NEW APPROACH TO THEORY OF SIGMA-DELTA ANALOG-TO-DIGITAL CONVERTERS. Valeriy I. Didenko, Aleksander V. Ivanov, Aleksey V.

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1 NEW APPROACH TO THEORY OF IGMA-DELTA ANALOG-TO-DIGITAL CONVERTER Valery I. Ddenko, Aleksander V. Ivanov, Aleksey V. Teplovodsky Department o Inormaton and Measurng Technques Moscow Power Engneerng Insttute (Techncal Unversty) Krasnokazarmennaa treet 4, 50, Moscow, Russa Ph: Fax: Abstract The man pont o new approach to the theory o sgma-delta modulaton s applcaton o a dscrete two-values dstrbuton law or a quantzaton nose nstead o a unorm dstrbuton law accepted beore. Due to new approach, the varance (standard devaton n the square) o the quantzaton nose becomes dependent on nput sgnal by parabolc uncton. The nose vs. requency s ound or derent nput sgnals takng nto account all requency range rom zero to hal o samplng requency. Usng dependence o standard devaton on nput sgnal and requency, derent characterstcs o sgma-delta modulaton can be predcted, ncludng NR. Derence between analytcal and smulaton results or new theory can be drven to any small value.. Introducton A quantzaton nose o sgma-delta modulaton was supposed to be descrbed by a unorm dstrbuton law as n the case o a quantzaton error or any ADC [-3]. Though ths approach was not evdent [] and gave gross errors [4], t was accepted n all papers known to the authors. As a result, the quantzaton nose or a modulator wth a smple comparator was ound as an addtve error wth a standard devaton σ = The comparator was represented as an ampler ollowed by a 3 nose voltage source. The quantzaton nose was supposed to have constant spectral densty ( whte nose ), though dscrete spces dependng on nput sgnal were mentoned []. Denton o equvalent comparator gan was gven usng smulaton [] dependng on sgnal value. For low requency ormula or NR was ound [] as: 3a (n + ) R n + NR = 0log () α π n where n s the modulator order, a s the nput sgnal ampltude, α s the gan o nose transer uncton, R s the oversamplng rato equal to rato o comparator samplng requency to samplng requency at the output o the dgtal lter [, 3]. The latter requency s usually supposed to be equal to double maxmum requency o nput sgnal. The value o α was usually supposed equal to [-4]. The standard devaton can be ound rom () as: n απ σ ( ) = () n+ 3(n + ) R 3 For the rst order modulator n =, σ ( ).05α R, 3 NR 0 log 0.456a R / α.. New approach to modellng o sgma-delta modulator The quantzaton nose o the sgma-delta modulator s a varable even or constant nput drect current sgnal. Ths varable s represented at the output o the modulator Y by one o two possble values: V REF or V REF. The rst value s usually shown as +, the second value s. Then the nput sgnal range s X. For quantzaton nose analyss, all modulator components (swtches, ntegrators, a comparator) are supposed to be deal. Then the expectaton o a modulator output must be equal to nput sgnal X and only two levels o error take place: Δ = --X or Y= -V REF and Δ = -X or Y=

2 V REF. At any random tme moment, the realzaton o Δ or Δ s a random case. Probablty o the rst case s P = 0.5(-X) whle probablty o the second case s P = 0.5(+X). Ths random quantty s descrbed by dscrete two-value dstrbuton law (not by unorm one!) that s shown n Fg.. ( Δ ) --X -X Δ Fgure. Dstrbuton law o the quantzaton nose or a sgma-delta modulator Two δ - unctons (Drac unctons) are shown n Fg. representng P and P correspondngly. The expectaton o the probablty law: M( Δ ) = - (+X)0.5(-X) + (-X)0.5(+X) = 0 (3) Varance o the probablty law: σ = (--X- M( Δ )) 0.5(-X) + (-X- M( Δ )) 0.5(+ X) = -X (4) Equatons (3) and (4) are true or any order o a modulator, both or a contnuous-tme one and a dscrete-tme modulator. The rst order contnuous-tme modulator s analyzed below n deault. In opposte to many prevous papers (see secton ), the varance gven by (4) depends on nput sgnal. It s gong to zero when nput sgnal s closed to ends o the range (X=±). For X = 0 t s equal to and three tmes more n comparson wth the value used beore. Derent ways can be used to check (4) by smulaton. Let s consder a sequence o N pulses at the modulator output ncludng N postve and N- negatve pulses. Evaluatons (5) and (6) can descrbe expectaton o nput sgnal and standard devaton correspondngly: X = ( N N ) / N (5) X σ = (6) I N corresponds to the perod o pulses at the output o the modulator, then X and σ. uch stuaton takes place, accordng to (5), product o N by 0.5(+X) gves the entre number o N (the entre number o N- = N - N correspondngly). uch sgnals are named n papers [, ] as dle tones wthout detaled analyss. Maxmum relatve delectons between X and X, σ and σ equal correspondngly to: X = σ = δ X = ± NX (7) ( X N ) δ σ N( X ) (8) Both relatve errors can be made as low as necessary the value N s large enough. I X ncreases, then more value N s necessary or gven relatve error. Equaton (4) corresponds to all requency range o nose rom 0 to 0.5. The latter value has no connecton wth the sample theorem, as s usually supposed [], and s explaned by maxmum possble rate o postve and negatve pulse change. For low (hgh) X hgh (low) requency o quantzaton nose s more typcal. I absolute value o X s dstrbuted unormly between 0 and, then the constant spectral densty o nose appled to the comparator output can be supposed (model o the modulator descrbed n secton s assumed, but value o nose evaluaton s ound by other way). The output spectral densty o the modulator s:

3 ωτ ) OUT = COMP ( + ( ) (9) where COMP and are the spectral densty and equvalent gan o the comparator correspondngly, τ - tme constant o an ntegrator. The varance o ths sgnal n requency range rom zero to any requency s: I πτ σ ( ) = COMP ( arctg ) (0) πτ = 0. 5, then (0) must be equal to (4). From ths condton COMP can be ound as: COMP = X πτ ( arctg ) πτ It s clear rom () that the varance o quantzaton nose or the comparator s more than or the modulator output. Ths act s explaned by nluence o negatve eedback at requency 0.5. One can nd rom (0) and (): πτ ( arctg ) πτ σ ( ) = ( X ) () πτ ( arctg ) πτ I requency bandwdth <<, then () s smpled: πτ 3 ( ) ( ) σ ( ) = ( X ) (3) πτ 3 ( arctg ) πτ The value o equvalent comparator gan can be chosen rom derent condtons. For example, a tme delay o the true modulator at low requency must be the same as or lnear model. The delay o the modulator or derent sgnal s ound wthn nterval rom zero tll. Average value o the delay s 0.5. To realze ths delay at low requency, the equvalent gan o the comparator must be equal to =. Then () and (3) are transormed correspondngly to: τ σ ( ) () R π ( arctg ) = ( X ) π R (*) π R( arctg ) π 3 σ ( ).5 ( X ) / R (3*) The coecent R n (*) and (3*) s the rato o maxmum nose requency at the output o the modulator 0.5 to the requency band where the quantzaton nose s consdered. The latter requency can be wthout any connecton wth requency o the nput sgnal. Thereore the coecent R s better to name as requency band rejecton nstead o oversamplng rato accepted now. I snusodal sgnal wth ampltude a s appled to the modulator nput, then the varance o quantzaton nose equals to:

4 πτ ( arctg ) πτ σ ( ) = ( 0.5a ) (4) πτ ( arctg ) πτ The NR at low requency or = τ can be ound approxmately as: 3 NR 0lg 0.9R a /( 0.5a ) (5) It s nterestng to say that derence o NR ound by new and ormal approaches s not so strong as derence o standard devatons. 3. Results o smulaton We used the applcaton program Matlab 6.5 and bult-n smulaton program mulnk 5 and the standard oversampled gma-delta scheme o A/D converter wth the rst order modulator shown n Fg.. The samplng requency or ths model o ADC s 5 04 Hz. Fgure. Oversampled gma-delta A/D Converter wth rst order modulator and approxmaton loop runs at 5 khz. Decmaton by 64 yelds nal 8 khz A/D rate To check the undamental equaton (4) we appled derent randomly chosen constant sgnals X whch taken at the nput sgnal range [0; ] to the nput o the modulator. Only postve range was used because the standard devaton s the even uncton o X. For each nput sgnal X we took sequence o samples rom the modulator output wth some number o ponts N. We calculated evaluaton o the nput sgnal expectaton by usng a standard Matlab uncton mean, whch s equvalent to (5). tandard devaton evaluaton was ound by usng a standard Matlab uncton std, whch s equvalent to (6). Maxmum delecton o calculated results rom X and σ gven by (4) were close to the errors predcted by (7) and (8). For example, at X = 0.97 and N = 4000, delecton o calculated σ rom σ gven by (4) s 0. %, delecton predcted by (8) s 0.3 %. The equaton (4) was checked by other method too. For derent X we bult the ampltude spectrum by usng Fast Fourer Transorm (FFT) wth N = 89 ponts or each X. The standard devaton evaluaton σ or X was calculated as the sum o root mean squares or all harmoncs: / = N U m, j j= where U m,j j th harmonc n spectrum. σ, (6)

5 Fgure 3. Theoretcal and smulaton σ vs. nput sgnal and relatve error between them The result s shown n Fg. 3. The maxmum derence between (6) and standard devaton calculated by (4) obtans 4.3% at X = By ncreasng number o ponts n FFT, the maxmum error wthn the whole nput range [0; ] decreases wth rate approxmately -0 db per decade and or N = 3768 t amounts only 0.4%. All prevous smulatons correspond to the whole requency range rom 0 to s /. We calculated σ by (6) n lesser requency range to check (). We appled derent constant sgnals X and bult relaton between σ and requency bandwdth. The results or X = {0.; 0.65; 0.95} are shown n Fg. 4. In ths gure we can see that the relaton or X nearest to (X = 0.95) has man jump to low requency and or small sgnal (X = 0.) ths jump placed to hgh requency. The spectrum o sgnal rom modulator output has man harmonc at hgh (low) requency or low (hgh) nput sgnals as was predcted n secton. σ, σ () Fgure 4. Theoretcal σ by () and practcal σ vs. requency bandwdth or some constant nput sgnals X

6 For some constant nput sgnals X, the derence between () and smulaton results can be huge (see Fg. 4), but the average value o many realzatons or derent X, unormly dstrbuted wthn the nput sgnal range [0; ], seems to be close to (). The results o such smulatons are represented n Table or derent number o ponts X (changed rom 0 to 500). From the output o modulator were taken N = 89 ponts or every X and the average values o standard devaton evaluaton was calculated or all X and gven requency. Number o ponts X eq eq /( τ s ) Average relatve error, % Root o relatve errors average squares sum, % Table. Relaton between parameters and number o ponts σ, σ Fgure 5. Theoretcal σ (thck lne) by () and average standard devaton evaluaton σ or derent X (thn lne) vs. requency bandwdth The equvalent gan o the comparator was chosen = τ n secton. Now we wll dene such value eq o ths coecent that gves the mnmum root o relatve errors average squares sum. The relatve errors were ound as delecton o the average values o standard devaton evaluaton and (). From Table some conclusons can be made. I the number o ponts X ncreases then the value o equvalent comparator gan eq s gong to the value = τ predcted n secton. The average relatve error (%) and the root o relatve errors average squares sum (%) are decreasng. Theoretcal and smulaton results or 500 ponts are shown n Fg. 5. In Fg. 6 the relatons between σ and requency bandwdth or snusodal sgnals wth ampltudes X = {0.; 0.65; 0.95} are shown. The relaton or ampltude nearest to (X = 0.95) has man jump n low requency and or small ampltudes (X = 0.) ths jump placed n hgh requency. The spectrum o sgnal rom modulator output has man harmonc at hgh (low) requency or low (hgh) ampltudes as was predcted n secton and smlar to results or constant nput sgnals. Due to ths act, NR or hgher (lower) ampltudes wll be less (more) than predcted by (5).

7 σ, σ (4) Fgure 6. Theoretcal σ (4) and practcal σ vs. requency bandwdth or a ew o nput snusodal sgnal wth ampltude X and requency 00 Hz To check the expectable delay o the modulator, the snusodal sgnals wth 00 Hz, khz and 0 khz requences n were appled to the nput. The phase delay φ between nput and output sgnals was measured by usng FFT. The tme delay o the modulator t m = φ/ π n at requency n dstngushed rom theoretcal (secton ) t m = T s / = / s less than 0.0 ppm. 4. Conclusons The oundaton o new approach to the theory o sgma-delta s applcaton o the dscrete two-level dstrbuton law nstead o the unorm dstrbuton law accepted beore. Due to ths new approach, the varance o the quantzaton nose s ound to have parabolc dependence on nput sgnal (4) nstead o addtve error as was supposed beore. New equatons or varance wthn requency range are ound. Delectons o these equatons receved by smulaton are decreasng to any neglgble level number o calculaton ponts unormly dstrbuted wthn the nput sgnal range [0; ] s large enough. For hgh (low) X low (hgh) requency o quantzaton nose s more typcal and NR wll be less (more). Reerences [] P. Benabes, P. Aldebert, R. Kelbasa, Analog-to-dgtal sgma-delta converters modellng or smulaton and synthess, Proceedngs o Internatonal Workshop on ADC Modellng and testng, Bordeaux, France, pp. 3-4, eptember 9-0, 999. [] ystem applcaton gude, Analog Devces techncal reerence books, U..A., 993. [3] Applcaton Note 870 Demystyng gma-delta ADCs, 005 Maxm Integrated Products. [4] A. J. Davs, G. Fsher, Behavoural modellng o sgma-delta modulators, Computer tandards & Interaces, 9 pp , 998. [5] V.I. Ddenko, A.L. Movchan, J.. olodov. Behavoural Modellng O Instrumentaton gma-delta ADC, Proceedngs o the IMEKO TC-4 3 th Internatonal ymposum on Measurements or Reseach and Industry Applcatons and the 9 th European Workshop on ADC Modellng and Testng, Athens Greece, 9 th eptember st October 004, Volume, pp