3. Sampled measurements

Transcription

1 3. Sampled meaurement By the end of thi ection you will be able to: Decribe the function of a finite aperture ampler Decribe the operation of A/D and D/A converter. Dicu the bandwidth and quantiation noie of A/D converter Decribe capabilitie and limitation of Digital Ocillocope Dicrete time meaurement Although thi i not neceary, we uually ample a ignal to convert a meaurement to a digital repreentation. A dicrete time meaurement then conit of the following operation: Signal conditioning to conform to the ampling theorem impoed retriction Sampling i.e. recording an intantaneou value of the ignal Quantiation namely approximating the ignal value by a finite reolution digital repreentation. In the following we will aume that ampling i performed repetitively at interval T, or a frequency f by a ample and hold circuit, which, therefore record the value of the ignal at n dicrete time t n = = nt. By ampling a ignal we therefore introduce a mapping between time f and the index n of the meaurement. The Sampling theorem tate that in order not to loe any information through the ampling proce the ignal mut atify ome contraint: Low-pa ampling A time varying ignal V () t with a Fourier tranform v( ) can be reproduced exactly from ample taken at a frequency f if it Fourier tranform vanihe for frequencie greater than f /. Thi i the mot common tatement of the Shannon Sampling Theorem, but in reality it i a pecial cae of it. The ampling theorem eentially ay that the amplitude and phae of an infinite duration inuoidal waveform of frequency f can be recovered only if we record more than two value of the ignal during each period. It i not enough to record exactly two value during a period. To ee thi let aume that by coincidence we ample exactly at the zero croing! We then get only the phae information, up toπ, but no amplitude information whatoever. If, on the other hand, we obtain infiniteimally more than ample per period we can calculate both the amplitude and the phae of the waveform. Indeed, ampling occur at a lightly different phae during each period. Then, the amplitude i: A= max{ vi } (1) Once the amplitude i determined the phae i obtained from the zero croing poition. Thi argument doe not clarify why the ampling frequency ha to be more than twice the greatet frequency in the ignal pectrum. From the low-pa ampling data the original ignal can be recovered by uing the following interpolation formula: CP Imperial College Autumn

2 () n= () = ( ) ( ), () = inc( π ) x t x n g t nt g t f t Note that an infinite number of ample required in eq. (). Thi make the interpolation formula in eq. () of little practical interet. The critical bandwidth f N = fb = f / i called the Nyquit frequency. Strictly peaking, ignal containing the Nyquit frequency in their pectrum cannot be recontructed from the ample at f. Likewie, a finite duration ignal require a higher than the Nyquit frequency to recontruct. Jut how much higher frequency than the Nyquit rate i required i eay to etimate. During the entire duration Δt we require at leat one ample more than twice the number of period to unambiguouly reolve both magnitude and phae. So we can write: 1 Namp = famp * Δ t > fupperδ t+ 1 famp > fupper + Δt All thi aume the ampling clock i clean, i.e. perfect. Real clock have jitter i.e. the nth ampling event happen at: tn = nt + δt. The mot naïve interpretation would ugget that the maximum poible ampling period mut atify the Nyquit criterion. Unfortunately, thi i not poible, ince the jitter uncertainty i uually Gauian, and the probability a ampling event occur a timeδt away from it ideal poition i given by: Thi mean that any time diplacement away from the ideal ampling intance i poible, epecially a the ample length i long. Sampling i of coure triggered by an ocillator. It can be hown that for many ocillator jitter i a random walk proce, i.e. diffuive. The rapidity of the diffuion proce i determined by a correlation time ξδ t expreed a a multiple ξ of the ampling period. The longer thi time i the more rapidly the ampling intant diffue away from their ideal location. In that cae, fampδ t ξ 1 f amp ξδt τ P( δt) = e f πξδt amp 1 P( δt) = e πτ The naïve counting argument can then be applied with thi decription of jitter, to etimate a lower bound for the ampling frequency which allow complete recontruction. Thi i not an entirely atifactory approach, a it doe not adequately account for whether it i poible to recontruct the ignal by not knowing the ampling intance. Indeed, it can be better done by ignal-to-noie ratio argument, i.e. by aking for the minimum ampling rate which will lead to a required ignal-tonoie ratio. δt τ Band-pa ampling A practical radio ignal ha frequency component in ome finite range of frequencie: fl + fh fl fi fh, fc =, fc = fh fl (3) CP Imperial College Autumn

3 The ampling ignal i, however, a erie of narrow impule with power in all the harmonic of the ampling frequency. The ampling proce itelf i ignal multiplication, i.e. mixing. The mixing proce map the input ignal into a ignal of mixing image. The downconverted image of the ignal band mixed with the m th ampling frequency image lie at: fl mfs fi mfs fh mfs. Similarly, the negative frequency component mut atify: fl + mfs fi + mfs + fh + mfs A long a thee image do not overlap the ignal can be recovered. Solving the overlap problem one can how that the minimum requirement for ignal recovery i that the even order mixed down band do not overlap. The image overlap problem i olved graphically in Figure 1. The general olution for large order mixing product m i: ( ) f mf < f m 1 f f f = f < f H S L S H L B S It can be hown that in general a band-pa ignal can be uniquely ampled at a Nyquit ampling rate which atifie: f < f < 4f (4) B N B The bet cae condition (lowet ampling rate) occur when the carrier frequency f c and the bandwidth f atify: B f + f = nf (5) c B B In the limit of mall fractional bandwidth we get that the Nyquit rate for band-pa ignal atifie: lim f f f N = fb (6) c B If the band pa ignal i ampled at a frequency f then the recontruction formula become: C (7) n= () = ( ) ( ), () = inc( π ) co( π ) x t x n g t nt g t f t f t A wa the cae with low pa ampling, the minimum ampling rate i one which guarantee two ample per period for all frequency component in the ignal. Except that now the bandpa character of the ignal guarantee that the ignal change very lowly over a number of period o that ample obtained during different conecutive period are a good a if they were obtained during one ingle period! Once again, a ignal of finite duration need a higher ampling rate than an infinite duration ignal to be completely recontructed. Jitter i a much more evere retriction in bandpa ampling than i in the lowpa cae. The iue with jitter i that if the ampling intance i uncertain then the ignal phae i uncertain at the time the ample wa taken. In bandpa ampling, though, two conecutive ample may have been obtained many period apart ( many i in fact of the order of the ratio of N = fu / fample ). And a rather inignificant time jitter may mean that the actual ampling intance i everal period apart from the ideal ample time! In more precie term we ay that all mixing product of the noie power of the ampler circuit are aliaed into the ignal band. The noie power i therefore amplified CP Imperial College Autumn

4 by the ame N = fu / fample factor which i our benefit in term of lower ampling frequency. The ignal to noie ratio in a band-pa ignal ampled at f ample I a factor N lower than the ame ignal ampled with the low-pa criterion, and the ame equipment. Band-pa ampling i ued in extremely high frequency application, uch a ampling ocillocope and radar receiver, where the fractional bandwidth (bandwidth to carrier ratio) i extremely mall. Figure 1: Allowed rate (white) and forbidden rate (grey) for band-pa ampling a a function of the maximum band frequency. In principle frequencie twice the Bandwith are ufficient Interpolation The interpolation formula in eq. () and (7) are not practical for two reaon. Firtly, they require an infinite number of ample. Second, and mot important, the interpolation function g(t) i not phyically realiable; it repreent a non caual filter (note that g(t) i defined for both poitive and negative value of time). A ample and hold or zero order hold approximate the ignal by aigning to it, during the interval between two ampling event, the lat ampled value. Thi i indeed an accurate decription of the ample and hold circuit preceding an A/D converter. The recontruction formula ugget the frequency domain repone of the ample-and-hold i ( ) inc( π ) g f = T ft (8) Subequent low pa filtering remove the dicontinuitie introduced by the ampling proce. A firt order hold approximate the ignal by it linear interpolation between ample: ( ) ( 1) ( ) ( ) x nt x n T x () t = x( ( n 1) T) + t nt T (9) CP Imperial College Autumn

5 The interpolator i equivalent to a linear filter with a frequency repone: ( ) inc ( π ) H f = T ft (10) Once again low pa filtering can improve the interpolation by removing the derivative dicontinuitie. 3.. Signal conditioning The ampling theorem dictate than any ignal to be ampled mut atify the Nyquit bandwidth criterion. The filter ued for conditioning i called an anti alia filter. Ideally we would require infinite attenuation at frequencie exceeding the Nyquit frequency. In practice the finite attenuation in the filter top band provide a contribution to the meaurement noie floor. A a rule, we wih to keep the out of band ignal which will be aliaed into the band to le than ½ LSB. We calculate then the break frequency and order of the antialiaing filter o that the aliaed component are le than ½ LSB, a hown in the illutration. Pa band Filter SNR MIN Filter Alia Figure : Illutration of the deign of an anti-aliaing filter 3.3. D/A converion F S We dicu the D/A converion firt becaue it i more traightforward to implement than A/D converion. The baic D/A converter tructure ue binary weighted current ource which are witched in and out of the circuit to repreent a binary number. In Figure 3 we how two uch tructure, the binary weighted reitive ladder and the R-R ladder. Both device are followed by a tranimpedance amplifier which um the current and convert it into a voltage. In the binary weighted converter of N bit, if the 0 th bit i LSB and N-1 bit the MSB, the reitance value are 1 given by N n 1 Rn = RN 1 o that the LSB reitor i N time bigger than the MSB. Thi introduce a major limitation of thi type of converter, a the error ariing from component tolerance mut be kept below ½ LSB. If η i the fractional component tolerance, the contraint on the maximum number of bit i: 1 N η < N < 1+ log ( η ) (11) Thi i a evere retriction, a even 1% component tolerance would retrict the length of a converter to about 6 bit. The R-R ladder alleviate thi problem omewhat, in that only two value of component are ued, and in general identical component can be manufactured to cloer tolerance, epecially on IC. The analyi of the R-R ladder i an exercie in deriving the Thevenin equivalent circuit by uperpoition, alternatively turning on and off the voltage ource repreenting the bit of the input digital data. CP Imperial College Autumn

6 V 4R LSB V MSB R R R R 4R R R R R LSB MSB Binary weighted R-R Ladder (a) Figure 3: Simple D/A converter. (a) binary weighted ladder. (b) R-R ladder Both converter ue an op-amp a a tranimpedance amplifier, and are conequently limited by the op-amp frequency repone and lew rate. A much fater converter can be made by directly witching binary weighted current, a hown in Figure 4. Scaled current ource are eay to implement on an IC, a they repreent a number of tranitor connected in parallel. The operation of a weighted current ource D/A converter i limited by the larger gate current drive required by the higher bit, and at high peed by the o-called injected charge, i.e. the gate current appearing in the channel of the device and contributing to the converter output. The limitation of thi converter are alleviated in the current teering DAC (Figure 5) where witche are ued to direct the caled current. V 4R (b) B3 B B1 B0 x8 x4 x x1 Vcc R R R LSB MSB Vout Thermometer coded Figure 4: A thermometer coded and a binary weighted current ource DAC. Vcc x8 x4 x x1 B3 B B1 B0 Vout Figure 5: Current teering DAC CP Imperial College Autumn

7 A common characteritic of the converter preented o far i the large DC power diipation. In high peed and low power application the reitor of the R-R ladder can be replaced by capacitor, (C-C/, repectively)and the tranimpedance amplifier by an integrator. It can be hown that if the integrator i ideal the circuit operation i identical to that of the R-R ladder. Operation of the capacitive R-R ladder i limited at lower frequencie by noie current. Finally, in high reolution and medium peed application, overampled feedback, ΣΔ, modulator are ued extenively. We will tudy thee later Quantiation or A/D converion - generalitie The final tep in a dicrete meaurement i the converion of the meaurement to a digital repreentation. Clearly not all value of the input ignal can be repreented digitally. The dicrepancy between the ignal and it digital repreentation called quantiation noie. Quantiation i performed by A/D converter which we examine later Quantiation An ideal A/D converter perform a truncation operation. Ideally it return the nearet integer to x yx = (1) q i.e. the converter expree the analogue input x a a multiple of the quatiation tep q, and then return the nearet integer to x/q, namely x 1 nq = int + (13) q Sometime the A/D operation i tated a x nq = int (14) q The difference between the two definition i an additive contant, i.e. an offet of half a quantization tep. When applied to a time varying ignal x(t) the two definition differ pectrally by an additive contant to the f=0 fourier component of x(t). In the preent analyi we will tick to the nearet integer definition a much a poible Quantiation Noie It follow from the definition of the A/D converion that the analogue input x can be written a: x = nq+ e (15) The quantity eq i called the quantiation error or quantiation noie. The quantiation noie i aumed to have a uniform probability denity function between ± q/, i.e. q q ( q) = 1/, / < q < / ( q ) = 0, q ( q/ w, q/ ) p e q q e q p e (16) The ideal average (mean quare) quantiation noie power i aumed to be white between f and f. The total power of the quantiation noie i: CP Imperial College Autumn

8 e E e e p e de de q / q q ( q) = q ( q) q = q = q 1 q / (17) over a bandwidth B = f. Note that, a uual in ignal proceing, a load impedance over which the power i diipated i implied. In a real calculation, if the converter the converter meaure voltage the actual quantiation noie power will be E ( eq ) q N = q R = 1R (18) The quantiation noie et a limit on the maximum ignal to noie ratio that can be achieved in a ampled data ytem. The quantiation noie i not correlated with any ource of thermal noie. We oberve that the total quantiation noie power i independent of the bandwidth. Thi implie that we can lower the power pectral denity of quantiation noie: Nq D( eq ) = (19) f by increaing the ampling frequency. Thi i in fact what we do with overampling. A general ignal i the uperpoition of a poibly infinite number of Fourier component. Focuing on a ingle The power of a inuoidal ignal of amplitude A i P = A. Such a ignal can be made to fit exactly in the range of an N bit converter by etting the amplitude to half the converter range: A= N q, we can calculate that the maximum S/N ratio, uually called the ignal to quantiation noie ratio SNQR i: N 1 ( ) ( ) ( ) SQNR = 10log 6 A q = 10log 3i = N db (0) A we hall ee later, the argument can be inverted. A converter operating at a particular SNQR i aid to be N-bit by inverting thi formula. Furthermore, we talk of the effective number of bit ENOB of the converter a the number of bit of an ideal converter which ha SNQR equal to the converter SNR after all ource of noie and uncertainty have been accounted for Sampling jitter The error in the time at which a ample i taken mut be mall enough o that the maximum error committed when ampling a pure frequency at the nyquit rate doe not exceed ½ leat ignificant bit. If thi condition i not oberved, then the equivalent number of bit of the converter i limited by timing jitter and not the converter. Quantitatively, the retriction on the RMS timing jitter i: ω A 1 1 AΔ t < Δ t < = n+ 1 n n+ 1 ω π f For example, conumer audio i 16 bit ampled at f = 44.1kHz. The allowed timing RMS jitter, above which ignificant reolution degradation occur, i approximately 55 pec. CP Imperial College Autumn

9 3.5. A/D converter Flah converter The implet conceptually and alo the fatet A/D converter i the Flah converter, hown in Figure 6. The input ignal i compared to all poible value in the converion range and a decoder elect and output the code. Flah converter word length i limited by component tolerance, and they alo have a high power diipation due to the big number of comparator (although the latter can eaily be implemented with CMOS gate). Figure 6: Flah converter Feedback converter Feedback converter compare the output of an internal D/A converter to the input. The comparion i ued to provide a uitable input to the D/A converter. The implet i the ingle lope ramp converter, hown in Figure 7, where a counter increment the D/A input until it exceed the ignal input. Such a converter i low and ha a code dependent converion time. A ingle lope converter may ue an analogue ramp generator (integrator) in the place of the counter. Figure 7: Single lope ramp converter A dual lope ramp converter (Figure 8) integrate the input ignal for a time t 1 and then ubtract from it the integral of a fixed voltage until the output reache again zero, which take a time t. If an intermediate output V int i reached after t 1, if τ i the integrator time contant then: t1 t t V = int Vin Vref Vin Vref τ = τ = t (1) CP Imperial College Autumn

10 The logic time t and provide a uitable input to the D/A converter. Thi type of converter i not only fater, but alo exhibit a maller code-dependent variation of the converion time. Furthermore, any nonlinearitie of the integrator cancel, at leat to the lowet order. Figure 8: Dual lope ramp converter A very popular (and much fater in multibit application) converter i the ucceive approximation converter (Figure 9). An N bit ucceive approximation converter ha a fixed converion time, of N+1 clock cycle. Thi allow contruction of 16 bit converter with le than 0 μ converion time. CP Imperial College Autumn

11 Figure 9: A ucceive approximation A/D converter 3.6. Overampling If we ample a ignal at a much higher than the Nyquit rate we hould in principle be able to ue the extra ample to obtain a higher reolution than the underlying converter. By traightforward overampling we can in principle gain 0.5 bit of reolution for every doubling of the ampling rate. To ee thi, we have to compute the power pectral denity of the SQNR. The ignal power occupie frequencie fb < fig < fb, and a a reult, the ignal power pectral denity i the total ignal power divided by (twice) the ignal bandwidth: A Pig = 4 fb The quantization noie power pectral denity i the total quantization noie power divided by (twice) the ampling frequency, a the quantiation noie occupie frequencie f < f < f : The SQNR i then: P N q = 4 f PSD SQNR = = PSD N q f B but we have already aumed that the ignal fit in the converter range exactly: A n = q So that SQNR i given, in term of the number of bit n, and the overampling ratio 6A f i given by: k M = = f /f PSD 6A f SQNR = = = 3 M = 3 PSDN q fb B n 1 n+ k we can then write the SQNR in term of an effective number of bit ENOB: ENOB = n + k /+1/ CP Imperial College Autumn

12 Since: 1 SQNR 3 ENOB = And clearly the effective number of bit increae by ½ bit for each bit of overampling. We have averaged M ucceive meaurement to average out the quantiation noie Dither From Ken Pohlmann "Principle of Digital Audio," 4th edition, page 46: "...one of the earliet ue of dither came in World War II. Airplane bomber ued mechanical computer to perform navigation and bomb trajectory calculation. Curiouly, thee computer (boxe filled with hundred of gear and cog) performed more accurately when flying on board the aircraft, and le well on ground. Engineer realized that the vibration from the aircraft reduced the error from ticky moving part. Intead of moving in hort jerk, they moved more continuouly. Small vibrating motor were built into the computer, and their vibration wa called 'dither' from the Middle Englih verb 'didderen,' meaning 'to tremble.' Today, when you tap a mechanical meter to increae it accuracy, you are applying dither, and modern dictionarie define 'dither' a 'a highly nervou, confued, or agitated tate.' In minute quantitie, dither uccefully make a digitization ytem a little more analog in the good ene of the word." To perform the averaging effectively we need to add ome noie, to make ure that the ignal and noie um croe frequently the converter deciion threhold. Such intentional noie i called dither. The required dither amplitude typically exceed the converter quantiation tep. More preciely, we chooe to repreent the ignal x by a random variable y = x+ ex obtained by adding to the ignal e x, a random variable repreenting the dither noie. After N meaurement, the ratio of the ignal and dither noie average i: x N x = = N () e Ne x So that y x+ O e / N (3) ( RMS ) Thi i the ame reult decribing averaged meaurement in the preence of noie. An important engineering problem remain, though: How can we generate, in hardware, thi dither noie component o that it i of the correct magnitude? I it alo poible to make the averaging proce converge more rapidly by giving the dither noie ome appropriate pectral characteritic? The anwer to both i in a very old engineering trick, ued originally to cramble pace communication to make them more robut to interference! CP Imperial College Autumn

13 3.7. Δ-Σ converter The Delta modulator hown in Figure 10 i a ignal to PWM converter. A a PWM ignal contain copie of the pectrum both at baeband, it i very eay to decode by retiming and low-pa filtering, alo hown in Figure 10 Figure 10: The Delta modulator (left) and demodulator (right) The igma delta converter further reduce the in-band quantiation noie in overampling by neting the converter inide a feedback loop. It turn out that thi arrangement alo automatically generate the dither noie required for the enhanced reolution! The converter conit of the igma delta modulator, (Figure 11), the output of which i a (poibly multiple bit) pule width modulated waveform. A ubequent digital filter interpret the output of the modulator. Clock + V in (analog) - Filter H() A/D Converter V out (PWM) D/A Converter Figure 11: A igma delta modulator The magic in the operation of the igma delta modulator lie in that the input ignal and the quantiation noie have different tranfer function. Auming the gain of the A/D and D/A converter are both unity, the ignal tranfer function i: H( ) G ( ) = (4) 1 + H( ) while the quantiation noie tranfer function i: GE ( ) = H (5) The total power of the quantiation noie i given by eq. (0), and it power pectral denity i: P e = E ( e )/ f = q 4f ( ) q q ( ) CP Imperial College Autumn

14 If we require the ignal to quantiation noie ratio at much a lower frequency f N (ince, after all we are ampling at a frequency that i much greater than the Nyquit frequency), the in-band ignal to quantiation noie ratio will approximately be: 1 3 ( ) ( ) N S i f H f N N f N = (6) E f H ( f ) To give a concrete example, conider that the converter are 1 bit wide (i.e the A/D converter i a comparator, and the D/A a witch) and that the filter i an ideal integrator. In term of the overampling ratio M = f f = k the SQNR i: N SQNR 3 3 6M 1.5 k + = = i (7) Which i the ame a the SNQR of a 1.5k+1 bit modulator. So the implet 1 t order modulator gain 1.5 bit reolution for every 1 bit overampling. Uing higher order filter and multiple loop we can make much bigger reolution gain with overampling ratio. The maximum effective reolution that can be achieved i N+1/ bit per bit, if an Nth order loop filter i ued. Sigma delta converter are very popular in digital audio, and other relatively high ample rate and high reolution application. They are frequently ued with a bandpa loop filter in conumer radio IF tage. CP Imperial College Autumn

15 3.8. Intrument uing ampled meaurement The digital torage ocillocope A digital ocillocope conit, a hown on Figure 1, of a ampler/a/d converter, and a digital timebae generator. For many application the digitiing ocillocope ha important advantage over it analog counterpart. Mot importantly it can trigger at the end of a waveform making eaier the obervation of tranient and one-off event. It can perform computation on waveform with it built-in proceing capability, for intance it can average a number of waveform and compute the (fat) Fourier tranform. It alo allow torage in memory of waveform and comparion to ubequent obervation. Since a ampler i central to the operation of the ocillocope, an important iue i obervation of the ignal bandwidth veru ampling rate retriction impoed by the Nyquit ampling theorem. Failure to do o reult into aliaing and the obervation of artefact. A digital ocillocope normally operate in Real-time ampling mode: all ample are collected equentially in a ingle period a the waveform i received. To omewhat relax the ampling theorem contraint equivalent-time ampling (or coherent ampling) may be employed, a hown in Figure 13: the waveform i recontructed from ample acquired over a number of cycle of the waveform. The ordering of conecutive ample in the time domain may be equential or random, and clearly, equivalent time ampling i effective only on periodic waveform. Figure 1: Block diagram of a digital ocillocope CP Imperial College Autumn

16 Sampling Technique Figure 13: Real veru equivalent time ampling. LeftL Sequential ampling. Right Random ampling There i an interrelation between the ampling rate f S, the weep time T X and the memory record length M: fstx = M (8) The record length doe not need to be equal, of coure, to the total available memory. A large memory allow more flexibility in chooing the diplayed weep time. The ampled data will appear a a et of dot on the creen. Thi open the poibility to viual aliaing, i.e. the implicit interpolation performed by the human eye may interpret the ignal a being at a different frequency. Often explicit interpolation will be employed, i.e. to ocillocope will join the dot With no interpolation, about 5 ample per period are required to recontruct a inuoidal waveform. With a linear interpolator thi i reduced to about 10 data point per cycle, and with a ine interpolator the waveform can be accurately recontructed with a few a.5 ample per period (cloe to the Nyquit rate). Interpolation may introduce diffraction effect, e.g. ringing when oberving teep dicontinuitie. Digital filter can be ued to minimie uch artefact at the cot of little additional ampling. Figure 14: Diplay, and viual artefact. CP Imperial College Autumn

17 When the interpolation method i accounted for, the ueable torage bandwidth i defined for digital torage ingle event capture a: USB = Maximum ample rate x (1/C) C depend on the number of ample per cycle, which depend on the method of interpolation. For a dot diplay (no interpolation) C=5 Linear interpolation C=10 Sine interpolation C=.5 For repetitive ignal, USB = full cope bandwidth (ince equivalent time ampling can be ued). The Ueful Rie Time of a digital cope i approximately T R =1.6 ample period, a illutrated in Figure 15. Actual bandwidth and rie time of a DSO will therefore change with the timebae etting (ample rate). But USB and T R give an indication of the fatet ignal which can be captured. Figure 15: Ueful rietime of a digital cope. CP Imperial College Autumn

18 A digital cope allow arithmetic operation on the data acquired. Averaging conecutive weep i a common way to enhance the ignal to noie ratio, and hence the effective number of bit. All the ame, the effective number of bit of the A/D converter front end can be defined a the width of a converter whoe quantiation error equal the actual noie floor of the converter when ued to digitie a ine wave N A. The effective number of bit (ENOB) combine variou factor into a ingle meaure of performance, which meaure the digitie accuracy veru frequency. The difference between the actual and effective width of the converter i called the number of lot bit. If E Q i the quantiation noie power denity, N A LB = log (9) E Q A the noie floor and ditortion often rie rapidly with frequency, the effective number of bit correpondingly reduce harply at higher frequencie. Enemble averaging can be ued to increae n ocillocope' reolution. If K waveform are averaged, the ignal to noie (in power!) ratio will increae by a factor of K, o the effective reolution will increae by δ N = 1logK bit. To minimie the obviou memory requirement to tore many waveform, ubequent waveform are added to a running average of previou one, effectively implementing an IIR filter Sampling Ocillocope At microwave frequencie real time, or even equivalent time ampling become impractical. Yet ocillocope that can diplay ignal to frequencie up to 40GHz exit. Thee exploit aliaing to ample the ignal. The waveform to be oberved i effectively bandpa ampled, o the bandpa ampling criteria now apply, i.e. the ignal need to have a retricted bandwidth. A the underlying ampling rate may be quite large, thi i not a eriou retriction. More eriou retriction arie from the need to operate the ampling gate on very high frequency ignal. Both the aperture (capture time) and the timing jitter (uncertainty in time poition) of the ampler need to be mall compared to the highet frequency oberved. Finally, it i neceary to ue preamplifier (intead of attenuator in conventional cope) which can everely retrict the intrument' dynamic range. CP Imperial College Autumn