5.5 Sampling. The Connection Between: Continuous Time & Discrete Time

Transcription

1 5.5 Sampling he Connection Between: Continuou ime & Dicrete ime Warning: I don t really like how the book cover thi! It i not that it i wrong it jut ail to make the correct connection between the mathematic and phyical reality!!!! Follow thee note and you ll get it

2 Sampling i Key Part o CD Scheme Sampled & Digitized muic on a Compact Dic What enure that we can perectly recontruct the muic ignal rom it ample???!!!! Record Create a equence o ample i.e., a equence o number Microphone 2 Amp Sample & Digitize Code Burn bit into CD Value 0 - Play ime Laer Senor Decode Recontruct Amp Value ime Speaker Original? 2

3 Sampling i Key Part o Many Sytem Sytem that ue Digital Proc. DSP generally get a continuou-time ignal rom a enor a cont.-time ytem modiie the ignal an analog-to-digital converter ADC ample the ignal to create a dicrete-time ignal A dicrete-time ytem to do the Digital Proceing and then i deired convert back to analog uing a digitalto-analog converter DAC Analog Electronic ADC DSP Computer DAC C- C- Sytem C- D- D- Sytem D- C- 3

4 I Sampling i Valid Can Perectly Recontruct rom Sample xt C- ADC x[n] D- DAC xˆ t C- Clock x ˆ t x t Can we make:??? I we can then we can proce the ample x[n]a an alternative to proceing xt!!! 4

5 Practical Sampling-Recontruction Set-Up Analog-to-Digital Converter Digital-to-Analog Converter xt ADC DAC x [n] x n Sample at t n Hold Sampling Interval F / Sampling Rate Pule Gen Clock at t n ~ x t C LPF xˆ t Value ime Value ime 5

6 You learn the circuit in an electronic cla Here we ocu on the why, o we need math model We tart in a little dierent place than the book but we end up with the ame reult but a little eaier to ee how/why Math Model or Sampling ADC Math Modeling the ADC i eay. x[n] xn, o the n th ample i the value o xt at t n x[ n] x t x n t n Note: the book ue the impule ampling model o Eq or the ADC but that ha no connection to a phyical ADC we ll ee later that it doe have a phyical connection to the DAC! 6

7 Math Model or Recontruction DAC Math Model or the DAC conit o two part: converting a D equence o number into a C pule train moothing out the pule train uing a lowpa ilter C LPF x[n] Pule Gen ~ x t ht Hω xˆ t n ~ x t x n p t n xˆ t Xˆ ω ~ x t h t ~ X ω H ω Prototype Pule pt ~ x t xt t t 7

8 Impule Sampling Model or DAC Now we have a good model that handle quite well what REALLY happen inide a DAC but we impliy it!!!! o Eae Analyi: Ue p t δ t Why????. Becaue delta unction are EASY to analyze!!! 2. Becaue it lead to the bet poible reult ee later! 3. We can eaily account or real-lie pule later!! p t δ t n ~ x t x n δ t n In thi orm thi i called the Impule Sampled ignal. Now.. Uing property o delta unction we can alo write n ~ x t x t δ t n 8

9 Sampling Analyi p. Analyi will be done uing the Impule Sampling Math Model xt Sample at t n ADC Hold x[n] xn Impule Gen DAC ~ x t ht Hω xˆ t ~ x t x t δ t n x t δ t n xt δ t Impule rain t Impule Sampled Note: we are uing the impule ampling model in the DAC not the ADC!!! ~ x t x t δ t 2 3 t t 9

10 Sampling Analyi p. 2 Goal Determine Under What Condition We Get: Recontructed C Original C x ˆ t x t Approach:. Find the F o the ignal 2. Ue Freq. Repone o Filter to get 3. Look to ee what i needed to make ~ x t ~ X ˆ ω X ω H ω X ˆ ω X ω 0

11 Sampling Analyi p. 3 Step #: Hmmm well δ t i periodic with period o we COULD expand it a a Fourier erie: δ So what are the FS coeicient??? c k / 2 δ / 2 t e [ ] jk2πf t e t 0 jk2πf t t c ke Period ec k Fund. Freq F / Hz jk2πf t / 2 δ t e / 2 So an alternate model or δ t i jk2πf t Only one delta inide a ingle period By iting property o the delta unction!!! δ t k e jk2πf t

12 2 Sampling Analyi p. 4 So we now have. k F t jk k F t jk e t x e t x t t x t x π π δ 2 2 ~ By requency hit property o F each term i a requency hited verion o the original ignal!!! So uing the requency hit property o the F give: + k kf X X ~ Extremely Important Reult the bai o all undertanding o ampling!!! [ ]! +! ~ F X F X X F X F X X Original F Shited Replica

13 Sampling Analyi p. 5 So the BIG hing we ve jut ound out i that: the impule ampled ignal inide the DAC ha a F that conit o the original ignal F and requency-hited verion o it where the requency hit are by integer multiple o the ampling rate F hi reult allow u to ee how to make ampling work By work we mean: how to enure that even though we only have ample o the ignal, we can till get perect recontruction o the original ignal. at leat in theory!! he igure on the next page how how. 3

14 xt Sample at t n ADC Hold Sampling Analyi p. 6 x[n] xn Impule Gen DAC C LPF ~ x t xˆ t A X B A/ B ~ X 2F F F B F B 2F o enure that the replica don t overlap the original. we need F B B or equivalently F 2B When there i no overlap, the original pectrum i let unharmed and can be recovered uing a C LPF a een on the next page. 4

15 xt Sample at t n ADC Hold Sampling Analyi p. 7 x[n] xn Impule Gen DAC C LPF ~ x t xˆ t A X B A/ B ~ X 2F F H F 2F A Xˆ Xˆ X i F 2B 5

16 Sampling Analyi p. 8 What thi analyi ay: Sampling heorem: A bandlimited ignal with BW B Hz i completely deined by it ample a long a they are taken at a rate F 2B. Impact: o extract the ino rom a bandlimited ignal we only need to operate on it properly taken ample! hen can ue a computer to proce ignal!!! xt Sample at t n Hold x[n] xn Computer Extracted Inormation hi math reult publihed in the late 940! i the oundation o: CD, MP3, digital cell phone, etc. 6

17 Some Sampling erminology F i called the ampling rate. It unit i ample/ec which i oten equivalently expreed a Hz he minimum ampling rate o F 2B ample/ec i called the Nyquit Rate. Sampling at the Nyquit rate i called Critical Sampling. Sampling ater than the Nyquit rate i called Over Sampling Note: Critical ampling i only poible i an IDEAL lowpa ilter i ued. o in practice we generally need to chooe a ampling rate omewhat above the Nyquit rate e.g., 2.2B ; the choice depend on the application. 7

18 Aliaing Analyi: What i ample are not taken at enough??? ADC DAC xt x[n] xn Sample at t n Hold X Impule Gen C LPF ~ x t xˆ t B B ~ X aliaing 2F F F H 2F I F < 2B... Xˆ X Xˆ Called aliaing error o enable error-ree recontruction, a ignal bandlimited to B Hz mut be ampled ater than 2B ample/ec 8

19 Aliaing Analyi: What i the ignal i NO BANDLIMIED??? ADC DAC xt x[n] xn Sample at t n Hold X Impule Gen C LPF ~ x t xˆ t ~ X F F For Non-BL Aliaing alway happen regardle o F value All practical ignal are Non-BL!!!! o we chooe F to minimize the aliaing to an level acceptable or the peciic application 9

20 Practical Sampling: Ue o Anti-Aliaing Filter In practice it i important to avoid exceive aliaing. So we ue a C lowpa BEFORE the ADC!!! F 44. khz ADC Microphone Amp C Anti-Aliaing LPF C X Sample & Digitize Code Dicrete-ime Burn bit into CD -20 khz 20 khz H aa AA Filter X aa 20 khz -22 khz ~ X aa 20 khz khz Minimal Aliaing khz 20

21 Summary o Sampling Math Model or Impule Sampling ay he F o the impule ampled ignal ha pectral replica paced F Hz apart hi math reult drive all o the inight into practical apect heory ay or a BL d with BW B Hz It i completely deined by ample taken at a rate F 2B hen Perect recontruction can be achieved uing an ideal LPF recontruction ilter i.e., the ilter inide the DAC heory ay or a Practical Practical ignal aren t bandlimited o ue an Anti-Aliaing lowpa ilter BEFORE the ADC Becaue the A-A LPF i not ideal there will till be ome aliaing Deign the A-A LPF to give acceptably low aliaing error or the expected type o ignal 2