Nonuniform sampling AN1

Transcription

1 Digial Alias-free Signal Processing Applicaion Noes Nonuniform sampling AN1 Sepember 2001

2 1 Inroducion To process signals digially, hey obviously have o be presened in he appropriae digial forma. Therefore he original analog signal, before processing, has o be convered ino a digial one, i.e. i has o be digiized. Once a signal is digiized, he feaures of he obained digial signal, good as well as bad, are fixed and nohing can be done o change hem. In an ideal world, hese feaures would exclusively dep and acually would copy he feaures of he original analog signal. The realiy is differen. The wo basic operaions of any analog-digial conversion, namely, sampling and quanizaion, impac he characerisics of he digial signal subsanially. The characerisics of he analog signal a he ADC inpu and of he digial signal a is oupu are jus similar raher han idenical. How large and significan he differences beween hem are deps on he digiizing mehods and heir implemenaions applied. Le us illusrae ha by wo simple examples: Aliasing or oally indisinguishable represenaion of differen frequencies by one and he same daa se migh occur only in resul of periodic sampling. When sampling is nonuniform, frequencies are represened by a digial signal in a unique way. Spurious frequencies appearing in specra of digial signals are produced by he radiional fixed hreshold quanizaion. To remove hem, eiher sampling or quanizaion has o be irregularized or randomized. This applicaion guide describes he sampling procedure based on he use of nonuniform (non-equidisan) sampling. The applicaion of randomized quanizaion is described in a separae applicaion guide [1]. 2 Sampling Informaion carried by an analog signal can be represened in a digial form by a sequence of is insananeous values measured a discree ime insans. These signal readings are usually considered as signal sample values and he process of aking hem is referred o as sampling. The insans a which he samples are obained form a sream of evens, which can be depiced graphically as a sampling poin process. Characerisic feaures of he sampled signals o a large exen dep on he paerns of he poin processes generaed and used for sampling. When sampling is menioned in he conex of DSP, usually i is assumed ha i is deerminisic and uniform (equidisan). The model of sampling according o which signal samples are separaed by ime inervals wih a consan and known duraion is he mos popular. This is readily comprehensible because such a sampling approach appears o be he mos naural and obvious. I also has a number of aracive advanages. AN1. Nonuniform sampling 2

3 However, as was esablished relaively long ago, he applicaion of periodic sampling alone does no suffice. The periodic sampling model is no applicable when flucuaions in sampling insans canno be ignored or when signal samples can be obained only a nonuniform or even random ime inervals. Sudies have indicaed ha randomness in sampling is no always harmful; random irregulariies in sampling someimes can even be beneficial. These irregulariies, if properly inroduced, provide, for insance, such a useful effec as he suppression of aliasing. And such sampling iself usually is considered as nonuniform. Deping of he required performance specificaion of he signal processing sysem o be developed, given boh in funcional and performance qualiy erms, sampling can be adaped by aking he following decision: eiher periodic or nonuniform sampling has o be chosen. While periodic sampling is preferable, in high frequency signal processing cases, he required periodic sampling rae migh be oo high. Then usage of nonuniform sampling migh be beer. However special and more complicaed signal processing algorihms have o be used in ha case. Therefore adaping he sampling operaion o he specific signal processing condiions comes down o a rade-off beween he complicaions caused eiher by high sampling rae or more complex algorihms. Periodic sampling is preferable whenever he specrum of he signal can be resriced as required by he Sampling Theorem. Firs, periodic sampling is he simples mehod of performing his procedure and is easy o implemen. Second, periodic sequences of signal samples are well suied o digial processing. Noe only ha, many highly efficien fas algorihms are applicable. Randomized sampling may prove o be more profiable when i is undesirable or even impossible o pre-filer signals before heir analog digial conversion, when he signal o be processed conains componens a frequencies exceeding half of he sampling rae accepable under he given specific condiions. However he essence of randomized sampling is aking he signal sample values a unknown random ime insans. Therefore applicaion of randomized sampling is limied o he relaively seldom-me cases where he informaion abou he exac sampling insans is no relevan. Pseudorandomized sampling is he mos ofen used nonuniform sampling based anialiasing echnique. The indicaions for is usage are he same as for he randomized sampling excep ha he sampling insans in his case are known wih high precision. The significance of he sampling operaion is deermined by he fac ha many essenial digial signal characerisics, impacing he whole signal processing process subsanially, dep on he mehods and echniques used o perform i. In he radiional DSP case, he only way o reduce his ofen undesirable depence is increasing he sampling frequency. However, he possibiliies hen are poor and limied. In addiion, his approach in many cases produces an increased number of bis requiring more complicaed hardware for he subsequen processing of he obained in his way digial signals. Deliberae inroducion, when necessary, of an elemen of randomness ino he sampling operaion helps a lo in obaining he flexibiliy essenial for adapaion of he analog-digial conversion o he condiions of he given specified signal processing AN1. Nonuniform sampling 3

4 ask. Therefore he applicaion of nonuniform sampling is more flexible hen use of radiional uniform sampling. 3 Principles of nonuniform sampling The properies of nonuniformly sampled signals are mainly defined by he mode of generaion of poin sreams used for he implemenaion of sampling, ha is, he selecion of poins in ime for signal readou. Alhough here are a relaively large variey of known non-equidisan spaced poin processes, only a few of hem acually have he characerisics required for highperformance sampling. The available experise in applicaion of various random poin sreams suggess ha he mos advisable echnique for producing sampling insans is based on addiive random poin process. I is really well suied for he purposes of deliberae sampling randomizaion. This poin process has such remarkable properies and is so flexible ha i suis random sampling applicaions very well. Specifically, sampling carried ou in his way ensures ha all pars of any inpu signal are sampled wih equal and consan probabiliy. Therefore such sampling is signal-indepen. And i provides for unbiased (no sysemaic errors) esimaion of signal parameers, including specral parameers. However, here are several oher poin processes which should also be considered because hey are conneced o relaively frequenly observed and imporan effecs such as flucuaion of uniform sampling insans or random selecion of uniformly spaced samples. 3.1 Uniform sampling wih jier Flucuaion of sampling insans is a fairly common occurrence. I can even be said ha i is always presen. In he case of uniform sampling wih jier, signal samples { k } are aken a ime momens k = kt + τ k, T > 0, where T is a period of uniform sampling, bu { τ k } is a family of idenically disribued indepen random variables wih zero mean. This sampling scheme is illusraed in Figure 1. I shows he probabiliy densiy funcions p k () of ime inervals ( k 0). As can be seen from Figure 1d, his paricular funcion has muliple peaks. Noe ha as increases he peaks shown do no decrease. To undersand he meaning of he funcion p (), imagine ha a narrow ime window is moved along he -axis. Under he condiion 0, he funcion p () a any arbirary ime insan is equal o he probabiliy ha one of he sampling poins will fall wihin his window. Therefore if a signal is sampled a he insans which are deermined by he saisical relaionship illusraed by Figure 1d some pars of he signal will be sampled wih a higher probabiliy han ohers. This is obviously undesirable, as i will lead o he signal processing errors. There is an excepion. If he ime inervals ( k 0) are disribued uniformly in he inervals ( kt ± 0.5T ) respecively, hen he resuling sampling poin densiy funcion is consan. Bu in fac his mehod of generaing nonuniform sampling poin sream has a number of subsanial disadvanages, which preven is wide applicaion: AN1. Nonuniform sampling 4

5 The random variables { τ k } should be disribued sricly uniformly wihin he given inervals Time inervals beween any wo sampling insans may be very shor. a) p( ) b) p( ) c) p( ) d) p() 0 Figure 1. Probabiliy densiy funcions characerizing periodic sampling wih jier. a), b), c) probabiliy densiy funcions of a sum of 1, 2 and 3 ime inervals, respecively; d) resuling sampling poin densiy funcion Addiive nonuniform sampling In he case of addiive nonuniform sampling, signal samples are aken a ime momens k+1 = k + τ k, where { τ k } is a family of idenically disribued indepen posiive random quaniies. Such a nonuniform poin sream can be easy implemened o suppress he overlapping effec. The degree of randomizaion can be varied hrough appropriae selecion of only one parameer - σ / µ, where µ and σ are he mean value and sandard deviaion of he inervals beween sampling poins, respecively. Obviously he mean sampling rae is equal o 1 / µ. The addiive random sampling scheme iself is illusraed by Figure 2. Probabiliy densiy funcion of ime inerval ( k 0 ) = τ 1 + τ τ k in his case can be calculaed as pk ( ) = pk 1( )* pτ ( ), where he aserisk * denoes he composiion operaion, and p ( ) = p 1( τ ) is a densiy funcion of { τ k } disribuion. On he grounds of cenral limi heorem in saisics: As he random variable ( k 0 ) = τ 1 + τ τ k represens he ne resul of a linear sum of saisically indepen consiuen variables, hen whaever probabiliy densiy funcion hese variables may have, he probabiliy densiy AN1. Nonuniform sampling 5

6 of ( k 0 ) = τ 1 + τ τ k will approach he normal form as k approaches infiniy. As can be seen from Figure 2, he sampling poin densiy funcion p ( ) = p ( k ) in k= 1 he case of addiive nonuniform sampling wih increasing will always o become fla. The value of his consan level is equal o 1 / µ. When nonuniform sampling is seleced for applicaion in a given case, an appropriae sampling rae has o be se up. The crieria for he choice of an appropriae sampling rae, in he case of nonuniform sampling, compleely differ from hose commonly used for periodic sampling. The specral componen of he signal wih highes frequency o be nonuniformly sampled and processed acually is no a crierion hen. The mean sampling rae is calculaed by evaluaing he number of signal samples needed and he longes ime inerval during which hose samples have o be acquired for one signal realizaion. While he minimum of he signal samples o be aken a periodic sampling acually is equal o he required number of hem in he case of nonuniform sampling, excessive samples ofen are aken a periodic sampling jus o avoid aliasing. a) p( ) b) p( ) c) p( ) d) p() 0 µ Figure 2. Probabiliy densiy funcions characerizing addiive random sampling. a), b), c) probabiliy densiy funcions of a sum of 1, 2 and 3 ime inervals, respecively; d) resuling sampling poin densiy funcion. 1/µ AN1. Nonuniform sampling 6

7 4 Advanages 4.1. Avoiding aliasing Assume ha a digial daa se, represening a signal sample value sequence, is given. I is shown graphically in Figure 3. These signal samples migh be used for reconsrucion of he original sinusoidal signal hey belong o. The sine funcion 1 (black) is found o be fiing he daa. However, if he reconsrucion process is coninued, i becomes clear ha here are oher sinusoids a differing frequencies, which can be drawn exacly hrough he same sample value poins like he firs one (doed curves). All hese sinusoids are aliases and overlapping of hem is aliasing. Aliasing resuls in an uncerainy. Indeed, look a hese sine waves of differen frequencies. Which of hem is he righ one? Evidenly impossible o say if no addiional informaion is supplied. The effec of aliasing, of course, is well known. As well as he means how o avoid i. I has been proved ha here will be no uncerainy if all specral componens above a cerain frequency are filered off he original signal by a lowpass filer. (Under cerain condiions aliasing would no occur if he bandwidh of he signal does no exceed half of he sampling frequency no maer how high is he cenral frequency of he signal). Then i is needed o sample wih frequency a leas wice higher han he highes specral componen presen in he signal. However, while he saisfacion of his requiremen resolves he aliasing problem, such compulsory filering off of he signal componens wih upper frequencies also imposes a very drasic limiaion on applicaion of digial signal processing in frequency domain. The achievable bandwidh of digial sysems hen is deermined by he achievable sampling rae. The laer obviously deps on he microelecronic echniques used. To widen he bandwidh, more cosly and more power consuming chips have o be used. Original Sample ime Alias Figure 3. Illusraion of he aliasing effec aking place in he case of periodic sampling. I is very emping o do somehing and o eliminae his limiaion. Tha would open up a broad area of new exciing digial signal processing applicaions. Apparenly, if here would be some oher way how o avoid aliasing, digial processing of signals would be applicable in a much broader frequency range. So he quesion is: is here AN1. Nonuniform sampling 7

8 such a possibiliy? Forunaely, he answer is affirmaive. Applicaion of nonuniform sampling offers his. To see how nonuniform sampling helps in avoiding aliasing, look a Figure 4. The lower frequency sine funcion is again sampled and he corresponding daa se is obained. However he disances beween he sampling insans along he ime axis differ. And ha proves o be very useful. Indeed, i can be easily seen ha only one sine funcion can be drawn exacly hrough he indicaed poins represening he sample value sequence aken from he firs sinusoid. Oher frequencies simply do no go hrough hem. ime Sample Aliases in periodic Figure 4. Only one sine funcion goes exacly hrough he nonuniformly spaced sample values. This effec can be easily checked. Sudies of i would confirm he fac ha, in he case of correcly performed nonuniform sampling, sinusoids wih differen frequencies are represened by differen daa ses. Therefore nonuniformly sampled signals have no compleely overlapping aliases like hose observed a periodic sampling. Consequenly, i can be expeced ha applicaion of nonuniform sampling should open up he possibiliy of disinguishing all specral componens of he signal, even if heir frequencies subsanially exceed he mean sampling rae Applicaion example: oscillogram The nonuniform sampling is well applicable for obaining oscillograms in cases where repeiious signals are observed. I allows o view and measure waveform characerisics of signals in a specral region which more imes exceeds he value of mean sampling rae. Le us illusrae ha by an example. The very simple procedure based on overlapping of signal samples, aken from he number of consecuive signal periods, o he one signal period. Figure 5 illusraes he oscillograms obained by such a procedure in wo cases when uniform and nonuniform sampling of signal is used. A waveform reconsruced from uniformly sampled signal is shown in Figure 5a, while a waveform, reconsruced from sample values of a pseudorandomly sampled signal, is shown in Figure 5b. An addiional linear inerpolaion is applied for beer visual percepion. As can be seen from he oscilogram form from uniformly sampled signal, he signal samples are concenrae ino few poins. The number and posiion of hese poins deps on he raio beween signal and sampling period. In he case of addiive nonuniform sampling he sampling poin densiy funcion is consan. Tha means ha AN1. Nonuniform sampling 8

9 he poin densiy funcion of oscillogram also is consan and does no deps on he signal and sampling frequencies. The waveform obained in he oscillogram is fuzzy. There are wo reasons for ha. Firs, he discree signal usually is corruped by some noise, for insance, by quanizaion noise. The value of SNR is 20 db in he presened example of oscillogram. Second, he fuzziness of he oscillogram is due o flucuaions or jier of he sampling insans. The sandard deviaion of jier is ~1% of mean sampling inerval beween wo samples. Special smoohing procedure can be used o remove his fuzziness in oscillogram. a) b) Figure 5. Example of oscillograms. The frequency of he highes signal componen 10.3 imes exceeds he mean sampling rae a) uniform samplin; b) addiive nonuniform sampling Applicaion example: specrogram The erm specral analysis covers a wide area of diverse signal analysis echniques. When signal specra are considered on he basis of complex exponenial funcions, he discree Fourier ransform (DFT) is he mos versaile analyical ool. The expecaion of he specrum of a properly randomly sampled signal coincides wih he specrum of he respecive analog signal even if he mean sampling rae is considerably below he upper frequency of he signal specrum. Thus nonuniformiy of sampling for specral analysis applicaions seems o be very useful, as he following example indicaes. AN1. Nonuniform sampling 9

10 Suppose ha a specrogram of a composie wideband signal (few sinusoids and noise) has o be esimaed. The Figure 6 shows he specrogram resuls obained by very N 1 simple formula S( f ) = xn exp( j2π f n ). The specrograms given here display Θ n= 1 he specrum of a noisy signal conaining hree sinusoidal componens a he indicaed frequencies. The Signal o Noise Raio (SNR) is 10 db. In periodic sampling ( n = nt ) case (Figure 6a) here are aliases for signal componens. The second specrogram (Figure 6b) was obained simply by changing he sampling mode from uniform o addiive nonuniform. As can be seen, nonuniformiy of sampling in his case efficienly improves he specrogram, allowing o deermine rue signal componens a frequencies exceeding he half of mean sampling rae. The increased background noise level in he second specrogram is he disadvanage of he use of nonuniform sampling. To suppress i he more complicaed signal processing procedures should be applied for solving of specral analysis asks. 2 a) S() db 0-5 True componens f/f sampling Nyquis frequency limi b) S() db f/f sampling Nyquis frequency limi Figure 6. Example of specrograms of composie noisy signal. a) uniform sampling; b) addiive nonuniform sampling. AN1. Nonuniform sampling 10

11 5 Implemenaion specifics 5.1 Pseudorandomized sampling In general, each of he aken signal samples has o be represened by wo numbers: he value of he sample and he ime insan when i has been aken. There are no problems for periodic sampling as he sampling inervals are consan. Only he sample values are measured hen and iming is aken care of jus by couning he signal samples aken. The siuaion is quie differen wih nonuniform sampling. The duraion of sampling inervals in ha case is varying and each sampling even akes place a a specific ime insan. How o obain he informaion exacly when each signal sample has been aken is quie a demanding engineering problem. Especially if he required ime resoluion is aken ino accoun. Indeed, he period, for example, of 1 GHz signal is 1 nanosecond. To sample such a signal, he ime resoluion obviously has o be a few picoseconds. More informaion abou pracical implemenaion of such sampling can be found in [2]. The nonuniform sampling iming echnique is based on aking he signal samples a exacly predeermined ime insans sored in memory. A ypical sample value block is shown in Figure 7. As can be seen, he disances beween he signal samples vary. However, he sampling process is well conrolled. Each sampling even akes place a a ime insan deermined prior o sampling. Figure 7. A block of nonuniformly aken samples. The sampling ime inervals are given, apparenly, by finie digial numbers. Therefore he value of he leas significan bi, measured in ime unis, represens he discreeness of he pseudorandomized sampling process or, in oher words, he period of he hidden periodiciy presen in his process. Due o his periodiciy, aliasing is observed also in he case of pseudorandomized sampling. However he frequency deermining aliasing hen is he frequency of his hidden periodiciy raher han he mean sampling frequency. As he former ypically exceeds he laer many imes, he usage of pseudorandomized sampling usually leads o avoidance of aliasing in he classical sense of his erm. AN1. Nonuniform sampling 11

12 The possible approach of generaing pseudorandom pulse sreams is illusraed in Figure 8. The pseudorandom pulse sream is formed by aking each ni -h pulse from he high-frequency periodic pulse sequence generaed by sabilized clock. The variables n i, which are acually pseudorandom, are generaed by wo pseudorandom number generaors PRNG 1 and 2. These pseudorandom numbers are fed ino one of he wo couners. This couner is swiched o he high-frequency sabilized clock generaor ha begins o coun he clock pulses. Deping on how his couner has been prese i is sooner or laer filled o is capaciy. When he couner is overfilled, a pulse for he oupu sequence is formed. The nex pulse is obained by repeaing his cycle wih he oher couner and PRNG. The mos valuable propery of such a pseudorandom pulse sream is, ha i is fully deermined by he clock frequency and by he sequence of pseudorandom numbers used. This means ha: Such pulse sreams can be reproduced a will whenever his is necessary. The each pulse of sream will always appear a an exacly predeermined insan. Descripions of paricular pseudorandom pulse sream can be sored in compuer memory and his informaion hen used for calculaing he ime insans when i is necessary. PRNG1 Sabilized clock Couner 1 Couner 2 Pulse former Oupu Conrol PRNG2 Figure 8. Block diagram of he pseudorandom pulse sreams generaor. The lising of MATLAB funcion is given in he Appix 1. This funcion is provided for simulaion of nonuniform sampling. The possible use and parameers descripion is done in he help secion of funcion. AN1. Nonuniform sampling 12

13 5.2 Impac of sampling jier As i was shown, he randomness presen a sampling migh play a significan posiive role. However ha is no he case always. Random flucuaions of sampling insans or jier migh impac he analog-digial conversions and he subsequen signal processing quie srongly in a bad way. And such jier is a fairly common occurrence. I can even be said ha jiering happens always wih more or less noiceable harmful effecs due o i. How seriously jier affecs he precision of processing resuls deps on he kind of processing being performed and, of course, on he magniude of he sampling insan flucuaions. The essence of he harmful sampling insan jier is random deviaions of he real sampling insans from heir corresponding prese values. When a sample value is aken from a signal a a ime insan a lile bi earlier or laer han he predeermined insan, which is used a calculaions, an error is inroduced a calculaions. To see how sensiive algorihms can be o such sampling insan flucuaions, look a Figure 9. a) b) Figure 9. Specral analysis of a signal sampled using he addiive pseudorandom sampling, number of processed samples N=4096: wihou jier; (b) jier wih sandard deviaion 20 psec, mean sampling rae 80 Msamples/s. AN1. Nonuniform sampling 13

14 As i can be seen from Figure 9, jier error inroduces raher srong background noise ino he specral analysis resuls. Experimens have shown ha, in pracice, sampling jier presens he major error source of signal analysis. Therefore elecronic devices performing signal digiizing on he basis of nonuniform sampling have o ensure sufficienly precise iming. A presen momen, ha is possible for he frequency band up o few gigaherz. 6 References 1. DASP Applicaion Noe AN2. Randomized quanizaion. Sepember DASP Applicaion Noe AN3. Fundamenals of hardware design for DASP. Sepember 2001 (in preparaion). AN1. Nonuniform sampling 14

15 Appix 1. Lising of he MATLAB funcion for calculaion of ime insans in he case of nonuniform sampling. funcion = nonunif(n, F_s, SM, mode, 0, opion, J); %TNONUNIF Generaion of ime insans for nonuniform sampling. % % =nonunif(n, F_s, SM) produces N nonequidisanly % spaced ime insans wih mean sampling rae F_s and raio SM % beween mean value and sandard deviaion of he ime inervals. % If SM=0, he uniform sampling poin sream is generaed. % % =nonunif(n, F_s, SM, 'mode') % Nonuniformiy can be based on wo differen calculaion ways: % mode='per' he approach of periodic poin sream wih jier is % used; % mode='add' he approach of addiive random poin sream is used. % Defaul mode is 'add'. % % =nonunif(n, F_s, SM, 'mode', 0) % Parameer 0 deermine he value of firs ime insan (0), % he defaul value is (0)=0; % % =nonunif(n, F_s, SM, 'mode', 0, opion) % Two opions for random number disribuion are possible: % opion='unif' - ime inervals are uniformly disribued; % opion='norm' - ime inervals are normally disribued; % Defaul opion is 'unif'. % % =nonunif(n, F_s, SM, 'mode', 0, opion, J) % J - he random number generaor is se o is J-h sae. % I allows o reduplicae he generaed sampling poin sream. error(nargchk(3,7,nargin)); if nargin<4, mode = 'add'; elseif ~srcmp(mode, 'per')&~srcmp(mode, 'add'); error('invalid mode'); if nargin>5 if ~srcmp(opion, 'unif')&~srcmp(opion, 'norm'); error('invalid disribuion opion'); else opion='unif'; if nargin>6 v=version; if v(1) > 4 rand('sae', J); randn('sae', J); else rand('seed', J); randn('seed', J); AN1. Nonuniform sampling 15

16 (1)=0; if srcmp(opion, 'unif'); if srcmp(mode, 'per'); =[1:N]./F_s+sqr(6)*SM./F_s*(rand(1,N)-.5); =-(1); elseif srcmp(mode, 'add'); for ii=2:n, (ii)=(ii-1)+1./f_s+sqr(12)*sm./f_s*(rand-.5); elseif srcmp(opion, 'norm'); if srcmp(mode, 'per'); =[1:N]./F_s+sqr(2)*SM./2./F_s*(randn(1,N)); =-(1); elseif srcmp(mode, 'add'); for ii=2:n, (ii)=(ii-1)+1./f_s+sm./f_s*(randn); if nargin>4 =+0; AN1. Nonuniform sampling 16