EEO 40 Digial Proceing Pro. Mark Fowler Noe Se #3 Baic Sampling heory Reading Aignmen: Sec. 6. o Proaki & Manolaki /9
Sampling i Key o Much o oday echnology Analog Elecronic ADC DSP Compuer DAC C- C- Syem C- D- D- Syem D- Cell Phone, CD, Digial Muic, Radar, Medical Imaging, ec.! C- he ir ep o ee ha hi i poible: Can we recover he ignal rom i ample???!!! x C- ADC x[n] D- Clock DAC xˆ C- xˆ x Can we make:? I we can hen we can proce he ample x[n] inead o x!!! 2/9
Pracical Sampling-Reconrucion Se-Up Analog-o-Digial Converer Digial-o-Analog Converer x ADC DAC x [n] = x n Sample a = n Hold = Sampling Inerval F = / = Sampling Rae Pule Gen Clock a = n ~ x C LPF xˆ Value 2 0 - -2 2 0 0.2 0.4 0.6 0.8 ime Value 0 - -2 0 0.2 0.4 0.6 0.8 ime 3/9
Mah Model or Sampling ADC You learn he circui in an elecronic cla Here we ocu on he why, o we need mah model Mah Modeling he ADC i eay. x[n] = xn, o he n h ample i he value o x a = n x[ n] x x n n Analog-o-Digial Converer ADC x x [n] = x n Sample a = n Hold = Sampling Inerval F = / = Sampling Rae 4/9
Mah Model or Reconrucion DAC Mah Model or he DAC coni o wo par: convering a D equence o number ino a C pule rain moohing ou he pule rain uing a lowpa iler x[n] Pule Gen ~ x C Filer H xˆ Filer made rom ranior/opamp, Reior & Cap ~ x x n p n n ˆ H Prooype Pule p ~ x x -2-2 3 5/9
Impule Sampling Model or DAC Now we have a good model ha handle quie well wha REALLY happen inide a DAC bu we impliy i!!!! o Eae Analyi: Ue p Why????. Becaue dela uncion are EASY o analyze!!! 2. Becaue i lead o he be poible reul ee laer! 3. We can eaily accoun or real-lie pule laer!! p ~ x x n n n In hi orm hi i called he Impule Sampled ignal. Now.. Uing propery o dela uncion we can alo wrie ~ x x n n 6/9
x Sample a = n ADC Hold Sampling Analyi p. Analyi will be done uing he Impule Sampling Mah Model x[n] = xn Impule Gen DAC ~ x h H xˆ ~ x x x n n x Impule rain Impule Sampled -2 - ~ x x Noe: we are uing he impule ampling model in he DAC no he ADC!!! -2-2 3 2 3 7/9
Sampling Analyi p. 2 Goal = Deermine Under Wha Condiion We Ge: Reconruced C = Original C ˆ x x Approach:. Find he F o he ignal 2. Ue Freq. Repone o Filer o ge 3. Look o ee wha i needed o make ~ x ~ ˆ H ˆ 8/9
Sampling Analyi p. 3 Sep #: Hmmm well i periodic wih period o we COULD expand i a a Fourier erie: So wha are he FS coeicien??? c k / 2 / 2 e jk 2F e 0 jk 2F c ke Period = ec k Fund. Freq = F = / Hz jk 2F / 2 e / 2 So an alernae model or i jk 2F Only one dela inide a ingle period By iing propery o he dela uncion!!! k e jk 2F 9/9
Original F Sampling Analyi p. 4 So we now have. k F jk k F jk e x e x x x 2 2 ~ By requency hi propery o F each erm i a requency hied verion o he original ignal!!! So uing he requency hi propery o he F give: k kf ~ Exremely Imporan Reul he bai o all underanding o ampling!!! 2 2 ~ F F F F Shied Replica Ue FS Reul 0/9
Sampling Analyi p. 5 So he BIG hing we ve ju ound ou i ha: he impule ampled ignal inide he DAC ha a F ha coni o he original ignal F and requency-hied verion o i where he requency hi are by ineger muliple o he ampling rae F : ~ 2F F F 2F Original F hi reul allow u o ee how o make ampling work By work we mean: how o enure ha even hough we only have ample o he ignal, we can ill ge perec reconrucion o he original ignal. a lea in heory!! he igure on he nex page how how. /9
x Sample a = n ADC Hold Sampling Analyi p. 6 x[n] = xn Impule Gen DAC C LPF ~ x xˆ B A A/ B ~ kf k 2F F o enure ha he replica don overlap he original. we need F B B or equivalenly F 2B F B F B 2F When here i no overlap, he original pecrum i le unharmed and can be recovered uing a C LPF a een on he nex page. 2/9
x Sample a = n ADC Hold Sampling Analyi p. 7 x[n] = xn Impule Gen DAC C LPF ~ x xˆ B A A/ B ~ kf k 2F F H F 2F A ˆ ˆ i F 2B 3/9
Sampling Analyi Reul Wha hi analyi ay: Sampling heorem: A bandlimied ignal wih BW = B Hz i compleely deined by i ample a long a hey are aken a a rae F 2B ample/econd. Impac: o exrac he ino rom a bandlimied ignal we only need o operae on i properly aken ample hen can ue a compuer o proce ignal!!! x Sample a = n Hold x[n] = xn Compuer Exraced Inormaion hi mah reul publihed in he lae 940! i he oundaion o: CD, MP3, digial cell phone, ec. 4/9
Aliaing Analyi: Wha i ample are no aken a enough??? ADC DAC x x[n] = xn Sample a = n Hold Impule Gen C LPF ~ x xˆ B B ~ 2F aliaing F F H 2F I F 2B... ˆ ˆ Called aliaing error o enable error-ree reconrucion, a ignal bandlimied o B Hz mu be ampled aer han 2B ample/ec 5/9
Aliaing Analyi: Wha i he ignal i NO BANDLIMIED??? ADC DAC x x[n] = xn Sample a = n Hold Impule Gen C LPF ~ x xˆ ~ F F For Non-BL Aliaing alway happen regardle o F value All pracical ignal are Non-BL!!!! o we chooe F o minimize he aliaing o an level accepable or he peciic applicaion 6/9
Pracical Sampling: Ue o Ani-Aliaing Filer In pracice i i imporan o avoid exceive aliaing. So we ue a C lowpa BEFORE he ADC!!! F = 44. khz ADC Microphone Amp C Ani-Aliaing LPF C Sample & Digiize Code Dicree-ime Burn bi ino CD -20 khz 20 khz H aa AA Filer aa 20 khz -22 khz ~ aa 20 khz -22.05 khz Minimal Aliaing 22.05 khz 7/9
Some Sampling erminology F i called he ampling rae. I uni i ample/ec which i oen equivalenly expreed a Hz. he minimum ampling rae o F = 2B ample/ec i called he Nyqui Rae. Sampling a he Nyqui rae i called Criical Sampling. Sampling aer han he Nyqui rae i called Over Sampling Sampling lower han he Nyqui rae i called Under Sampling Noe: Criical ampling i only poible i an IDEAL lowpa iler i ued. o in pracice we generally need o chooe a ampling rae omewha above he Nyqui rae e.g., 2.2B ; he choice depend on he applicaion. 8/9
Summary o Sampling Mah Model or Impule Sampling inide he DAC ay he F o he impule ampled ignal ha pecral replica paced F Hz apar hi mah reul drive all o he inigh ino pracical apec heory ay or a BL d wih BW = B Hz I i compleely deined by ample aken a a rae F 2B hen Perec reconrucion can be achieved uing an ideal LPF reconrucion iler i.e., he iler inide he DAC heory ay or a Pracical Pracical ignal aren bandlimied o ue an Ani-Aliaing lowpa iler BEFORE he ADC Becaue he A-A LPF i no ideal here will ill be ome aliaing Deign he A-A LPF o give accepably low aliaing error or he expeced ype o ignal 9/9