Study of the Variane in the Histogram Test of ADCs F. Corrêa Alegria and A. Cruz Serra Teleommuniations Institute and Department of Eletrial and Computer Engineering Instituto Superior Ténio, Tehnial University of Lisbon, Portugal. Laboratorio de edidas Eletrias, IST, Av.Roviso Pais, 1049-001 Lisboa, Portugal Telephone: +351 1841789, Fax: +351 1841767,E-mail: falegria@lx.it.pt, aserra@ist.utl.pt Abstrat In this paper an overview of the unertainty in the results obtained with the histogram test of Analog to Digital Converters is presented. The effet of phase noise, input equivalent additive noise and random phase differene on the number of ounts of the umulative histogram is shown. Theoretial and experimental results are presented. I. INTRODUCTION The histogram test method is a tool widely used for the haraterization of analog to digital onverters (ADCs). This test gives information about the deterministi harateristis of the onverters through the transfer funtion from whih the gain, offset, transition voltages and ode bin widths an be obtained. A signal with a nown amplitude density funtion is used to stimulate the onverter. Usually a sinusoidal stimulus signal is used beause it is the signal that an be more easily generated with a low distortion. Several samples are taen and the onverter transfer harateristi is determined by omparing the output odes experimentally obtained with the ones expeted from an ideal onverter. The results of the haraterization suffer from errors aused by different non idealities in the test setup, namely the stimulus signal amplitude, offset, frequeny, additive noise, phase noise, and in the sampling signal frequeny and phase noise. Also the presene of additive noise and of aperture unertainty in the ADC itself, that are not haraterized with this test method, have an adverse influene in the results obtained. The errors referened in the previous paragraph an be lassified in two types: deterministi errors, that results in a inauray of the results, and an be orreted if determined; and random errors that lead to an unertainty in those same results and an not be orreted but should be speified. The results an thus be seen as random variables with a statistial distribution that an be onsidered normal. The nowledge of its mean and variane allow the determination of a onfidene interval that ontains the exat value of the ADC harateristi with a ertain degree of onfidene. In the following setions the determination of the variane of the transition voltages is studied. II. TRANSITION VOLTAGES DETERINATION There are two approahes that an be taen in the determination of the transition voltages, namely, using the histogram or the umulative histogram. An histogram is onstruted by ounting the number of samples reeived in eah of the output odes and the umulative histogram is onstruted by ounting the number of samples whose output ode is equal to or smaller than eah of the possible output odes. The two approahes produe, ideally, the same results, however, the use of the histogram, leads to an iterative omputation with the onsequent aumulation of errors. The umulative histogram is thus traditionally used. The probability distribution of the transition voltages an thus be determined from the probability distribution of the number of ounts of the umulative histogram. The error soures mentioned in the previous setion are usually ombined into three effets: i) Phase noise that translates the effet of the funtion generator phase noise, lo generator phase noise and ADC aperture unertainty; ii) Input-equivalent noise whih ombines the noise of the stimulus signal generator and the additive noise of the ADC [1]; iii) Random phase differene whih is due to the fat that the samples are aquired at a onstant rate but the start of the aquisition is in no way related with the stimulus signal and thus the value of the stimulus signal in the sampling instants is not always the same for different realizations of the test. This allows the sample phases to be uniformly distributed allowing the determination of the expeted output odes from the stimulus signal amplitude distribution funtion. Stimulus signal and sampling signal frequeny errors will ause the samples phases to stop being uniformly distributed. III. VARIANCE In 1984 Doernberg et al. studied the unertainty in the determination of the ode bin width []. That wor too into aount only the effet of input-equivalent noise with a variane muh greater than the ideal ode bin width. In those onditions the sampling voltage was onsidered uniformly distributed inside eah ode bin. Later on, in 1994, Blair produed great advanements in this field [3]. In his wor, both the effets of input-equivalent noise and random phase differene between the stimulus signal and the sampling lo were onsidered. Unertainty expressions for the ode bin width as well as for the transition voltages were presented and inluded in the de fato standard for ADC testing [4]. The effet of phase noise, not taen into aount by Blair, was studied in 1999 by Chiorbolli et al. [5]. An addition to
the expressions from Blair was made to tae into aount this effet. The analysis of the unertainty of the ode bin widths and transition voltages done in the wors referred, too into aount eah of the three effets separately and the results obtained were added to eah other. In [6] a unified approah was presented that studies the influene of these effets together providing a more preise view of the ontribution of eah effet to the unertainty of the transition voltages. It was shown that the probability density funtion of the number of ounts of the umulative histogram an be determined from the number of aquired samples () and from the probability that a sample belongs to a lass of the umulative histogram (p ). The expression presented for the mean was π = p ( ) d γ γ π (1) and for the variane was σ = +σ σ π π = p ( γ) 1 p ( γ) dγ σ ϕ π 1 π π j +ϕ j= 0 π σ p ( ) = d p d ϕ ϕ γ γ π 0 The probability that a sample belongs to a lass of the umulative histogram depends, in turn, on the normalized input-equivalent noise standard deviation (σ n ), whih is equal to the standard deviation of the input-equivalent noise (σ) divided by the stimulus signal amplitude (A), on the phase noise (σ ) and on the normalized transition voltage (U[]) and sample phase (γ). ( γ) aos( y) +γ πn aos( y) +γ πn U [ ] + 1 1 σ ( u y) σ n= e + e σn 1 πσnσ 1 y p = e dydu (3) The expression for the variane was divided into two terms named the mean of the onditional variane ( ) and the σ variane of the onditional mean ( σ ). A. ean of the onditional variane The dependene of on the standard deviations of phase σ noise and input-equivalent noise, determined in [6], is represented in Fig. 1. () Fig. 1 Representation of the term divided by the number of aquired σ samples () for a transition voltage equal to the stimulus signal offset (U[+1]=0) as a funtion of the standard deviation of the phase noise (σ ) and of the standard deviation of the input-equivalent noise divided by the stimulus signal amplitude (σ n). This term, as an be seen from equation (), is diretly proportional to the number of samples (). For small values of the standard deviations this terms inreases proportionally to the standard deviation. In partiular, in the absene of one of the noises soures, the rates of inrease are equal. σ σ n 0 n ϕ σ σ = 0 π π (4) σ σ 0 ϕ σ σ n = 0 π π In [3] the same rate was obtained in the ase of the presene of input-equivalent noise alone (1.13/π). For large values of standard deviation the term approahes /4. This σ behavior was overlooed in [3] for the ase of the umulative histogram (and the transition voltages). There the dependeny on the standard deviation was onsidered always linear. The same ourred in [5] in the ase of phase noise exlusively. B. Variane of the onditional mean The dependene of the term σ on the number of samples and the transition voltage is more ompliated then the term. Fig. represents this term in the ase of σ absene of phase noise. In the ase of absene of inputequivalent noise the dependene on the standard deviation of the σ is similar to the one depited in Fig..
to onsidering that effet was equal for all values of standard deviation. C. Variane of the number of ounts The variane of the number of ounts is determined by adding the terms and σ σ ϕ. The result obtained an be seen in Fig. 4. Fig. Representation of the term σ in the absene of phase noise (σ =0) and 5 aquired samples as a funtion of the normalized transition voltage (U[+1]). For small values of the standard deviation of the noises soures the term σ depends strongly on the transition voltage. The number of ars seen in Fig. is equal to the number of samples. The maximum value of this term ours in the absene of phase noise and it is equal to ¼. If there is an error in the stimulus signal frequeny or in the sampling frequeny the value of this term an be higher as was seen in [9]. As the standard deviation of noise inreases the term σ dereases until it reahes 0 for high values of the standard deviation (Fig. 3). Fig. 3 Representation of the term σ for a transition voltage equal to the stimulus signal offset (U[+1]=0) and 5 aquired samples as a funtion of the standard deviation of the phase noise (σ ) and of the standard deviation of the input-equivalent noise divided by the stimulus signal amplitude (σ n). For small values of the standard deviations this terms dereases proportionally to the standard deviation with the same rate as the term. σ In [3] the effet of the random phase differene between the sampling lo and the stimulus signal was onsidered separately from the effet of input-equivalent noise. This lead Fig. 4 Representation of the term σ for a transition voltage equal to the stimulus signal offset (U[+1]=0) and 5 aquired samples as a funtion of the standard deviation of the phase noise (σ ) and of the standard deviation of the input-equivalent noise divided by the stimulus signal amplitude (σ n). For high values of standard deviation, the variane of the number of ounts approahes /4 as was the ase for term. For low values of standard deviation the variane in σ the number of ounts beomes onstant and equal to ¼. D. Variane of the Transition Voltages In [6], an asymptotially approximate expression was presented for the variane of the transition voltages: ( ) 1 1 A Aπ σ +σ σt max, min, (5) 4 4 π π There maximum relative error in using equation (5) is lower than 17%. Note that this is the error of the determination of the error interval of the transition voltages, determined with the histogram method, and not of the transition voltages themselves. There is no exat expression, however, more exat but more ompliated expressions ould be determined. The expression presented in [5] is now reprodued, with the notation used in this paper, for omparison with the proposed expression. Aπ σ 1 1 σt + σ + A π 4 (6) π π π This expression has a greater error than the one proposed, namely the variane inreases to infinite when the standard deviation tend to infinite instead of approahing /4. This behavior is aounted for with the minimum funtion in equation (5).
Altough in the traditional histogram test, where a full sale stimulus signal is used, the standard deviation of the input equivalent is generally muh smaller than the stimulus signal amplitude ( σn 1 ), there is a test, using small amplitude stimulus signals [7, 8], where the standard deviation of the input-equivalent noise may be omparable to the stimulus signal amplitude. The effet of the random phase differene between the stimulus signal and the sampling lo is better aounted for with the maximum funtion in equation (5) instead of being added to the effets of the other fators (term ¼ in equation (6)) as was done in [3], ausing an overestimation of the variane. IV. FREQUENCY ERRORS To guarantee that all odes have equal opportunity to be stimulated the number of samples must be aquired during an integer number of periods of the input signal (D). Denoting by the number of samples aquired, the frequeny of the stimulus signal (f) and the frequeny of the sampling lo (f s ) must satisfy the following relation ρ= f D f =. (7) s This relation, however, does not guarantee that the sample phases are evenly distributed. To ahieve that the numbers D and must be mutually primes [3-5]. In pratie the referred frequenies do not verify (7) exatly, ausing the sample phases not to be uniformly distributed as it was originally intended. In [9] a study of the influene of errors in both frequenies in the variane of the umulative histogram was presented. In Fig. 5 a three dimensional representation of presented in the ase of absene of noise. σ is the normalized transition voltage U[] (transition voltage divided by the stimulus signal amplitude). Analyzing Fig. 5 we find that when ρ is a rational number with a denominator of 5 () and a numerator mutually prime with 5, we have a minimum in the maximum value (over α) of the variane (ρ: 0.; 0.4; 0.6 and 0.8). Also when the value of ρ is one of the elements of the Farey sequene [10] of order 4 (-1) we have a loal maximum of the variane (ρ: 0.5; 0.33; 0.5; 0.66 and 0.75). The Farey sequene of order P is omposed by the rational numbers, in inreasing order, with a denominator equal to or smaller than P. This value of the variane of the number of ounts of the umulative histogram, represented in Fig. 5, is the value that should be used in (5), instead of the first term ¼, when there are frequeny errors. The value of ¼, also used in (6), is the maximum value (over α) of the variane in the absene of frequeny errors. In Fig. 6 the variane of the number of ounts of the umulative histogram is represented as a funtion of the phase interval length α in the absene of frequeny error (dashed line) in the ase =7. Seven idential paraboli ars an be seen in the figure as was expeted. The maximum value of the variane is ¼ in the absene of frequeny error. Even in the presene of frequeny error the variane is equal to or lower than ¼ if ρ 1 (8) ρ D as was demonstrated in [11]. This ase is depited with a solid line in Fig. 6. Fig. 6 - Representation of σ as a funtion of α for D= and =7 in the ases of ρ=d/ (dashed line) and ρ=d/+1/ (solid line). Fig. 5 - Representation of σ for different values of ρ and α (Μ=5). The variable α represents the length of the phase interval within whih the samples belong to a ertain lass of the umulative histogram. The value of α is diretly related to The results represented in Fig. 6 are equivalent to the ones of Figure of [11] validating the approah taen in this paper. Also to validate the wor presented in [9, 1] real tests of ADCs were performed [1]. In Fig. 7 experimental results are shown. In that test 5 samples were aquired with a sampling lo of 100Hz. A triangular shapes stimulus signal with 30Hz of frequeny was used. This onditions orrespond to the presene of frequeny errors beause the relation between
the stimulus signal frequeny and the sampling lo frequeny (ρ) do not satisfy (7): 30,000 3 1.5 ρ= = =. (9) 100, 000 10 5 In Fig. 7 one an observe a good aordane between the experimental values, represented by vertial bars (that translate the unertainty of the estimation of the varianes), and the theoretial values (solid line). σ 0,7 0,6 0,5 0,4 0,3 0, 0,1 0 0 0, 0,4 0,6 0,8 1 α Fig. 7 Representation of the experimental results of the variane of the number of ounts of the umulative histogram (dots) in the ase of 50% frequeny error (=5). Vertial bars are used to represent the 99.9% onfidene interval. The theoretial value of the variane is represented by a solid line. V. CONCLUSIONS An overview of the study of the unertainty in the determination of the transition voltages with the histogram method was presented highlighting the wor of the authors in this partiular subjet [6, 9, 1]. An expression, for the determination of the referred unertainty, whih is asymptotially approximate but more aurate than the one used in the de fato standard [4] and subsequently expanded [5] was determined. The use of the presented expression allows for a better estimation of the unertainty of the transition voltages and for a more exat dimensioning of the number of samples required leading to a faster test. An analytial expression for the variane in the ase of frequeny errors is not urrently nown. The value ¼ an be used for small frequeny error. The possibility of determining the variane of the number of ounts of the umulative histogram for every value of ρ allows the determination of the unertainty of the results when the test of an ADCs was performed, for whatever reason, with frequenies that do not satisfy (8). The wor presented here is subjet of onsideration for inlusion in the IEEE standard for ADC haraterization. There is however still wor to be done in this subjet, namely the study of the unertainty of the ode bin widths and the determination of an analytial expression for the variane in the ase of frequeny errors. VI. ACKNOWLEDGENT This wor was sponsored by the Portuguese researh program PRAXIS XXI, projet entitled Analog to Digital onverters haraterization, ref. P/EEI/13170/1998, whose support the authors gratefully anowledge. VII. REFERENCES [1] Paolo Carbone and Dario Petri, Noise Sensitive of the ADC Histogram Test, Proeedings of the IEEE Instrumentation and easurement Tehnology Conferene, St. Paul, innesota, ay 18-0, 1988, pp.88-91. [] Joey Doernberg, Hae-Seung Lee and David A. Hodges, Full-Speed Testing of A/D Converters, IEEE Journal of Solid-State Ciruits, vol. SC-19(6), pp. 80-87, Deember 1984. [3] Jerome Blair, Histogram easurement of ADC Nonlinearities, IEEE Trans. on Instr. and eas., vol. 43, pp. 373-383, June 1994. [4] IEEE std. 1057-1994, IEEE Standard for digitizing waveform reorders, The Institute of Eletrial and Eletronis Engineers, In., New Yor, De. 1994. [5] G. Chiorboli and C. orandi, About the Number of Reords to be Aquired for Histogram Testing of A/D Converters using Synhronous Sinewave and Clo Generators, in Proeedings of the 4th Worshop on ADC odeling and Testing, Bordeaux, Frane, 9-10 September 1999, pp. 18-186. [6] F. Corrêa Alegria, A. Cruz Serra, Unertainty in the ADC Transition Voltages determined with the Histogram ethod, submitted for publiation in the Proeedings of the IEKO 001 Congress, September 13-14, 001, Lisbon, Portugal. [7] F. Alegria, P. Arpaia, A. Cruz Serra, P. Daponte, ADC Histogram Test by Triangular Small-Waves, aepted for publiation on the proeedings of the 18 th IEEE ITC Conferene, ay 001, Budapest, Hungary. [8] F. Alegria, P. Arpaia, A. Cruz Serra, P. Daponte, ADC Histogram Test using Small-Amplitude Input Waves, aepted for publiation on easurement, Elsevier. [9] F. Corrêa Alegria and A. Cruz Serra, Influene of Frequeny Errors in the Variane of the Cumulative Histogram, aepted for publiation on the IEEE Trans. on Instrum. and eas. [10] R.Graham, D.Knuth and O.Potashni, Conrete athematis. nd edition, Addison-Wesley, 1994. [11] Paolo Carbone and Giovanni Chiorboli, ADC Sinewave Histogram Testing with Quasi-hoerent Sampling, Proeedings of the 17 th IEEE ITC Conferene, vol. 1, pp. 108-113, ay 000, Baltimore, USA. [1] F. Corrêa Alegria and A. Cruz Serra, Variane of the Cumulative Histogram of ADCs due to Frequeny Errors, aepted for publiation on the proeedings of the 18 th IEEE ITC Conferene, ay 001, Budapest, Hungary.