dr. uri mahlab1. 2 communication system dr. uri mahlab3 + channel noise channel noise block diagram...

Dr. Uri Mahlab 1

Dr. Uri Mahlab 2

Communication system

Dr. Uri Mahlab 3

++ Channel noiseChannel noise

Block diagram of an Binary/M-ary signalingBlock diagram of an Binary/M-ary signaling schemescheme

++

Dr. Uri Mahlab 4

Block diagram DescriptionBlock diagram Description

{d{dkk}={1,1,1,1,0,0,1,1,0,0,0,1,1,1}}={1,1,1,1,0,0,1,1,0,0,0,1,1,1}

TTbb

TTbb

ForFor

ForFor

Dr. Uri Mahlab 5

Block diagram Description Block diagram Description (Continue - 1)(Continue - 1)

{d{dkk}={1,1,1,1,0,0,1,1,0,0,0,1,1,1}}={1,1,1,1,0,0,1,1,0,0,0,1,1,1}

Dr. Uri Mahlab 6


{d{dkk}={1,1,1,1,0,0,1,1,0,0,0,1,1,1}}={1,1,1,1,0,0,1,1,0,0,0,1,1,1}

100110100110

Dr. Uri Mahlab 7

TimingTiming


{d{dkk}={1,1,1,1,0,0,1,1,0,0,0,1,1,1}}={1,1,1,1,0,0,1,1,0,0,0,1,1,1}

TTbb

100110100110

tt

tt

Dr. Uri Mahlab 8

InformationInformation

sourcesourcePulsePulse

generatorgeneratorTransTransfilterfilter

) ) X(tX(tTimingTiming


HHTT(f)(f)

TTbb 2T2Tbb

3T3Tbb 4T4Tbb

5T5Tbb

tt6T6Tbb

Channel noise n(t)Channel noise n(t)

++

tt

ReceiverReceiverfilterfilter

HHRR(f)(f)

tt

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TTbb 2T2Tbb

3T3Tbb 4T4Tbb

5T5Tbb

tt6T6Tbb

tt

tt

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TimingTiming

++


++

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Block diagram DescriptionBlock diagram Description

TTbb 2T2Tbb

3T3Tbb 4T4Tbb

5T5Tbb

tt6T6Tbb

tt

1 0 0 0 1 01 0 0 0 1 0

1 0 0 1 1 01 0 0 1 1 0

tt

tt

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Typical waveforms in a binary PAM system

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TimingTiming

++

++

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Explanation of PExplanation of Prr(t)(t)

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The element of a baseband binary PAM system

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For and is the total time delay in the For and is the total time delay in the system, we get. system, we get.

tt

tt

tt

tmY

tt11

tt22 tt33

ttmm

The input to the A/D converter isThe input to the A/D converter is

Dr. Uri Mahlab 18tt

tmY

tt11

tt22 tt33

ttmm

The output of the A/D converter at the sampling timeThe output of the A/D converter at the sampling time

ttm m =mT=mTbb+t+tdd

TTbb 2T2Tbb

3T3Tbb 4T4Tbb

5T5Tbb

tt6T6Tbb

Dr. Uri Mahlab 19tt

tmY

tt11

tt22 tt33

ttmm

ISI - ISI - IInter nter SSymbolymbolIInterferencenterference

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Explanation of ISIExplanation of ISI

tt

ff ff

tt

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Explanation of ISI - ContinueExplanation of ISI - Continue

tt

ff ff

tt

TTbb 2T2Tbb

3T3Tbb 4T4Tbb

5T5Tbb

tt6T6Tbb

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Dr. Uri Mahlab 23

-The pulse generator output is a pulse waveform

If kth input bit is 1

if kth input bit is 0

-The A/D converter input Y(t)

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Dr. Uri Mahlab 25

5.2 BASEBAND BINARY PAM SYSTEMS

- minimize the combined effects of inter symbol interference and noise in order to achieve minimum probability of error for given data rate.

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5.2.1 Baseband pulse shaping

The ISI can be eliminated by proper choice of received pulse shape pr (t).

Doe’s not Uniquely Specify Pr(t) for all values of t.

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TheoremTheorem

ProofProof

To meet the constraint, Fourier Transform Pr(f) of Pr(t), shouldTo meet the constraint, Fourier Transform Pr(f) of Pr(t), shouldsatisfy a simple condition given by the following theoremsatisfy a simple condition given by the following theorem

Dr. Uri Mahlab 28

b

b

T2/1

T2/1

bbbr df)fnT2jexp(T)nT(p

Which verify that the Pr(t) with a transform Pr(f) Which verify that the Pr(t) with a transform Pr(f) Satisfy Satisfy ____________________________

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The condition for removal of ISI given in the theorem is calledThe condition for removal of ISI given in the theorem is calledNyquist (Pulse Shaping) CriterionNyquist (Pulse Shaping) Criterion

TTbb 2T2Tbb-T-Tbb-2T-2Tbb

11

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The Theorem gives a condition for the removal of ISI using a Pr(f) withThe Theorem gives a condition for the removal of ISI using a Pr(f) witha bandwidth larger then rb/2/. a bandwidth larger then rb/2/.

ISI can’t be removed if the bandwidth of Pr(f) is less then rb/2.ISI can’t be removed if the bandwidth of Pr(f) is less then rb/2.

TTbb 2T2Tbb

3T3Tbb 4T4Tbb

5T5Tbb

tt6T6Tbb

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pr )t(

Particular choice of Pr(t) for a given application

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A Pr(f) with a smooth roll - off characteristics is preferable A Pr(f) with a smooth roll - off characteristics is preferable over one with arbitrarily sharp cut off characteristics.over one with arbitrarily sharp cut off characteristics.

Pr(f)Pr(f) Pr(f)Pr(f)

Dr. Uri Mahlab 33

In practical systems where the bandwidth available for In practical systems where the bandwidth available for transmitting data at a rate of rtransmitting data at a rate of rbb bits\sec is between r bits\sec is between rbb\2 to r\2 to rbb Hz, a class of pHz, a class of prr(t) with a (t) with a raised cosine frequency raised cosine frequency characteristiccharacteristic is most commonly used.is most commonly used.A raise Cosine Frequency spectrum consist of a flat amplitude portion and a roll off A raise Cosine Frequency spectrum consist of a flat amplitude portion and a roll off portion that has a sinusoidal form.portion that has a sinusoidal form.

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raised cosine frequency characteristicraised cosine frequency characteristic

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The BW occupied by the pulse spectrum is The BW occupied by the pulse spectrum is B=rB=rbb/2+/2+. . The minimum value of B is rb/2 and the maximum value is The minimum value of B is rb/2 and the maximum value is rrbb..

Larger values ofLarger values of imply that moreimply that more bandwidth is required for a bandwidth is required for a given bit rate, however it lead for faster decaying pulses, whichgiven bit rate, however it lead for faster decaying pulses, whichmeans that synchronization will be less critical and will not cause means that synchronization will be less critical and will not cause large ISI.large ISI.

=rb/2 leads to a pulse shape with two convenient properties. =rb/2 leads to a pulse shape with two convenient properties. The half amplitude pulse width is equal to Tb, and there are zero The half amplitude pulse width is equal to Tb, and there are zero crossings at t=3/2Tb, 5/2Tb…. In addition to the zero crossingcrossings at t=3/2Tb, 5/2Tb…. In addition to the zero crossing at Tb, 2Tb, 3Tb,…...at Tb, 2Tb, 3Tb,…...

SummarySummary

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5.2.25.2.2

Optimum transmitting and receiving Optimum transmitting and receiving filtersfilters

The transmitting and receiving filters are chosen to provideThe transmitting and receiving filters are chosen to providea propera proper

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-One of design constraints that we have for selecting the filters -One of design constraints that we have for selecting the filters is the relationship between the Fourier transform of pis the relationship between the Fourier transform of prr(t) and (t) and ppgg(t).(t).

In order to design optimum filter Ht(f) & Hr(f), we will assume that Pr(f), In order to design optimum filter Ht(f) & Hr(f), we will assume that Pr(f), Hc(f) and Pg(f) are known. Hc(f) and Pg(f) are known.

Where tWhere tdd, is the time delay Kc normalizing constant., is the time delay Kc normalizing constant.

Portion of a baseband PAM systemPortion of a baseband PAM system

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If we choose Pr(t) {Pr(f)} to produce Zero ISI we are left If we choose Pr(t) {Pr(f)} to produce Zero ISI we are left only to be concerned with noise immunity, that is will chooseonly to be concerned with noise immunity, that is will choose

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Noise ImmunityNoise Immunity

Problem definition:Problem definition:

For a givenFor a given : :•Data Rate - Data Rate - •Transmission power -Transmission power -•Noise power Spectral Density -Noise power Spectral Density -•Channel transfer function -Channel transfer function -•Raised cosine pulse -Raised cosine pulse -

Choose Choose

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Error probability CalculationsError probability Calculations

At the m-th sampling time the input to the A/D is:At the m-th sampling time the input to the A/D is:

We decideWe decide::

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A=aKcA=aKc

The noise is assumed to be zero mean Gaussian at the receiver inputThe noise is assumed to be zero mean Gaussian at the receiver inputthen the output should also be Zero mean Gaussian with variance Nothen the output should also be Zero mean Gaussian with variance No

given by:given by:

Dr. Uri Mahlab 42bb

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00

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-A-A AA

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UU

Q(u)Q(u)

dz=dz=

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0N

A

PPerrorerror decreases as decreases as increaseincrease

Hence we need to maximize the signalHence we need to maximize the signalto noise Ratioto noise Ratio

Thus for maximum noise immunity the filter transfer functions HThus for maximum noise immunity the filter transfer functions HTT(f)(f)and Hand HRR(f) must be xhosen to maximize the SNR(f) must be xhosen to maximize the SNR

Dr. Uri Mahlab 48

Optimum filters design calculationsOptimum filters design calculations

We will express the SNR in terms of HWe will express the SNR in terms of HTT(f) and H(f) and HRR(f) (f)

We will start with the signal:We will start with the signal:

The PSD of the transmitted signal is given by::The PSD of the transmitted signal is given by::

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And the average transmitted power ST isAnd the average transmitted power ST is

The average output noise power of nThe average output noise power of n00(t) is given by:(t) is given by:

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The SNR we need to maximize is The SNR we need to maximize is

Or we need to minimize Or we need to minimize

Dr. Uri Mahlab 51

Using Schwartz’s inequality Using Schwartz’s inequality

The minimum of the left side equaity is reached whenThe minimum of the left side equaity is reached when

V(f)=Const*W(f)V(f)=Const*W(f)

If we choose :If we choose :

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whenminimized is 2

The filter should have a linear phase response in a total time delay of tdThe filter should have a linear phase response in a total time delay of td

Dr. Uri Mahlab 53

Finally we obtain the maximum value of the SNR to be:Finally we obtain the maximum value of the SNR to be:

Dr. Uri Mahlab 54

For AWGN with For AWGN with

andand

ppgg(f) is chosen such that it does not change much over the (f) is chosen such that it does not change much over the bandwidth of interest we get. bandwidth of interest we get.

Rectangular pulse can be used at the input of HRectangular pulse can be used at the input of HTT(f).(f).

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5.2.3 Design procedure and Example

The steps involved in the design procedure.

Example:Design a binary baseband PAM system to transmit data at a a bit rate of 3600 bits/sec with a bit error probability less than .10 4

The channel response is given by:

The noise spectral density is HzwattfGn /10)( 14

Dr. Uri Mahlab 56

Solution:

HzwattfG

HzB

p

bitsr

n

e

b

/10)(

2400

10

sec/3600

4

4

If we choose a braised cosine pulse spectrum with6006/ br

Dr. Uri Mahlab 57

We choose a pg(t)

We choose

)()()()()(

)10)(3600( 31

fpfHfHfHfp

K

rRcTg

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Plots of Pg(f),Hc(f),HT(f),HR(f),and Pr(f).

Dr. Uri Mahlab 59

To maintain a 410eP

For Pr(f) with raised cosine shape

1)( dffPr

And hence dBmST 23)10)(3600)(06.14( 10

Which completes the design.

Dr. Uri Mahlab 60

In order to transmit data at a rate of rb bits/sec with zero ISI PAM data transmission system requires a bandwidth of at least rb

/2 HZ.

+Binary PAM data transmission at a rate of rb bits/sec with zero ISI

1.such filters are physically unrealizable

2. Any system with this filters will be extremely sensitive to perturbations.

204

5.3

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Dr. Uri Mahlab 62

The duobinary signaling schemes use pulse spectra Pr(f):

The duobinary scheme utilizes controlled amounts of ISI for transmitting data at a rate of rb /2 HZ. The shaping filters for duobinary are easier to realize than the ideal rectangular filters.

The pulse response Pr(t): tPr

Dr. Uri Mahlab 63

The output Y(t) of the receive filter can be written as:

The Am' s can assume one of two values +/-A depending on whether the m th input bit is 1 or 0. Since Y(tm) depends on Am & Am-1 assuming no noise :

+2A if the m th and (m-1st)bits are both 1's

0 if the m th and (m-1st)bits are different

-2A if the m th and (m-1st)bits are both zero

If the output is sampled at tm= mTb/2+ td than it is obvious that in the absence of noise:

Y(tm) =

5.3.1

Dr. Uri Mahlab 64

Transmitting and receiving filters for optimum performance

Since no is a zero mean Gaussian random variable with a

variance No we can write pe as:

The receiving levels at the input to the A/D converter are 2A, 0, and -2A with probabilities 1/2, 1/4. The probability of bit error pe is given by:

For the direct binary PAM case :

5.3.2

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The integral can be evaluated as :

Where /2=Gn(f) is the noise power spectral density the probability of error is:

Dr. Uri Mahlab 66

M-ARY SIGNALING SCHEMES

In baseband binary PAM we use pulses with one of 2 possible amplitude, In M-ary baseband PAM system we allowed M possible levels (M>2) and there M distinct input symbols.

ss rT 1During each signaling interval of duration the source is converted to a four-level PAM pulse train by the the pulse generator.

The signal pulse noise passes through the receiving filter and is sampled by the A/D converter at an appropriate rate and phase.

the M-ary PAM scheme operating with the preceding constraints can transmit data at a bit rate of rs log2M bit/sec and require a minimum bandwidth of rs/2 HZ.

5.4

Dr. Uri Mahlab 67

5.7 MISCELLANEOUS TOPICS

5.7.1 Eye Diagram The performance of baseband PAM systems depends on the amount of ISI and channel noise.

The received waveform with no noise and no distortion is shown in Figure 5.20a the “open” eye pattern results

Figure 5.20b shows a distorted version of the waveform the corresponding eye pattern.

Figure 5.20c shows a noise distorted version of the received waveform and the corresponding eye pattern.

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Dr. Uri Mahlab 69

In the figure 5.21 we see typical eye patterns of a duobinary signal

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If the signal-to-noise ratio at the receiver is high then the following observations can be made from the eye pattern shown simplified in Figure 5.22:

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1. The best time to sample the received waveform is when the eye is opening is largest.

2. The maximum distortion is indicated by the vertical width of the two branches at sampling time.

3. The noise margin or immunity to noise is proportional to the width of the eye opening.

4. The sensitivity of the system to timing errors is revealed by the rate of closing of the eye as sampling time is varied.

5. The sampling time is midway between zero crossing.

6. Asymmetries in the eye pattern indicate nonlinearities in the channel.

Dr. Uri Mahlab 72

5.7.2 synchronization

Three methods in which this synchronization can be obtained are:

1. Derivation of clock information from a primary or secondary standard.

2. Transmitting a synchronizing clock signal.

3. Derivation of the clock signal from the received waveform itself.

An example of a system used to derive a clock signal from the received waveform is shown in figure 5.23.

To illustrate the operating of the phase comparator network let us look at the timing diagram shown in figure 5.23b

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Dr. Uri Mahlab 74

5.7.3 Scrambler and unscrambler

Scrambler: The scrambler shown in figure 5.24a consists a “feedback” shift register.

Unscrambler:The matching unscrambler have a ”feed forward” shift register structure.

In both the scrambler and unscrambler the outputs of several stages of of shift register are added together modulo-2 and the added to the data stream again in modulo-2 arithmetic

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Dr. Uri Mahlab 76

Scrambler affects the error performance of the communication system in that a signal channel error may cause multiple error at the output of the unscrambler.

The error propagation effect lasts over only a finite and small number of bits.

In each isolated error bit causes three errors in the final output it must also be pointed out that some random bit patterns might be scrambled to the errors or all ones.

dr. uri mahlab1. 2 communication system dr. uri mahlab3 + channel noise channel noise block diagram...

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