Chapter 4. Sampling of Continuous-Time Signals

Transcription

1 Chapte 4 Sampling of Continuous-Time Signals 1

2 Intodution Disete-time signals most ommonly ou as epesentations of sampled ontinuous-time signals. Unde easonable onstaints, a ontinuous-time signal an be quite auately epesented by samples taken at disete points in time. Impotant topis inlude: The poess of peiodi sampling in some detail Aliasing, whih ous when the signal is not band limited o when the sampling ate is too low. Continuous-time signal poessing an be implemented though a poess of sampling, disete-time poessing, and the subsequent eonstution of a ontinuous-time signal. analog signal digital signal digital signal Sampling DSP D/A analog signal 2

3 Disete-Time Signals Filte is used fo noise emoval and anti aliasing Thee is also a low pass filte afte D/A analog signal digital signal digital signal A/D DSP D/A analog signal analog signal Low pass Filte Sample & Hold Quantize Coding digital signal 3

4 Peiodi Sampling The typial method of obtaining a disete-time epesentation of a ontinuous-time signal is though peiodi sampling A sequene of samples, x[n], is obtained fom a ontinuous-time signal x (t) aoding to the elation: x [ n] x ( nt ) n T sampling peiod f 1/ T sampling fequeny The sampling fequeny an also be expessed by adians pe seond: s 2 /T s In a patial setting, the sampling is implemented by an A/D onvete. Impotant onsideations inlude quantization of the output samples, lineaity of quantization steps, the need fo sample-and-hold iuits, and limitations on the sampling ate. 4

5 Sampling with a Peiodi Impulse Tain A moe onvenient epesentation of sampling inlude an impulse tain modulato followed by onvesion of the impulse tain to a sequene. The impulse tain: s ( t ) ( t nt ) n (a) Mathematial epesentation of oveall system not phisial (b) xs(t) fo two sampling ates () Output fo two sampling ates (d) x[n] ontains no expliit infomation about the sampling ate 5

6 Fequeny Domain Repesentation of Sampling x ( t ) x ( t ) s ( t ) x ( t ) ( t nt ) ( Modulation) s n x ( t ) x ( nt ) ( t nt ) ( Shifting popety ) s n Conside the Fouie tansfom of X s (t): Fouie Fouie If s( t ) S ( j ) and x ( t ) X ( j ) S( j) s( t) e jt 2 S( j) ( ks) whee s 2 / T is the sampling ate in adians/s. T k 1 1 X s ( j ) X ( j )* S ( j ) X j ( k s ) 2 T The Fouie tansfom of x s (t) onsists of peiodially epeated opies of Fouie tansfom of x (t) 6 dt k C

7 Fequeny Domain Repesentation S(jΩ) is onvolved with X s (j Ω ) If Ω s > 2Ω n the eplias of X (jω) do not ovelap, and x (t) an be eoveed with an ideal lowpass filte If Ω s < 2Ω n the opies of X (jω) ovelap, and is no longe eoveable. of Sampling 7

8 Fequeny Domain Repesentation of Sampling By applying the ontinuous-time Fouie tansfom to equation We obtain x ( t ) x ( nt ) ( t nt ) s n X ( j ) x ( nt ) e S n j Tn j jn onsequently x [ n] x ( nt ) and X ( e ) x [ n] e n 1 2 ( ) X j k ( j ) X ( e j ) X ( e j T X e X j s ) T T k T T j X ( e ) is a fequeny-saled vesion of X s (jω) when ω=ωt Fequeny Ω=Ωs in Xs(jΩ) is nomalized to ω=2π fo j X ( e ) Ω=Ωs=2πf=2π/T ω=ωt=2πt/t=2π The time axis is nomalized by a fato of T so the fequeny axis is nomalized by a fato of f s = 1/7 8

9 Exat Reovey of Continuous-Time (a) epesents a band limited Fouie tansfom of x(t) whose highest nonzeo fequeny is. N (b) epesents a peiodi impulse tain with S fequeny. fom Its Samples () shows the output of impulse modulato in the ase S N N S 2N 9

10 Exat Reovey of Continuous-Time If Ω s > 2 Ω n X C ( j) doesn t ovelap theefoe x(t) an be eoveed fom xs(t) with an ideal low pass filte with gain T and utoff fequeny H ( j) N C S N It means X ( j ) X ( j ) fom Its Samples C = 10

11 Aliasing Distotion (a) epesents a band limited Fouie tansfom of x(t) Whose highest nonzeo fequeny is. N (b) epesents a peiodi impulse tain with S fequeny. () shows the output of impulse modulato in the ase S N N S 2N 11

12 Aliasing Distotion In this ase the opies of X C ( j) ovelap and is not longe eoveable by lowpass filteing theefoe the eonstuted signal is elated to oiginal ontinuous-time signal though a distotion efeed to as aliasing distotion. 12

13 Suppose Example: The effet of aliasing in the sampling of osine signal x ( t ) os( t ) X (jω)=π(δ(ω Ω 0 )+ δ(ω+ω 0 ) 0 13

14 Nyquist Sampling Theoem Sampling theoem desibes peisely how muh infomation is etained when a funtion is sampled, o whethe a band-limited funtion an be exatly eonstuted fom its samples. Sampling Theoem: Suppose that x ( t ) X ( j ) is band-limited to a fequeny inteval X ( j) 0 fo C, i.e., Then x(t) an be exatly eonstuted fom equidistant samples 2 T whee is the sampling peiod, f 1/ is the sampling s fequeny (samples/seond), N N x [ n] x ( nt ) x (2 n / ) 2 / s 0 X s s 2 / s T s C C ( j) N s s T s is fo adians/seond. N 14

15 Ovesampled Suppose that is band-limited: x ( t ) X ( j ) C X A C ( ) j Then if T is suffiiently small, X ( e ) appeas as: S N 0 N A j X ( e ) T s Condition: T 0 NTS N S T T o T o 2 N S N S N S S N 15

16 Citially Sampled Citially sampled: T o 2 N S S N A T s j X ( e ) Aoding to the Sampling Theoem, in geneal the signal annot be eonstuted fom samples at the ate TS / N. This is beause of eos will ou if X ( N ) 0, the folded fequenies will add at Conside the ase: and note that fo 2 2 T S 0. x ( t ) A sin( Nt ) Aj ( N ) ( N ) /. N x ( nt ) A sin( nt ) A sin( n) 0 (fo all n) s s 16

17 Undesampled (aliased) If sampling theoem ondition is not satisfied T o 2 A T s j X ( e ) N S S N The fequenies ae folded - summed. This hanges the shape of the spetum. Thee is no poess wheeby the added fequenies an be disiminated - so the poess is not evesible. Thus, the oiginal (ontinuous) signal annot be eonstuted exatly. Infomation is lost, and false (alias) infomation is eated. If a signal is not stitly band-limited, sampling an still be done at twie the effetive band-limited. 17

18 Reonstution of a Bandlimited Signal Figue(a) epesents an ideal eonstution system. Ideal eonstution filte has the gain of T and utoff fequeny N C S N A onvenient and ommonly used hoie of utoff fequeny is: C S / 2 / T fom Its Samples 18

19 Reonstution of a Bandlimited Signal fom Its Samples Given x[n], we an fom an impulse tain x s (t): x ( t ) x [ n] ( t nt ) S n x ( t ) x [ n] h ( t nt ) n sin( t / T ) h () t t / T sin( ( t nt ) / T ) x ( t ) x [ n] ( t nt ) / T n 19

20 Reonstution of a Bandlimited x ( t ) x [ n] h ( t nt ) n Signal fom Its Samples h Fo all intege (0) 1 x ( mt ) x ( mt ) values of m. independent fom the h ( nt ) 0 n 1, 2,... sampling peiod T. Theefoe the esulting signal is an exat eonstution of x(t) at the sampling times. the fat that, if thee is no aliasing, the low pass filte intepolates the oet eonstution between the samples follows fom the fequeny-domain analysis of the sampling and eonstution poess. 20

21 Ideal D/C Convete The popeties of the ideal D/C onvete ae most easily seen in the fequeny domain. j Tn x ( t ) x [ n] h ( t nt ) X ( j ) x [ n] H ( j ) e n n j Tn X ( j ) H ( j ) x [ n] e n j T X ( j ) H ( j ) X ( e ) 21

22 Disete-Time Poessing of Continuous-Time Signals A majo appliation of disete-time systems is in the poessing of ontinuous-time signals. We know fom the pevious setions x [ n] x ( nt ) j 1 2 k X ( e ) X C ( j ( )) T T T k sin( ( t nt ) / T ) y ( t ) y [ n] ( t nt ) / T n jt Y ( j ) H ( j ) Y ( e ) jt TY ( e ), / T 0, othewise 22

23 LTI Disete-Time systems If the disete time system in the pevious slide is an LTI system we have: j j j Y ( e ) H ( e ) X ( e ) j T j T Y ( j ) H ( j ) H ( e ) X ( e ) Y ( j) H ( j) Y( e jt 1 2 k Y ( j ) H ( j ) H ( e ) X C ( j ( )) T k T IF X then ideal lowpass filte H (jω) C ( j ) 0 fo / T anels the fato 1/T and selet only the tem fo k=0 jt H( e ) X ( j), / T Y ( j) 0, / T jt ) 23

24 LTI Disete-Time systems In geneal if the disete-time system is LTI and if the sampling fequeny is above the Nyquist ate assoiated with the band width of the input x(t), then the oveall system will be equivalent to a LTI ontinuous-time system with an effetive fequeny esponse given by: jt H ( e ) X C ( j ), / T Y ( j ) H eff ( j ) X C ( j ) 0, / T H eff jt H ( e ), / T ( j) 0, / T 24

25 Example: Ideal Continuous-Time Lowpass Filteing Using a Disete-Time Lowpass Filte Conside the LTI disete-time system with fequeny esponse of: The fequeny esponse is peiodi (2π) 1, j C H( e ) 0, C The fequeny esponse of oveall system is: The effetive fequeny esponse is the same as an ideal lowpass filte with utoff fequeny Ω=ω/T C 1, T Heff ( j) C 0, T 25

26 Example: Ideal Continuous-Time Lowpass Filteing Using a Disete-Time Lowpass Filte FT of a bandlimited signal FT of output of DTS FT of sampled input in CT FT of output of DTS FR of filte FT of disete samples and FT of DTS FT of output 26

27 Example: Ideal Continuous-Time Lowpass Filteing Using a Disete-Time Lowpass Filte The ideal lowpass disete-time filte with disetetime utoff fequeny ω has the effet of an ideal lowpass filte with utoff fequeny Ω = ω / T. The utoff fequeny depends on both ω and T. A vaiable ontinuous-time lowpass filte an be implemented by a fixed disete-time lowpass filte but vaying the sampling peiod T, 27

28 Example: Disete-Time Implementation of an Ideal Continuous-Time Bandlimited Diffeentiato The ideal ontinuous-time diffeentiato system is d y ( t ) [ x ( t )] H C ( j ) j dt Fo poessing bandlimited signals, it is suffiient that j, / T Heff ( j) 0, / T Theefoe the oesponding disete-time system has fequeny j esponse: H ( e ) j / T with peiod 2 0, n 0 1 j jn n os n sin n h[ n] e d os( n) 2 2 T n T, n 0 nt 28

29 Example: Disete-Time Implementation of an Ideal Continuous-Time Bandlimited Diffeentiato If this system has the input x ( t ) os( t ) / T x [ n] os( Tn) jt 1 X ( e ) [ ( 0 k s) ( 0 k s)] T k j X ( e ) ( ) ( ) T j j j j Y ( e ) H ( e ) X ( e ) [ ( 0) ( 0)] T j j0 j0 Y ( e ) ( 0) ( 0) T T jt Y ( j ) TY ( e ) j ( ) j ( ) d y ( t ) 0sin( 0t ) [ x ( t )] dt ( / T) T ( ) 29

30 Impulse Invaiane If the desied ontinuous-time system has bandlimited fequeny esponse then how to hoose j so that H eff He ( ) H ( j ) H ( j ) eff jt H ( e ), / T ( j) 0, / T j H ( e ) H ( j / T ), T be hoosen suh that H ( j ) 0 / T HC ( j) h[ n] Th ( nt ) the impulse esponse of the disete-time system is a saled, sampled vesion of h(t) In this ase the disete-time system is said to be an impulse-invaiant vesion of the ontinuous time system. 30

31 Impulse Invaiane We have: x[n] = x (nt), by eplaing x[n] and x (n) by h[n] and h (t) : h[n] = h (nt) Sine H (jω) = 0 Ω π/t Modifying above equation fo sale fato T we have: h[n] = Th (nt) Now the disete-time system is said to be an impulse-invaiant vesion of the ontinuous-time system. 31 k j T k T j H T e H )) 2 ( ( 1 ) ( ) ( 1 ) ( T j H T e H j ) ( ) ( T j H e H j

32 Example: A disete-time lowpass filte obtained by impulse invaiane We want to obtain an ideal lowpass disete-time filte with utoff fequeny. we an do this by sampling a ontinuous-time ideal lowpass filte with utoff fequeny / T / T 1, H ( j) 0, sin( t ) h () t t sin( nt ) sin( n ) h[ n] Th ( nt ) T nt n 32

33 Changing the sampling ate using disete-time poessing We have seen that a ontinuous-time signal an be epesented by a disete-time signal. x [ n] x ( nt ) It is often neessay to hange the sampling ate of x[n] and obtain a new disete-time signal suh that x [ n] x ( nt ) One appoah is to eonstut x () t and then esample it with peiod T, but it is of inteest to onside methods that involve only disete time opeations. 33

34 Sampling ate edution by an intege fato Disete-time sample o ompesso x [ n] x [ nm ] x ( nmt ) d If X then is an exat C( j ) 0 fo N xd [ n] epesentation of x ( t ) iff / T / MT T MT N Downsampling: the opeation of eduing the sampling ate (inluding any filteing). 34

35 Fequeny domain elation between the input and output of the ompesso j 1 2k x [ n] x ( nt ) X ( e ) X C( j ( )) T T T j 1 2 x d [ n] x ( nmt ) X d ( e ) X C ( j ( )) MT MT MT i km k, 0 i M 1 M 1 j k 2 i X d( e ) X C( j ( )) M i 0 T k MT T MT M 1 j 1 j ( / M 2 i / M ) X d ( e ) X ( e ) M i 0 k 35

36 Downsampling without Aliasing Example: M=2 36

37 Downsampling with aliasing 37

38 Downsampling with pefilteing to avoid aliasing 38

39 Ineasing the sampling ate by an intege fato We will efe to the opeation of ineasing the sampling ate upsampling x i[ n] x [ n / L] x ( nt / L) n 0, L, 2 L,... The system on the left is alled a sampling ate expande. Its output is x e x [ n / L], n 0, L, 2 L,... [ n] 0, othewise x [ n] x [ k ] [ n kl ] e k The system on the ight is a lowpass disete-time filte with utoff fequeny / L and gain L. 39

40 Ineasing the sampling ate by an intege fato j jn jlk jl X e ( e ) x [ k ] [ n kl] e x [ k ] e X ( e ) n k k This system is an intepolato beause of it fills in the missing samples. 40

41 Ineasing the Sampling Rate By an Intege Fato x e x [ n / L ], n 0, L, 2 L,... [ n] 0, othewise x [ n] x [ k ] [ n kl ] e k sin( n/ L) h [ n] i n/ L sin( ( n kl ) / L ) x i [ n] x [ k ] ( n kl ) / L k theefoe x i [ n] x [ n / L ] x ( nt / L ) x ( nt ) n 0, L, 2 L,... If the input sequene x [ n] x ( nt ) was obtained by sampling without aliasing then x i[ n] x ( nt ) is oet fo all n, and x i [ n] is obtained by ovesampling of x () t. 41

42 Linea Intepolation In patie ideal lowpass filtes an not be implemented exatly. In some ases, simple intepolation poedue ae adequate. h lin 1 n / L, n L [ n] 0, othewise H lin j ( e ) 1 sin( L / 2) L sin( / 2) 2 x [ n] x [ k ] h [ n k ] lin e lin k k x [ k ] h [ n kl ] lin 42

43 Changing the Sampling Rate by a Nonintege Fato By ombining deimation and intepolation it is possible to hange the sampling ate by a nonintege fato. The intepolation and deimation filte an be ombined togethe. 43

44 Changing the Sampling Rate by a Nonintege Fato 44