Chapter 2. Sampling. 2.1 Sampling. Impulse Sampling 2-1

Transcription

1 2- Chaper 2 Sampling 2. Sampling In his chaper, we sudy he represenaion o a coninuous-ime signal by is samples. his is provided in erms o he sampling heorem. Consider hree coninuous-ime signals x (), x 2 (), and x 3 ()shown in Figure 2.. We can choose a sampling rae o sec. such ha x (k ) = x 2 (k ) = x 3 (k ), or k =, ±, ±2,.. In general, here are an ininie number o signals ha can generae a given se o samples. he quesions is: under wha condiions, a signal x() can be recovered uniquely rom is samples? We now show ha i a signal is bandlimied, hen i can be recovered exacly rom is samples aken a a sampling rae ha is a leas wice he highes requency componen in he signal. his is given in erms o he Shanon Sampling heorem. We will show ha using ideal sampling (impulse rain sampling) and using pulse (pracical case) sampling. 2..a Impulse Sampling Consider an inpu signal x() wih Fourier ransorm X() = or > W Hz. he signal is muliplied by he impulse rain o obain a sampled signal x s () as shown in Figure 2.2 (noe: sill x s () is a coninuous-ime signal).

2 2-2 CHAPE 2. SAMPLING x 2 () x () x 3 () Figure 2.: hree coninuous-ime signals x() X x s() P P () = k= δ( k ) Figure 2.2: Muliply x() by he impulse rain o obain a sampled signal x s () x s () = x()p () (2.) X s () = X() P () (2.2) = X( n/ ) (2.3) X s () k= = s k= X( n s ), (2.4) where s = (2.5) ha is, X s () is an a periodic uncion o requency consising o a summaion o shied replicas o X(), s called by s. We shall denoe s by he sampling requency in Herz(Hz).

3 2.. SAMPLING 2-3 o examine he recovery issue we will examine hree cases: s > 2W ; s = 2W ; s < 2W ; Over sampling Perec sampling Under sampling as shown in Figure 2.3 X() W W 2 s s s 2 s () Over Sampling X s () 2 s s s 2 s (2) Perec Sampling X s() 3 s 2 s s s 2 s 3 s (3) Under Sampling X s () aliasing eec 4 s 3 s 2 s s s 2 s 3 s 4 s Figure 2.3: hree cases o sampling For exac reconsrucion o a bandlimied signal wih highes requency componen a = W H z, we need o sample he ime-signal a a rae s 2W H z as shown in Figure 2.4.

4 2-4 CHAPE 2. SAMPLING h() x() X x s () Ideal econsrucion Filer x r () P () X() A W W X s() 2 s s s 2 s s + W H() s W W < c < s W c c A X r() W W Figure 2.4: econsrucion 2..b Pulse Sampling x s () = p()x() (2.6)

5 2.. SAMPLING 2-5 x() X x s () p() p() α 2 2 Figure 2.5: Pulse Sampling p() = {, n < α/2, elsewhere he pulse rain p() is a periodic uncion, hence p() = C k = k= α 2 α 2 = sin(παk) πk C k e j 2π k (2.7) 2π j e k d = { } ( ) 2π j j 2πk e k α 2 2π +j e k α 2 (2.8) (2.9)

6 2-6 CHAPE 2. SAMPLING Hence, sin παk C k = α παk p() = x s () = = k= ( k= k= = αsinc(αk) (2.) αsinc(αk)e j 2π k (2.) C k e j 2π k ) x() (2.2) C k x()e j 2πk C k α = 2 α α 3 α 2 α α α 2 α 3 α k Figure 2.6: Eec o α on C k Bu, x() X() (2.3) x( ) e j2π X() (2.4) e ±j2π x() X( ) (2.5)

7 2.. SAMPLING 2-7 Hence, X s () = X s () = C k X( k ) (2.6) k= k= C k X( k s ), (2.7) Where Again, we have hree cases: s = (2.8) X() W W X s () c 2 c c c c 2 2 s s s 2 s s + W s W Figure 2.7: Sampling wih s < 2W s > 2W ; s = 2W ; s < 2W ; Over sampling Perec sampling Under sampling For exac reconsrucion, we mus, again, have s 2W and use an ideal lowpass iler wih bandwidh such ha: c > W and ( s W ) > c. his is saed in erms o he

8 2-8 CHAPE 2. SAMPLING sampling heorem. he Sampling heorem Le x() be a bandlimied signal wih X() = or > W H z. hen, x() is uniquely deermined by is samples x(n ), n =, ±, ±2,... i s > 2W, where s =, by passing he sampled uncion hrough a lowpass iler wih cuo requency c such ha: W < c < s W, and wih a Gain o. he handou on signal reconsrucion provides deails o he process. Now, we wan o pu he samples inside he compuer memory. o do ha, we have o quanize he samples according o he available accuracy. hereore, digiizing a signal (image) involves wo processes: sampling and quanizaion. Quanizaion is done by assigning a value o each sample according he wordlengh n used; or example, i you have n-bis wordlengh, we can assign q = 2 n quanizaion levels. he value q is aken o be equal o he range o x(); i.e., q = x max () x min () (2.9) Once he signal is quanized (aer being sampled), all wha we have in he compuer is a se o numbers. On hese numbers, we are going o build he heory o discree-ime signals and sysem. 2.2 Commens on Signal econsrucion In he ollowing, we sudy wo imporan orms o daa reconsrucion: inerpolaion which is suied or daa ransmission sysems and exrapolaion which is more suied or eedback sysems. Adoped rom: Elmer G.Gilber, Noes on sampled-daa sysems, Compuer, Inormaion and Conrol Program, he Universiy o Michigan, Ann Arbor, 983.

9 2.2. COMMENS ON SIGNAL ECONSUCION a Inerpolaion Inerpolaion is he process o iing a uncion o ime hrough he daa poins x[n] = x(n ) o a daa sequence {x[n]}. Any such uncion y i () is called an inerpolaion uncion o he daa sequence. I is generally no convenien o i a polynomial hrough he daa poins because he degree o he polynomial mus be one less han he number o daa poins, which is usually very large. Hence, we ake a dieren approach. Le he inerpolaion uncion be given by y i () = n= x[n]h i ( n ) (2.2) his inerpolaion uncion is linear in x[n] and will pass hrough he daa poins x[n] = x(n ) i h i () = (2.2) h i (k ) =, k is an ineger (2.22) he orm o he inerpolaion uncion y() depends on he orm o he inerpolaiongeneraing uncion h i () in (2.2). Figure 2.8 shows hree examples o he uncion h i () (and y i ()) ou o many oher possibiliies. he irs wo generaing uncions are he irs members o a class o generaing uncions such ha: d N h i = piecewise consan uncion, (2.23) dn h i () =, N, N. (2.24) For N =, zero-order inerpolaion, y i () is piecewise consan and or N =, irs-order inerpolaion, y i () is coninuous and piecewise linear. For he second-order inerpolaion (N = 2) y i () is coninuous and piecewise quadraic, ec. hus, by inroducing more complex uncions h i (), smooher inerpolaion uncions are obained. he inerpolaion uncion generaed by sinc(/ ) (someimes called he cardinal inerpolaion uncion) h i () = sin(π/ ) π/ = sinc(/ ) (2.25) is he smoohes o all inerpolaions in he sense ha y i () is bandlimied wih bandwidh 2 s, where s = is he sampling requency in Herz. he irs wo inerpolaion uncions in Figure 2.8 are no bandlimied. Wih he cardinal inerpolaion uncion, he uncion y i () has he propery y i () = x() i x() is bandlimied wih bandwidh < s /2. he oher

10 2- CHAPE 2. SAMPLING inerpolaions reconsruc x() exacly only i x() has a specialized uncional orm. For example, in he case o he irs-order inerpolaion, y i () = x() only i x() is a piecewise coninuous linear uncion o, wih all slope disconinuiies a = n. y i () h i () (a) 2 2 y i () h i () (b) y i () h i () = sinc( ) (c) Figure 2.8: Generaing Funcions and heir esuling Inerpolaions : (a) Zero-order Inerpolaion, (b) Firs-order Inerpolaion, (c) Cardinal Inerpolaion From pracical poin o view, he inerpolaion uncion described above canno be realized in he on-line (real ime) reconsrucion o daa sequences. o see he naure o his diiculy, consider he irs-order inerpolaion uncion (Figure 2.8.b). Beore one can begin he inerpolaion in he inerval n (n + ), one mus know x[n]andx[n + ]. hus x[n + ] mus be known one ull sample period beore i becomes available. I is possible o achieve a irs-order on-line inerpolaion i a delay o seconds is inroduced, ha is, i y i ( ) is generaed. he delay required or zero-order inerpolaion is clearly 2 ; or

11 2.2. COMMENS ON SIGNAL ECONSUCION 2- he higher order i is N. hus, he smooher he inerpolaion becomes, he larger he required delay or on-line realizabiliy. he inerpolaion produced by he cardinal generaing uncion may no begin unil all daa poins are available since he ails o h i () exends o =. For daa ransmission links where he daa is inspeced some ime aer i is received, he inerpolaion delay is no a serious limiaion. I, however, he inerpolaion process occurs inside a eedback loop, he delay may seriously aec he sabiliy o he eedback sysem. hus, in a eedback sysem a dieren orm o daa reconsrucion called exrapolaion is generally employed. 2.2.b Exrapolaion Exrapolaion is he process o iing a uncion o ime o daa poins received in he pas, so ha his daa sequence may represen he daa sequence unil he nex daa poin is received. Usually, he exrapolaion uncion y e () is aken o be segmens o polynomials in. Zeroorder and irs-order exrapolaions are shown in Figure 2.9. he generalizaions o higher order exrapolaions is sraighorward, being based on he Gregory-Newon exrapolaion ormula. I is ineresing o noe ha y e () has jump disconinuiies even or higher-order exrapolaions. However, or slowly varying daa sequences he magniude o he jump generally becomes less as he order o he exrapolaion increases. O course, an n h order exrapolaor will reconsruc x() exacly i y e () is an n h degree polynomial in. A ormula o he orm (2.2) also holds or exrapolaion, ha is, y e () = n= x[n]h e ( n ) (2.26) he naure o he exrapolaion generaing uncions h e () is quie dieren rom he inerpolaion generaing uncions (see Figure 2.9):. Zero-order exrapolaor: h e () = { < <,. (2.27) 2. Firs-order exrapolaor: h e () = ( + / ) < ( / ) < 2 <, 2. (2.28) he reader should show ha (2.28) is indeed correc.

12 2-2 CHAPE 2. SAMPLING y e() h e () (a) 2 2 y e() h e() 2 (b) 2 Figure 2.9: Exrapolaion Generaing Funcions and heir esuling Exrapolaions: (a) Zero-Order Hold (b) Firs-Order Hold Since exrapolaion begins anew each ime a new daa poin is received, i seems reasonable ha he delay inroduced by his daa reconsrucion process should no increase wih he order o he exrapolaion. Perhaps he bes way o seeing his, and also o comparing he processes o inerpolaion and exrapolaion, is o give (2.2) and (2.26) a requency domain inerpreaion. his is done in Figure 2.. I he uncion x() is passed hrough an impulse sampler and hen hrough a linear sysem wih impulse response h i (), he response o he linear sysem is y i (). he specrum o y i () is obained rom X s () by Y i () = X s () H i (), (2.29) where H i () is he ranser uncion corresponding o he impulse response h i () and is given by H i () = h i ()e j2π d. (2.3)

13 2.2. COMMENS ON SIGNAL ECONSUCION 2-3 For exrapolaion he resuls are he same excep he subscrip i is replaced by e. Impulse Sampler Linear Sysem ime domain Frequency domain x() x s () X() X s () Impulse response h i () y i () Y i () = H i ()X s () Figure 2.: epresenaion o Inerpolaion as Filering o x s () where x s () is Obained by Impulse Sampling In he case o he cardinal inerpolaion uncion we can easily show ha h i () = { s /2 > s /2. (2.3) hus, he linear sysem is an ideal low-pass iler and he inerpolaion uncion is an ideal low-pass iler. I is, hereore, apparen why he cardinal inerpolaion is an exac reconsrucion o x() when x() is bandlimied wih bandwidh less han 2 s. For he zero-order inerpolaor i is easy o show ha: H i () = sin(π ) π = sinc( ). (2.32) hus, or his ype o inerpolaion he siuaion shown in Figure 2. resuls. Figure 2.2. shows he gain and phase shi associaed wih inerpolaions and exrapolaions o he zero- and irs-order ype. In order o make he inerpolaion realizable o on-line reconsrucion o sampled-daa, appropriae delays have been inroduced. In each case i can be seen ha H() has a lowpass characer which ends o rejec he specral componens o X s () a requencies greaer han 2 s which are inroduced by sampling. he delayed inerpolaion o zero-order is exacly he same as he zero-order exrapolaion. he Fourier ransorm o he zero- and irs-order inerpolaor/exrapolaor are

14 2-4 CHAPE 2. SAMPLING e X() e X() H i () e Y i () Figure 2.: Frequency Domain Inerpreaion o he Inerpolaion Process (Zero-Order) (a) Zero-order inerpolaor and exrapolaor: h ir () = h er () = h i ( /2) H ir () = H er () = H i ()e j2π/2 = sinc( )e j2π/2 (2.33) (b) Firs-order inerpolaor: h ir () = h i ( ) H ir () = H i ()e j2π = sinc 2 ( )e j2π (2.34) (c) Firs-order exrapolaor: h er () = h e ( ) H er () = ( + j2π ) sinc 2 ( )e j2π (2.35)

15 2.2. COMMENS ON SIGNAL ECONSUCION 2-5 H() (a) (b) (c) s 2 s 3 s H() s 2 s 3 s π (a) (b) (c) 2π Figure 2.2: Gain and Phase Shi or Zero-Order and Firs-Order Inerpolaions and Exrapolaors We added he exra scrip r in 2.33 and 2.34 o emphasize ha hese ormulas correspond o he realizable inerpolaor/ exrapolaor cases. As shown in Figure 2.2., i he phase lag o he irs-order exrapolaion is compared o ha o zero-order exrapolaion, i is seen o be less or low requencies bu greaer or high requencies. he irs-order exrapolaion produces less phase lag han he irs-order inerpolaion. Similar resuls are obained or higher-order inerpolaions and exrapolaions. Oher ypes o daa reconsrucion processes arise in sampled-daa sysems. Oen hey may be represened by he general procedures described above. For example, pulse modulaion may be approximaed by

16 2-6 CHAPE 2. SAMPLING y p () = h p ( n )x(n ), (2.36) n= where h p () = { α/2 α/2 > α/2. (2.37) x() x s() y p() Figure 2.3: Approximaion o x s () by y p () he approximaion is good i α << or x() varies lile in one sample period.

17 2.2. COMMENS ON SIGNAL ECONSUCION 2-7 See Figure 2.3. Anoher example is he use o a Buerworh lowpass iler or inerpolaion. I he lowpass iler is preceded by a zero-order exrapolaor (or zero-order inerpolaor), he resuling H i () has a gain given by [ ( H i () = sinc( ) + c ) 2k ] 2, (2.38) where c is he cuo requency o he iler and k he order o he iler. 2.2.c Implemenaion o Sampling and Daa econsrucion Processes he physical devices which implemen he processes o sampling and daa reconsrucion may assume dieren orms, depending on such hings as: he physical naure o he physical signal involved, he required accuracy, cos, weigh, reliabiliy, ec. Since a deailed reamen o hese maers is no easible in hese noes, le us consider in a limied way he special problems o ranserring coninuous daa, in he orm o volages, ino and ou o a digial compuer. A scheme or enering daa is shown in Figure 2.4. he sample-hold device picks up he samples o he inpu volage x(n ) and holds hem a shor ime so ha he analog-odigial conversion may be compleed. I conversion o x() is made direcly, he variabiliy o conversion ime wih he inpu volage would cause he sampling insans o be spaced non-uniormly in a complex way. he arrangemen shown in Figure 2.4 includes a very simple mechanizaion or he sample-hold device. he swich is closed or n β < n. During his period y() = x(). hus, when he swich is open, n < < n + β, y() = x(n ). I β <<, he sample-hold device has an oupu which approximaes ha o a zero-order exrapolaor. For low sampling raes he swich migh be a compuer operaed relay; or high sampling rae i migh be a ransisor circui. I is imporan ha he analog-o-digial converer (A/D) have a high inpu impedance so ha he capacior does no discharge appreciably when he swich is open.

18 2-8 CHAPE 2. SAMPLING Sample and Hold Sampling Conrol rom Digial Compuer Oupu Inpu Funcion + x() - Swich C Analog o Digial Converer logic levels o digial compuer Figure 2.4: Arrangemen or Enering Sampled Daa ino Digial Compuer here are a variey o ways in which he deails o he A/D may be mechanized. In principle, however, all converers consis o hree basic elemens: a volage source, conrolled by digial elemens; a comparaor, which deermines he sign o he dierence beween he inpu volage and he oupu o he volage source; and a conrol device, which operaes he digial elemens rom he comparaor oupu. he conrol device provides a sysemaic procedure or adjusing he volage source so ha is oupu maches he inpu volage. When he mach is obained he sae o he digial elemens is a digial represenaion o he inpu volage, which may be read ino he compuer. A bes, he accuracy o he conversion is deermined by he number o bis or decimal digis carried in he represenaion. his round-o error is called he quanizing error o he conversion. he eec o quanizing is shown graphically in Figure 2.5. Clearly, he inpu-oupu relaion is nonlinear. hus, any sysem which uilizes conversion equipmen is nonlinear and he heory o sampled-daa sysems is no applicable! Oen, however, he quanizing errors are so small ha he sair-sep relaion in Figure 2.5. may be approximaed by a linear one. he validiy o such approximaions should always be checked, eiher by urher analysis or by sysem simulaion. One analyic approach, which has me wih considerable success, is o approximae he quanizing error as a disurbing random noise. his analysis is ou o he scope o his course and will no be presened in hese noes. Le us now consider he process o daa reconsrucion. he ranserring o daa ou o he digial compuer involves digial-o-analog conversion (D/A). A schemaic represenaion o

19 2.2. COMMENS ON SIGNAL ECONSUCION 2-9 Digial represenaion o x x Maximum Quanizing Error x Figure 2.5: Inpu-Oupu elaion or Analog-o-Digial Converer (A/D) D/A is shown in Figure 2.6. ealizaion (a) employs a ladder nework, while realizaion (b) uilizes an operaional ampliier. he binary represenaion o x[n] is loaded ino he regiser o he converer a = n, and is held here unil = n + when he nex daa poin is enered. hus, neglecing he swiching ransiens and quanizing errors, he converer is a direc implemenaion o a zero-order exrapolaor (someimes called a zero-order hold). By combining several converers, exrapolaors o higher order may be realized. One realizaion o a irs-order exrapolaor (irs-order hold) is shown in Figure 2.7. A = n he swich S is closed or a very brie insan, discharging he capacior C. Furhermore, he irs converer is loaded wih x[n], and he second wih x[n ] (obained rom he regiser o he irs converer). For > n > he oupu ampliier produces x[n] + (x[n] + x[n ]) ( n )/ which is he desired exrapolaion. Provided he necessary addiional delay or realizabiliy is added, inerpolaors can be consruced in a similar way (he reader should obain a realizaion o he irs-order inerpolaor). In pracice, exrapolaors and inerpolaors o order higher han irs are rarely

20 2-2 CHAPE 2. SAMPLING (a) - + (b) + - sign bi y e() 3 y e() x[n] = ( ) = 7 28 Figure 2.6: Analog-o-Digial Converer (A/D) wih egiser, (a) Ladder Nework, (b) Operaional Ampliier employed. Exrapolaors o any order may be buil uilizing only one D/A converer. For deails he reader should consul he immense lieraure on his subjec.

21 2.2. COMMENS ON SIGNAL ECONSUCION 2-2 x[n] logic lines or loading o converer regiser D o A converer.... x[n]. y e () converer conrol rom digial compuer D o A converer C x[n ] C = S Swich conrol rom digial compuer Figure 2.7: Implemenaion o Firs-Order Exrapolaion