Testing methodologies for analog-to-digital converters

IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 41, NO. 5. OCTOBER 1992 595

Testing Methodologies for Analog-to-Digital Converters

Arnaldo Brandolini, Senior Member, IEEE, and Alessandro Gandelli, Member, IEEE

Abstract-The paper presents a theoretical approach and an experimental test system devoted to introduce a set of parameters based on Walsh functions and conformed to characterize the transfer function of analog-to-digital converters. On the basis of previous work [l], the authors present an enhanced system that provides better accuracy in the evaluation of the performance of conversion devices under dynamic conditions. The theory covers an introductive approach to generalize the conversion processes and employs a powerful purpose-oriented tool to understand their in-depth operativity. The error parameters are defined by mathematical algorithms based on Walsh functions and related transform, while their properties are correlated to a standard reference input, a triangular waveform provided by the system. Moreover, this methodology opens the way towards the introduction of standard techniques in testing conversion devices.

I. INTRODUCTION HE evaluation of static and dynamic performance in T analog-to-digital converters requires the complete and

correct definition of significant parameters related to the physical behavior of the device [2]-[4]. Several interesting and significant works have been published to present correctly the theory of dynamic ADC testing, and these authors introduce new and very sophisticated techniques to provide meaningful results.

Reviewing the papers that influenced the development of our work, several interesting considerations can be provided.

1) E. A. Sloane, in his works devoted to the Walsh transform application in ADC testing in 1980 [5] and 1981 [6], presents a way to compute classical ADC error parameters using Walsh techniques. He summarizes the main properties of Walsh functions looking more at the specific search for connections with existing theory in- stead of defining a new, autonomous and complete mathematical tool to study in-depth ADC characteristics,

2) M. Vanden Bossche, J . Schoukens, and J. Renne- boog focus their investigation on differential (DNL) and integral nonlinearities (INL) [7 ] . They examine classical diagnostic methods based on Fourier transform techniques to provide better ways for DNL and INL estima- tion and the Walsh transform test to identify bit failures.

Manuscript received October 22, 1991; revised May 1 , 1992. The authors are with the Dipartimento di Elettrotecnica, Politecnico di

IEEE Log Number 9202489. Milano, 20133 Milano, Italy.

Input sine wave is applied as standard reference to the system under test.

3. T. M. Souders and D. R. Flach [8] design an automated test set to characterize ADC behavior using a 20-b DAC as standard reference. The procedure developed for calibrating ADC’s is capable of accurately characterizing the device under test around the transition level. The input signal is a stationary voltage level, and the sampling frequency is fixed (generally 10 kHz). In this way only a full static characterization can be provided.

4) R. A. Malewski, T. R. McComb, and M. M. C. Collins [9] start their work remarking that “there is pres- ently no generally accepted method for testing a digitiz- er’s dynamic performance, however, many manufacturers employ the equivalent bit test technique. ” They propose to adopt a different version of the equivalent bit test procedure that seems good to characterize devices working with impulse input signals and high sampling frequency.

Looking at these previous works and at their results, we can first clearly affirm that the best representation of the ADC transfer function distortions is reached testing the devices varying both the input signal frequency and the sampling frequency over their whole range.

The wealth of information deduced from the trend of every significant error parameter as a function of the sampling rate and the input-signal frequency can be appreci- ated if compared with parameters calculated under set conditions, for example, at the maximum rate operating frequency only [3]. The proposed representation is partic- ularly more helpful when it comes to evaluating the in- strument’s optimal operating ranges.

11. DYNAMIC ANALYSIS OF THE A/D CONVERTER TRANSFER FUNCTION

The characterization of the A/D converter transfer function (T.F.)’ implies the examination of the correct test conditions for the A/D conversion devices in order to avoid any loss of essential information.

Because the input signal spectrum and the sampling frequency are parameters introducing significant effects during the device test, the T.F. can only be evaluated correctly if information on these parameters is entered in the

‘In the following text transfer function (T.F.) indicates a widespread relationship between the input and output of the conversion system. This definition is not limited to linear systems.

0018-9456/92$3.00 0 1992 IEEE

596 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 41, NO. 5, OCTOBER 1992

dynamic test procedures. The first hurdle is therefore to choose the most appropriate input signal spectrum for a given test condition. Of the large number of possibilities available two significant signal families are known to solve this problem.

The first family is constituted by sinusoidal waveforms providing, therefore, input signals with only one frequency component. This approach greatly simplifies test procedures because the spectral content in the output can be easily correlated with the input when the system is linear. This methodology, however, in the presence of nonlinear phenomena, typical in A/D conversion systems, cannot correctly define the system response when more frequencies are superimposed in the input signal.

The second family is based on multifrequency spectrum input signals. In practice this implies that, in the presence of nonlinear systems, the response of the system is more complete and significant. It is also possible to simplify the test procedures by identifying particular types of input signals, as in the case of triangular waveform. This waveform presents, in fact, constant first derivative in the sem- iperiod and so can be looked upon as being one of the best test signals for A/D systems.

We can summarize, as this stage, by stating that, on the basis of experimental test results, the first solution gives the best compromise when the system under test is completely linear. In nonlinear systems signals different from sine waves are required to correctly define the system: The best category of waveform pertains to the constant derivative type.

We can now continue by implementing the second mentioned method of dynamical analysis using the dedi- cated and powerful technique based on Walsh functions and transforms.

111. APPLICATION OF THE WALSH FUNCTIONS AND

WALSH TRANSFORM IN SEARCH FOR THE TRANSFER FUNCTION

The Walsh functions and the Walsh transform [lo], [ 1 11 represent mathematical tools that are both effective and of relative computational simplicity in the search for the transfer function representation. While, in general, they may be difficult to handle from an analytical point of view, they do, on the other hand, lend themselves quite natu- rally to implementation in binary logic systems (such as computers). They also produce effective results and require only very limited expenditure on hardware struc- tures of the computers used to perform the computation. If we consider a normalized and ideal T.F. of an M-bit converter, we can express its trend as a function of a limited number of Walsh functions (WAL, (k ,u ) ) appropriately weighted by means of the Walsh coefficients, X,. Thus, we have:

M

Qid (U) C Xk Waw [(2k - 11, ul k = O

where u defines the domain of the Walsh functions that for the classic T.F. results: u E 0 .+ 0.5 (Fig. 1). Note

1.5 1.5 WAL(0.u)

Fig. 1 . Example of a 2-bit A/D T.F. and its Walsh components.

that only M + 1 Walsh functions are sufficient for the complete definition of an ideal transfer function.

It is also possible to define any real T.F. using a finite2 number of Z of Walsh functions (with 2M maximum value for Z) as follows:

2 - 1

Q, (U) = C Xk Waw ( k , U). k = O

This second relationship includes also the ideal T.F. representation shown above if it is considered as a real T.F. without errors (in this case all the coefficients except those powers of 2 are nulls).

The above described way of representation character- izes the exact trend of the transfer function and, at the same time, suggests an interesting test methodology for A/D converters. Using the Walsh transform, the output of A/D systems can be processed in order to obtain the specific coefficients Xk characterizing every Walsh function WAL,(k,u) (Fig. 1). It is also possible to correlate the trend of the T.F. with the time/amplitude trend of a triangular-waveform input signal in order to obtain a sampled representation of the T.F. (Fig. 2). In this case the T.F. Qre(u) is defined by u E 0 + 1 and is characterized by the symmetrical trend.

When the discrete Walsh transform is used to analyze the T.F. the first objective is to establish the resolution intended for the characterization of the T.F. trend. Clearly, the minimum number of points to be considered, for an M-bit converter, is 2M. In practice it is advisable to increase this number, for instance, by multiplying it by a power of 2 , in order to improve the accuracy. Increasing the number of points in the interval covered by every quantization step allows the differences between ideal and real T.F. to be defined in more detail. The number of points d sampled for each ideal quantization step of the T.F. is called characteristic resolution. The maximum

'It is important to note that the proposed representation defines the T.F. using a finite number of coefficients. On the contrary the representation using Fourier transform requires an infinite number of functions to build the T.F. We can also note that the Walsh coefficients (+ 1, - 1 ) are not affected by quantization errors like the Fourier coefficients, represented by truncated transcendental numbers.

BRANDOLINI AND GANDELLI: TESTING METHODOLOGIES FOR ADC's 591

0.5 1.0

Fig. 2. Triangular waveform after 3-bit AID conversion. In this case the output signal represents, for U E 0 + 0.5, the T.F. Q ( u ) considering a positive-slope input signal; in the second interval, U E 0.5 + 1.0, it represents again the T.F. Q(u) resulting from a negative-slope input signal.

significant number of points d is consequently dependent on the accuracy which we intend to establish for the input signal. The total number of sampled points is consequently equal to:

N, = d2'.

Following the definition of the ideal and real transfer function we can now introduce the errorfunction A ( U ) of the T.F. as the difference:

Walsh coefficients defining the Walsh transform of A ( U )

(error coeficients) can be directly calculated from the difference of every pair of Xk obtained transforming Qre(u) and Qid(u) when the sum of the second function has been integrated to Z elements adding proper zero terms for the Walsh function having position different by powers of two. The error coeficients, organized in an error coeficient vector, comprise the whole and existing information on the errors introduced by the conversion system.

IV. ERROR CLASSIFICATION

There are many significant error parameters obtainable through the analysis of the error coeficient vector, and a global view calls for a detailed description of the mathematical properties on which the theory is based [ 121 , [ 131, [ 141. It is thus possible to use the coefficients both for an overall evaluation of the properties of the T.F. and to carry out a local search for any imperfections presented as performed in previous works by other authors [5], [7].

In this work we propose the introduction of several different classes of T.F. error parameters, such as these given below:

parameters defining the global accuracy (class I); parameters defining the presence of code oscillations in the single quantization step (class 11); parameters defining the position of the local distortions (class 111); parameters defining local and global hysteresis (class IV);

parameters defining very large single code errors

The main purpose of this work is to analyze the methodology and show the system built and the software developed [ 161, [ 181 to compute the parameters related to the above-mentioned errors. The following parameters represent just the first result of this research. It is in fact possible to find other mathematical relationships to define error parameters and their behavior using the same methodology.

(class V).

V. ERROR ANALYSIS AND INTERPRETATION Let us now examine the main parameters covered by

1) The mean error coeficients, are defined as our analysis [ 131, [ 141.

(Appendix 1):

where a = N , / j - 1 represents the mean value of the error distributed on an interval of j successive samples of the T.F., starting from sample i = N,(7 - l ) / h , {T = 1, 2, * - , h ) , where h is the number of uniform intervals in which is divided the T.F., and j and h are powers of two. The relationship can be generalized to an interval extended from sample p to sample q, where p and q are different from powers of two, using the following analytic expression:

j iEi, , j , + j 2 E i 2 , j 2 + * . + j k E i k , j t [email protected] =

where the sum is extended to the k binary intervals which compose the interval p -+ q.

Using mean error coeficients computed by changing the location (p) and the width (q - p) of the interval any distortion in the T.F. mainly distributed up (+) or down (-) the ideal trend can be detected and quantified (Fig. 3). The identification and analysis of the distorted intervals are performed through successive approximation tests working on smaller and smaller intervals. The totality of the mean error coeficients obtained by varying interval location and extension can be arranged as a matrix defining completely the T. F. errors.

It is interesting to note that classical AID conversion error parameters can be defined by means of the Ei,j coefficients. When the extension q - p is limited to one or a few samples, a local search for errors, like diferential nonlinearity and code omission (i.e., classes I11 and V), can be performed.

A better and simpler characterization of the T.F. follows, however, by the joint use of different types of error parameters. The reason for which we. introduce more parameters to define the error trend is justified by obtaining a synthetic representation of the T.F. distortions greatly significant for human operators.

598 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 41, NO. 5. OCTOBER 1992

- ideal

- \ real

-

Fig. 3 . Examples of T.F. errors influencing the mean error coefficients. In the p + q interval, extended to four ideal steps, there is a partial balance of the mean error. In the p ' + q interval there is no error balance. In the p " + q interval the mean value error concerns only one ideal step, and it can be assumed as differential nonlinearity.

VI. CORRECT SAMPLING OF THE TEST SIGNAL: THE CIRCULAR SAMPLING

To correctly apply our test method based on Walsh transform the proper input-signal frequency and sampling frequency sets must first be determined. Understandably, when working with algorithms requiring a number of samples power of two, the set of frequencies is undoubtedly limited.

For example, if a conversion system having 10 MHz as maximum sampling frequency (f,) and 8 bits of resolution (M) is considered, the highest frequency ( fr) for the input triangle waveform, using a characteristic resolution d of 4 points per step, results in:

fc fr = ~ = 4882.8 [Hz]. d 2 M ' '

The set of test signal frequencies may be determined con-

so on for the other tests. An analogous criterion may be adopted to choose sampling frequencies.

In this way only low-frequency input-signal waveforms can be applied, and a poor and nonuniform point distribution for characterizing every test is obtained. This fact results in an incomplete survey of the T.F. because the

2) The quadratic residues, R,,j are defined as (Appen- sequently by dividing by the maximum frequencyfr and

dix I):

( (i ;,1)j)12 X,WAL, n , -

N, / j R . . =

' 9 J

where /3 = kN,/j + n represents quadratic sums of the differences (error coeflcients) between the real and ideal T.F. distributed on the interval constituted by j successive samples starting from sample i (where i and j must be powers of two).

The quadratic residues allow one to detect and enhance the absolute values of the errors in any interval of the T.F. and differ from the mean value errors that can generally recognize just polarized errors. The Ri,j defines not only the global accuracy (I class) but can be used to detect the existence of step oscillations (I1 class) or in general local distortion having null E;,, (I11 class). It is possible to demonstrate that if Ri,j > 1 /j, then an error greater than two digital levels occurs.

More information on other error parameters, used in our experimental system, can be found in Appendix I1 where odd coeflcient sums, even coeflcient sums, and the shi' invariant power spectral points are introduced. An example of complete analysis performed on a 2-b extension amplitude of a simulated T.F. is shown in Appendix 111.

All of the above-described parameters provide a highly significant index of the conformity of the transfer function with respect to the ideal trend. The values of the parameters increase considerably due both to large local defects and small deformations uniformly distributed throughout the transfer function. In both cases this pinpoints a dete- rioration in the ideal transferfunction of the converter and suggests more detailed analysis of the T.F. Overall, the evaluation of the converter accuracy is undoubtedly more complete than that supplied by the classic differential and integral linearity or other classic parameters.

high-frequency domain is out of the analysis limits. A possible answer to this handicap can be found using the circular sampling [14]. This method introduces more T.F. sampling points, and this fact guarantees more test conditions, more evenly distributed in the plane (f,, fr), and also provides higher maximum waveform frequency.

If the input signal shown in Fig. 4 is considered, where T, is taken to be the sampling period, T the required ideal sampling period that covers the imposed characteristic resolution, T, the input-signal period, T,* (different from T and T,) the time after which the point having the cor- respondent position of that following in the ideal sampling the actual value is sampled, we therefore have the following relationship:

T, = NIT T,* = N2T, -k T T, = N,T

where N1 and N2 are integer numbers. On the basis of the previous relationships we may therefore deduct:

NZN, + 1 T,* = (N2NC + l )T = T C f

NI Since also T,* is a multiple of T,, we have:

NIN, + 1 = K

NI where K is a positive integer.

Finally, we obtain:

N1K - NZN, = 1 .

The previous equation may be solved if N1 and N , are primes of each other [15]. If N, is a power of two, the solution requires N1 to be an odd number.

BRANDOLINI AND GANDELLI: TESTING METHODOLOGIES FOR ADC’s 599

I ? 10 Y a

t

I I 1 34 5 C l l Y l I ( I Z ~ H ~ l C # I 11 13 15 I

Fig. 4. The circular sampling technique applied to the quantized triangular input signal.

In this way we can properly sample the input signal (and consequently the T.F.) without using in practice high sampling frequency. The sampled points obtained using this methodology need to be reordered to perform correct Walsh transform computation. The circular sampling is more general than the classic equivalent-time sampling, that can be assumed as a limit case of this methodology.

VII. THE TEST SYSTEM The test system is based on a computer suitably

equipped with the required interfaces to give complete versatility for the possible connections of A/D conversion systems. Also, by means of one of these interfaces, test structure for the integrated ADC’s can be connected, that not only has a wide range of fast memories but is also equipped with all the auxiliary devices required for timing, interfacing, and powering the integrated circuits examined. The system is completed by a programmable waveform generator (managed by the computer) which supplies the input signal to the devices under test.

The main factors which condition the correct evolution of the test are: the input standard signal to the conversion system and the sampling time. The relationship between the frequency of the input signal and the sampling time is clarified by the considerations described previously for the convenient sampling of the triangular waveform. Errors in generating the above-mentioned signals are reflected by a decrease in test system accuracy.

The standard signal generation is the fulcrum for the achievement of the best accuracy, and the distortion introduced in the real behavior of the standard signal may be very detrimental. In fact, because of the high sensitivity of the test method, any distortion introduced in the standard input signal results in a wrong evaluation of final parameters. For this reason the minimum accuracy of the standard signal should be at least double that of the one chosen to examine the device. Besides, the input signal should have an amplitude equal to the rated sweep of the converter being checked.

It should now be clear that the necessary characteristics for input signal and sampling timing are practically at- tainable only with the use of digital waveform synthesiz- ers with high DAC resolution. Any other standard generator may therefore be more difficult and more expensive to realize. The experimental setup built by the authors is equipped with a waveform generator by Analogic Co., model 2020/2000, whose performance is guaranteed by proper calibration and maintenance procedures. The global accuracy (including linearity) is guaranteed better than 0.025 % .

Another main problem arises with the choice of the correct field in which the dynamic performance of ADC’s under test is analyzed. Generally speaking, the characterization of an integrated converter can be obtained by changing the range of the input-signal spectrum from about 1 Hz to 130% of the maximum operating frequency stated in the component’s specifications. Here an excel- lent reading may be achieved by employing the logarith- mic distribution of the sampling frequencies. The test ex- ecution can be entirely managed by the computer, thus making performance of the test completely automatic.

VIII. TEST ACCURACY AND EXPERIMENTAL RESULTS Analyzing the final results the following consideration

can be provided. A) The choice of a constant-derivative standard signal

(the triangular waveform) guarantees the better results in terms of accuracy and sensitivity because every quantization step is explored with the same number of points. Therefore, the input signal is synthesized starting from digital values and using a high-resolution, low-noise and high-linearity DAC . The quantization error introduced by the device does not influence the test because of the higher resolution of the DAC generating the standard input signal with respect to the ADC under test. The characteristic resolution, d , influences, on the other hand, the final accuracy of the result.

B) The choice of global parameters obtained, through computation, by the error coeficient vectors allows one to define completely the accuracy of the conversion pro- cess, looking at both global and local characterization of T.F. errors. The method accuracy can be increased to the required level in function of the standard input-signal accuracy. No other parameters influence the final degree of accuracy. It is evident, in this way, that this test methodology results in a more complete characterization of ADC T.F. if compared with the traditional test methods using classical parameters unable to provide the T.F. re- construction.

The above considerations are justified by the rational approach introduced to define and study the T.F. and by the injective relationships existing between the T.F. and the mathematical parameters used for its definition. Pre- vious considerations suggest a massive industrial appli-

600 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 41, NO. 5 , OCTOBER 1992

TABLE I QUADRATIC RESIDUES R,,4096 (TOP) A N D MEAN ERROR COEFFICIENT

(BOTTOM) DETECTED BY VARYING THE SIGNAL A N D THE SAMPLING FREQUENCIES DURING THE TEST OF A N %BIT ADC, TESTED WITH d = 8

(4096 SAMPLES).

1000 Hz

3000 Hz

5000 Hz

8000 Hz

10000 Hz

15 30 61 122 244 488 Hz Hz Hz Hz Hz Hz

0.145 0.099 0.029 0.048 0.081 0.092 0.271 0.268 0.245 0.256 0.274 0.378

0.269 0.255 0.237 0.248 0.265 0.380 0.085 0.075 0.023 0.181 0.094 0.060 0.259 0.243 0.246 0.287 0.276 0.396

0.253 0.226 0.251 0.259 0.279 0.375

0.253 0.226 0.236 0.265 0.263 0.363

-0.022 -0.078 -0.015 0.028 0.011 0.062

0.064 -0.040 0.031 0.027 0.073 0.038

0.064 -0.040 0.061 -0.089 0.067 0.113

cation of automatic test systems based on the described test methodology because it is certainly adapted to guar- antee high-level certification procedure. Considering the ADC as a measuring transducer it is in fact possible to define and standardize accuracy classes related to the above-mentioned error parameters or other correlated parameters satisfying the industrial qualification require- ments. In our testing system prototype the aforemen- tioned methodology has been successfully applied in testing different A/D converters and conversion systems so that high-accuracy dynamic characterization of these devices can be obtained [ 161, [ 181.

Table I presents the resume of quadratic residues and mean value errors evaluated as functions of the sampling frequency and input-signal frequency. These parameters have been obtained by testing an 8-bit ADC with characteristic resolution d = 8. Values tabulated are computed starting from sample 1 and considering the interval covering all the samples (i.e., R1,4096, E1.4096) .

Mean value errors are expressed in terms of LSB, quadratic residues as LSB2. For example, the value of E,,4096 computed forfs = 122 Hz andf, = 5 kHz equal to 0.181 means that an error distributed on the whole T.F. extension exists, therefore comparable to an offset error of 0.181 LSB.

Mean value errors can be compared with quadratic residues to recognize the entity of the single errors in the T.F. Because:

~ 1 , 4 0 9 6 = X ; + X: + * + X ~ ~ - I

NcE;,4096 = xi + xi -k . . . + xi it is possible to demonstrate that if R1.4096 N$:,4W6 then the error distribution is uniform and mainly polarized (and in this case it shows a significant offset error and an integral nonlinearity error). On the contrary, the condition R1.4096 > NcE:,4096 reveals the presence of significant oscillations in the error distribution.

In Table I the experimental results validate the first statement for the ADC under test.

IX. CONCLUSION Original matter presented and discussed in this work

can be summarized as:

i)

ii) iii)

iv)

v)

vi)

vii)

the definition of A/D converter T.F. by means of Walsh transform coefficients; the definition of the error coeflcient vector; the general classification of A/D conversion errors; several examples of error parameters and their interpretation; the introduction of time-equivalent circular sampling; the choice of the triangular waveform as standard input signal; the hardware built and the software developed to implement the above-mentioned test methodology.

Indeed, the precise purpose of this study was to enhance the accuracy of ADC testing methods and their application to the systems based on A/D conversion, including the checking and testing of intrinsic characteristics of more complex digital instruments, already performed with conventional measuring apparatus.

The characterization of the devices tested using this method is more complete than that obtained by conventional testing methods. To be more specific, the parameters defined can be evaluated as the sampling rate and the input-signal spectrum change, and no! in just one given operating state. Moreover, investigation on the transfer function with regard to both local and overall properties is possible by means of a limited number of elements defined by rational numbers. Furthermore, the relative ease of applying this new approach may lead to the introduction of similar systems for the certification of components used in industry [ 171, [ 181, [20].

APPENDIX I

Deduction of the Mean Error Coeficients Let the mean error of j successive samples in the T.F.

domain ( U ) , starting from sample i = Nc(7 - l ) / h , (7 =

1 , 2, , h } , where h is the number of uniform intervals in which is divided the T.F. , with j and h powers of two, be given by:

i f i - 1

E;,; k = i

where xk are the time samples resulting from A ( U ) .

representation we obtain: Substituting the xk with the inverses of their transform

4.; = ( 1 / j [HI XwJ

where [HI is the Walsh-Hadamard matrix [lo], X, the vector of the Walsh coefficients X, and J the unitary vec-

BRANDOLINI AND GANDELLI: TESTING METHODOLOGIES FOR ADC's

I C

; E s

tor with N , elements, all nulls except those from element i to i + j - I , which are equal to 1.

By means of the Walsh-Hadamard properties [lo], [19], solving the matrix-vector product, it results in:

I I I I I I I I l l 1 l l I l I l l I I !A 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I l I I I I I I I I I I I I

I l 1 1 1 1 1 1 l l 1 1 l I I I 1 1 1 1 1 1 1 1 1 l I I l l I l 1 1 l 1 l 1 1 I 1 1 l 1 1 1 1 1

= X,I

where I is a partition of [HI resulting in a transposed vector function of i a n d j . Rewriting the same expression in series form, we obtain:

a

= C X,WAL, n = O

Deduction of the Quadratic Residues The quadratic residues can be evaluated considering the

sum of all the squares, appropriately weighted, of the mean error coefficients included in the interval covered by the j samples.

It results in this way:

where a = N c / j - 1; p = k N c / j + n.

j = N,, the quadratic residue becomes: In this particular case when, for example, i = 1 and

Nr - 1

which corresponds to the quadratic sum of the Walsh error coefficients, resulting in a proportional value to the absolute sum of all the error areas.

APPENDIX I1 In the following appendix we resume and show several

other parameters, usually employed in measurement system analysis performed on digital energy meters [16], 1181.

The odd coeflcient sums, Gi,j and even coeficient sums, Di,j, are defined as:

F F / \

where L

D ; , ~ = C x,, WAL, n = O

L

~t~ = C X~WAL,,, n = O

( = 2n + [(h + l ) N , ] / j a n d ~ = 2n + 1 + [ (h + lYVpl/j

IDEAL T.F D I

60 1

I REAL T.F. D I G I T A L

C 0 D E S

vi"! 1 I I . l I I I I I I I I I I I I I I l l I l l

I I I I I I I I I I I I I I l l l I l l l l l l l l I l I 1 1 l l 1 1 1 1 1 1 1 1 1 1 1 1 1 1 l 1 1 1 1 1 1 l I I I I

I I

l 2 3 4 5 6 1 8 9 1 0 1 1 1 2 1 3 1 4 1 S 1 6 I 2 3 ~ 5 6 1 8 9 1 0 l 1 1 2 1 3 1 4 1 5 1 6 SAMPLED POINTS - SAMPLED POINTS

Fig. 5 . The 2-bit extension interval of the T.F. used to show error coefficient properties.

s ~ ~ I ~ ~ I 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

IINTERVAL ZINTERVALS 4INTERVALS . 16lNTERVALS P - - - .

Fig. 6 . Error vector trend and the binary intervals used to test T.F. properties.

TABLE 11 ERROR COEFFICIENTS AND WALSH COEFFICIENTS FOR THE T.F. OF FIG. 5

Walsh Coefficients Sample A ( U ) = ere ( U ) - Q,d (U)

1 3 0.1875 2 0 0.5625 3 2 0.3125 4 0 0.1875 5 0 - 0.0625 6 - 1 0.5625 7 0 -0.4375 8 2 0.1875 9 0 0.5625

10 - 1 - 0.0625 1 1 - 1 -0.0625 12 0 -0.1875 13 1 0.5625 14 1 0.1875 15 -3 0.4375 16 0 0.0625

extended to j samples ( j being a power of two) and

i = 1 + k j { k = 0, 1 , , L; h = 1, 2 , , j ;

L = N , / ( 2 j ) - 1).

The even coeficient sums reveal the presence of hysteresis in this section i + ( i + j - 1) with respect to the

c - I I

are the partial hysteresis sums starting from sample i and section ( N , - i - j + 1) + ( N , - i) if Di,; is different

602

TABLE 111 MEAN VALUE ERRORS A N D QUADRATIC RESIDUES COMPUTED FOR INTERVALS SHOWN I N FIG. 6

1 Interval 2 Intervals 4 Intervals

E , , , , = 0.1875 E l , 8 = 0.7500 E , . , = 1.2500 R , , , , = 1.9375 = 1.1250 R I , , = 0.8125

E2,4 = 0.2500 R 2 , , = 0.3125

E2.8 = -0.3750 E3.4 = -0.5000 R2,8 = 0.8125 R3,4 = 0.1250

E4,4 = -0.2500 R4,, = 0.6875

8 Intervals 16 Intervals

E l , 2 = 1.5000 E , , , = 3.0000 = 0.5625 R , , = 0.5625

E 2 , , = 0.0000 R 2 , , = 0.0000

R 2 , 2 = 0.2500 R ? , , = 0.2500

R 4 , , = 0.0000 E3,2 = -0.5000 E , , , = 0.0000 R, ,2 = 0.0625 R S , , = 0.0000

R , , , = 0.0625

= 0.2500 R7,, = 0.0000 E8, I = 2.0000 R 8 , , = 0.2500

R s , z = 0.0625 R 9 , , = 0.0000

R , , , , = 0.0625 E6.2 = -0.5000 E , , , , = -1.0000 R 6 , 2 = 0.0625 R I , , , = 0.0625

E,, , , = 0.0000 R I , , , = 0.0000

E , , 2 = 1.0000 E , , . , = 1.0000 R7,2 = 0.1250 R , , , , = 0.0625

EI4, , = 1.0000 R , , , , = 0.0625

Eg.2 = -1.5000 E , , , , = -3.0000 R8,2 = 0.5625 R , , , , = 0.5625

R , , , , = 0.0000

E 2 , 2 = 1.0000 E , , , = 2.0000

E , . , = 0.0000

E,,, = -1.0000

E 4 . , = 1.0000 E 7 , , = 0.0000

E 5 . 2 = -0.5000 E9.1 = 0.0000

E,,,, = -1.0000

E , , , , = 0.0000

from zero, and the error can be studied using the difference:

A . . = G . . - D . . 1.J 1.J 1 , J '

If Ai, j is equal to zero, only one of the two symmetrical sections examined differs from the ideal value. When Ai, j is different from zero, both the samples differ from their ideal value (this fact happens also without the presence of hysteresis).

Shift invariant power spectral points [lo], U ( k ) , are defined as:

U(0) = x;

U(k) = c (x; + X',)

U(P) = X i ?

L

i = 1

where: p = 2k(2i - I ) - 1 ; U = 2k(2i - 1); p = log2 N, + 1; L = (log2 N,)/2k-1. They represent the most concise way to show the global accuracy of the conversion system. They result from condensing the error se- quence information into a limited number of components.

sampled with characteristic resolution d = 2. In Fig. 5 the ideal and real T.F. trends are shown.

In Fig. 6 intervals in which the analysis of the T.F. can be performed are represented. In the example all the results are referred to only one sampling frequencyf, and only one signal frequency f,.

Table I1 reports the 16 values of the differences A ( u ) = Qre(u) - Qid(u) and the 16 Walsh coefficients computed from these differences.

In Table I11 the values for mean error coeficients and quadratic residues computed in all the binary intervals shown in Fig. 6 are collected.

Looking at Table I11 it can be assumed that the first half of the T . F. is more distorted than the second part (RI, 8 >

In particular, the first and the last quarters are more affected by errors than the second and the third quarters (R,,4 and R4,4 > R2,4 and R3,4). Mean value errors suggest that the real T.F. mainly stays over the ideal T.F. in the first half and under the ideal T.F. in the second half.

Odd and even coeficient sums and the differences, A i , j , evaluated for extensions of one sample, are presented in Table IV.

R2,8).

The Di. coefficients reveal that only samples 4 and 5 show absence of hysteresis in the first half of the T.F. (i.e., they present the same value of their specular points in the second half of the T.F.). The combined analysis of

APPENDIX I11 The purpose of this appendix is to present the test re-

sults for a 2-bit extension interval of a simulated T.F.

BRANDOLINI AND GANDELLI: TESTING METHODOLOGIES FOR ADC’s

TABLE IV ODD AND EVEN COEFFICIENT SUMS AND THE DIFFERENCES A,,,

Sample G,. I D,, I 4. I

0.28125 0.28125 0.28125 0.03125 0 0.125 0.03125 0.125

0.28125 0.28125 0.03125 0.03125 0 0 0.03125 0.125

0 0 0.25 0 0 0.125 0 0

TABLE V SHIFT INVARIANT POWER SPECTRAL POINT U(@.

Spectrum Point u(k) 0.0352 1.1562 0.3906 0.3516 0.0039

Di, I and Ai, provides the information that in the samples 0, 1 , 3 ,4 , 6 , 7 only one of the specular points differs from the ideal value of the T.F. , while the samples 2 and 5 (with or without hysteresis) are both different from the ideal T.F. amplitude. Errors in sample 2 are greater than in sample 5 because A2, > As, I .

To complete the test results Table V shows the shift invariant power spectral points. They are a concise way to present errors in the T.F. and are independent from the starting phase of the sampling procedure. The global analysis shows error oscillations more significant than error polarization; in fact, it results in R I , 16 = 1.9375 > N$,, 16

= 0.5625. Moreover, the hysteresis is distributed on the whole T.F.

ACKNOWLEDGMENT The authors would like to thank the reviewers for their

helpful comments.

I

I

I

I

[16] A. Brandolini and A. Gandelli, “Teoria delle catene di conversione analogico-numerica-Fase I ,” Contratto di Ricerca ENEL-CRA/Po- litecnico di Milano, Milano, 1989.

[17] T. R. McComb, J. Kuffel, and R. Malewski, “Measuring characteristics of the fastest commercially available digitizers,” IEEE Trans.

[18] A. Brandolini and A. Gandelli, “Teoria delle catene di conversione analogico-numerica-Relazione conclusiva,” Confratto di Ricerca ENEL-CRA/Politecnico di MiIano, Milano, 1990.

[19] F. Schipp, W. R. Wade, and P. Simon, Walsh Series, An Introduction to Dyadic Harmonic Analysis.

[20] B. E. Peetz, ”Dynamic testing of waveform recorders,” IEEE Trans. Instrum. Meas., vol. IM-32, Mar. 1983.

PWRD, vol. PWRD-2, July 1987.

Bristol, U.K.: Adam Hilger, 1990.

603

REFERENCES

A. Brandolini, E. Carminati, and A. Gandelli, “A new approach for AID converter testing,’’ IEEE InstrumentationlMeasurement Tech- nology Conference IMTC 88, San Diego, CA. Analog-Digital Conversion Handbook. Analog Devices, Prentice- Hall, 1986. Dynamic Performance Testing of A fo D Converters. Hewlett-Pack- ard, Product Note 5 180A-2. B. M. Gordon, The Analogic Data-Conversion System Digest. Wakefield, MA: Analogic Corporation. 1981. E. A. Sloane, “Application of Walsh functions to converter testing,” 1980 IEEE Test Conference, CH 1608-9180, 1980. E. A. Sloane, “A system for converter testing using Walsh transform techniques,” 1981 IEEE Test Conference, CH 1693-1/81, 1981. M. Vanden Bossche, J. Schoukens, and J. Renneboog, “Dynamic testing and diagnostic of A/D converters,” IEEE Trans. CAS, vol.

T. M. Souders and D. R. Flach, “An automated test set for high resolution analog-to-digital and digital-to-analog converters,” IEEE Trans. Instrum. Meas., vol. IM-28, Dec. 1979. R. A. Malewski, T . R. McComb, and M. M. C. Collins, “Measuring properties of fast digitizers employed for recording HV impulses,” LEEE Trans. Instrum. Meas., vol. IM-32, 1983. D. F. Elliott and K. R. Rao, Fast Transform. Algorithms, Analyses, Applications. K. G . Beauchamp, Application of Walsh and Related Functions. New York: Academic Press, 1984. A. Gandelli, Teoria delle catene di conversione analogico-numerica, Ph.D. Thesis, I1 ciclo, Milan-Rome, 1988. G. P. Baldi and G. G. A. Ceiner, Verijca dei convertitori analogico- numerici mediante 1 ‘us0 della trasformata di Walsh, Ms.D. Thesis, Politecnico di Milano, Milano, Italy, Mar. 27, 1991. G. D’Antona and A. Gandelli, “Circular sampling techniques to test A/D converters using Walsh transform,” submitted for publication to Signal Processing. G. W. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. Oxford, England: Oxford Science Publications, Oxford University Press, Oxford, 1988.

CAS-33, Aug. 1986.

New York: Academic Press, 1982, ch. 8.

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Testing methodologies for analog-to-digital converters