Baseband transmission2003

8/3/2019 Baseband transmission2003

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Baseband Transmission and Line

Coding

Presented by:

T.F.Lai

Member of IEEE, IET & WWF

Electrical Project Consultant (http://electricalprojecttestimonial.blogspot.com/)


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Outline of the Presentation

Introduction

Baseband Centre Point Detection

Baseband binary error rates in Gaussian noise

Multilevel baseband signaling

Error accumulation over multiple hops

Unipolar signaling

Polar signaling

Dipolar signaling


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Introduction

This chapter addresses two issues central to baseband signal transmission,

digital signal detection and line coding.

The analysis of detection processes is important since this leads to the

principle objective measure of digital communications system quality, i.e.

the probability of symbols being detected in error, as quantified byprobability error.

Appropriate coding of the transmitted symbol pulse shape can minimize

the probability of error by ensuring that the spectral characteristic of the

digital signal are well matched to the transmission channel.

The line code must also permit the receiver to extract accurate PCM bitand word timing signals directly from the received data, for proper

detection and interpretation of the digital pulses.


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Baseband centre point detection

The detection of digital signals involves two processes:

Reduction of each received voltage pulse (i.e. symbol) to a single

numerical value.

Comparison of this value with a reference voltage (or, for multisymbol

signaling, a set of reference voltage) to determine which symbol was

transmitted.

In the case of symbols represented by different voltage levels the simplest

way of achieving 1 is to sample the received signal plus noise; 2 is then

implemented using one or more comparators.

In the case of equiprobable, binarym symbols (zero and one) represented

by two voltage levels (e.g. 0V and 3V) intuition tells us that a sensible

strategy would be to set the reference, Vref(Figure 6.1(b)), mid-way

between the two voltage levels (i.e. at 1.5V).


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Baseband centre point detection

Decisions would then be made at the centre of each symbol period on the

basis of whether the instantaneous voltage (signal plus noise) is above or

below this reference.

Sampling the instantaneous signal plus noise voltage somewhere near the

middle of the symbol period is called centre point detection, Figure 6.1(a).

The noise present during detection is often either Gaussian or

approximately Gaussian.


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Baseband Binary Error Rates in

Gaussian Noise Figure 6.2(a) shows the pdf of a binary information signal which can take

on voltage levels V0 and V1 only.

Figure 6.2(b) shows the pdf of a zero mean Gaussian noise process, Vn(t),

with RMS value W V.

Since the process has zero mean the RMS value and standard deviationare identical.

Figure 6.2(c) shows the pdf of the sum of the signal and noise voltage.

Figure 6.2(c) is perhaps no surprise we can think of the quasi-DC

symbol voltage biasing the mean value of the noise to V0 and V1 .

Figure 6.2(c) is thus the convolution of 6.2(a) and (b).

For equiprobable symbols the optimum decision level is set at (V0 + V1)/2.


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Gaussian Noise


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Gaussian Noise Given that the symbol 0 is transmitted (i.e. voltage level V0) then the

probability, Pe1, that the signal plus noise will be above the threshold at

the decision instant is given by twice the shaded area under the curve

p0(Vn) in Figure 6.2(c):

------------------- (1)

Using the change of variable x = (Vn V0)/2W this becomes:

------------------------ (2)

n2

)VV(

2/)VV(

1e dve2

1P

2

20n

10

W

g

TW!

g

W

T!

22/)VV(

2x

1e

01

dxe1

P


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Gaussian Noise The incomplete integration in equation 2 cannot be evaluated analytically

but can be recast as a complementary error function, erfc(x), defined by:

----------------------- (3)

Thus equation 2 becomes:

----------------------- (4)

Alternatively, since erfc(z) and the error function, erf(z), are related by:

erfc(z) = 1 erf(z) -------------------------------------- (5)

g

(

T!z

2x

dxe2

)z(erfc

W

! 22

VV

erfc2

1

P01

1e


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Gaussian Noise Then Pe1 can also be written as:

------------------------- (6)

The advantage ofusing error (or complementary error) function in the

expression for Pe1 is that this function has been extensively tabulated.

If the digital symbol one is transmitted then the probability Pe0, that the

signal plus noise will be below the threshold at the decision instant is:

--------------- (7)

-

W

!

22

VVerfc1

2

1P 011e

n2

)VV(2/)VV(

0e dve2

1P

2

2

1n10

W

g TW!


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Gaussian Noise It is clear from the symmetry of this problem that Pe0 is identical to both

Pe1 and the probability of error, Pe, irrespective of whether a one or zero

was transmitted.

Noting that the probability of error depends on only symbol voltage

difference and not absolute voltage levels, Pe can be rewritten in terms of(V = V1 V0:

--------------------------- (8)

Equation (8) is valid for both unipolar signaling and polar signaling.

-

W

(!

22

Verfc1

2

1Pe


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Gaussian Noise(Figure 6.3)


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Gaussian Noise Whilst the x-axis of Figure 6.3 is clearly related to signal to noise ratio it is

not identical to it for all pulse levels and shapes.

This is partly because it involves the signal only at the sampling instant

(whereas a conventional SNRuses a time averaged signal power) and

partly because the use of(V neglects any transmitted DC component. For NRZ, unipoar, rectangular pulse signaling, Figure 6.4, the normalised

peak signal power is Speak = (V2 and the average signal power is S = (V2/2.

The normalised Gaussian noise power is N = W2.

We therefore have:

------------------------ (9)

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peak N

S2

N

SV

!

!

W

(


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Gaussian Noise Where (S/N)peak indicates peak signal power divided by average noise

power.

Substituting for (V/W in equation (8) we therefore have:

---------------------- (10)

Or

----------------------------(11)

-

!

2/1

peak

eNS

221erf1

21P

-

!

2/1

e N

S

2

1

erf12

1

P


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Gaussian Noise (Figure 6.4/6.5)


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Gaussian Noise For NRZ, polar, rectangular pulse signaling with the same voltage spacing

as in the unipolar, Figure 6.5, the peak and average signal powers are

identical, Speak = S = ((V/2)2 and we can therefore write:

------------------------ (11)

Substitute into equation (8) we now have:

------------------------- (12)

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peak N

S2N

S2

V

!

!W

(

-

!

2/1

e N

S

2

1

erf12

1

P


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Gaussian Noise Whilst probabilities of symbol and bit error are the quantities usually

calculated, it is the symbol error rate (SER) or bit error rate (BER) which is

usually measured.

These are simply the number of symbol or bit errors occurring per unit

time measured over a convenient period. The symbol error rate is clearly related to the probability of symbol error

by:

SER = PeRs

Where Rs is the symbol rate in symbol/s or baud.

Bit error rate is related to the probability of bit error by:

BER = PbRsn = PbRb


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Multilevel baseband signaling

Figure 6.6 shows a schematic diagram of a multilevel or multisymbol

signal.

If the number of equally spaced, allowed levels is M then the symbol plus

noise pdfs look like those in Figure 6.7 (drawn for M=4).

The probability of symbol error for the M-2 inner symbols (i.e. any but thesymbols represented by the lowest or highest voltage level) is now twice

that in the binary case.

This is because the symbol can be in error if the signal plus noise voltage is

too high or too low.


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Multilevel baseband signaling P

eM|

inner symbol= 2P

e-------------------------------------------- (13)

The symbol error for the two outer levels is identical to that for the binarycase,

PeM|outer symbol = Pe -----------------------------------------------(14)

Once again assuming equiprobable symbols, the average probability ofsymbol error is:

-------- (15)

Substitute for Pe from equation 8,

-------------------- (16)

eeeeM PM

)1M(2P

M

2P2

M

2MP !!

-

W(!

22Verf1

M1MPeM


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All signal transmission media (e.g. cables, waveguides, optical fibres)

attenuate signals to a greater or lesser extent.

This is even the case for the space through which radio waves travel.

For long communication paths attenuation might be so severe that the

sensitivity of normal receiving equipment would be inadequate to detectthe signal.

In such cases the signal is boosted in amplitude at regular intervals along

the transmitter receiver path.

The equipment which boost the signal is called repeater and the path

between adjacent repeaters is called a hop. A repeater along with its associated, preceding, hop is called a section.

Thus long distance communication is usually achieved using multiple hops.


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Repeaters used on multi-hop links can be divided into essentially two

types.

These are amplifying repeaters and regenerative repeaters.

For analogue communication linear amplifiers are required.

For digital communications either type of repeater could be used butnormally regenerative repeaters are employed.

Figure 6.8 shows a schematic of an m-hop link.


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If a binary signal with voltage levels s(/2 is transmitted then the voltagereceived at the input of the first amplifying repeater is sE(V/2 + n1(t)whereE is the linear voltage attenuation factor (or voltage gain factor < 1)

and n1(t) is the random noise (with standard deviation, or RMS value, W)

added du

ring the first hop. In a well designed system the voltage gain, GV, of the repeater will be just

adequate to compensate for the attenuation over the first hop, GV = 1/E.

At the output of the first repeater the signal is restored to its original level

(s(V/2) but the noise signal is also amplified to a level GVn1(t) and willnow have an RMS value of G

VW.

The probability of error for m-hop link is therefore given by:

-

W

(!

m22

Verf1

2

1|P hopsme


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Unipolar signaling

Unipolar signaling (also called on-off keying) refers to a line code in which

one binary symbol (denoting a digital 0, for example) is represented by the

absence of a pulse and other binary symbol (denoting a digital 1) is

represented by the presence of a pulse.

There are two common variations on unipolar signaling, namely non-return to zero and return to zero.

In the former case the duration X of the mark pulse to the duration T0 of

the symbol slot.

In the latter case X is less than T0.

Both NRZ and RZunipolar signals have a non-zero average DC level.

Transmission of these signals over links with either transformer or

capacitor coupled AC repeaters results in the removal of this line and the

consequent conversion of the signals to a polar format.


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Unipolar signaling

Furthermore, since the continuous part of the RZ and NRZ signal spectrum

is non-zero at 0Hz then AC coupling results in distortion of the transmitted

pulse shapes.

If the AC coupled lines behave as high pass RC filters (which is typically the

case) then the distortion takes the form of an exponential decay of thesignal amplitude after each transition.

This effect, referred to as signal droop, is illustrated in Figure 6.13:


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Polar signaling

In polar signaling systems a binary one is represented by a pulse g1(t) and

a binary zero by the opposite (or antipodal) pulse g0(t) = -g1(t).

Polar signaling has a significant power advantage over unipolar signaling.

Fundamentally this is because the pulses in a unipolar scheme are only

orthobonal whilst the pulses in polar scheme are antipodal. Another way of explaining the difference in performance is to observe that

the average or DC level transmitted with unipolar signals contains no

information and is therefore wasted power.

Polar binary signaling also has the advantage that, providing the symbols

are equiprobable, the decision threshold is 0V. This means that no automatic gain control is required in the receiver.


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DiPolar signaling

Dipolar is designed to produce a spectral null at 0Hz.

This makes it especially well suited to AC coupled transmission lines.

The symbol interval, T0, is split into positive and negative pulses each of

width T0/2, Figure 6.11.

Baseband transmission2003

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