APPLICATION NOTE AN95091 INTRODUCTION UNDERSTANDING EFFECTIVE BITS Toy Girard, Sigatec, Desig ad Applicatios Egieer Oe criteria ofte used to evaluate a Aalog to Digital Coverter (ADC) or data acquisitio system is the effective umber of bits achieved. The effective umber of bits provides a meas to evaluate the overall performace of a system. However, like ay other parameter, a uderstadig of the theory behid the effective bits, ad differet methods used to determie effective bits is ecessary to properly compare compoets ad systems. The itet of this applicatio ote is to provide the basic theory behid effective bits, describe differet methods used to determie effective bits, ad to explai the limitatios i its usage. NUMBER OF BITS VERSES EFFECTIVE BITS The umber of bits i a data acquisitio system is ormally specified as the umber of bits of the digitizer. Ay data acquisitio system, or ADC, has iheret performace limitatios. Whe evaluatig a acquisitio system the effective umber of bits provided by the system ca useful i determiig if the system is right for the applicatio. There are may sources of error i a acquisitio system. Cosiderig a system i terms of effective bits, all error sources are icluded. Evaluatio of system performace is made without the eed to cosider the idividual error sources, all of which may ot be characterized by the maufacturer. THE PERFECT SYSTEM A perfect data acquisitio system is oe i which the captured aalog sigal is free from oise ad distortio. The system is free from ay frequecy depedet performace characteristics, up to the maximum samplig rate ad the badwidth limit of the system. I this system the oly limitatio to system performace would be the quatizatio error iheret i all samplig systems. Quatizatio error occurs because a aalog sigal is sampled ad its level represeted by a fiite umber. The output of a ADC has 2 levels, where is the umber of coverter bits. For example, a 8 bit ADC has 2 8, or 256 levels. The bit which represets the smallest voltage chage is called the least sigificat bit (lsb). Each sample has a quatizatio error of up to ±0.5 of the lsb due to the differece betwee the true aalog voltage level ad the level represeted by the ADC output. 1
The quatizatio error defies a base lie oise level which limits the systems ability to resolve small sigals i a perfect system. This is commoly referred to as Sigal to Noise ratio (S/N) which is the ratio of the Root Mea Square (RMS) of the largest sigal divided by the RMS value of the oise. The maximum sigal level is 2 bits ad the oise is 1 bit. The higher the resolutio of a system, the larger the maximum sigal level with respect to the 1 bit quatizatio oise. The result is that higher resolutio systems have a lower base lie oise level resultig i a greater dyamic rage. The SNR of a ideal system S is foud usig the maximum RMS sigal level N ideal (V RMS ) ad the quatizatio error ( QE RMS ). The mea square of the quatizatio error is foud by itegratig over the quatizatio voltage error rage as follows: QE MS + V / 2 q E V de V q = = 12 V / 2 q 2 2 q The RMS value of the quatizatio error is foud by takig the square root of the mea square error: QE RMS 2 Vq Vq = = 12 12 where V q is the quaitizatio voltage ad is equal to 1 bit. The ideal sigal to oise ratio ca the be foud as follows: V RMS = 2 2 2 QE RMS = 1 12 2 S VRMS 2 2 2 12 2 3 = = = = 2 1225. N ideal QE 1 RMS 2 2 2 12 I terms of decibels: S 20 log 10 ( 2 1255. ) db N ideal or S N ideal (. 602+ 176. ) db where is the umber of bits. 2
THE REAL SYSTEM No system is ideal. Every coverter has o-liearity, oise from referece voltages, ad aperture jitter. Every system adds oise to the aalog sigal beig sampled, creates sigal distortio (harmoics), ad has some amout of clock jitter. All these elemets, ad more, cotribute to system performace degradatio with respect to the ideal system. I effect, all error sources combie to limit the effective resolutio of a system. Because differet error sources are depedet upo various operatig coditios, the effective umber of bits is also depedet upo the operatig coditios. The amplitude of the test sigal ca chage the level of distortio. Sigal amplitudes ear full scale should ormally be used. Typically the frequecy of the samplig clock has little affect o the effective umber of bits. The geerally accepted method is to use the systems maximum samplig frequecy for effective bits characterizatio. The frequecy ad amplitude of the iput aalog sigal have the largest affect o the effective umber of bits. This is geerally due both to distortio of the sigal by the aalog frot ed ad the system oise icrease due to clock jitter as the sigal frequecy is icreased. I state-of-the-art systems i terms of samplig rate ad aalog badwidth, the effective umber of bits geerally decreases sigificatly as the sigal frequecy icreases. SYSTEM PERFORMANCE MEASUREMENTS The effective umber of bits is determied by samplig a spectrally pure siusoidal wave ad determiig the RMS sigal ad oise levels recorded by the system. I this case oise refers to aythig which is ot sigal, this icludes quatizatio oise, oise from the iput sigal, ad ay distortio of the sigal. Ofte data acquisitio system maufacturers specify a performace parameter called SINAD. SINAD is the ratio of the fudametal SINusoidal sigal power acquired to the total Noise Ad Distortio. Because this parameter cotais oise ad distortio, it ca be used directly to calculate effective bits. Whe Sigal to Noise Ratio (SNR) is specified, it is ofte the ratio of the fudametal siusoidal sigal power to the oise, ot icludig the harmoics. Figure 1 shows the relatioship of SINAD to SNR i a typical applicatio. Usig SNR to calculate the effective bits of a system is acceptable i some applicatios. For example a commuicatios system where the badwidth of iterest is small ad it is kow that o harmoics, icludig those above the Nyquist limit which will be aliased, will fall withi that bad. 3
db 44 42 40 38 36 34 32 30 28 DATA ANALYSIS SINAD VS SNR 1 10 100 1000 Sigal SINAD SNR FIGURE 1 There are two data aalysis methods used to determie SINAD, ad therefore effective bits. Oe method uses time domai data aalysis ad the other uses frequecy domai data aalysis. Time domai aalysis uses a complex algorithm to fit a perfect sie curve to the acquired time domai data poits. The algorithm must match the sie-wave frequecy, phase, amplitude, ad offset to the data. The perfect wave is the subtracted from the data ad the remaiig data is oise ad distortio. This is the method commoly used by some ADC maufacturers. Frequecy domai aalysis uses a FFT to covert time domai data to the frequecy domai. The appropriate FFT frequecy bis are the selected to determie SINAD. Sigatec, ad some ADC maufacturers, use the frequecy domai aalysis method. Sigatec uses a 4096 poit FFT with a Blackma-Harris widow to covert the data to the frequecy domai. The eergy from a sigal spreads over a umber of bis i the FFT. This makes selectio of bis for sigal calculatios importat whe usig the frequecy domai method. A improper bi selectio will affect the SINAD/effective bits calculatio. Usig a umber of frequecy bis which is greater tha the umber which actually cotai sigal eergy will result i summig oise eergy ito the sigal, resultig i a erroeously high sigal level. Usig too few frequecy bis to represet the fudametal will result i a fudametal power which is too low ad a oise level which is too high. The first 4 bis of the FFT represet the dc term caused by offset shift ad the FFT widowig fuctio, ad are excluded from aalysis. The 9 bis cetered 4
aroud the peak frequecy are summed to determie the fudametal sigal power ad the remaiig bis are cosidered oise for the SINAD calculatio. Figures 2, 4, ad 6 show the frequecy domai plots of data acquired with Sigatec s DA500A, 500MHz digitizig rate, 500MHz aalog badwidth data acquisitio system. Figures 3, 5, ad 7 are exploded views of showig the frequecy bis at the fudametal frequecy. The shaded bis are the oes used to calculate the level of the fudametal frequecy. Amplitude (db) FFT RESULT (25MHz) -10-20 -30-40 -50-60 -70-80 0 50 100 150 200 250 FIGURE 2 FFT BINS (25MHz) 24.0 24.3 24.5 24.8 25.0 25.3 25.5 25.8 26.0 FIGURE 3 Amplitude (db) FFT RESULT (95MHz) -10-20 -30-40 -50-60 -70-80 0 50 100 150 200 250 FIGURE 4 FFT BINS (95MHz) 93.0 93.4 93.7 94.1 94.5 94.8 95.2 95.6 95.9 FIGURE 5 Amplitude (db) FFT RESULT (230MHz) -10-20 -30-40 -50-60 -70-80 0 50 100 150 200 250 FIGURE 6 FFT BINS (230MHz) 228.0 228.5 229.0 229.5 230.0 230.5 231.0 FIGURE 7 5
The selectio of the frequecy used for SINAD ad effective bits test is importat. The sigal frequecy must be selected such that the harmoic terms are sufficietly separated from the fudametal so as ot to be cosidered part of the sigal. The relatioship of the sigal frequecy to the samplig frequecy must be selected such that the harmoics, which may be above the Nyquist limit, do ot alias oto the fudametal. This would cause the harmoic eergy to sum with the sigal eergy resultig i a erroeously high sigal level ad a lower oise level. The resultig effective bits value is sigificatly better tha the system actually provides. Figures 8 ad 9 illustrate the importace of frequecy selectio. The fudametal frequecy is 166.4MHz. This results i a 2d harmoic of 332.8MHz which is above the Nyquist limit, so it will alias to 167.2MHz. If the fudametal frequecy was slightly higher, the 2d harmoic will fold directly o top of the fudametal. Amplitude (db) -10-20 -30-40 -50-60 -70 FFT RESULT (166.4MHz) -80 0 50 100 150 200 250 FIGURE 8 70 60 50 40 30 20 10 FFT BINS (166.4MHz) Fudametal 2d Harmoic 0 165.4 165.9 166.4 166.9 167.4 167.8 FIGURE 9 Whe values for SINAD are icluded i product specificatios, a simple calculatio yields the effective umber of bits. The equatio S (. 602+ 176. ) db was N ideal previously show to represet the ideal Sigal to Noise ratio, assumig a full scale iput sigal, ad a oise value which icludes distortio. This equatio ca be used for a S effective bits calculatio by replacig the theoretical with the measured SINAD N ideal ad lettig equal the effective umber of bits rather tha the actual digitizer bits. Solvig this equatio for effective bits yields: EffectiveBits = SINAD 176. 602. where SINAD is give i db ad the test sigal for SINAD is ear full scale. 6
Figure 10 shows the effective bits whe usig SINAD ad SNR for the calculatio. The decrease i effective bits as the sigal frequecy icreases is typical of high speed, high badwidth systems. The decrease i effective bits from the SNR calculatio, labeled A i Figure 10, is due to digitizer clock ad aperture jitter. The differece i the decrease betwee the SNR ad the SINAD effective bits, labeled B i Figure 10 is due to sigal distortio. 7 EFFECTIVE BITS Number Eff. Bits 6.5 6 5.5 5 A B 4.5 1 10 100 1000 SINAD SNR FIGURE 10 Some maufactures add a correctio factor to the SINAD term whe calculatig effective bits to correct for the differece i dyamic rage betwee the amplitude of the test sigal ad the maximum full scale amplitude of the coverter. I a high speed, large badwidth system addig this correctio factor is iappropriate. The ature of the correctio i the calculatio assumes that uused dyamic rage would be pure sigal with o oise or distortio. I real life high speed systems this will ot occur ad the calculatio will yield a ufairly large effective umber of bits. For these reasos Sigatec does ot iclude this correctio factor i ay calculatios or graphs give i this documet. DROPPING BITS A commo miscoceptio regardig effective bits ad system performace is that ay resolutio beyod the effective bits is useless. For example, i a 8 bit system with 6 effective bits, the 2 least sigificat bits are useless ad ca be disregarded i data aalysis. This is ot true. As discussed above, the effective umber of bits, or SINAD, represets the overall system performace. I other words, these parameters are the combiatio of all error sources. If the total error is broke ito two parts, quatizatio error ad all other errors, the total system oise is give by: 7
Error = E + E 2 2 q s where E q is the quatizatio error ad E s is the RMS values of all other errors. Usig a digitizer with fewer bits, or droppig the bits from processig icreases the quatizatio error. The equatio above shows that as the quatizatio error icreases, the overall error icreases, icreasig SINAD ad decreasig the effective umber of bits. If the oise due to system oise ad error is ot sigificatly greater tha quatizatio error, the removig bits from the data will icrease the overall oise level, which reduces data quality. Figure 11 shows data which was acquired with Sigatec s DA500A data acquisitio board. The graph shows four SINAD curves, oe each for data acquired with 8, 7, 6, ad 5 digitizer bits. The y-axis grid lies are positioed at the SINAD values correspodig to a perfect 8, 7, 6, 5, ad 4 bit system. At low frequecies where the errors itroduced by the system are small relative to the quatizatio error, each of the trucated data sets closely matches its respective effective bit value. However, at higher frequecies where the sigal error icreases, there is a decrease i the SINAD value. 48 SINAD WITH TRUNCATED BITS 42 db 36 30 24 1 10 100 1000 8 Bits 7 Bits 6 Bits 5 Bits FIGURE 11 8
EVALUATING RESULTS As previously stated, the effective umber of bits achieved by a data acquisitio system is a reflectio of the systems overall performace. Evaluatio of a system s effective bits show that if a applicatio requires a sigal to be resolved to true eight bits of resolutio, for example, the a digitizer with a resolutio higher tha eight bits is required. Whe usig effective bits to evaluate system performace, the method used to determie the umber is importat. The method must allow for the iclusio or exclusio of various parameters depedig o the specifics of the applicatio. For example SINAD, which icludes distortio terms, should ormally be used to derive effective bits, but i some applicatios usig SNR may be appropriate. It is also importat to evaluate the effective bits at a frequecy ad amplitude appropriate for the applicatio. The effective bits, ad other performace parameters, should be specified at a particular frequecy or set of frequecies. This is importat due the ormal decrease i performace as the sigal frequecy icreases i high performace systems. The effective umber of bits for a high performace system caot be assumed to be the same as the effective umber of bits for the ADC used i the system. Therefore the effective umber of bits, alog with all other performace parameters must be specified for the etire system. Referecig ADC maufactures specificatios i system data sheets provides iformatio which is ot useful to the system user. Whe evaluatig a system for a particular applicatio it is ofte ecessary to look beyod the effective bits parameter. While the effective umber of bits gives a idicatio of overall system performace, it may ot be the best parameter to use whe cosiderig a system for a particular applicatio. Other parameters such as Spurious Free Dyamic Rage (SFDR) or Total Harmoic Distortio (THD) are more appropriate for some applicatios. Ofte it is importat to evaluate characteristics such as harmoic distortio or SNR as a fuctio of sigal frequecy. SUMMARY The effective umber of bits achieved by a data acquisitio system ca provide a user with a overall isight ito the systems performace. However, as with ay other performace parameters, the user should uderstad the methods used by the maufacturer to determie the effective umber of bits. I critical applicatios it is geerally ecessary to look beyod the umber effective bits to other performace specificatios which idetify specific system limitatios. 9