Chapter 4 Receiver Design - Temple Universitysilage/Chapter4MS.pdf · EE4512 Analog and Digital Communications Chapter 4 Chapter 4 Receiver Design • Examining Thermal ... Nyquist,

EE4512 Analog and Digital Communications Chapter 4

Chapter 4Chapter 4

Receiver DesignReceiver Design


Chapter 4Chapter 4

Receiver DesignReceiver Design•• Probability of Bit ErrorProbability of Bit Error

•• Pages 124Pages 124--149149


•• Probability of Bit ErrorProbability of Bit Error

The low pass filtered and sampled PAM signal results in The low pass filtered and sampled PAM signal results in an expression for the an expression for the probability of bit errorprobability of bit error PPb b (S&M p. (S&M p. 124124--127). 127). AA is the amplitude at the sampling point and is the amplitude at the sampling point and γγis the attenuation of the channel (0 is the attenuation of the channel (0 ≤≤ γγ ≤≤ 11))

P{ P{ ithith bit in error } = P(bbit in error } = P(bii = 0) P{ n= 0) P{ noo[ (i[ (i--1)1)TTbb + + TTbb/2 ] /2 ] < < ––γγA }A }+ + P(bP(bi i = 1) P{ n= 1) P{ noo[ (i[ (i--1)1)TTbb + + TTbb/2 ] /2 ] ≥≥ γγA }A }


•• Review of Probability and Stochastic ProcessesReview of Probability and Stochastic Processes(S&M p. 127(S&M p. 127--132)132)

Probability distribution functionProbability distribution function FFXX(a) = P{ (a) = P{ XX = a }= a }

Probability density functionProbability density function ffXX(x) = d F(x) = d FXX(x) / (x) / dxdx

Mean (or expected value)Mean (or expected value) µµX X = = ∫∫ x x ffXX(x)(x) dxdx∞∞

VarianceVariance σσXX22 == ∫∫ (x (x –– µµXX) ffXX(x)(x) dxdx

––∞∞EE { ({ (X X –– µµXX)2 2 }}


•• Review of Probability and Stochastic ProcessesReview of Probability and Stochastic Processes(S&M p. 127(S&M p. 127--132)132)

Joint probabilityJoint probabilitydistribution functiondistribution function FFX,YX,Y(a(a, b) = P{ , b) = P{ XX = a and = a and YY = b }= b }

Joint probabilityJoint probabilitydensity functiondensity function ffX,YX,Y(x, y)) = (x, y)) = ∂∂22 FFX,YX,Y(x, y) / (x, y) / ∂∂xx ∂∂yy

Conditional probabilitiesConditional probabilities P { P { XX > a and event Z } => a and event Z } =P { event Z } P{ X > a | event Z }P { event Z } P{ X > a | event Z }


Chapter 4Chapter 4

Receiver DesignReceiver Design•• Examining Thermal NoiseExamining Thermal Noise

•• Pages 132Pages 132--136136


•• JohnsonJohnson––Nyquist noiseNyquist noise or or thermal noisethermal noise is the is the electronicelectronicnoisenoise generated by the thermal agitation of the charge generated by the thermal agitation of the charge carriers (usually the carriers (usually the electronselectrons) inside an ) inside an electrical electrical conductorconductor at equilibriumat equilibrium.

This thermal noise was first measured by This thermal noise was first measured by John B. JohnsonJohn B. Johnsonat at Bell LabsBell Labs in in 19281928. He described his findings to . He described his findings to Harry Harry NyquistNyquist, also at Bell Labs, who was able to explain the , also at Bell Labs, who was able to explain the results. results.

1984-1995Harry Nyquist Harry Nyquist 18891889--19761976


•• Thermal (or Gaussian) noise is approximately Thermal (or Gaussian) noise is approximately whitewhite,,meaning that the meaning that the power spectral densitypower spectral density is equal is equal throughout the throughout the frequency spectrumfrequency spectrum. Additionally, the . Additionally, the amplitude of the signal has very nearly a amplitude of the signal has very nearly a GaussianGaussianprobability density functionprobability density function with mean with mean µµnn = 0.= 0.

S&M Figure 4S&M Figure 4--33µµnn = 0 = 0 σσnn = 1= 1


•• Since thermal noise has a Since thermal noise has a GaussianGaussian probability density probability density functionfunction the probability that a noise voltage the probability that a noise voltage n(tn(t) at time t) at time toowill be will be less than or equal to a thresholdless than or equal to a threshold ––γγA is (S&M Eq. A is (S&M Eq. 4.27):4.27):

and the probability that and the probability that aa noise voltage noise voltage n(tn(t) at time t) at time to o will be will be greater than a threshold greater than a threshold γγAA is (S&M Eq. 4.28):is (S&M Eq. 4.28):

) (−

−∞

≤ − −

∫γA 2

no X 2

nn

(x -µ )1P{ n(t γA } = F γA) = exp dx2σ2π σ

) (∞

> −

∫2

no X 2

nγA n

(x -µ )1P{ n(t γA } = 1 F γA) = exp dx2σ2π σ


•• The probabilistic properties ofThe probabilistic properties ofthermal noise do not change withthermal noise do not change withtime (time (stationaritystationarity). Thermal noise). Thermal noiseis an is an insidious propertyinsidious property ofofcommunication systems that limitscommunication systems that limitsthe speed of reliable data transmissionthe speed of reliable data transmissionand the detection of weak signals. and the detection of weak signals.


•• A A MMATLABATLAB and and Simulink Simulink simulation verifies the spectral simulation verifies the spectral characteristics of thermal noise and the performance of characteristics of thermal noise and the performance of lowlow--pass filtered pass filtered ffcutoff cutoff = 11.25 kHz thermal noise. = 11.25 kHz thermal noise.

MS Figure 1.11MS Figure 1.11


•• Thermal noise PSD = Thermal noise PSD = NNoo, | f | , | f | →→ ∞∞ MS Figure 1.12MS Figure 1.12

• Thermal noise LPF PSD = Thermal noise LPF PSD = NNoo, | f | < 11.25 kHz, | f | < 11.25 kHz

NNooNoNNoo (single(single--sided spectrum)sided spectrum)

11.25 kHz11.25 kHzNo


•• Thermal noise PSD = Thermal noise PSD = NNoo, | f | , | f | →→ ∞∞ MS Figure 1.12 MS Figure 1.12

• Thermal noise autocorrelation MS Figure 1.14Thermal noise autocorrelation MS Figure 1.14

uncorrelated

uncorrelated


•• Thermal noise LPF PSD = Thermal noise LPF PSD = NNoo, | f | < 11.25 kHz , | f | < 11.25 kHz

• LPF thermal noise autocorrelation MS Figure 1.14LPF thermal noise autocorrelation MS Figure 1.14

uncorrelated

correlated


•• Simulink Simulink Histogram block fromHistogram block from the the Statistics, Signal Statistics, Signal Processing Processing BlocksetBlockset is analogous to the is analogous to the pdf. pdf. The Vector The Vector Scope block from the Scope block from the Sinks, Signal Processing Sinks, Signal Processing BlocksetBlockset, , displays the histogram. displays the histogram. SimulinkSimulink blocks to calculate the blocks to calculate the mean mean µµ and variance and variance σσ22 are in are in Statistics.Statistics.

NoiseHistogram.mdlNoiseHistogram.mdl


•• Thermal noise histogram displayThermal noise histogram display

• LPF LPF ffcutoffcutoff = 1 kHz thermal noise histogram display= 1 kHz thermal noise histogram display

Gaussian pdfGaussian pdf

pdf remains pdf remains Gaussian Gaussian after LPFafter LPF

µµ = 10= 10σσ22 = 1= 1

µµ = 10.42= 10.42σσoo

22 = 0.64= 0.64


•• LPF thermal noise has an average normalized power:LPF thermal noise has an average normalized power:

σσoo2 2 = = NNoo ffcutoffcutoff (S&M p. 135)(S&M p. 135)

The probability that the The probability that the ithith--bit is received in error is:bit is received in error is:

) )−

−∞

∞

< − ≥

∫

∫

i o i oγA 2

oi 2

oo

2o

i 2oγA o

P(b = 0) P{ n(t γA } + P(b = 1) P{ n(t γA } =

(x -µ )1P(b = 0) exp dx +2σ2π σ

(x -µ )1 P(b = 1) exp dx2σ2π σ


•• If and only ifIf and only if the binary thresholds are symmetrical (the binary thresholds are symmetrical (––γγA A and and γγA), A), the the P(bP(bii = 0) + = 0) + P(bP(bii = 1) = 1 and the Gaussian = 1) = 1 and the Gaussian normal normal pdfspdfs are are symmetricalsymmetrical the the probability that the probability that the ithith--bitbitis received in error becomes (S&M p. 136):is received in error becomes (S&M p. 136):

) )∞

< − ≥

∫

i o i o

2o2oγA o

P(b = 0) P{ n(t γA } + P(b = 1) P{ n(t γA } =

(x -µ )1 exp dx2σ2π σ

γA-γA


Chapter 4Chapter 4

Receiver DesignReceiver Design•• Gaussian ProbabilityGaussian ProbabilityDensity Function, Probability ofDensity Function, Probability ofBit ErrorBit Error

•• Pages 137Pages 137--149149


•• Gaussian (normal)Gaussian (normal)probabilityprobabilitydensitydensityfunction (pdf)function (pdf)


•• Gaussian (normal)Gaussian (normal)probabilityprobabilitydistributiondistributionfunction function


•• Gaussian (Normal) Probability Distribution Gaussian (Normal) Probability Distribution

Abraham de Abraham de MoivreMoivre was a was a FreFrenchnchmathematicianmathematician famous for famous for de de Moivre'sMoivre'sformulaformula, which links , which links complex numberscomplex numbersand and trigonometrytrigonometry, and for his work on, and for his work onthe the normal distributionnormal distribution and and probabilityprobabilitytheorytheory in 1734. He wrote a book onin 1734. He wrote a book onprobability theoryprobability theory entitled entitled The DoctrineThe Doctrineof Chancesof Chances which was said to be highlywhich was said to be highlyprized by gamblers. 166prized by gamblers. 16677--17541754

GaussGauss rigorously justified and extended the work in rigorously justified and extended the work in 18091809..


•• Gaussian (Normal) Probability Distribution Gaussian (Normal) Probability Distribution

Johann Carl Friedrich Gauss was aJohann Carl Friedrich Gauss was aGermanGerman mathematicianmathematician and and scientistscientistwho contributed significantly to manywho contributed significantly to manyfields, including fields, including number theorynumber theory,,geometrygeometry, , electrostaticselectrostatics, , astronomyastronomyand and opticsoptics. .

17771777--18551855


•• Gaussian Gaussian pdfspdfs

µµ = 0, = 0, σσ = 1= 1S&M Figure 4S&M Figure 4--6a6a

µµ = 1.6, = 1.6, σσ = 1= 1S&M Figure 4S&M Figure 4--6b6b


•• Gaussian Gaussian pdfspdfs

µµ = 0, = 0, σσ = 2= 2S&M Figure 4S&M Figure 4--6c6c

µµ = 1, = 1, σσ = 2= 2S&M Figure 4S&M Figure 4--6d6d


•• The The probability of bit errorprobability of bit error is the is the areaarea under the Gaussianunder the Gaussianpdf from the threshold pdf from the threshold aa to to ∞∞ which could be tabulated. which could be tabulated. However, the probability of bit error is determined by three However, the probability of bit error is determined by three independent variables (independent variables (aa, , µµ and and σσ) and this would be an ) and this would be an unwieldy table.unwieldy table.

S&M Figure 4S&M Figure 4--77

Q-function


•• Rather, construct a Rather, construct a single tablesingle table with with µµ =0 and =0 and σσ = 1 for the= 1 for theprobability of bit error as the area under the Gaussianprobability of bit error as the area under the Gaussianpdf as a function of the threshold pdf as a function of the threshold a a only which is known as only which is known as the the complementary errorcomplementary error or or QQ--functionfunction..


Q-function


•• The The QQ--functionfunctionfor for µµ =0 =0 and and σσ = 1= 1as a functionas a functionof theof thethreshold athreshold ais listed inis listed inTable 4Table 4--11(S&M(S&Mp. 141) andp. 141) andAppendix BAppendix B(MS p. 185(MS p. 185--186)186)


•• Reading theReading theQQ--functiofunctionntabletablecorrectlycorrectly

••What isWhat isQ(1.82)?Q(1.82)?

•• What isWhat isQ(2.63)?Q(2.63)?

•• What isWhat isQ(3.18)?Q(3.18)?


•• Reading theReading theQQ--functiofunctionntabletablecorrectlycorrectly

•• Q(1.82) =Q(1.82) =0.03440.0344

•• Q(2.63) =Q(2.63) =0.00430.0043

•• Q(3.18) =Q(3.18) =0.00070.0007


•• What if What if µµ ≠≠ 0? It can0? It canbe shown that thebe shown that theQQ--function tablefunction tableremains valid ifremains valid ifthe thresholdthe thresholdvariable in the tablevariable in the tableis changed from is changed from aato to a a –– µµ. Note that. Note thatthe areas under thethe areas under theGaussian Gaussian pdfspdfs are theare thesame.same.

S&M Figure 4S&M Figure 4--8a8aFigure 4Figure 4--8b8b

a a –– µµ

a


•• What if What if µµ ≠≠ 0 and0 andσσ ≠≠ 1? It can also1? It can alsobe shown that thebe shown that theQQ--function tablefunction tableremains valid ifremains valid ifthe thresholdthe thresholdvariable in the tablevariable in the tableis first changed fromis first changed fromaa to to a a –– µµ and and ……

S&M Figure 4S&M Figure 4--9a9aFigure 4Figure 4--9b9b

a a –– µµ

a


•• …… then the thresholdthen the thresholdvariable in thevariable in theQQ--function tablefunction tableis changed fromis changed froma a –– µµ to (to (a a –– µµ) / ) / σσ..Note that argumentNote that argumenton the on the xx--axisaxis isiscompressed by compressed by σσand the and the yy--axisaxis isisexpanded byexpanded by σσ..

S&M Figure 4S&M Figure 4--9b9bFigure 4Figure 4--9c9c

a a –– µµ

(a –µ) / σ


•• For simple baseband PAM the probability of bit error For simple baseband PAM the probability of bit error PPb b is is expressed by the Qexpressed by the Q--function:function:

noise margin of sampled value= Q

average normalized noise power at the input to the single point sampler

bP

binary rectangular PAM data with AWGNbinary rectangular PAM data with AWGNsampling atsampling at

TTbb/2/2


Chapter 4Chapter 4

Receiver DesignReceiver Design•• Optimal Receiver: The MatchedOptimal Receiver: The MatchedFilter or Correlation ReceiverFilter or Correlation Receiver

•• Pages 149Pages 149--161161


•• The simple baseband PAM receiver structure is: The simple baseband PAM receiver structure is:

But is this the But is this the best that there isbest that there is? What about sampling an ? What about sampling an odd numberodd number of times (like 3) during each bit time of times (like 3) during each bit time TTbb? ?

TTbb


•• Although sampling can be increased to a very large, odd Although sampling can be increased to a very large, odd number of samples during number of samples during TTbb, there is an , there is an optimal wayoptimal way: :

Since LPF in PAM improved performance, assume that Since LPF in PAM improved performance, assume that the optimal processing is a the optimal processing is a linear filterlinear filter H(fH(f) (S&M p. 150)) (S&M p. 150)


•• After development (S&M p. 150After development (S&M p. 150--153) the optimal 153) the optimal processing is a linear filter processing is a linear filter HHoo(f(f))

The optimal linear filter The optimal linear filter HHoo(f(f) has an impulse response ) has an impulse response hhoo(t(t) and is known as a ) and is known as a matched filtermatched filter since the processing since the processing is matched to input signal is matched to input signal s(ts(t):):

∗

∗

−

−−

o-1 -1

o o

o

H (f) = k S (f) exp( j 2π f i )h (t) = F { H (f) } = k F { S (f) exp( j 2π f i ) }h (t) = k s(i t)

b

b

b

T

TT impulse responseimpulse response

matchedmatched to to s(ts(t))


•• The impulse response of the optimum filter The impulse response of the optimum filter hhoo(t(t) is a ) is a scaled (by scaled (by kk), time delayed (by i), time delayed (by iTTbb) and time reversed ) and time reversed (function of i(function of iTTbb–– t):t):

−o bh (t) = k s(iT t)


•• When optimum processing is used the argument inside When optimum processing is used the argument inside the Qthe Q--function is maximized (S&M p. 153function is maximized (S&M p. 153--154) and the 154) and the probability of bit error probability of bit error PPbb is:is:

where where EEbb is the is the energy per bit energy per bit of the received signal.of the received signal.

σ ∗

p

2

2p

o

| r(i ) h(i ) | = Q maximum

| r(i ) * h(i ) | = Q maximumσ

2 = Q

b bb

b bb

bb

T TP

T TP

EP N

S&M Eq. 4.58S&M Eq. 4.58


•• The optimum filter The optimum filter HHoo(f(f) is equivalent to the ) is equivalent to the correlationcorrelationreceiverreceiver (S&M p. 155(S&M p. 155--156).156).

Optimum FilterOptimum Filter −oh (t) = k s(i t)bT

Correlation ReceiverCorrelation Receiver


•• Since the optimum filter Since the optimum filter HHoo(f(f) and the correlation receiver ) and the correlation receiver are equivalent, with sare equivalent, with s11(t) = (t) = s(ts(t) for a matched filter and) for a matched filter andr(tr(t) = ) = γγ s(ts(t) where ) where γγ is the communication channel is the communication channel attenuation, attenuation, the energythe energy--perper--bit bit EEbb is (S&M p. 156,is (S&M p. 156,Eq 4.62):Eq 4.62):

∫ ∫i i

2 2

(i-1) (i-1)

= γ s(t) γ s(t) dt = γ s (t) dtb b

b b

T T

bT T

E


•• The expected or mean value The expected or mean value aaii(i(iTTbb) is the output of the ) is the output of the correlation receiver when correlation receiver when r(tr(t) = ) = γγ ssii(t(t) and ) and n(tn(t) = 0 where ) = 0 where γγis the communication channel attenuation.is the communication channel attenuation.

∫i

i i 1(i-1)

a (i ) = γ s (t) s (t) dt b

b

T

bT

T S&M Eq. 4.67S&M Eq. 4.67


•• PPbb = Q( = Q( √√(2 (2 EEbb / / NNoo) ) and the ratio ) ) and the ratio EEbb / / NNoo can be can be expressed in dB: 10 logexpressed in dB: 10 log1010 ((EEbb / / NNoo ). The resulting plot of ). The resulting plot of PPbb verses verses EEbb / / NNoo inin dB is a characteristic of dB is a characteristic of binarybinarysymmetricsymmetricPAM withPAM withAWGN.AWGN. S&M Figure 4S&M Figure 4--1414


•• Symmetric binary PAMSymmetric binary PAMimplies that the twoimplies that the twotransmitted signals fortransmitted signals forbinary 1 and binary 0binary 1 and binary 0s(ts(t)) and the resultingand the resultingoutputs outputs a(iTa(iTbb)) from thefrom thecorrelation receivercorrelation receiverare equal inare equal inmagnitude butmagnitude butopposite in sign:opposite in sign:

ssbibi=1=1(t) = (t) = –– ssbibi=0=0(t)(t)


Threshold = 0


•• The probability of bitThe probability of biterror for error for equallyequally--likelylikelybinary symmetricbinary symmetricPAM is the sum ofPAM is the sum ofthe the error regionserror regionsshown.shown.

P(bP(bii=0) = =0) = P(bP(bii=1) = 0.5=1) = 0.5



•• The binary PAM signals are symmetrical and the The binary PAM signals are symmetrical and the threshold is 0 (threshold is 0 (equidistantequidistant from the means or expected from the means or expected values values ±± a(ia(iTTbb) )) ). The error regions are equal in area.. The error regions are equal in area.



•• The probability of bitThe probability of biterror does noterror does notminimize if theminimize if thecorrelation receivercorrelation receiverthreshold isthreshold ismisadjusted misadjusted ((ττ ≠≠ 0).0).With a misadjustedWith a misadjustedthreshold the apriorithreshold the aprioriprobabilities are nowprobabilities are nowimportant since theimportant since thearea of the errorarea of the errorregions are no longerregions are no longerequal.equal.



•• The correlation receiver is also known as the The correlation receiver is also known as the integrateintegrate--andand--dump dump which describes the process.which describes the process.

matched filter or matched filter or correlation receivercorrelation receiver



Chapter 2Chapter 2

Baseband Modulation and Baseband Modulation and DemodulationDemodulation•• Optimum Binary BasebandOptimum Binary BasebandReceiver: The CorrelationReceiver: The CorrelationReceiverReceiver

•• Pages 40Pages 40--4141


•• The correlation receiver can be simulated in The correlation receiver can be simulated in SimulinkSimulink::



•• The The SimulinkSimulinkIntegrate and DumpIntegrate and Dumpblock is in the block is in the CommComm Filters,Filters,CommunicationsCommunicationsBlocksetBlockset


•• The parameters of the The parameters of the SimulinkSimulink Integrate and DumpIntegrate and Dumpblock are the integration period and offset in simulation block are the integration period and offset in simulation samples.samples.

ffsimulationsimulation = 50 kHz= 50 kHzTTsimulationsimulation = 1/= 1/ffsimulationsimulation ==

0.02 msec0.02 msec

TTbb = 1 msec= 1 msec

Integration period =Integration period =TTbb//TTsimulatiosimulationn = 1/ 0.02 == 1/ 0.02 =50 samples50 samples

Offset = 0 samplesOffset = 0 samples


•• The complete binary symmetrical rectangular PAM digital The complete binary symmetrical rectangular PAM digital communication system with BER analysis and optimum communication system with BER analysis and optimum receiver.receiver.


TransmitterTransmitter ReceiverReceiver

BERBER


•• Observed BER as a function of SNR for binary Observed BER as a function of SNR for binary rectangular PAM in a LPF simple receiver (rectangular PAM in a LPF simple receiver (LPFLPF) and the ) and the optimum correlation receiver (optimum correlation receiver (CRCR) with normalized signal ) with normalized signal power = 25 W (MS Table 2.3 p. 27 and Table 2.6 p. 43).power = 25 W (MS Table 2.3 p. 27 and Table 2.6 p. 43).

SNR dB AWGN SNR dB AWGN σσ22 VV22 BER (BER (LPFLPF)) BER (BER (CRCR))∞∞ 00 00 003.983.98 1010 00 000.960.96 2020 00 00−−3.523.52 5050 1 1 ×× 1010--44 00−−6.026.02 100100 6.3 6.3 ×× 1010--33 1 1 ×× 1010--44

−−9.039.03 200200 3.63 3.63 ×× 1010--22 6.4 6.4 ×× 1010--33

−−13.0113.01 500500 1.185 1.185 ×× 1010--11 6.02 6.02 ×× 1010--22

−−16.0216.02 10001000 1.345 1.345 ×× 1010--11


•• The The energy per bitenergy per bit EEb b = 2.5 = 2.5 ×× 1010--22 VV22--sec (S&M Eq. 4.62, sec (S&M Eq. 4.62, p. 156) for rectangular p. 156) for rectangular ±± 5 V 5 V PAM with the channel PAM with the channel attenuation attenuation γγ = 1:= 1:

The observed bit error rate (BER) can be compared to The observed bit error rate (BER) can be compared to the theoretical probability of bit error the theoretical probability of bit error PPbb (S&M Eq. 4.58, p. (S&M Eq. 4.58, p. 154) to validate the basic simulation.154) to validate the basic simulation.

∫ ∫i i

2 2

(i-1) (i-1)


b b

T T

bT T

E

2 = Q bb

o

EPN


•• The The ratioratio of the energy per bit of the energy per bit EEb b to the noise power to the noise power spectral densityspectral density NNoo (E(Eb b // NNoo)) rather than the SNR is the rather than the SNR is the conventional metricconventional metric used for BER performance. used for BER performance.

∫ ∫i i

2 2

(i-1) (i-1)


b b

T T

bT T

E

2 = Q bb

o

EPN

S&M Eq. 4.62S&M Eq. 4.62

S&M Eq. 4.58S&M Eq. 4.58


•• The BER and The BER and PPbb comparison (MS Table 2.7, p. 44): comparison (MS Table 2.7, p. 44):

Table 2.7Table 2.7 Observed BER and Theoretical Observed BER and Theoretical PPbb as a as a Function of Function of EEb b / N/ Noo in a Binary Symmetrical Rectangular in a Binary Symmetrical Rectangular PAM Digital Communication System with Optimum PAM Digital Communication System with Optimum ReceiverReceiver

EEb b / / NNoo dB BERdB BER PPbb∞∞ 00 001010 00 4.05 4.05 ×× 1010--66

88 00 2.06 2.06 ×× 1010--44

66 2.5 2.5 ×× 1010--33 2.43 2.43 ×× 1010--33

44 1.35 1.35 ×× 1010--22 1.25 1.25 ×× 1010--22

22 3.95 3.95 ×× 1010--22 3.75 3.75 ×× 1010--22

00 8.09 8.09 ×× 1010--22 7.93 7.93 ×× 1010--22


Chapter 4Chapter 4

Receiver DesignReceiver Design•• Correlation Receiver for AsymmetricCorrelation Receiver for AsymmetricPAM, Optimum Thresholds,PAM, Optimum Thresholds,Synchronization Synchronization

•• Pages 162Pages 162--173173


•• Asymmetric PAMAsymmetric PAMsignals do not havesignals do not haveequal equal means ormeans orexpected values ofexpected values ofthe output of thethe output of thecorrelation receiver:correlation receiver:

| a| a22(i(iTTbb) | ) | ≠≠ | a| a11(i(iTTbb) |) |



•• The optimumThe optimumthreshold threshold ττoptopt isisagain again equidistantequidistantbetween the meansbetween the meansor expected values ofor expected values ofthe output of thethe output of thecorrelation receiver:correlation receiver:

S&M Eq. 4.71 S&M Eq. 4.71

Here | aHere | a22(i(iTTbb) | + ) | + ττoptopt = a= a11(i(iTTbb) ) –– ττopt opt and the threshold is and the threshold is equidistant from the means or expected values.equidistant from the means or expected values.


= 2 1

opta (i )+ a (i )τ

2b bT T


•• If the If the aprioriaprioriprobabilitiesprobabilities are notare notequal then theequal then theoptimum thresholdoptimum thresholdττoptopt is not equidistantis not equidistantfrom the means orfrom the means orexpected value ofexpected value ofoutput of theoutput of thecorrelation receiver.correlation receiver.An asymmetric binaryAn asymmetric binaryPAM signal is shown:PAM signal is shown:

aa22(i(iTTbb) ) ≠≠ aa11(i(iTTbb))



•• The optimumThe optimumthreshold threshold ττoptopt wherewherethe apriorithe aprioriprobabilities areprobabilities are(1 (1 –– MM) and ) and MM(which sums to 1) is:(which sums to 1) is:

if M = 0.5 then:if M = 0.5 then:

[ ]− −

=−

2 2 2o 2 1 b

opt2 1

2 σ ln ( / (1 ) ) + a (i ) a (i )τ2 a (i ) a (i )

b

b b

M M T TT T

= 2 1

opta (i )+ a (i )τ

2b bT T

S&M Eq. 4.85S&M Eq. 4.85

S&M Eq. 4.71S&M Eq. 4.71


•• Asymmetric PAM withAsymmetric PAM withan optimum thresholdan optimum thresholdττoptopt has a probabilityhas a probabilityof bit error of bit error PPbb::


[ ]

( ){ }

2

22

− −

−

∫

1 2

o

1 2o

o

i2

1 2(i-1)

a (i ) a (i )= Q2 σ

a (i ) a (i ) = Q σ =

4 σ 2

= Q where = γ s (t) s (t) dt2

b

b

b bb

b b ob

Td

b do T

T TP

T T NP

EP EN

S&M Eq. 4.78S&M Eq. 4.78and Eq. 4.79and Eq. 4.79


•• The optimum correlation receiver for asymmetric binaryThe optimum correlation receiver for asymmetric binaryPAM uses the difference signal sPAM uses the difference signal s11(t) (t) –– ss22(t) as the (t) as the reference: reference:



•• The optimum correlation receiver can be The optimum correlation receiver can be reconfiguredreconfigured as as anan alternatealternate but but universal universal structure structure which can be used for which can be used for both asymmetric or symmetric binary PAM signals:both asymmetric or symmetric binary PAM signals:



•• If the If the aprioriaprioriprobabilitiesprobabilities are notare notequal then theequal then theoptimum thresholdoptimum thresholdττoptopt is not equidistantis not equidistantfrom the means orfrom the means orexpected value ofexpected value ofoutput of theoutput of thecorrelation receiver.correlation receiver.An asymmetric binaryAn asymmetric binaryPAM signal is shown:PAM signal is shown:

aa22(i(iTTbb) ) ≠≠ aa11(i(iTTbb))



•• The optimumThe optimumthreshold threshold ττoptopt wherewherethe apriorithe aprioriprobabilities areprobabilities are(1 (1 –– MM) and ) and MM(which sums to 1) is:(which sums to 1) is:

if M = 0.5 then:if M = 0.5 then:

[ ]− −

=−

2 2 2o 2 1

opt2 1

2 σ ln ( / (1 ) ) + a (i ) a (i )τ2 a (i ) a (i )

b b

b b

M M T TT T

= 2 1

opta (i )+ a (i )τ

2b bT T

S&M Eq. 4.85S&M Eq. 4.85

S&M Eq. 4.71S&M Eq. 4.71


•• The probabilityThe probabilityof bit errorof bit error PPbb thenthenisis::

S&M Eq. 4.86S&M Eq. 4.86

( ) ( )

− − −

− − −

1 opt opt 2

o o

2 2

1 opt opt 2

a (i ) τ τ a (i )= Q + (1 ) Q

σ σ

a (i ) τ τ a (i )= Q + (1 ) Q

2 2

b bb

b bb

o o

T TP M M

T TP M M

N N


•• If the aprioriIf the aprioriprobabilities areprobabilities areequal (equal (MM = 0.5):= 0.5):

and and PPbb becomes:becomes:

( ) ( )

( )

− −

2 2

1 opt opt 2

21 2

a (i ) τ τ a (i )= 0.5 Q + 0.5 Q

2 2

a (i ) - a (i )= Q = Q

2 2

b bb

o o

b b db

o o

T TP

N N

T T EPN N

= 2 1

opta (i )+ a (i )τ

2b bT T

S&M p. 168S&M p. 168



•• The BER and PThe BER and Pbb comparison (MS Table 2.8, p. 49): comparison (MS Table 2.8, p. 49):

Table 2.8Table 2.8 Observed BER and Theoretical Observed BER and Theoretical PPb b as a as a Function of Function of EEd d / N/ Noo in an Asymmetrical Binary in an Asymmetrical Binary Rectangular PAM Digital Communication System with Rectangular PAM Digital Communication System with Optimum Receiver.Optimum Receiver.

EEd d / N/ Noo dBdB BERBER PPbb∞∞ 00 001212 2.5 2.5 ×× 1010--33 2.53 2.53 ×× 1010--33

1010 1.28 1.28 ×× 1010--22 1.25 1.25 ×× 1010--22

88 3.59 3.59 ×× 1010--22 3.75 3.75 ×× 1010--22

66 8.05 8.05 ×× 1010--22 7.93 7.93 ×× 1010--22

44 1.334 1.334 ×× 1010--11 1.318 1.318 ×× 1010--11

22 1.856 1.856 ×× 1010--11 1.872 1.872 ×× 1010--11

00 2.362 2.362 ×× 1010--11 2.394 2.394 ×× 1010--11


•• Since for binary rectangular PAM Since for binary rectangular PAM EEdd = 4 = 4 EEbb (S&M p. 168)(S&M p. 168)the BER performance for asymmetric PAM is comparable the BER performance for asymmetric PAM is comparable to symmetric PAM if a to symmetric PAM if a EEd d / N/ Noo is is reducedreduced by 6 dBby 6 dB(10 log (4) = 6) : (10 log (4) = 6) :

EEdd / / NNoo dBdB BERBER PPbb

Asymmetric PAM Asymmetric PAM 1010 1.28 1.28 ×× 1010--22 1.25 1.25 ×× 1010--22

(MS Table 2.8, (MS Table 2.8, 88 3.59 3.59 ×× 1010--22 3.75 3.75 ×× 1010--22

p. 49) p. 49) 66 8.05 8.05 ×× 1010--22 7.93 7.93 ×× 1010--22

EEb b / / NNoo dBdB BERBER PPbb

Symmetric PAMSymmetric PAM 44 1.35 1.35 ×× 1010--22 1.25 1.25 ×× 1010--22

(MS Table 2.7, 2(MS Table 2.7, 2 3.95 3.95 ×× 1010--22 3.75 3.75 ×× 1010--22

p. 44) p. 44) 00 8.09 8.09 ×× 1010--22 7.93 7.93 ×× 1010--22


Chapter 2Chapter 2

Baseband Modulation and Baseband Modulation and DemodulationDemodulation•• The Correlation Receiver forThe Correlation Receiver forBaseband AsymmetricalBaseband AsymmetricalSignalsSignals



•• The optimum threshold The optimum threshold ττopt opt requires the additive Gaussian requires the additive Gaussian noise variance noise variance σσ22 as processed by the correlation as processed by the correlation receiver or receiver or σσoo

22::

[ ]− −

=−

2 2 2o 2 1

opt2 1

2 σ ln ( / (1 ) ) + a (i ) a (i )τ2 a (i ) a (i )

b b

b b

M M T TT T


σσoo


•• The optimum threshold The optimum threshold ττopt opt alsoalso requires the apriori requires the apriori probabilities Pprobabilities P11 = = MM and Pand P0 0 = = MM –– 11::

Here PHere P1 1 = 0.4998 = = 0.4998 = MM and Pand P0 0 = 1 = 1 –– MM = = 0.50040.5004

[ ]− −

=−

2 2 2o 2 1

opt2 1

2 σ ln ( / (1 ) ) + a (i ) a (i )τ2 a (i ) a (i )

b b

b b

M M T TT T


MeanMean


•• The Mean andThe Mean andVariance blocksVariance blocksare in the are in the Statistics,Statistics,Signal ProcessingSignal ProcessingBlocksetBlockset


•• The parameter of theThe parameter of theMean block can be setMean block can be setfor for running meanrunning mean or anor anaverage over theaverage over thesimulation period. Thesimulation period. Theresult can be shown byresult can be shown bya Display block from thea Display block from theSimulink Simulink BlocksetBlockset..



•• The parameter of theThe parameter of theVariance block can be setVariance block can be setfor for running variancerunning variance or anor anaverage over theaverage over thesimulation period. Thesimulation period. Theresult can be shown byresult can be shown bya Display block from thea Display block from theSimulink Simulink BlocksetBlockset..


•• The The SimulinkSimulink simulation for binary asymmetrical PAM simulation for binary asymmetrical PAM with BER analysis has a with BER analysis has a thresholdthreshold adjustment based on adjustment based on the estimate of the estimate of ττoptopt from measurements of from measurements of σσoo and and M.M.


ThresholdThreshold

ReferenceReference

[ ]− −

=−

2 2 2o 2 1

opt2 1

2 σ ln ( / (1 ) ) + a (i ) a (i )τ2 a (i ) a (i )

b b

b b

M M T TT T


•• The The SimulinkSimulink simulation for the alternative and universal simulation for the alternative and universal structure for the correlation receiver:structure for the correlation receiver:




•• Synchronization (not considered here) provides Synchronization (not considered here) provides timing timing recovery recovery or the exact beginning and end of a bit time or the exact beginning and end of a bit time TTbb::


start of start of TTbb??


•• Synchronization of both Synchronization of both symbol timesymbol time and carrier frequency and carrier frequency and phase is one of the advanced topics in EE4542 and phase is one of the advanced topics in EE4542 Telecommunications EngineeringTelecommunications Engineering. The effect is called . The effect is called jitterjitter..


Chapter 4Chapter 4

Receiver DesignReceiver Design•• MultiMulti--level PAM (Mlevel PAM (M--ary PAM)ary PAM)

•• Pages 200Pages 200--206206


•• MultiMulti--level (Mlevel (M--ary) PAM is another means to minimize the ary) PAM is another means to minimize the bandwidth required for a data transmission rate bandwidth required for a data transmission rate rrbb b/sec. b/sec. Rather than transmitting a binary signal in a bit time Rather than transmitting a binary signal in a bit time TTbb, , send a multisend a multi--level (usually a powerlevel (usually a power--ofof--2) signal during the 2) signal during the same period called the same period called the symbol timesymbol time TTSS..

A A multimulti--state comparatorstate comparator determines the received symbol determines the received symbol which is then decoded to the received bits. which is then decoded to the received bits.

S&M Figure 4S&M Figure 4--4848M = 4M = 4


•• The threeThe threeoptimumoptimumthresholds ifthresholds ifthe apriorithe aprioriprobabilitiesprobabilitiesare equallyare equallylikelylikely(P(Pii = 0.25)= 0.25)are:are:

=

=

=

1 2opt1

2 3opt2

3 4opt3

a (i )+ a (i )τ2

a (i )+ a (i )τ2

a (i )+ a (i )τ2

S S

S S

S S

T T

T T

T T

S&M Figure 4.49S&M Figure 4.49

S&M Eq. 4.136S&M Eq. 4.136


•• TheTheprobabilityprobabilityof symbolof symbolerrorerror PPss, , where M = 2where M = 2nn

is theis thenumber ofnumber oflevels,levels,can becan beshown to be:shown to be:where:where:

S&M Eq. 4.140S&M Eq. 4.140{ } ∫i

2

j k(i-1)

= γ s (t) - s (t) dts

s

T

d,symbolT

E

( ) −

s

2 M 1P = Q

M 2d,symbol

o

EN

S&M Eq. 4.139S&M Eq. 4.139

M = 4M = 4


•• There areThere are2 (M 2 (M –– 1)1)errorerrorregionsregionsdue to due to onlyonlyadjacentadjacentregions regions beingbeingmisinterpreted with M = 2misinterpreted with M = 2nn equally probable symbols. The equally probable symbols. The probability of occurrence probability of occurrence PPjj for a misinterpreted symbol is for a misinterpreted symbol is also equally likely and is:also equally likely and is:

SVU Eq. 2.41SVU Eq. 2.41( )−j1P =

2 M 1

M = 4M = 4


•• The amplitudes of the MThe amplitudes of the M--ary (M = 4) rectangular PAM ary (M = 4) rectangular PAM signal is signal is ±± 3A and 3A and ±± A. The energy difference for a A. The energy difference for a symbol symbol EEd,symbold,symbol = 4= 4γγ2 2 AA2 2 TTSS (S&M Eq. 4.140):(S&M Eq. 4.140):

The average received energy per symbol The average received energy per symbol EEavg,symbolavg,symbol ==55γγ22 AA22 TTSS (S&M Eq. 4.141)(S&M Eq. 4.141)

{ } ∫i

2

j k(i-1)

= γ s (t) - s (t) dts

s

T

d,symbolT

E

S&M Figure 4S&M Figure 4--50 modified50 modified11

10

01

00M = 4M = 4


•• Then Then EEd,symbold,symbol = 0.4 = 0.4 EEavg,symbolavg,symbol and substitute (M = 4):and substitute (M = 4):

s

0.4 3P = Q2

avg,symbol

o

EN

s

3P = Q2 2

d,symbol

o

EN S&M Eq. 4.139bS&M Eq. 4.139b

S&M Eq. 4.142aS&M Eq. 4.142a


10

01

00M = 4M = 4


•• The average energy per bit The average energy per bit EEbb = 0.5 = 0.5 EEavg,symbolavg,symbol since M =4 since M =4 and there are two bits per symbol:and there are two bits per symbol:

s

0.4 3P = Q2

avg,symbol

o

EN

S&M Eq. 4.142aS&M Eq. 4.142a

S&M Eq. 4.142bS&M Eq. 4.142b

s 0.8 3P = Q

2b

o

EN


10

01

00M = 4M = 4


•• The symbols transmitted can have 1 bit in error 4 / 6 of The symbols transmitted can have 1 bit in error 4 / 6 of the time and 2 bits in error 2 / 6 of the time:the time and 2 bits in error 2 / 6 of the time:

Transmitted DiTransmitted Di--BitBit Received DiReceived Di--BitBit Bits In ErrorBits In Error0000 0101 110101 0000 110101 1010 221010 0101 221010 1111 111111 1010 11

4 2= P(1 of 2 bits in error) + P(2 of 2 bits in error)6 64 1 2 2= + =6 2 6 3

b,4-level

b,4-level s s s

P

P P P P S&M Eq. 4.143S&M Eq. 4.143


•• The probability of bit error The probability of bit error PPb,4b,4--symbolsymbol = 2 = 2 PPss / 3 but can that / 3 but can that be improved? Change the assignment of symbols to dibe improved? Change the assignment of symbols to di--bits as a bits as a Gray code Gray code and there is only 1 bit in error for and there is only 1 bit in error for each of the six error regions and each of the six error regions and PPb,4b,4--symbolsymbol == PPss / 2/ 2

Transmitted DiTransmitted Di--BitBit Received DiReceived Di--BitBit Bits In ErrorBits In Error0101 0000 110000 0101 110000 1010 111010 0000 111010 1111 111111 1010 11

6= P(1 of 2 bits in error) 66 1 1= = 6 2 2

b,4-level

b,4-level s s

P

P P P


•• The simple change to a Gray coded symbol The simple change to a Gray coded symbol improvesimproves the the probability of bit error Pprobability of bit error Pbb::


10

00

01

=

=

0.8 0.8 2 3= Q Q3 2

0.8 0.8 1 3 3= Q Q2 2 4

b bb

o o

b bb

o o

E EPN N

E EPN N

Gray codedGray coded

Straight binaryStraight binarycodedcoded


•• Frank Gray was a researcher at Frank Gray was a researcher at Bell LabBell Labs who made s who made numerous innovations in television and is remembered for numerous innovations in television and is remembered for the the Gray codeGray code. The . The Gray codeGray codeappeared in his 1953 patent andappeared in his 1953 patent andis a is a binary systembinary system often used inoften used inelectronicselectronics. Gray also conducted. Gray also conductedpioneering research on thepioneering research on thedevelopment of development of televisiontelevision. He. Heproposed an early form of theproposed an early form of theflying spotflying spot scanning system forscanning system forTV cameras in 1927, and helpedTV cameras in 1927, and helpeddevelop a twodevelop a two--way mechanicallyway mechanicallyscanned TV system in 1930.scanned TV system in 1930. Frank GrayFrank Gray

18941894--19641964


Chapter 2Chapter 2

Baseband Modulation and Baseband Modulation and DemodulationDemodulation•• Multilevel (MMultilevel (M--ary) Pulseary) PulseAmplitude ModulationAmplitude Modulation



•• The The SimulinkSimulink simulation of a 4simulation of a 4--level straight binary (level straight binary (not not Gray codedGray coded) rectangular PAM system with BER analysis:) rectangular PAM system with BER analysis:


bit to symbol converterbit to symbol converter symbol to bit convertersymbol to bit converter


BERBER


•• Although Although SimulinkSimulinkprovides a Bit toprovides a Bit toInteger Integer blockblock and and an an Integer to BitInteger to Bitblock from theblock from theUtility Blocks,Utility Blocks,CommunicationCommunicationBlocksetBlockset, , thesetheseblocks use a blocks use a vectorvectorof bits rather thanof bits rather thanfrom and to a from and to a serialserialbitstreambitstream. .


•• A bit to symbol converter A bit to symbol converter Simulink Simulink subsystem can be subsystem can be configured from configured from DownsamplerDownsampler blocks and a blocks and a Delay Delay block block from the from the Signal OperationsSignal Operations, , Signal Processing Signal Processing BlocksetBlocksetand the and the Gain Gain and and Sum Sum blocks from the blocks from the SimulinkSimulink, , Math Math Operations Operations toto produce the 4produce the 4--level PAM signal (0, 1, 2 or level PAM signal (0, 1, 2 or 3) from the input 3) from the input didi--bit bit (00, 01, 10, or 11) with the most (00, 01, 10, or 11) with the most significant bit (MSB) first. significant bit (MSB) first.



•• The The Simulink Simulink bit to symbol converter can be easily bit to symbol converter can be easily reconfigured reconfigured toto produce an 8produce an 8--level PAM signal (0, 1, 2, level PAM signal (0, 1, 2, ……7) from the input 7) from the input tritri--bit bit (000, 001, (000, 001, …… 111) with the 111) with the most significant bit (MSB) first. most significant bit (MSB) first.



•• A symbol to bit converter A symbol to bit converter Simulink Simulink subsystem can be subsystem can be configured from configured from ConstantConstant blocks, blocks, Pulse Generator Pulse Generator blocks blocks and a and a Multiport Switch Multiport Switch block from the block from the Signal RoutingSignal Routing, , Simulink Simulink BlocksetBlockset toto produce the produce the didi--bitbit ((MSB first) from theMSB first) from the44--level PAM signal. level PAM signal.



•• A symbol to bit converter A symbol to bit converter Simulink Simulink subsystem can be subsystem can be easily reconfigured easily reconfigured toto produce the produce the tritri--bitbit ((MSB first) from MSB first) from an 8an 8--level PAM signal. level PAM signal.



•• Binary data to 4Binary data to 4--level PAM converterlevel PAM converter



•• Transmit binary data, Transmit binary data, rrbb = 1 kb/sec= 1 kb/sec

• Transmit 4Transmit 4--level PAM signal, level PAM signal, rrSS = 500 symbols/sec= 500 symbols/sec

0101

11

LSB LSB MSBMSB


•• MM--ary PAM transmits n bits per symbol (M = 2ary PAM transmits n bits per symbol (M = 2nn) but has the ) but has the same rectangular pulse shape as binary PAM. The same rectangular pulse shape as binary PAM. The normalized power spectral density for Mnormalized power spectral density for M--ary PAM PSDary PAM PSDMM has has the same the same sincsinc shape as that for binary PAM PSDshape as that for binary PAM PSDBB but uses but uses the the symbol timesymbol time TTSS rather than the rather than the bit timebit time TTbb::

( )( )

2M avg

2B

PSD (f) = A sinc π f

PSD (f) = A sinc π fS S

b b

T T

T T


•• Observed BER and theoretical Observed BER and theoretical PPbb as a function of as a function of EEbb / N/ No o in a 4in a 4--levellevel rectangular straightrectangular straight--binary coded PAM digital binary coded PAM digital communication system with optimum receivercommunication system with optimum receiver (MS Table (MS Table 2.11 p. 58).2.11 p. 58).

EEbb / / NNoo dB dB BERBER PPbb∞∞ 00 001010 2.7 2.7 ×× 1010--33 2.4 2.4 ×× 1010--33

88 1.28 1.28 ×× 1010--22 1.25 1.25 ×× 1010--22

66 3.42 3.42 ×× 1010--22 3.75 3.75 ×× 1010--22

44 7.52 7.52 ×× 1010--22 7.78 7.78 ×× 1010--22

22 1.291 1.291 ×× 1010--11 1.320 1.320 ×× 1010--11

00 1.842 1.842 ×× 1010--11 1.867 1.867 ×× 1010--11


•• The MThe M--ary PAM PSD uses the average amplitude ary PAM PSD uses the average amplitude AAavgavg, , where where PPjj is the apriori probability of occurrence of the is the apriori probability of occurrence of the amplitude amplitude AAjj..

= ∑M

2 2avg j j

j=1A A P


•• The MThe M--ary (M = 4) PAM PSD can be determined by a ary (M = 4) PAM PSD can be determined by a Simulink Simulink modelmodel

Fig239.mdlFig239.mdl



•• The The SimulinkSimulink simulation of a 4simulation of a 4--level straight binary (level straight binary (not not Gray codedGray coded) rectangular PAM system measures the BER) rectangular PAM system measures the BERperformance.performance. MS Figure 2.35MS Figure 2.35


BERBER


• The rectangular MThe rectangular M--ary pulse width ary pulse width is the entire is the entire symbol symbol timetime TTSS and is and is optimumoptimum in the bandwidth sense. The in the bandwidth sense. The bandwidth is similar to that for binary PAM but with bandwidth is similar to that for binary PAM but with TTbb = = TTS S (cf. S&M Table 3(cf. S&M Table 3--1 p. 86).1 p. 86).

Table 2Table 2--9 Bandwidth of a Binary Rectangular M9 Bandwidth of a Binary Rectangular M--ary PAM ary PAM Signal as a Percentage of the Total Power (MS p. 53)Signal as a Percentage of the Total Power (MS p. 53)

Bandwidth (Hz) Percentage of Total PowerBandwidth (Hz) Percentage of Total Power1/1/TTSS 90%90%1.5/1.5/TTSS 93%93%2/2/TTSS 95%95%3/3/TTSS 96.5%96.5%4/4/TTSS 97.5%97.5%5/5/TTSS 98%98%


•• PSDPSDMM M = 4, M = 4, rrbb = 1 kb/sec, first null bandwidth = 500 Hz= 1 kb/sec, first null bandwidth = 500 Hz

• PSDPSDBB rrbb = 1 kb/sec, first null bandwidth = 1 kHz= 1 kb/sec, first null bandwidth = 1 kHz

500 Hz500 Hz

1 kHz


Chapter 2Chapter 2

Baseband Modulation and Baseband Modulation and DemodulationDemodulation•• Performance of MPerformance of M--ary Pulseary PulseAmplitude ModulationAmplitude Modulation

•• Gray Encoded DataGray Encoded Data



•• The The SimulinkSimulink simulation of a 4simulation of a 4--level straight binary level straight binary Gray Gray encodedencoded rectangular PAM system with BER analysis:rectangular PAM system with BER analysis:


bit to Gray encoded bit to Gray encoded symbol convertersymbol converter

Gray encoded symbol Gray encoded symbol to bit converterto bit converter


BERBER


•• The M = 4 level symbol (0, 1, 2 and 3) is Gray encoded by The M = 4 level symbol (0, 1, 2 and 3) is Gray encoded by a a Lookup Table Block Lookup Table Block from the from the Simulink Simulink BlocksetBlockset..

M = 4 symbolM = 4 symbol

bit to Gray encoded bit to Gray encoded symbol convertersymbol converter



•• The M = 4 level symbolThe M = 4 level symbol(0, 1, 2 and 3) is Gray(0, 1, 2 and 3) is Grayencoded by a encoded by a LookupLookupTable Block Table Block from thefrom theSimulink Simulink BlocksetBlockset..


•• The M = 4 level symbol (0, 1, 2, and 3) is Gray encoded by The M = 4 level symbol (0, 1, 2, and 3) is Gray encoded by a a Lookup Table Block Lookup Table Block by by mappingmapping [0, 1, 2, 3] to [0, 1, 3, 2][0, 1, 2, 3] to [0, 1, 3, 2]

Gray coding 00 → 0001 → 0110 → 1111 → 10


•• The M = 4 level Gray encoded symbol is decoded as diThe M = 4 level Gray encoded symbol is decoded as di--bits bits by a by a Lookup Table Block Lookup Table Block from the from the Simulink Simulink BlocksetBlockset..

M = 4 symbolM = 4 symbol

Gray encoded symbol Gray encoded symbol to bit converterto bit converter



•• The M = 4 level symbol (0, 1, 2, and 3) is Gray decoded by The M = 4 level symbol (0, 1, 2, and 3) is Gray decoded by a a Lookup Table Block Lookup Table Block by by mappingmapping [0, 1, 2, 3] to [0, 1, 3, 2][0, 1, 2, 3] to [0, 1, 3, 2]

Gray coding 00 → 0001 → 0110 → 1111 → 10


•• PSDPSDMM M = 4, M = 4, rrbb = 1 kb/sec, first null bandwidth = 500 Hz= 1 kb/sec, first null bandwidth = 500 Hz

500 Hz500 Hz

•• The PSD of a Gray coded rectangular PAM signal is the The PSD of a Gray coded rectangular PAM signal is the samesame as a straightas a straight--binary rectangular PAM signal. The binary rectangular PAM signal. The coding coding does notdoes notchange the PSD.change the PSD.The PSD is onlyThe PSD is onlyaffected by the affected by the pulsepulseshapeshape and and data ratedata rate. .


•• The The SimulinkSimulink simulation of a 4simulation of a 4--level straight binary level straight binary Gray Gray encodedencoded rectangular PAM system measures the BER rectangular PAM system measures the BER performance.performance. MS Figure 2.42MS Figure 2.42


BERBER


•• Observed BER and theoretical Observed BER and theoretical PPbb as a function of as a function of EEbb / N/ No o in a 4in a 4--levellevel rectangular Gray coded PAM digital rectangular Gray coded PAM digital communication system with optimum receivercommunication system with optimum receiver (MS Table (MS Table 2.12 p. 60).2.12 p. 60).

EEbb / / NNoo dB dB BERBER PPbb∞∞ 00 001010 1.7 1.7 ×× 1010--33 1.8 1.8 ×× 1010--33

88 8.6 8.6 ×× 1010--33 9.3 9.3 ×× 1010--33

66 2.92 2.92 ×× 1010--22 2.81 2.81 ×× 1010--22

44 5.93 5.93 ×× 1010--22 5.84 5.84 ×× 1010--22

22 9.56 9.56 ×× 1010--22 9.90 9.90 ×× 1010--22

00 1.441 1.441 ×× 1010--11 1.404 1.404 ×× 1010--11


•• Observed BER as a function of Observed BER as a function of EEb b / / NNoo for binary for binary rectangular PAM for straightrectangular PAM for straight--binary coded (binary coded (SBCSBC) and ) and Gray coded (GC) data (MS Table 2.11 p. 58 and MS Gray coded (GC) data (MS Table 2.11 p. 58 and MS Table 2.12 p. 60).Table 2.12 p. 60).

EEb b / / NNoo dBdB BER (BER (SBCSBC)) BER (BER (GCGC))∞∞ 00 001010 2.7 2.7 ×× 1010--33 1.7 1.7 ×× 1010--33

88 1.28 1.28 ×× 1010--22 8.6 8.6 ×× 1010--33

66 3.42 3.42 ×× 1010--22 2.92 2.92 ×× 1010--22

44 7.52 7.52 ×× 1010--22 5.93 5.93 ×× 1010--22

22 1.291 1.291 ×× 1010--11 9.56 9.56 ×× 1010--22

00 1.842 1.842 ×× 1010--11 1.441 1.441 ×× 1010--11

Although a slight improvement, Gray coding improves Although a slight improvement, Gray coding improves BER performance without an increase in BER performance without an increase in EEbb..


End of Chapter 4End of Chapter 4

Receiver DesignReceiver Design

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