CHAPTasdadaddsaddasdasdsadadadasd dsadsdsa asdER 6.pdf

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1. COMMUNICATION
SYSTEMS
SAMPLING AND ANALOG- DIGITAL CONVERSION183

2. SAMPLING AND ANALOG-DIGITAL
CONVERSION
•ChapterNo.6ofTextBookRecommended
•Thislecture willbe uploaded on yourslate accounts aswellforyour
convenience.
•But I suggestyouto attendthe onlinelectures at time toavoidanyloss of
understandingtheparticulartopic.
•Foranyquery,youcancontactmeon
[email protected]

3. Sampling
•The process ofconvertingcontinuoustimesignalsintoequivalentdiscrete
timesignalsiscalledsampling.
•Thevalueof signalis measuredatcertain intervals in time.Each
measurementiscalledasample.
•Sufficient number ofsamplesmust betaken,sothattheoriginalsignalcan
bereconstructedproperly.
•Numberofsamples tobetakendepends on themaximumfrequencyof
signal.185

4. Sampling186

5. Sampling187

6. Sampling Frequency and Sampling Interval
•ItisalsocalledSamplingRateanddenotedby????????????
????????????.
•Samplingratedenotesthenumberofsamplestakenpersecond.
•Forproperreconstructionof signalfromsamples????????????
????????????>2????????????.
•Gapbetweenthesamplesof signalsis termed as Sampling Interval/Period
denotedby????????????
????????????.ItisReciprocalof samplingfrequency/rate,thus:
????????????
????????????=
1
????????????
????????????188

7. Sampling Theorem
SamplingTheoremcanbedefinedintermsof frequencyas:
The samplingtheorem canbe definedasthe conversion ofan analogsignal
intoadiscreteformbytakingthe sampling frequency????????????
????????????as twiceor equalto
bandwidth????????????of signali.e.
????????????
????????????≥2????????????189

8. Sampling Theorem
SamplingTheoremcanbedefinedintermsoftimeas:
The samplingtheorem canbe definedasthe conversion ofan analogsignal
intoadiscreteformbytakingthe sampling period/interval????????????
????????????less thanor
equaltoreciprocaloftwicethebandwidthof signali.e.
????????????
????????????≤
1
2????????????190

9. Three Cases for Sampling Rate
Threepossiblecasesbasedonsamplingrateare:
i.????????????
????????????>2????????????
ii.????????????
????????????<2????????????
iii.????????????
????????????=2????????????191

10. Case 1: ????????????
????????????>2????????????192

11. Case 2: ????????????
????????????<2????????????193

12. Case 3: ????????????
????????????=2????????????194

13. NyquistRate and NyquistInterval
•Fordistortionlessreconstructionof signalhenceitfollowsthat:
????????????
????????????≥2????????????
•Thecaseinwhich????????????
????????????=2????????????,samplingrateisreferredtoasNyquistRate????????????
????????????
•Likewise when????????????
????????????=2????????????, samplinginterval/periodis referred asNyquist
Interval.195

14. Example
Calculate the NyquistInterval for the following signal.
????????????????????????=2??????????????????????????????????????????????????????????????????????????????+8??????????????????????????????????????????????????????????????????????????????
????????????
????????????=
1
????????????
????????????
=
1
2∗????????????
????????????=
????????????
????????????
2????????????
=
500????????????
2????????????
=250????????????????????????
????????????
????????????=
1
2∗????????????
=
1
2∗250
=0.002????????????????????????????????????196

15. Example
A signal????????????(????????????)band-limited to3????????????????????????????????????is sampled atarate33% hi gherthanthe
Nyquistrate.Calculatetheactualsamplingrateof signal.
????????????
????????????=2????????????=2∗3000=6????????????????????????????????????
????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????,????????????
????????????=33%ℎ????????????????????????????????????????????????????????????????????????????????????????????????????????????
????????????
????????????
????????????=33%∗????????????
????????????
????????????
????????????=1+0.33∗6000=7980????????????????????????197

16. Quantization
•The process ofconvertingcontinuoustimesignalsintoequivalentdigital
timesignalsiscalledQuantization.
•Thedigitizationofanalogsignalsinvolvesthe rounding off of thevalues
whichareapproximatelyequaltotheanalogvalues.Themethodofsampling
choosesafewpointsontheanalogsignalandthenthesepointsarejoinedto
round off thevaluetoanear stabilized value.Sucha processis called
asQuantization.198

17. Quantization
•AnalogSignal:Continous(x-axis),Continous(y-axis)
•DiscreteSignal:Discrete(x-axis),Continous(y-axis)
•DigitalSignal:Discrete(x-axis),Discrete(y-axis)
Sampling Quantization
Analog
Signal
Discrete
Signal
Digital
Signal199

18. Quantization200

19. Quantization Levels
•Amplituderange(−????????????
????????????,????????????
????????????)isdividedinto????????????numberof uniform spaced
intervals.These????????????numberlevelsarecalledquantizationlevels.
•
NumberofQuantizationlevelsdependon????????????i.e.numberof bits/sample.
????????????= 2
????????????
So,ifnumberofbitspersampleis2thennumberofquantizationlevelsis:
????????????= 2
2
=4201

20. Magnitude of Quantization Levels
•Magnitude ofquantization levelsor widthbetweenlevels canbedefined as:
∆????????????=
2????????????
????????????
????????????
•Insome cases, itmight be:
∆????????????=
????????????
????????????????????????????????????−????????????
????????????????????????????????????
????????????
Where????????????
????????????=−????????????
????????????,then????????????
????????????????????????????????????=????????????
????????????and????????????
????????????????????????????????????=−????????????
????????????
∆????????????=
????????????
????????????−(−????????????
????????????)
????????????
=
2????????????
????????????
????????????202

21. Errors in Quantization
•Quantizedsamplesaretransmittedinbinarypulses.
•Atreceiverside,somepulsesmaybedetected incorrectly. Hencethereare
twosourcesoferrorinthisscheme.
i.PulseDetectionerror
ii.QuantizationError
•Pulsedetection isverysmallandnegligible, sotheerrorwhicharisesisonly
thequantizationerror.203

22. Quantization Error
•Quantizederrorinquantizationisrepresentedby????????????(????????????).
•????????????(????????????)istheundesiredsignalwhichactsasnoisesoitisalsoreferredasQuantization
Noise.
•
Maximumquantizationerroris
±∆????????????
2
i.e. quantizationerrorlies intherangeof
(
−∆????????????
2
,
+∆????????????
2
)
where
∆????????????=
2????????????????????????
????????????204

23. Power of Quantization Error
•Asquantizationerrorlies intherange of(
−∆????????????
2
,
+∆????????????
2
),sopowerof quantization errorcanbe
calculatedas:
????????????
????????????=
1
∆????????????
�
−∆????????????
2
+∆????????????
2
????????????
2
????????????
????????????
????????????
????????????=
∆????????????
2
12
=
????????????
????????????
2
3????????????
2
????????????
????????????ismeansquare value ofquantizationerror orquantizationnoise.Quantizationnoiseisalso
denotedby????????????
????????????.
????????????
????????????=????????????
????????????=
????????????
????????????
2
3????????????
2205

24. Example
A signal????????????????????????band-limited to3????????????????????????????????????is sampledatarate 33% hig herthan
Nyquistrate.Ifthelevelsarespacedwithaspacing scale0f 0.75thenhow
manylevelscanbedefined?206

25. Example
Asignal????????????????????????band-limitedto3????????????????????????????????????issampledat arate33%high erthanNyquistrate.Maximumacceptable
errorinsampleamplitude is0.5%of peak amplitude of signal.Howmanyquantization intervals/ levelsare
neededtoquantizethesignal?
Maximum acceptable error=
∆????????????
2
=0.5%????????????
????????????
∆????????????
2
=
????????????
????????????
????????????
0.5%????????????
????????????=
????????????
????????????
????????????
????????????=
????????????
????????????
0.5%????????????
????????????
????????????=
1
0.5%
=200207

26. Signal to Noise Ratio (SNR)
Itistheratioofpowerof signaltothepowerof noisei.e.
????????????????????????????????????=
????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????
????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????
=
????????????
????????????
????????????
????????????
Forexample:
If a 20W signal is quantized and the noise signal arises having 50W power then
????????????????????????????????????=
20
50
=
0.4208

27. Signal to Noise Ratio (SNR)
SNRcanbecalculated usingthe generalformula:
????????????
????????????
????????????
????????????
=????????????.????????????
2
Asweknow that????????????= 2
????????????
????????????
????????????
????????????
????????????
=????????????.(2
????????????
)
2
????????????
????????????
????????????
????????????
=????????????. 2
2????????????
To find number of bits/sample ????????????= 2
????????????
????????????=????????????????????????????????????
2????????????209

28. Signal to Noise Ratio (SNR)
Ifthesignalisuncompressed:
????????????=
3.????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????
????????????
????????????
2
Ifthesignaliscompressed:
????????????=
3
[ln1+????????????]
2210

29. Signal to Noise Ratio in dB
SNRindBscalecanbecalculatedas:
(
????????????
????????????
????????????
????????????
)
????????????????????????=10log(
????????????
????????????
????????????
????????????
)
(
????????????
????????????
????????????
????????????
)
????????????????????????=????????????+6????????????????????????????????????
where
????????????=10????????????????????????????????????
10????????????211

30. Transmission Bandwidth (????????????
????????????)
Minimum bandwidthrequireto transmitthe quantized signal.Itisdenotedby
????????????
????????????. ????????????
????????????=????????????????????????
????????????=
????????????
????????????
????????????
So,
????????????
????????????
????????????????????????
=????????????.(2)
2
????????????
????????????
????????????212

31. Channel Capacity(????????????)
Channelcapacityisameasureofmaximuminformationratethatachannelcan
transmit. Itismeasuredinbitspersecond(bps).
????????????=????????????×????????????
????????????
????????????=????????????×2????????????
????????????=2×????????????????????????
As , ????????????
????????????=????????????????????????
????????????
????????????=
????????????
2
????????????????????????213

32. Pulse Code Modulation (PCM)
PCMismostwidelyusedtechniqueamongpulsemodulationschemesi.e.Pulse
Position Modulation (PPM), Pulse Amplitude Modulation(PAM),Pulse Width
Modulation(PWM)etc.
Analog
Signal
Sampling Quantization
Discrete
Signal
Digital
Signal
Encoding
Code-
word214

33. Example
Asignal????????????(????????????)band-limitedto3????????????????????????????????????is sampledatarate33%higher thant he Nyquistrate.If the
signal is quantized using4quantizationlevels,calculatethechannel capacity and transmission
bandwidth.
????????????
????????????= 2????????????= 2∗3000= 6????????????????????????????????????
????????????
????????????=1+0.33∗6000=7980????????????????????????
????????????= 4
????????????=????????????×????????????
????????????
????????????=????????????????????????????????????
2????????????×7980= 2×7980=15960????????????????????????????????????????????????/????????????????????????????????????
????????????
????????????=
????????????
2
=
15960
2
=7980????????????????????????215

34. Example
A50????????????sinusoidalsignalofamplitud e5????????????is quantized using64qu antizationintervals.
Compute the(????????????????????????????????????)
????????????????????????.
(
????????????
????????????
????????????
????????????
)
????????????????????????=????????????+6????????????????????????????????????
????????????=10????????????????????????????????????
10????????????
????????????=
3.????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????
????????????
????????????
2
=
3.(50????????????)
5
2
= 6
????????????=64
????????????=????????????????????????????????????
2????????????= 6216

35. ????????????=10????????????????????????????????????
10????????????
????????????=10????????????????????????????????????
106=100.778=7.78
(
????????????
????????????
????????????
????????????
)
????????????????????????=????????????+6????????????????????????????????????
(
????????????
????????????
????????????
????????????
)
????????????????????????=7.78+66
(
????????????
????????????
????????????
????????????
)
????????????????????????=79.78????????????????????????217

36. Example
Asignal????????????(????????????)hasbandwidthof4????????????????????????????????????istransmittedusing binarycompanded
PCMwith????????????=100.Computethe(????????????????????????????????????)
????????????????????????ifthesignalisquantizedusing64
levels.
(
????????????
????????????
????????????
????????????
)
????????????????????????=????????????+6????????????????????????????????????
????????????=????????????????????????????????????
2????????????=6
????????????=
3
[ln1+????????????]
2
=
3
[ln101]
2
=0.1408218

37. ????????????=10????????????????????????????????????
10????????????
????????????=10????????????????????????????????????
100.1408=10−0.8513=−8.513
(
????????????
????????????
????????????
????????????
)
????????????????????????=????????????+6????????????????????????????????????
(
????????????
????????????
????????????
????????????
)
????????????????????????=−8.513+6(6)
(
????????????
????????????
????????????
????????????
)
????????????????????????=63.487????????????????????????219

38. Example
Asignal????????????(????????????)hasbandwidthof4????????????????????????????????????istransmittedusing binarycompandar
with????????????=100.Computethe(????????????????????????????????????)
????????????????????????ifnumberofbitsperquantizedsample
is8.
(
????????????
????????????
????????????
????????????
)
????????????????????????=????????????+6????????????????????????????????????
????????????=8
????????????=
3
[ln1+????????????]
2
=
3
[ln101]
2
=0.1408220

39. ????????????=10????????????????????????????????????
10????????????
????????????=10????????????????????????????????????
100.1408=10−0.8513=−8.513
(
????????????
????????????
????????????
????????????
)
????????????????????????=????????????+6????????????????????????????????????
(
????????????
????????????
????????????
????????????
)
????????????????????????=−8.513+68
(
????????????
????????????
????????????
????????????
)
????????????????????????=39.487????????????????????????221

40. Example
Asignal????????????(????????????)hasbandwidthof4????????????????????????????????????istransmittedusing binarycompanded
PCMwith????????????=100.Comparethe(????????????????????????????????????)
????????????????????????for????????????=64and????????????=256.
Hint:Compareboth(????????????????????????????????????)
????????????????????????andstatetheresultwhichishigherintermsof theirSNR
ratio.
(
????????????
????????????
????????????
????????????
)
????????????????????????=????????????+6????????????????????????????????????
????????????=
3
[ln1+????????????]
2
=
3
[ln101]
2
=0.1408222

41. •For ????????????=64????????????=6
(
????????????
????????????
????????????
????????????
)
????????????????????????=27.49????????????????????????
•For ????????????=256????????????=8
(
????????????
????????????
????????????
????????????
)
????????????????????????=39.49????????????????????????223

42. ????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????(
????????????
????????????
????????????
????????????
)
????????????????????????=39.49????????????????????????−27.49????????????????????????=12????????????????????????
12????????????????????????=10log(
????????????
????????????
????????????
????????????
)
1.2????????????????????????=log(
????????????
????????????
????????????
????????????
)
????????????
????????????
????????????
????????????
=16
Therefore, SNR of L=256 is 16 times higher than SNR of L=64224