分数域信号与信息处理及其应用 (16).pdf

lEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT,VOL.37,NO.2.JUNE 1988 245 Digital Spectra of Nonuniformly Sampled Signals:Fundamentals and High-speed Waveform Digitizers Abstract-This is the first paper of a series describing theories and applications of a newly developed digital spectrum analysis technique for a class of nonuniformly sampled signals.The overwhelming majority of the digital signal processing theories developed so far have always assumed uniform sampling.However,in practice,nonuniform sampling occurs in many data acquisition sys-tems due to imperfect timebase,and the errors it introduces cannot be ignored.In this paper,we first derive a digital spectral representation of a nonuniformly sampled signal,and then present a detailed spec-trum analysis of a nonuniformly sampled sinusoid.It is found that the spectrum of a nonuniformly sampled sinusoid comprises uniformly spaced line spectra;in addition,the signal-to-noise ratio is obtained in closed form.We then apply the theories to analyze the harmonic distortion in-troduced in high-speed waveform digitizers due to timebase errors.Specifically,we analyze waveform digitizers which utilize interleaving1 multiplexing and random equivalent time sampling techniques to ex-tend their capabilities with the current monolothic A/D converter technology.Theoretical results are confirmed by the experimental re-sults with a real waveform digitizer.I.INTRODUCTION HIS IS THE first paper of a series describing theories T and applications of a newly developed digital spec-trum analysis technique for a class of nonuniformly sam-pled signals.The overwhelming majority of the digital signal pro-cessing theories developed so far have always assumed that the digital signal at hand is obtained through sam-pling an analog waveform at uniform time intervals.Only a handful of papers on some aspects of nonuniform sam-pling theory have been published 1-3.However,in practice,nonuniform sampling occurs in many data ac-quisition systems due to imperfect sampling timebase,and the errors it introduces are usually dominating and cannot be ignored in precision instrumentation.Also,in some applications nonuniform sampling may be introduced in-tentionally to achieve certain desirable functions 4.In this paper we will concentrate on basic theory de-velopment and understanding.We first describe nonuni-formly sampled signals and then derive a digital spectral representation of the sampled signal.A closed-form spec-Manuscript received April 20,1987;revised September 18,1987.The author was with Tektronix Inc.,Beaverton,OR 97077.He is now with Physikalisch-Technische Bundesanstalt,Bundesallee 100,D-3300 Braunschweig,Germany.IEEE Log Number 8820518.tral representation of nonuniformly sampled sinusoidals and their signal-to-noise ratios will then be derived.Some implications of theoretical significance will then be dis-cussed.Finally,we apply the developed theories to ana-lyze timebase related harmonic distortions of high-speed waveform digitizers.11.NONUNIFORMLY SAMPLED SIGNALS Let g(t)be an analog signal with its Fourier transform G(w )(the superscript a is used to indicate an analog transform)bandlimited to(-1/2 T,1/2 T).The signal g(f)is sampled in such a structured way that the sampling time instances are not necessarily uniformly spaced in time,but have an overall period MT,see Fig.1.The sam-pled data sequence is then treated as if it were obtained by sampling another function g(t),also bandlimited to(-1/2T,1/2 T),at uniform rate 1/T.We are interested in finding the representation of the digital spectrum of(t)in terms of the Fourier transform,G(w),of g(t).111.DIGITAL SPECTRAL REPRESENTATION The basic principle used to derive the representation of a digital spectrum is to decompose the original sampled data sequence S=g(to),g(t l),g(f21,.*,g(tm),e,g(tM),g(tM+I),-1 into M subsequences So,S1,.*,as follows:S o =goo),g(t M)g(f 2 M L *-1 S I =d t l),g(tM+I),g(t 2 M+I),*.I&-I=d f M-l),&2M-I),g(t 3 M-1),*-1.It is clear that the mth subsequence S,is obtained by uni-formly sampling the signal g(t +t,)at the rate 1/MT.To reconstruct the original sequence S,we first insert(M -1)zeros between samples in all subsequences S,for m=OtoM-1,i.e.-S m =g(trn),0,0,*(M -1 zeroes),g(tM+,),0,0,*0018-9456/88/0600-0245$01.OO 0 1988 IEEE 246 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT,VOL.37.NO.2,JUNE 1988 sampled signal g(rnT),i.e.,tm=rnT,or rm=0,equiv-alently.By substituting rnTfor tm in(2),or 0 for rm in(4),we have(5)W G(w)=(A)c(!-Mi e-k m(2 a/M)T k=-w M m=O MT*Gaw-k(g).4 MT b-I I I I l l I1 I I Ill The summation term C e-jkm(2n/M)in(5)is equal to M for 0 1 12 3 M M I(M.?M d k=0,M,2M,.,and zero otherwise.Therefore,we Fig.1.Nonuniformly sampled signal.and then shift the subsequence 3,rn positions to the right,for rn=0 to M-1,i.e.-SmzPm=(rn zeros),g(tm),(M -1 zeroes),g(t M+m),-1 where z-is the unit delay operator,and finally sum up all the subsequences to obtain the original sequence M-1 s=c Smz-m m=O The digital spectrum,G(w),of S can then be represented by the summation of those of Smz-m,for rn=0 to M-1,PI(2)Let rm be the ratio of rnT-tm to the average sampling period T,i.e.,let t,=rnT-rmT(3)then we can rewrite(2)as m G(w)=(A)c T k=-m -l-Mi e-jw-k(2a/MT)rmT e-jkm(2n/M)M m=O*G a w -k(&).(4)Equations(2)and(4)are two general representations of the digital spectrum of a nonuniformly sampled signal.IV.UNIFORMLY SAMPLED SIGNALS As an example,let us apply(2)and(4)to derive the familiar digital spectrum representation of a uniformly have G(w)=(!)Gw-k(2n/T).(6)T k=-m Equation(6)is the well-known digital spectrum represen-tation of a uniformly sampled signal 5.V.NONUNIFORMLY SAMPLED SINUSOIDALS Since the sinusoidal signal plays a fundamental role in signal analysis,it is worthwhile to examine its properties in detail.Let us first derive its spectral representation by applying(4)derived in Section 111.For a sinusoidal e,with the frequencyf,where w,=27rf0,the Fourier trans-form is given by G(w)=2 d(w -a,).(7)Substituting(7)into(4),we have M-l m G(w)=(&)c c 2n6 m=O k=-m .w-w,-k(2a/M)e-Jm2f/fe-Jk(2/M)(8)where f,is the average sampling frequency 1/T.Let us defineasequenceA(k),k=0,1,2,*as follows:-,M-1,M,*M-l r.1 Then we can rewrite(8)as m G(w)=(L)T k=-m (10)Equations(9)and(10)are the complete spectral represen-tation of a nonuniformly sampled sinusoidal signal.Let us now explore some important properties of the digital spectrum of a nonuniformly sampled sinusoidal signal.First,from(9),it is seen that the sequence A(k)is periodic on k with the period M,hence the spectrum G(w)given by(10)is periodic on w with the period equal to 2 n/T =2n&,i.e.,the average sampling radian fre-quency.Furthermore,one period of the spectrum com-prises M line spectra uniformly spaced on the frequency axis f,with neighboring spectral lines separated by the JENQ:SPECTRA OF SAMPLED SIGNALS 247 amount off,/M.The main signal component is located at f,and with magnitude 1 A(0)1,while the mth spectral line islocatedatf,+(m/M)f,and withmagnitude IA(m)l.The spectrum is depicted in Fig.2.The window shown in Fig.2 includes one period of G(f).It is also noted that the relative strength among different spectral lines as represented by I A(k)I s are exaggerated to indicate that all A(k)s are,in general,different;however,in practice where r,is small,we have,from(9),I A(k)1=I A(M-k)I.Secondly,it is noted from(9)that the finite sequence A(k),k=0,1,M-11 is the dis-crete Fourier transform(DFT)of the sequence (1/M)e-r2h/h,m=0,1,-,M-13;therefore,by Parsevals theorem,the sum of the square of I A(k)I for k=0,1,*,M-1 is equal to M times the sum of the square of I(1/M)e-ir2f,/f,1 which is unity,Le.M-1 k=O c IA(k)12=1.(11)This is a very important property because it greatly sim-plifies the calculation of the signal-to-noise ratio,S/N For the remainder of this section,we will utilize(9)-(12)to derive the signal-to-noise ratios,S/N,of four spe-cial but interesting nonuniform sampling cases.A.Offset Sampling Suppose we sample a sinusoidal signal with both edges of the sampling clock.The clock pulses typically do not have an exact 50-percent duty cycle,hence the signal is not uniformly sampled.In this case,we call it offset sampling.It is clear that this is a special case of our model,i.e.,with M=2.From(9)and the fact that ro=0 by definition,we have and from(12),we have S/N=20 loglo(ICOt(wf/f,)I)db.(14)Assuming that the average sampling frequency f,is greater than the Nyquist frequency,i.e.,f,/f,30 db.Notice that the second term in(24)varies from-12.95 to-15.96 db(3.01-db change)as the degree of interleaving M changes from two to infinity,which means that S/N is quite in-sensitive to M.From the first term of(24),we can see that the S/N drops 20 db per decade change of either the signal frequency f,or the sampling error standard devia-tion u,.If we apply the rule of thumb-one effective bit for each 6 db of signal-to-noise ratio,then(3)says that the variation on effective bits is less than bit for the en-tire range of M,while greater than 3.3 bits per decade change of either f,or a,!It is also noted that if the effec-tive sampling rate is increased by adding more parallel sampling and A/D converter subsystems,f,increases while the percentage error standard deviation u,also in-creases proportionally,hence the signal-to-noise ratio re-mains relatively constant.B.Waveform Digitizers with Random Equivalent Time Sampling If the input signal to the waveform digitizer is periodic,then a technique called the random equivalent time sam-pling can be used to increase the effective digitizing rate at low cost(only one sampling and A/D converter sub-system is needed in contrast to 10 subsystems in inter-leaving/multiplexing scheme to increase the effective sampling rate 10 times).A simplified block diagram of a waveform digitizer with random equivalent time sampling is depicted in Fig.8.The system clock is a free running clock typically runs at the highest speed that the sample/hold and A/D can be operated properly.When a trigger event occurs,a digitized waveform record which is part of a complete waveform record is stored in the temporary waveform store,and the trigger interpolator measures the time offset between the trigger event and the systems sampling clock.A decision is then made,based on the measured time offset,either to discard the waveform rec-ord stored in the temporary waveform store or to place the record in a proper position in the waveform memory which will hold the complete acquired waveform.A typical acquisition process goes as follows.Let us assume that the sampling clock is running at 100 Ms/s,and we want to obtain an equivalent 1000-Ms/s digitiz-ing rate.To achieve that,we have to acquire 10 waveform records at lOO-Ms/s rate in sequential order which are time staggered by 1 ns each and then interleave them to-gether just like the multiplexing back end of the parallel waveform digitizer described above.To approximate this process,we first place the first acquired waveform record in the waveform memory and record the time offset be-tween the sampling clock and the trigger event and then restart the acquiring process.If the second waveform rec-ord has a time offset“close enough”to one of the re-maining 9 staggered time instants the waveform record is accepted and placed in the appropriate position in the waveform memory,otherwise,the waveform record is discarded.This process is repeated until 10 waveform records have been obtained and accepted into the wave-form memory.Here we used the phrase“close enough”to refect the reality that there is a zero probability of ob-taining a waveform record with perfect time offset(due to the asynchronous nature between the sampling clock and the input signal frequency),and hence it would take for-ever to obtain 10 waveform records with time offsets ex-actly 1 ns apart.Therefore,some compromise between the acquisition speed and the sampling time accuracy has to be made.It is our interest to find out how does the compromise on sampling timing accuracy affect the per-formance of the waveform digitizer.If a random equivalent time sampling waveform digi-tizer uses the criterion that a waveform record is accepted if the absolute value of its time offset is less than rT,where Tis the desirable sampling period,then,by assuming that the sampling time error is uniformly distributed in(-rT,rT),we can utilize the uniform random sampling model developed in Section V to analyze the situation.For the case where M waveform records are needed to complete a waveform acquisition process the signal-to-noise ratio is given by the following equation:)db.(M -1)sinc(2 n r ,/)+1(M-1)1 -sinc2(2 n r f o/)S/N=10 log,(25)For a reasonably large S/N,(3)applies with u,=r/&JENQ:SPECTRA OF SAMPLED SIGNALS 25 1 Fig.9.Digital spectrum of digitized sine wave(100 MS/s).The digital spectrum of the sinusoid eJwof digitized by a random equivalent time sampling waveform digitizer has exactly the same structure as that of the sinusoid digitized by a waveform digitizer using interleaving/multiplexing.There is one significant difference between the two digi-tizers though;in the parallel(interleaving/multiplexing)case,the sampling time error r,T is independent of the degree of interleaving M hence increasing the sampling rate by increasing M will not change S/N much;while in the sequential(random equivalent time sampling)case,the sampling time error rT is inversely proportional to M.From the previous analysis(see Section V)we know that the signal-to-noise ratio is relatively insensitive to the de-gree of interleaving M,and is quite sensitive to the sam-pling timing error rT,hence it can be expected that the signal-to-noise ratio(or equivalently the effective bits)should improve as the sampling rate increases for a wave-form digitizer operating in the random equivalent time sampling mode.Figures 9 and 10 show the spectra of a digitized sine wave digitized by a random equivalent time sampling waveform digitizer for M=5 and M=10,respectively.In both cases,the frequency of the sine wave is 6 MHz and the sampling and A/D subsystem is oper-ated at 20 Ms/s.In Fig.9 the effective sampling rate is 100 Ms/s,while in Fig.10 the effective sampling rate is 200 Ms/s.It is seen that the power of distorted harmonics are smaller for M=10 case,and is about 6 db less than of the M=5 case.These experimental results also con-firm the predictions of theoretical analysis on the structure of the line spectra.VII.CONCLUSIONS In this paper we have derived a digital spectral repre-sentation of a class of nonuniformly sampled signals.Spectrum analysis of a nonuniformly sampled sinusoid is Fig.10.Digital spectrum of digitized sine wave(200 MS/s).carried out in detail.It is found that the spectrum of a nonuniformly sampled sinusoidal signal comprises a se-quence of line spectra uniformly spaced on the frequency axis.The coefficient of each line spectrum is obtained in closed form.It is also shown that the