1 hs INTstd, fq ; T 1 , T 2 j

2 G˜ 1 hs INTstd, 0; T 1 , T 2 j g

H O Hs f dS e 1 O Hs f dS e 3 D˜ H O Hs f dS e 1N

5 z0

q

j 2 p fq t

q

q52N

J

1N

j 2 p fq t

q

G˜ 1 hs INTstd, f; T 1 , T 2 j

q52N

1N

1`

E w st ; T , T d g hs

5

q

1

1

2

1

INTstd, t je

2j 2 p ft

q

dt

q 52N

q

j 2 p fq t

, fq ; T 1 , T 2

J

(8)

2` 1`

5

E G hs 1

INT

std, f 9 jW1s f 2 f 9; T 1 , T 2ddf 9

(6)

2`

It should be noted that the purely dynamic nonlinearly controlled transfer function: D˜ hs INTstd, f; T 1 , T 2 j 5 G˜ 1 hs INTstd, f; T 1 , T 2 j

where G˜ 1 hs INTstd, 2 f; T 1 , T 2 j 5 G˜ *1 hs INTstd, f; T 1 , T 2 j and f 9 is the convolution integral variable in the frequency domain. Assuming an input signal with a finite discrete spectrum and recalling that the linear with memory input block is characterised by the transfer function Hs fd, the signal s INTstd at its output (Fig. 4) can be expressed as follows:

OS 1N

q52N

O Hs f dS e 1N

j 2 p fq t 5 INT, q e

s INTstd 5

q

q

j 2 p fq t

(7)

q52N

with Hs0d 5 1, where SINT, q are the spectral components of the signal s INTstd and Sq the correspondent ones of the input signal s(t). By considering Eq. (5) and taking into account Eqs. (6) and (7), we can assume as the finite-memory non-linear model of a S / H-ADC device a system whose output u˜ std is defined through the following relationship:

2 G˜ 1 hs INTstd,0; T 1 , T 2 j

(9)

with D˜ hs INTstd, 2 f; T 1 , T 2 j 5 D˜ * hs INTstd, f; T 1 , T 2 j, completely characterises the non-linear dynamic effects. In particular it can be noted that this transfer function, which weights the contribution to the output of each harmonic component of the input signal (Eq. 8), depends both on the instantaneous value of the signal s INTstd and on the frequency f. Besides, D˜ h ? j is null for f 5 0 (in fact D˜ hs INTstd, 0; T 1 , T 2 j 5 0); thus D˜ h ? j describes uniquely the non-linear dynamic effects. Therefore it can be concluded that the non-linear effects cannot be described uniquely by the static characteristic z 0 hs INTstd j of the device and by the full-frequency bandwidth (due to the slew-rate), since also the non-linear dynamic effects must be taken into account at the high frequencies.

270

D. Mirri et al. / Measurement 25 (1999) 265 – 283

Under the hypothesis of short duration of the non-linear dynamic effects with respect to the period of the input signal and negligible quantization error, the S / H-ADC device can therefore be mathematically modelled through three parameters, i.e. the linear transfer function Hs fd, the static characteristic z 0 hs INTstd j and the non-linearly controlled finitememory transfer function D˜ hs INTstd, f; T 1 , T 2 j. It is important to observe that the first two parameters are already given by the manufacturers, while the third one, which has a relevant effect at the high frequencies, represents the main contribution of this work.

be estimated through the second expression of Eq. (8) provided that the memory interval is not truncated and successively linearized with respect to the variable s INTstd around the DC value S0 :

OS 11

ustd(z 0 hs INTstd j 1

INT, q

q521

e j 2 p fq t f G1 hS0 , fq j

2 G1 hS0 , 0 jg dz ss9d D O Hs f dS e FS]] ds9 11

0

j 2 p fq t

(z 0 hS0 j 1

q

q

q521 q±0

s 95S 0

1 G1 hS0 , fq j 2 G1 hS0 , 0 j g

O Hs f dS e 11

3. Model identification

5 z 0 hS0 j 1

q

j 2 p fq t

q

q521

A hS0 , fq j

q±0

The parameters of the proposed mathematical model can be evaluated by using the S / H-ADC device to be characterised as the acquisition data system of an instrument (Fig. 4) which measures the RMS value of an input test signal by conveniently processing the sampled data [9]. The sampling strategy adopted was of a random type, previously introduced by the authors, which is characterised by a bandwidth limitation which is due uniquely to the S / H circuit [10,11]. Under the hypothesis that both the quantization error, the non-ideal effects associated to the sampling strategy and the numerical data processing errors are negligible, the output Sˆ of the instrument is an estimate of the RMS value of the signal ustd at the output of the non-linear block with short memory in Fig. 4, signal which is globally described by the first expression of Eq. (5) being the memory of a real S / H-ADC not limited also if it can be considered practically limited in some unknown interval. However the series of that expression can be truncated to the first order term when the alternate component of the test signal is imposed sufficiently small. By applying a DC input signal S0 of different amplitude, the static characteristic z 0 hS0 j of the S / HADC device can be directly measured (in fact Hs0d 5 1). Successively, a sinusoidal input signal with a DC component S0 (offset) is used as input signal. Under the hypothesis of an alternate component amplitude so small that the non-linear effects on it are negligible, also if sufficiently great to disregard the quantization error, the output signal can

O BhS , f jS e 11

5 z 0 hS0 j 1

0

q

j 2 p fq t

q

(10)

q521 q±0

where: B hS0 , f j 5 Hs fd A hS0 , f j

(11)

with: A hS0 , f j 5

FS

D G

dz 0ss INTd ]]] ds INT

2 G1 hS0 , 0 j

s INT 5S 0

1 G1 hS0 , f j (12)

The quantity A hS0 , f j is a small-signal offset-dependent transfer function which describes the behaviour of the non-linear block when a small AC signal with an offset S0 is applied at the input; the quantity B hS0 , f j represents instead a globally equivalent smallsignal offset-dependent transfer function whose amplitude can be directly deduced as the ratio between the RMS value of the input sinusoidal component, measured with a reference instrument, and the corresponding RMS one measured by an instrument which uses as acquisition data system the S / H-ADC device which must be characterised. By recalling that we assumed Hs0d 5 1, we can write:

S

dz 0ss INTd B hS0 , 0 j 5 A hS0 , 0 j 5 ]]] ds INT

D

s INT 5S 0

(13)

Therefore the quantity B hS0 , 0 j 5 A hS0 , 0 j, which

D. Mirri et al. / Measurement 25 (1999) 265 – 283

takes into account the non-linear effects at very low-frequency, coincides with the slope of the static characteristic. The difference G1 hS0 , f j 2 G1 hS0 , 0 j represents instead the variations of the non-linear effects in the high frequency range. Since the nonlinear effects when the offset is null are negligible,3 it results G1 h0, f j 5 G1 h0, 0 j. Therefore we can write: A h0, f j 5 A h0, 0 j 5 B h0, 0 j

(14)

By substituting into Eq. (11) with S0 5 0 we obtain: B h0, f j 5 Hs fd A h0, f j 5 Hs fdB h0, 0 j

(15)

271

1`

E G hS , f 9 jfWs f 2 f 9; T , T d 2 Ws f 9; T , T dgdf 9

5

1

0

1

1

2

(18) with Ws 2 f; T 1 , T 2d 5 W *s f; T 1 , T 2d. By observing that the following identity can be written because each of the two members corresponds to two waveforms simply translated one with respect to the other: 1`

1`

E Ws f 2 f 9; T , T ddf 9 5 E Ws f 9; T , T ddf 9 1

2

1

2`

Therefore from Eqs. (11) and (15) we have:

2

2`

2

(19)

2`

Eq. (18) can be rewritten as follows: 1`

B hS0 , f j A hS0 , f j 5 ]]]B h0, 0 j B h0, f j

(16)

E fG hS , f 9 j

D˜ hS0 , f; T 1 , T 2 j 5

1

0

2`

The difference G hS0 , f j 2 G hS0 , 0 j, which takes into account the non-linear effects in the high frequency range, can now be deduced from Eqs. (12), (13), (16): G1 hS0 , f j 2 G1 hS0 , 0 j 5 A hS0 , f j 2 A hS0 , 0 j B hS0 , f j 5 ]]]B h0, 0 j 2 B hS0 , 0 j B h0, f j (17)

2 G1 hS0 , 0 j g fWs f 2 f 9; T 1 , T 2d 2 Ws f 9; T 1 , T 2dg df 9

(20)

and, by using the result of Eq. (17), we can deduce the final relationship which relates the parameter D˜ h ? j of the model to the adopted measurement procedure: 1`

B hS , f 9 j E F]]] B h0, 0 j B h0, f 9 j 0

D˜ hS0 , f; T 1 , T 2 j 5

2`

By recalling Eq. (6), the non-linearly controlled purely dynamic transfer function D˜ h ? j of the model (Eq. (9)) can be expressed as follows:

G

2 B hS0 , 0 j fWs f 2 f 9; T 1 , T 2d 2 Ws f 9; T 1 , T 2dg df 9

D˜ hS0 , f; T 1 , T 2 j

5 G˜ 1 hS0 , f; T 1 , T 2 j 2 G˜ 1 hS0 , 0; T 1 , T 2 j

1`

E G hS , f 9 jWs f 2 f 9; T , T ddf 9

5

1

0

1

2

2` 1`

2

E G hS , f 9 jWs 2 f 9; T , T ddf 9 1

0

1

2

2`

3

This is a direct consequence of the hypothesis that the small signal frequency response at zero offset of the entire system is described by the transfer function H( f ) of the linear block while the corresponding gain depends on the non-linear block because H(0)51.

(21)

where f 9 is the convolution integral variable. This result shows that the non-linearly controlled purely dynamic transfer function D˜ hs INTstd, f, T 1 , T 2 j of the model can be experimentally measured by evaluating the frequency response of the globally equivalent small-signal offset-dependent transfer function B hS0 , f j for a sufficiently large set of DC values S0 and by selecting a memory interval (T 1 , T 2 ) which adequately smooths the random ringing of the experimental results. It is important to observe that the ratio B hS0 , f j /B h0, f j is not affected by the relative errors common to B hS0 , f j and B h0, f j; therefore in any further study it is convenient to express the final measurement results as a function of the small signal offset-dependent transfer function A hS0 , f j of the

D. Mirri et al. / Measurement 25 (1999) 265 – 283

272

non-linear block which is proportional to the above ratio (see Eq. (16)). In the particular case of a rectangular window, Eq. (21) becomes:

B hS0 , f j A hS0 , f j 5 ]]]B h0, 0 j B h0, f j

P f j2pf 1 z sS d g P f j2pf 1 p s0d g 5 KsS d]]]]]]]]]] P f j2pf 1 p sS d g P f j2pf 1 z s0d g P z s0d ]]] (24) P p s0d n

0

D˜ hS0 , f; T 1 , T 2 j 5

EF

2`

s 51 m

r 51

1`

sT 1 1 T 2d

m

B hS0 , f 9 j ]]] B h0, 0 j 2 B hS0 , 0 j B h0, f 9 j

0

r

0

r

r 51 n

s

s 51

n

G

s 51 m

3 e 2j p f 9sT 2 2T 1d f e j p fsT 2 2T 1dsincss f 2 f 9d

sT 1 1 T 2dd 2 sincs f 9sT 1 1 T 2ddg df 9

s

r 51

(22)

s

r

with A hS0 , 2 f j 5 A* hS0 , f j. For f 5 0 we obtain:

P z sS d ]]] A hS , 0 j 5 B hS , 0 j 5 KsS d P p sS d n

Due to the hypothesis of an input signal with an alternate component amplitude sufficiently low in order to neglect the non-linear effects on it, the globally equivalent small-signal offset-dependent transfer function B hS0 , f j for any prefixed value of the DC component S0 is a rational function and can be expressed as follows:

P f j2pf 1 z sS d g B hS , f j 5 KsS d]]]]]] P f j2pf 1 p sS d g n

0

s 51 0 m r 51

s

r

0

(23)

0

where n , m, B hS0 , 2 f j 5 B * hS0 , f j, the zeros z ssS0d and the poles prsS0d are real positive or complex (in complex conjugate pairs) with a positive real part; this last property is a consequence of the hypothesis of stability and causality [12] for the poles, and of minimum phase for the zeros. In fact the system is only apparently anticipative [1,2,9]. The effect of non-linearity in the adopted measurement procedure is to make poles, zeros and the quantity K of Eq. (23) functions of the DC component S0 of the input signal. The small signal offset-dependent transfer function of the non-linear block A hS0 , f j, which, according to Eq. (16), is proportional to the ratio of the two functions B hS0 , f j and B h0, f j, is characterised by an equal number of poles and zeros and it can be expressed as follows:

0

0

0

s 51 m

r 51

s

r

0

(25)

0

Consequently A hS0 , 0 j can be directly measured for each DC offset S0 either by considering the lowfrequency response of the system, or deduced, according to Eq. (13), as the slope of the static characteristic. By collecting the quantity 2p f in each term of the product it can be easily verified that:

P z s0d A h0, 0 j lim A hS , f j 5 KsS d ]]] 5 KsS d ]] Ks0d P p s0d n

f →6`

0

0

s 51 m

r 51

s

0

(26)

r

where the last passage is due to Eq. (24) for S0 5 0. The measurement procedure allows to measure directly the static characteristic z 0 hS0 j and the modulus uB hS0 , f ju of the globally equivalent small-signal offset-dependent transfer function for a convenient number of offsets S0 and frequencies f, so that the modulus u A hS0 , f ju of the purely non-linear offsetdependent memory effects can be deduced (Eq. (16)). Under the reasonable hypothesis that B hS0 , f j is a minimum-phase transfer-function, the phase of the ratio modulus uB hS0 , f ju /uB h0, f ju can be deduced from its frequency dependence through the Bode formula, which can be applied also to the function A hS0 , f j (see Appendix A). It can be shown that the phase /A hS0 , f j can be deduced from the modulus

D. Mirri et al. / Measurement 25 (1999) 265 – 283

of the function A hS0 , f j through the following relationship (A19):

E 0

lnu A hS0 , f 9 ju ]]]] df 9 f 92 2 f 2

(27)

It should be noted that the phase in the point sS0 , f 9d is theoretically proportional to the integral with respect to f 9 in the complete frequency range of lnu A hS0 , f 9 ju weighted by the function 1 /s f 9 2 2 f 2d which becomes lower and lower as f 9 moves away from f. The globally equivalent small-signal offset-dependent transfer function B hS0 , f j, which is the measured quantity, becomes for f greater than a certain value fmax too small to be measured; therefore the frequency interval which can be used to deduce the phase /A hS0 , f j is superiorly limited to the value fmax . It can be shown (see Appendix B) that, by using the frequency interval in which B hS0 , f j can be measured, this phase can be estimated with the following approximated formula (Eq. (B5)) f max

2f /AsS0 , fd 5 ] p

ln AsS , f 9d E ]]]] df 9 f 9 2f u

u

0

2

2

0

fmax 1 f 1 1 ] lnu A hS0 , fmax juln]] p fmax 2 f 1 esS0 , f, fmaxd

extrapolated interval s fmax # f 9 # 1 `,d is not greater that its observed maximum value M in the measurement interval:

1`

2f /A hS0 , f j 5 ] p

273

(28)

In order to find a superior limit to the error es ?d in Eq. (28) it can be assumed, for the supposed regularity of the function lnu A hS0 , f 9 ju in its definition range, that the maximum of the modulus of the derivative of lnu A hS0 , f 9 ju with respect to f 9 in the

U

U

dlnu A hS0 , f 9 ju ]]]] # M df 9

for 0 # f 9 # fmax

(29)

In this hypothesis it can be shown (Eq. (B11)): p (30) uesS0 , f, fmaxdu , ]M 4 Now the parameters of the large-signal non-linear model (i.e. the functions z 0 h ? j and D˜ h ? j) can be computed for any value of s INTstd and f by using suitable interpolation formulae.

4. Measurement procedure for model identification and experimental results On the basis of the model identification process outlined in the previous section, the measurement procedure can be synthesised as follows. Firstly, a variable amplitude DC signal S0 is applied to the instrument of Fig. 5 in which the S / H-ADC device under test is used as input acquisition system. Due to the hypothesis of Hs0d 5 1, in this way the static characteristic z 0 hS0 j of the non-linear system with short memory can be measured. Successively, a small sinusoidal input signal at different frequencies and with a DC component S0 variable in the definition range is applied to the same instrument of Fig. 5. In this way the modulus of the equivalent smallsignal offset-dependent transfer function of the entire system B hS0 , f j can be measured and that of the non-linear block A hS0 , f j derived (Eq. (16)). Under

Fig. 5. Block diagram of the instrument used to characterise the non-linear dynamic effects of a S / H-ADC device.

274

D. Mirri et al. / Measurement 25 (1999) 265 – 283

the hypothesis of a minimum phase system, the phase can be obtained from the frequency dependence of u A hS0 , f ju through a modified Bode formula and the required purely dynamic non-linearly controlled transfer function D˜ hS0 , f; T 1 , T 2 j, which weights the contribution of each spectral component of the input signal (Eq. (8)), can be derived through Eq. (21), or Eq. (22) in the particular case of a rectangular window. The acquisition system under test uses the Date SHM361 with an acquisition time of 20 ns, an aperture delay of 12 ns, a linearity error of 0.15%, a droop-rate of 10 mV/ ms, a full-power bandwidth of 60 MHz (maximum values); the successive approximation analog to digital converter is the 16-bit BurrBrown ADS7805P with a maximum non-linearity error of 64LSB and a conversion time of 8 ms. In order to reduce the effect of the S / H droop rate due to the relatively high time interval between two successive samples, the adopted ADC was of a sampling type. The output of the S / H circuit was doubled through an amplifier circuit in order to

obtain a full-scale input for the ADC when the input signal is at full-scale for the S / H (2.5 V). The prefixed minimum lag between two successive sampling instants was T c 510 ms. The DSP is a 32 bit floating point (TMS320C31) unit; the board has an expansion interface (DSPlink bus) used to transfer the data from the acquisition system to the DSP. Fig. 6 shows the measured values and the corresponding 6th order polynomial approximation of the static characteristic z 0 hS0 j graphically represented in terms of the deviation DS0 with respect to the input DC value S0 in the range 62.3 V; the non-linearity error turns out to be equal to 0.1%, within the value indicated by the manufacturer. Fig. 7 shows the modulus of the small-signal transfer function Hs fd of the linear block (Fig. 4); from this figure it can be deduced that the small-signal bandwidth of the entire system under test is |90 MHz. In Fig. 8a the modulus u A hS0 , f ju of the purely non-linear offsetdependent transfer function (approximated with a 6th order polynomial with respect to both S0 and f ) of the non-linear block is plotted as a function both of

Fig. 6. Deviation DS0 of the static characteristic with respect to the DC input signal S0 .

D. Mirri et al. / Measurement 25 (1999) 265 – 283

275

Fig. 7. Modulus uHs fdu of the transfer function of the linear block of Fig. 1 as a function of frequency f.

S0 and f; Fig. 8b shows its phase angle as a function both of S0 and f deduced from Eq. (27). The frequency superior value is limited to 60 MHz in order to reduce the error introduced in Eq. (27) by the minimum measurable value of uB hS0 , f ju; by 24 using Eq. (29) we obtain M 5 41 3 10 and, from Eq. (30), we deduce a superior limit of the modulus of the error equal to 32 3 10 24 rad. Finally Fig. 9a and b shows respectively the real and imaginary plot of the non-linearly controlled purely dynamic transfer function D˜ hs INTstd, f j as a function both of s INTstd and f (Eq. (21)), while Fig. 10 shows the modulus of the same function. Fig. 11 gives the contour plot of D˜ hs INTstd, f j as a function both of the signal s INTstd and the frequency f. From this figure it results that the non-linear dynamic effects, at full-power or at small-amplitude operation when a great DC component is present, become relevant at 50 MHz. This result must be compared with the full-power bandwidth of the S / H device (60 MHz), which takes into account uniquely the slew-rate effect. Besides this

plot puts in evidence the non-perfect symmetric behaviour of the device as regards the non-linear purely dynamic effects. It is important to observe that the non-linear dynamic effects are not yet taken into account by the manufacturers of the S / H-ADC devices though they are relevant at high frequencies both for full-power signals, or small-amplitude signals in the presence of a great DC component.

5. Conclusions A new model has been proposed for the characterisation of the non-linear dynamic effects of a S / H-ADC device and a new mathematical approach has been given to describe its non-ideal dynamic behaviour. This approach is based on the only mild hypothesis of a relatively small duration (both with respect to the inverse of the signal bandwidth and the minimum sampling interval) of the memory effects associated with dynamic non-linear phenomena in

276

D. Mirri et al. / Measurement 25 (1999) 265 – 283

Fig. 8. Modulus u A hS0 , f ju (a) and phase angle /A hS0 , f j (b) of the transfer function of the purely non-linear offset-dependent memory effects as a function both of S0 and f.

D. Mirri et al. / Measurement 25 (1999) 265 – 283

Fig. 9. Real (a) and imaginary (b) plot of the non-linearly controlled transfer function D˜ hs INT (t), f j as a function of s INT and f.

277

278

D. Mirri et al. / Measurement 25 (1999) 265 – 283

Fig. 10. Modulus of the non-linearly controlled transfer function uD˜ hs INTstd, f ju as a function of s INTstd and f.

Fig. 11. Contour plot of some discrete values of the modulus of D˜ hs INTstd, f j.

D. Mirri et al. / Measurement 25 (1999) 265 – 283

S / H devices. The proposed parameters to characterise these devices are the static characteristic, which depends on the input signal amplitude, the linear transfer function and the non-linearly controlled purely-dynamic transfer function, which depends also on the frequency of the input signal. This transfer function weights the contribution of each harmonic component of the input signal. The given measurement procedure can be carried out directly in the circuit in which the S / H-ADC device must operate and requires RMS measurements under small signal sinusoidal excitation subject to a given additive DC bias. Such measurements provide a complete characterisation of the non-linear dynamic model which predicts the actual response of the S / H-ADC device under any other non-sinusoidal operating condition. This paper shows that the non-linear effects of a S / H-ADC device cannot be described uniquely by its static characteristic and by the fullpower frequency; there are also the non-linear purely-dynamic effects which are a function both of signal amplitude (included the DC component) and frequency. In order to characterise these purelydynamic effects a non-linearly controlled transfer function is introduced; it represents the weight which must be applied to each spectral component of the input signal. The measurement procedure to deduce this transfer function is described and the measurements results for a commercial device are given. From these results it can be concluded that the non-linear purely-dynamic effects must be taken into account together with the full-power bandwidth, already given by the manufacturers; in fact they become relevant in the same frequency range. It must be pointed out that the non-linear purely-dynamic effects are relevant at the high frequencies also in small signal conditions when a large DC component is present.

Appendix A

where:

P f j2pf 1 z sS d g P f j2pf 1 p s0d g A hS , f j 5 ]]]]]]]]]] P f j2pf 1 p sS d g P f j2pf 1 z s0d g n

1

P z s0d A hS , f j 5 KsS d ]]] A hS , f j P p s0d

r 51

0

r 51

r

0

0

r

0

r 51 n

s 51

r

s

(A2) with A 1 hS0 , 2 f j 5 A *1 hS0 , 0 j 5 1. Further lim A 1 hS0 , f →6`

f j and A 1 h0, f j 5 A 1 h0, f j 5 1 and consequently:

lim lnA 1 hS0 , f j 5 0

(A3)

f →6`

The sign of the real coefficient KsS0d in Eq. (A1) is equal to that one of B hS0 , 0 j (Eq. (25)) because the n m ratio s 51 z ssS0d / r 51 prsS0d is real and positive (see comments after Eq. (23)). The sign of B hS0 , 0 j (and of KsS0d) is normally positive due to the positive slope of the static characteristic (Eq. (13)). Consequently the coefficient of A 1 hS0 , f j in Eq. (A1) is real and positive. So the phase of A hS0 , f j coincides with that one of A 1 hS0 , f j. By recalling that A 1 hS0 , f j 5u A 1 hS0 , f jue js/ A 1 hS 0 , f jd, the relationship between lnu A 1 hS0 , f ju and the phase /A 1 hS0 , f j can be deduced. To this end the well-known relationship for a complex quantity must be taken into account:

P

P

lnA 1 hS0 , f j 5 lnu A 1 hS0 , f ju 1 js/A 1 hS0 , f jd

(A4)

By remembering that lnu A 1 hS0 , f ju 5 1 / 2ln f A 1 hS0 , f j A 1* hS0 , f jg and that A 1 hS0 , 2 f j 5 A *1 hS0 , f j, we conclude that lnu A 1 hS0 , f ju is a real and even function of f; therefore also its Fourier inverse transform, denoted by a pstd, is a real and even signal of t. Similarly the phase of A 1 hS0 , f j multiplied by j, i.e. j/A 1 hS0 , f j, is an imaginary and odd function of f due to the fact that /A 1 hS0 , f j 5 2 /A 1 hS0 , 2 f j; therefore its Fourier inverse transform, denoted by a dstd, is a real and odd function of t. As a conclusion it can be observed that the inverse transform of lnA 1 hS0 , f j, a 1std, can be subdivided in two real signals, odd and even respectively, as follows: a 1std 5

s

1

s

1`

n

0

m

s 51 m

0

Eq. (24) can be rewritten as follows:

s 51 m

279

(A1)

E lnsA hS , f jde 1

0

j 2 p ft

df 5 a pstd 1 a dstd

(A5)

2`

Now we try to demonstrate that, under the condition that poles and zeros are real and positive or in

D. Mirri et al. / Measurement 25 (1999) 265 – 283

280

complex conjugate pairs with positive real parts, a 1std is a causal signal, i.e. it is null for t , 0. Through an integration by parts of the first expression of Eq. (A5) it can be written: 1`

1 a 1std 5 2 ]] j2p t

E e H]dfd flnsA hS , f jdg J df j 2 p ft

1

0

2`

(A6) because the first term of the integration by part is null due to Eq. (A3). By considering the logarithm of Eq. (A2) and deriving it, Eq. (A6) becomes:

where F 21 h ? j is the inverse Fourier Transform and ustd is the well-known Heaviside unit step function. Therefore the generic contribution of each term of the sums with real and positive parameter ucu is causal. In the second case the generic contribution w 2std to Eq. (A7), due to two terms which have complex conjugate parameters with real positive part, appears in the following form: 1`

w 2std

1 5] 2p t

1 E e F]]]] js f 2 f d 1 c j 2 p ft

0

2`

a 1std 5 1 2] 2p t

1`

1 1 ]]]] js f 1 f0d 1ucu

e ]]] df O E 3 z sS d n

s 51

2`

j 2 p ft

s

1`

0

jf 1 ]] 2p

1`

1`

OE

OE

m e j 2 p ft e j 2 p ft ]]] df 6 ]]] df 1 prs0d prsS0d r 51 r 51 2` jf 1 ]] 2` jf 1 ]] 2p 2p m

1`

2

s 51

2`

j 2 p ft

(A7)

s jf 1 ]] 2p

1. when the generic contribution w 1std to the sums of Eq. (A7) is due to an addendum which has at the denominator a real positive parameter (zero or pole divided by 2p ) denoted by ucu; 2. when the generic contribution w 2std is due both to an addendum which has at the denominator a complex parameter with real positive part indicated by ucu 1 jf0 together with its complex conjugate ucu 2 jf0 . Obviously both ucu and f0 are functions of S0 . By considering the first case, a generic term of the sums in Eq. (A7) assumes, apart from the sign, the following form: 1`

e E ]] df jf 1 c

2`

H

u u

e E ]] df jf 1 c 4 j 2 p ft

u u

1`

1 5 ] coss2p f0 td pt

e E ]] df jf 1 c j 2 p ft

u u

2`

Two different cases can be separately taken into account:

1 w 1std 5 ] 2p t

1`

1e

j 2 p ft

2`

2j 2 p f 0 t

j 2 p ft

df

e E ]] df jf 1 c

2`

e O E ]]] df z s0d 4 n

3

1 5 ] e j 2 p f0 t 2p t

G

u u

5 2 coss2p f0 tdw 1std

(A9)

Therefore also w 2std is causal as w 1std. As a conclusion, Eq. (A7) shows that the function a 1std is causal being the sum of causal contributions. Now, by recalling the last expression of Eq. (A5), since a 1std is causal, for t $ 0 we have: a ds 2 td 5 2 a ps 2 td and a dstd 5 a pstd. Therefore it results: a 1std 5 2a pstdustd

(A10)

By remembering that a 1std is the inverse Fourier transform of ln A 1 hS0 , f j (see Eq. (A5)) and by using well known properties of the Fourier transform Fs ?d, it can be deduced that: ln A 1 hS0 , f j 5 F f a 1std; f g 5 2F f a pstd; f g * F f ustd; f g

u u

1`

J

1 1 1 5 ] F 21 ]]]]; t 5 ] e 22 pucut ustd t t j2p f 1 2pucu (A8)

52

E F fa std; f 9 gF fustd; f 2 f 9 g df 9 p

2`

(A11)

D. Mirri et al. / Measurement 25 (1999) 265 – 283

where * represents the convolution symbol, F f a pstd; f 9 g is the Fourier transform of a pstd and F f ustd; f 2 f 9 g that one of the Heaviside function [13] that is:

F

d s f 2 f 9d j F f ustd; f 2 f 9 g 5 ]]] 2 ]]] 2 2ps f 2 f 9d

G

(A12)

By substituting Eq. (A12) into Eq. (A11) it results:

281

1`

ln A hS , f 9 j 2f E ]]]] df 9 2 ] p f 9 2f P z s0d 1 3 ln KsS d ]]] E ]]] df 9 P p s0d f 9 2 f ln A hS , f 9 j 2f 5 ] E ]]]] df 9 p f 9 2f

2f /A 1 hS0 , f j 5 ] p

u

F f a std; f 9 g E ]]]] df 9 f 2f9

j ln A 1 hS0 , f j 5 F f a pstd; f g 2 ] p

2

0

n

3

0

4

s

s 51 m

r

r 51

1`

2

2

0

1`

u

u

0

2

1`

u

0

2

(A18)

2

0

p

2`

(A13) By comparing this equation with Eq. (A4) we conclude:

due to the fact that the second integral is null. Finally, by recalling that /A hS0 , f j 5 /A 1 hS0 , f j, the final expression can be written which relates the phase of A hS0 , f j with its modulus as a function of the frequency: 1`

F f a pstd; f g 5 lnu A hS0 , f ju

(A14)

/A hS0 , f j 5

ln A hS , f 9 j E ]]]] df 9 f 9 2f u

u

0

2

(A19)

2

0

and: for f $ 0.

1`

1 /A 1 hS0 , f j 5 ] p

ln A hS , f 9 j E ]]]] df 9 f92f u

1

u

0

(A15)

2`

Appendix B

By simple manipulations, this equation can be rewritten as follows: 1`

1 /A 1 hS0 , f j 5 ] p

ln A hS , f 9 j E ]]]] s f 9 1 fd df 9 f 9 2f u

1 2

u

0

2

2`

1`

2f 5] p

ln A hS , f 9 j E ]]]] df 9 f 9 2f u

1 2

u

0

(A16)

2

Since the globally equivalent small-signal offsetdependent transfer function B hS0 , f j, which is the measured quantity, becomes for f greater than a certain value fmax too small to be measured, the frequency interval which can be used to deduce /A hS0 , f j is superiorly limited to the value fmax . Therefore Eq. (A19) must be rewritten as follows: f max

0

for each f $ 0. Let us now come back to Eq. (A1) and consider the logarithms of the moduli; i.e.

P z s0d KsS d ]]] P p s0d n

lnu A 1 hS0 , f ju 5 lnu A hS0 , f ju 2 ln

3

0

s 51 m

r 51

s

r

4

2f /AsS0 , fd 5 ] p

ln AsS , f 9d E ]]]] df 9 1 DsS , f, f f 9 2f u

2

0

d

max

0

(B1) where the deviation D(?) is given by: 1`

2f DsS0 , f, fmaxd 5 ] p

ln AsS , f 9d E ]]]] df 9 f 9 2f u

u

0

2

2

f max

(A17) By substituting into the last expression of Eq. (A16) it results:

u

0

2

1`

1 5] p

f92f E ln AsS , f 9d d ln ]] f91f u

f max

0

u

(B2)

D. Mirri et al. / Measurement 25 (1999) 265 – 283

282

with 0 , f , fmax # f 9 , 1 `. The last passage in Eq. (B2) is due to the fact that:

can be assumed that u( d lnu A hS0 , f 9 ju / df 9)u # M for x9 [sx max , 1 `d. Consequently it results:

f92f 2f d ln ]] 5 ]]] df 9 f 9 1 f f 92 2 f 2

uesS0 , f, fmaxdu # ]MW( f, fmax )

1 p

Successively, through an integration by parts it results:

1 esS0 , f, fmaxd

(B3)

u

f max

(B4) By substituting Eq. (B3) into Eq. (B1) it can be written: f max

2f /AsS0 , fd 5 ] p

ln AsS , f 9d E ]]]] df 9 f 9 2f u

u

0

2

(B5)

Now by considering the modulus of the error from Eq. (B4) it results: u´sS0 , f, fmaxdu 1`

u

0

u

(B9)

1 f ]D O ]]] S f s2k 2 1d 1`

Ws f, fmaxd 5 2

k51

2

2k 21

max

2

p ,] 4

(B10)

uesS0 , f, fmaxdu , ]M.

fmax 1 f 1 1 ] lnu A hS0 , fmax ju ln ]] p fmax 2 f

d ln A hS , f 9 j f91f E U]]]] df 9 U ln ]] df 9 f92f

2s2k21d

it can be obtained by integrating the series term by term:

p 4

0

1 #] p

SD

f91f 1 f9 ]] ] ln ]] 5 2 2k 2 1 f92f f k 51

being f # fmax . Consequently:

2

1 esS0 , f, fmaxd

(B8)

and by using the series expansion: 1`

d A hS , f 9 j f91f E H]]]] df 9 J ln ]] df 9 f92f 0

f91f E ln ]] df 9 f92f

O

1`

u

1`

f max

where the final error is given by: 1 esS0 , f, fmaxd 5 ] p

where: Ws f, fmaxd 5

fmax 1 f 1 DsS0 , f, fmaxd 5 ] lnu A hS0 , fmax juln ]] p fmax 2 f

(B7)

(B6)

f max

It is important to emphasise that the maximum frequency fmax (and consequently ln( fmax )) in Eq. (B6) must be sufficiently greater than the maximum frequency f (and so ln( f )) in which the phase /A hS0 , f j is estimated. On the hypothesis that the maximum of the modulus of the derivative of lnu A hS0 , f 9 ju with respect to f 9 in the interval s fmax , 1 `d is not greater than its maximum value M in the measurement interval s 2 `, fmaxd, where 0 # f 9 # fmax , for the supposed regularity of the function lnu A hS0 , f 9 ju, it

(B11)

References [1] D. Mirri, G. Iuculano, F. Filicori, G. Vannini, Modeling of non-ideal dynamic characteristics in a S / H-ADC device, in: IEEE Instrumentation and Measurement Technology Conference (IMTC / 95), Boston, 1995, pp. 27–32. [2] D. Mirri, G. Pasini, F. Filicori, G. Iuculano, G. Neri, A non-linear dynamic modelling approach for the characterization and error compensation in sampling oscilloscope, Measurement 22 (3–4) (1997) 97–112. [3] F. Filicori, G. Vannini, V.A. Monaco, A non-linear integral model of electron devices for HB circuit analysis, IEEE Trans. Microwave Theory Tech. 40 (7) (1992) 1456–1465. [4] D. Mirri, G. Iuculano, F. Filicori, G. Vannini, G. Pasini, G. Pellegrini, A modified Volerra series approach for the characterization of nonlinear dynamic systems, in: IEEE Instrumentation and Measurement Technology Conference (IMTC / 96), Brussels, 1996, pp. 710–715. [5] T.A. Rebold, F.H. Irons, A phase plane approach to the compensation of high speed analog to digital converters, Proc. IEEE Int. Symp. Circuits Systems (1987) 455, May. [6] F.H. Irons, D.H. Hummels, S.P. Kennedy, Improved compensation for analog-to-digital converters, IEEE Trans. Circuits Systems 38 (8) (1991).

D. Mirri et al. / Measurement 25 (1999) 265 – 283 ´ a` la [7] V. Volterra, Lecons sur le functions de lignes professes Sorbonne en 1912, Paris, 1913. [8] H.W. Chen, Modelling and identification of parallel nonlinear systems: structural classification and parameter estimation methods, Proc. IEEE 83 (1) (1995) 39–66. [9] D. Mirri, G. Pasini, F. Filicori, G. Iuculano, G. Vannini, R. Rossini, Experimental evaluation of dynamic-nonlinearities in a S / H-ADC device, in: IMEKO TC-4 Symposium ‘Electrical Instrument in Industry’, Glasgow, 1997, pp. 137–140. [10] G. Iuculano, D. Mirri, F. Filicori, A. Menchetti, M. Catelani,

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A criterion for the performance analysis of synchronous and asynchronous sampling instruments based on non linear processing, IEE Proc. 139 (A4) (1992) 141–152. [11] D. Mirri, G. Iuculano, A. Menchetti, F. Filicori, M. Catelani, Recursive random sampling strategy for a digital wattmeter, IEEE Trans. Instrum. Measure. 41 (6) (1992) 979–984. [12] B.P. Lathi, Linear Systems and Signals, Berkeley-Cambridge Press, Carmichael, CA, 1992. [13] I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York, 1965.