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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)T1)Tbb + T+ Tbb/2 ] /2 ] < < ––γγA }A }+ + P(bP(bi i = 1) P{ n= 1) P{ noo[ (i[ (i--1)T1)Tbb + T+ Tbb/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 1889-1976


•• 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 SystemVue SystemVue simulation verifies the spectral simulation verifies the spectral characteristics of thermal noise and the performance of characteristics of thermal noise and the performance of Butterworth lowButterworth low--pass filtered fpass filtered fC C = 1 kHz thermal noise. = 1 kHz thermal noise.

SVU Figure 1.58SVU Figure 1.58


•• Thermal noise (Token 0, Sink 1) temporal display.Thermal noise (Token 0, Sink 1) temporal display.

• Thermal noise PSD = NThermal noise PSD = Noo, | f | , | f | →→ ∞∞ (SVU Figure 1.59)(SVU Figure 1.59)

NoNo

No (single-sided spectrum)


•• Thermal noise (Token 0, Sink 1) temporal display.Thermal noise (Token 0, Sink 1) temporal display.

• Thermal noise autocorrelation function (SVU Figure 1.61)Thermal noise autocorrelation function (SVU Figure 1.61)

uncorrelated


•• LPF thermal noise (Token 2, Sink 3) temporal display.LPF thermal noise (Token 2, Sink 3) temporal display.

• LPF fLPF fC C = 1 kHz thermal noise PSD = N= 1 kHz thermal noise PSD = Noo, | f | < f, | f | < fCC

No (single-sided spectrum)

σo2 = No fC


fC


•• LPF thermal noise (Token 2, Sink 3) temporal display.LPF thermal noise (Token 2, Sink 3) temporal display.

• LPF fLPF fC C = 1 kHz thermal noise autocorrelation function= 1 kHz thermal noise autocorrelation function

S&M Figure 5S&M Figure 5--2121


•• Thermal noise histogram displayThermal noise histogram display

• LPF fLPF fC C = 1 kHz thermal noise histogram display= 1 kHz thermal noise histogram display

Gaussian pdf

Gaussian pdfpdf remains Gaussian after LPF


•• Thermal noise histogram displayThermal noise histogram display

• LPF fLPF fC C = 1 kHz thermal noise histogram display= 1 kHz thermal noise histogram display

Gaussian pdf

Gaussian pdf

σn

σo


•• SystemVue SystemVue Sink CalculatorSink Calculator histogram analogous to the histogram analogous to the pdfpdf


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

σσoo2 2 = N= Noo ffC C (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. prized by gamblers. 1667-1754

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. .

1777-1855


•• 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. Rather, construct a unwieldy table. Rather, construct a single tablesingle table with with µµ =0 =0 and and σσ = 1 for the= 1 for theprobability of bitprobability of biterror as the areaerror as the areaunder the Gaussianunder the Gaussianpdf as a function ofpdf as a function ofthe threshold the threshold a a onlyonlyknown as theknown as theQQ--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).p. 141).


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

b

noise margin of sampled valueP = Q

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

SVU Figure 2.8

original binary data

data with AWGN

Sampling at Tb/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 best that there is? What about sampling an But is this the best that there is? What about sampling an odd number of times during each bit time Todd number of times during each bit time Tbb? ?

Tb


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

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

Tb


•• 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 b-1 -1

o o b

o b

H (f) = k S (f) exp( j 2π f iT )h (t) = F { H (f) } = k F { S (f) exp( j 2π f iT ) }h (t) = k s(iT 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 ), time delayed (by iTiTbb) and time reversed ) and time reversed (function of (function of iTiTbb–– tt):):

−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 Pprobability of bit error Pbb is:is:

where Ewhere Ebb is the is the energy per bit energy per bit of the received signal.of the received signal.

σ ∗

b bb

p

2b b

b 2p

bb

o

| r(iT ) h(iT ) |P = Q maximum

| r(iT ) * h(iT ) |P = Q maximumσ

2 EP = Q 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 −o bh (t) = k s(iT t)

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 Ebit Ebb is (S&M p. 156,is (S&M p. 156,Eq 4.62):Eq 4.62):

∫ ∫b b

b b

iT iT2 2

b(i-1)T (i-1)T

E = γ s(t) γ s(t) dt = γ s (t) dt


•• The expected or mean value The expected or mean value aaii(iT(iTbb) 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.

∫b

b

iT

i b i 1(i-1)T

a (iT ) = γ s (t) s (t) dt S&M Eq. 4.67


•• PPbb = Q( = Q( √√(2 E(2 Ebb / N/ Noo) ) and the ratio ) ) and the ratio EEbb / N/ No o can be can be expressed in dB: 10 logexpressed in dB: 10 log1010 (E(Ebb / N/ Noo ). The resulting plot of ). The resulting plot of PPbb verses Everses Ebb / N/ No in o in 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(iTa(iTbb) )) ). 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 89Pages 89--9494


•• The correlation receiver can be simulated in The correlation receiver can be simulated in SystemVueSystemVue..



•• The The SystemVueSystemVue integrateintegrate--andand--dumpdump token is in the token is in the Communications Library.Communications Library.


•• The parameters of the The parameters of the SystemVueSystemVue integrateintegrate--andand--dumpdumptoken are a token are a holdhold--value value output, output, zero offsetzero offset and an and an integration timeintegration time of one bit time Tof one bit time Tbb (1 msec here) .(1 msec here) .


•• 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.


Downsampler

BER Analysis

Delay

Threshold = 0


•• 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 (LPF) and the rectangular PAM in a LPF simple receiver (LPF) and the optimum correlation receiver (CR) with normalized signal optimum correlation receiver (CR) with normalized signal power = 25 W (SVU Table 2.3 p. 73 and Table 2.6 p. 93).power = 25 W (SVU Table 2.3 p. 73 and Table 2.6 p. 93).

SNR dBSNR dB AWGN AWGN σσ VV BER (LPF)BER (LPF) BER (CR)BER (CR)∞∞ 00 00 006.026.02 2.52.5 00 0000 55 00 00−−3.523.52 7.57.5 6 6 ×× 1010--44 00−−6.026.02 1010 4.5 4.5 ×× 1010--33 1 1 ×× 1010--44

−−7.967.96 12.512.5 1.89 1.89 ×× 1010--22 2.8 2.8 ×× 1010--33

−−9.549.54 1515 8.8 8.8 ×× 1010--33


•• 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 Pthe theoretical probability of bit error Pbb (S&M Eq. 4.58, p. (S&M Eq. 4.58, p. 154) to validate the basic simulation.154) to validate the basic simulation.

∫ ∫b b

b b

iT iT2 2

b(i-1)T (i-1)T

E = γ s(t) γ s(t) dt = γ s (t) dt

bb

o

2 EP = Q N


•• The BER and PThe BER and Pbb comparison (SVU Table 2.7, p. 94): comparison (SVU Table 2.7, p. 94):

Table 2.7Table 2.7 Observed BER and Theoretical PObserved BER and Theoretical Pb b as a as a Function of Function of EEbb/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

EEbb/N/Noo dBdB NNoo VV22--secsec BERBER PPbb∞∞ 00 00 001010 2.50 2.50 ×× 1010--33 00 4.05 4.05 ×× 1010--66

88 3.96 3.96 ×× 1010--33 00 2.06 2.06 ×× 1010--44

66 6.28 6.28 ×× 1010--33 2.2 2.2 ×× 1010--33 2.43 2.43 ×× 1010--33

44 9.96 9.96 ×× 1010--33 1.21 1.21 ×× 1010--22 1.25 1.25 ×× 1010--22

22 1.58 1.58 ×× 1010--2 2 3.91 3.91 ×× 1010--22 3.75 3.75 ×× 1010--22

00 2.50 2.50 ×× 1010--22 8.13 8.13 ×× 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(iT(iTbb) | ) | ≠≠ | a| a11(iT(iTbb) |) |



•• 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(iT(iTbb) | + ) | + ττoptopt = a= a11(iT(iTbb) ) –– 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 b 1 b

opta (iT )+ a (iT )τ

2


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


[ ]

( ){ }

2

2

− −

−

∫b

b

1 b 2 bb

o

1 b 2 bb

o

iT2d

b d 1 2o (i-1)T

a (iT ) a (iT )P = Q2 σ

a (iT ) a (iT ) P = Q

4 σ

EP = Q where E = γ s (t) s (t) dt2 N

S&M S&M EqsEqs. 4.78 and 4.79. 4.78 and 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(iT(iTbb) ) ≠≠ aa11(iT(iTbb))



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

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

[ ]− −

=−

2 2 2o 2 b 1 b

opt2 b 1 b

2 σ ln ( M / (1 M) ) + a (iT ) a (iT )τ2 a (iT ) a (iT )

= 2 b 1 b


2

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

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


•• The probabilityThe probabilityof bit error Pof bit error Pbb thenthenisis::

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

( ) ( )

− − −

− − −

1 b opt opt 2 bb

o o

2 2

1 b opt opt 2 bb

o o

a (iT ) τ τ a (iT )P = M Q + (1 M) Q

σ σ

a (iT ) τ τ a (iT )P = M Q + (1 M) Q

2 N 2 N


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

and Pand Pbb becomes:becomes:

( ) ( )

( )

− −

2 2

1 b opt opt 2 bb

o o

21 b 2 b d

bo o

a (iT ) τ τ a (iT )P = 0.5 Q + 0.5 Q

2 N 2 N

a (iT ) - a (iT ) EP = Q = Q2 N 2 N

= 2 b 1 b


2

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



•• The BER and PThe BER and Pbb comparison (SVU Table 2.8, p. 94): comparison (SVU Table 2.8, p. 94):

Table 2.8Table 2.8 Observed BER and Theoretical PObserved BER and Theoretical Pb b as a as a Function of Function of EEdd/N/Noo in an Asymmetrical Binary in an Asymmetrical Binary Rectangular PAM Digital Communication System with Rectangular PAM Digital Communication System with Optimum ReceiverOptimum Receiver

EEdd/N/Noo dBdB NNoo VV22--secsec BERBER PPbb∞∞ 00 00 001212 1.58 1.58 ×× 1010--33 2.8 2.8 ×× 1010--33 2.53 2.53 ×× 1010--33

1010 2.50 2.50 ×× 1010--33 1.19 1.19 ×× 1010--22 1.25 1.25 ×× 1010--22

88 3.96 3.96 ×× 1010--33 3.86 3.86 ×× 1010--22 3.75 3.75 ×× 1010--22

66 6.28 6.28 ×× 1010--33 8.34 8.34 ×× 1010--22 7.93 7.93 ×× 1010--22

44 9.96 9.96 ×× 1010--33 1.291 1.291 ×× 1010--11 1.318 1.318 ×× 1010--11

22 1.58 1.58 ×× 1010--2 2 1.867 1.867 ×× 1010--11 1.872 1.872 ×× 1010--11

00 2.50 2.50 ×× 1010--22 2.322 2.322 ×× 1010--11 2.394 2.394 ×× 1010--11


•• Since for binary rectangular PAM ESince for binary rectangular PAM Edd = 4 E= 4 Ebb (S&M p. 168)(S&M p. 168)the performance for asymmetric PAM is comparable to the performance for asymmetric PAM is comparable to symmetric PAM if a Esymmetric PAM if a Edd / N/ Noo is reduced by 6 dBis reduced by 6 dB(10 log (4) = 6) : (10 log (4) = 6) : EEdd/N/Noo dBdB NNoo VV22--secsec BERBER PPbb

1010 2.50 2.50 ×× 1010--33 1.19 1.19 ×× 1010--22 1.25 1.25 ×× 1010--22

88 3.96 3.96 ×× 1010--33 3.86 3.86 ×× 1010--22 3.75 3.75 ×× 1010--22

66 6.28 6.28 ×× 1010--33 8.34 8.34 ×× 1010--22 7.93 7.93 ×× 1010--22

EEbb/N/Noo dBdB NNoo VV22--secsec BERBER PPbb

44 9.96 9.96 ×× 1010--33 1.21 1.21 ×× 1010--22 1.25 1.25 ×× 1010--22

22 1.58 1.58 ×× 1010--2 2 3.91 3.91 ×× 1010--22 3.75 3.75 ×× 1010--22

00 2.50 2.50 ×× 1010--22 8.13 8.13 ×× 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

•• Pages 94Pages 94--100100


•• 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 b 1 b

opt2 b 1 b

2 σ ln ( M / (1 M) ) + a (iT ) a (iT )τ2 a (iT ) a (iT )


σσoo


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

Here PHere P1 1 = 0.4996 and P= 0.4996 and P0 0 = 0.5004= 0.5004


[ ]− −

=−

2 2 2o 2 b 1 b

opt2 b 1 b

2 σ ln ( M / (1 M ) ) + a (iT ) a (iT )τ2 a (iT ) a (iT )



•• The The SystemVueSystemVue simulation for binary asymmetrical PAM simulation for binary asymmetrical PAM with BER analysis has a threshold adjustment.with BER analysis has a threshold adjustment.


ThresholdReference


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


S&M Figure 4-21


Chapter 4Chapter 4

Synchronization and EqualizationSynchronization and Equalization•• Symbol SynchronizationSymbol Synchronization

•• Pages 241Pages 241--246246


•• Synchronization provides Synchronization provides timing recovery timing recovery or the exact or the exact beginning and end of a bit time Tbeginning and end of a bit time Tbb::

S&M Figure 4-21

Tb


•• The soThe so--called called early/late symbol synchronizerearly/late symbol synchronizer uses two uses two correlation receivers which integrate from a delay d to Tcorrelation receivers which integrate from a delay d to Tb b and 0 to Tand 0 to Tbb –– dd. The times 0 and T. The times 0 and Tbb are the current best are the current best estimate of the beginning and end of the bit time. Theestimate of the beginning and end of the bit time. Thedifference indifference inthe absolutethe absolutevalues of values of the integratorthe integratoroutputs is aoutputs is ameasure of themeasure of thetiming errortiming errorwhich is usedwhich is usedto adjust theto adjust thedelay d.delay d.

SVU Figure 4-8


•• The The SystemVueSystemVue simulation of the early/late symbol simulation of the early/late symbol synchronizer uses the Bit Synchronizer token from the synchronizer uses the Bit Synchronizer token from the Communication Library. Communication Library.

SVU Figure 4-9

Jitter

Jitter output Additive noise


•• Output of Comparator jitter output (Token 4, Sink 10)Output of Comparator jitter output (Token 4, Sink 10)

• Overlay of synch and loop error (Token 6, Sinks 7 and 8)Overlay of synch and loop error (Token 6, Sinks 7 and 8)

TTbb / 2/ 2delaydelay

2T2Tbb

delay


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 rbandwidth required for a data transmission rate rbb b/sec. b/sec. Rather than transmitting a binary signal in a bit time TRather than transmitting a binary signal in a bit time Tbb, , 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 symbol time Tsame period called the symbol time TSS..

A multiA multi--state comparator determines the received symbol state comparator determines the received symbol which is then decoded to the received bits. which is then decoded to the received bits.



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

=

=

=

1 S 2 Sopt1

2 S 3 Sopt2

3 S 4 Sopt3

a (iT )+ a (iT )τ2

a (iT )+ a (iT )τ2

a (iT )+ a (iT )τ2

S&M Figure 4.49S&M Figure 4.49

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


•• TheTheprobabilityprobabilityof symbolof symbolerrorerror PPsswhere M iswhere M isthe numberthe numberof levelsof levelscan becan beshown to be:shown to be:

where:where:

S&M Eq. 4.140S&M Eq. 4.140{ } ∫s

s

i T2

d,symbol j k(i-1) T

E = γ s (t) - s (t) dt

( ) −

d,symbols

o

E2 M 1P = Q

M 2NS&M Eq. 4.139S&M Eq. 4.139


•• There areThere are2 (M 2 (M –– 1)1)errorerrorregionsregionsdue to due to onlyonlyadjacentadjacentregions regions beingbeingmisinterpreted with M equally probable symbols. The misinterpreted with M 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


•• 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 Esymbol Ed,symbol d,symbol = 4= 4γγ2 2 AA2 2 TTS S (S&M Eq. 4.140):(S&M Eq. 4.140):

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

{ } ∫s

s

i T2

d,symbol j k(i-1) T

E = γ s (t) - s (t) dt

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

10

01

00


•• Then EThen Ed,symbold,symbol = 0.4 E= 0.4 Eavg,symbol avg,symbol and substitute (M = 4):and substitute (M = 4):

avg,symbols

o

0.4 E3P = Q2 N

d,symbols

o

E3P = Q2 2N S&M Eq. 4.139bS&M Eq. 4.139b

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


10

01

00


•• The average energy per bit EThe average energy per bit Eb b = 0.5 E= 0.5 Eavg,symbolavg,symbol since M =4 since M =4 and there are two bits per symbol:and there are two bits per symbol:

avg,symbols

o

0.4 E3P = Q2 N

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

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

bs

o

0.8 E3P = Q2 N


10

01

00


•• 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

b,4-level

b,4-level s s s

4 2P = P(1 of 2 bits in error) + P(2 of 2 bits in error)6 64 1 2 2P = P + P = P6 2 6 3

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


•• The probability of bit error PThe probability of bit error Pb,4b,4--symbolsymbol = 2 P= 2 Pss / 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 GrayGray--code 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 Peach of the six error regions and Pb,4b,4--symbolsymbol = P= Ps s / 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

b,4-level

b,4-level s s

6P = P(1 of 2 bits in error) 66 1 1P = P = P6 2 2


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


10

00

01

=

=

b bb

o o

b bb

o o

0.8 E 0.8 E2 3P = Q Q3 2 N N

0.8 E 0.8 E1 3 3P = Q Q2 2 N 4 N

GrayGray--codedcoded

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 Gray

1894-1964


Chapter 2Chapter 2

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

•• Pages 100Pages 100--112112


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

SVU Figure 2.33

ScalingSymbol to bit converter

4-level PAM Scaling


•• The algebraic polynomial scaling token is in the Function The algebraic polynomial scaling token is in the Function Library: y = 1.5 + 0.06 x Library: y = 1.5 + 0.06 x


•• The encoder/decoder symbol to bit token is in the The encoder/decoder symbol to bit token is in the Communications Library: Communications Library:


•• 44--level PAM signal (Token 0, Sink 23)level PAM signal (Token 0, Sink 23)

• Bit to symbol output (Token 6, Sink 20)Bit to symbol output (Token 6, Sink 20)


delay

11

+5


•• 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::

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..

( )( )

2M avg S S

2B b b

PSD (f) = A T sinc π T f

PSD (f) = A T sinc π T f

= ∑M

2 2avg j j

j=1A A P


•• PSDPSDMM M = 4, rM = 4, rbb = 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 Hz

1 kHz


End of Chapter 4End of Chapter 4

Receiver DesignReceiver Design

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