digital communications objectives to describe a binary digital transmission system to explain...

Digital CommunicationsDigital Communications

Objectives• To describe a binary digital transmission system• To explain intersymbol interference• To show how an eye diagram is produced and explain its

significance for system performance assessment• To show the pulses with zero intersymbol interference• To introduce the condition for zero intersymbol

interference• To introduce the term “error probability”• To introduce the concept of error control coding• To introduce the concept of line coding and to provide

illustrations of some practical line codes• To study digital carrier modulation schemes


Recall that PAM is the simplest pulse modulation system, and can beused to get a complete digital system. PAM signals quantized coded PCM signal

PCM signals consist of binary digits, i.e. a string of 1’s and 0’s.

These binary digits are then transmitted over the transmissionchannel

digital transmission


Binary digital transmission system

• When the signal arrives at the output of the channel, it is attenuated due to the loss and distorted due to limited bandwidth of the channel • The distortion can be partially corrected by the use of an equalizer.


Equalization

A method of signal recovery is to use a filter that has a transfer function which is the reciprocal of the channel transfer function Q()

So that the output spectrum G0() = G()Q()Heq() becomes G()/T. This technique of correcting the frequency response of a system for a known distortion is called equalization.

mw <w1/ ( )( )

0eq

QH

elsewhere


Equalization



• Threshold device can restore the original signal, but the timing of the transitions may be irregular.

• Causes of irregularity:1. transition time varies with different transmission channels

2. threshold circuit may be corrupted by the noise

Thus, a clock waveform is extracted from the signal and used to retime the data.


Intersymbol InterferenceIn practice, the digital pulses are not perfectly rectangular and the transmission medium is neither perfectly linear nor distortionless (because it has limited bandwidth).

A transmission channel has only limited bandwidth a small portion of signal spectrum is suppressed.

signal spectrum distortion spreading of signal pulse (dispersion).

Spreading of a pulse beyond its interval causes it interfereswith neighboring pulses intersymbol interference (ISI).


Intersymbol interference


Is it is possible to obtain zero ISI with limited channel bandwidth?

Condition for zero intersymbol interference (ISI)If the signal shape is p(t), then p(t) must satisfy

p(0) = non-zero constantp(nT) = 0, when n 0.

What kind of pulse shape satisfies the condition for zero ISI?


However, it is virtually impossible to achieve thiskind of pulse shape in practice, as it is the impulse response of an ideal lowpass filter.


Both the intersymbol interference and noise may cause errors, and hence affect the system performance.

How to evaluate the combination effect of noise and intersymbol interference on overall system

performance?

--- Use an eye pattern (eye diagram).

The eye pattern is the synchronized superposition of all possible signal patterns obtained in a particular signal interval.


The eye pattern (eye diagram)


The interior region of the eye pattern is called the eyeopening. (Eye opening is also the timing error allowed on the sampler at the receiver)

Little ISI, little noise, little jitter more open eye

An eye pattern provides useful information about the system performance.


The width of the eye opening defines the time interval over which the received signal can be sampled without error from ISI. (The best sampling time: when the vertical opening of the eye is the largest)

The height of the eye opening, at a specified sampling time, defines the system noise margin

The slope of the open eye indicates the sensitivity to the timing error.


Probability of error in transmissionIn a binary digital communication system, the twolevels (“1” and “0”) may be represented by

A & 0 (unipolar binary signal)or A & - A (polar binary signal)

In a polar binary signal transmission using pulse p(t), assume that the binary levels at the receiver are - A and A. The observed waveform y(t) contains random noise n(t), so that

y(t) = p(t) + n(t)

Digital CommunicatinsDigital Communicatins

At a given time t1, the possible receiver inputs are A + n(t1) (signal

present)or - A + n(t1) (signal absent)

The detection threshold is 0, i.e. if the sample value > 0, the digit is detected as “1”; if the sample value < 0, the digit detected as “0”.

There is a probability that the noise amplitude > signal amplitude, the error occurs when A + n(t1) is negative or – A + n(t1) becomes positive.


Distribution of signal + noise


Normally, the noise waveform has a Gaussian probability density function (pdf)

where is the rms value of the noise amplitude.

The probability of a one being interpreted as a zero (error):

The probability of a zero being interpreted as a one (another error):

2 2/ 21( )

2yp y e

20

2

1 ( )(0 /1) exp

22

y AP dy

2

20

1 ( )(1/ 0) exp

22

y AP dy


The total error probability is given byPe = P(1) P(01) + P(0) P(10)

where P(1) is the probability of a one being sent, and P(0) is that for a zero. Usually ones and zeros are equally likely, so

P(0) = P(1) = ½.

Substituting x = (y + A)/, and x = (y – A)/ respectively, we get

where T(A/) is known as the Gaussian tail function (sometimes called the complementary error function).

21(0 /1) (1/ 0) exp

22

( / )

A

xP P dx

T A


Thus,

Its values are given in tabular form in most mathematical handbooks. Alternatively, it is given in graph form as a function of the voltage SNR.

A standard norm for Pe is 10-9, that is, if the transmissionrate is 1 Gbit/s, there will be, on average, one error every second.

( / )eP T A


In binary PCM, each sample of the signal is represented by a codeword of, say, k bits. These bits are transmitted.

Receiver is used to recognize each codeword in order to reconstruct the samples, but errors may occur in the transmission as a result of noise.

Ways to improve the reliability: to increase the signal-to-noise ratio. to use appropriate coding.


Coding for digital transmission

Several techniques of coding for digital transmissionwill be discussed • source coding • error control coding• line coding

Source CodingConcerning with how should source messages be represented

using the minimum number of bits.

Example: Encoding English text

Seven bit ASCII (American Standard Code for Information Interchange) code

Character Binary code Character Binary codeSpace 0100000 A 10000019 0111001 a 1100001



Error control codingIn error control coding, the extra digits are introduced, in order to detect the presence of errors in the received pattern.

Parity checks

Sum check


Row and column parity check


Line CodingThe digital data need to be coded into electrical pulses for the purpose of transmission over the channel. This process is called line coding or transmission coding.

There are two major categories of line codes: • return-to-zero (RZ), and • nonreturn-to-zero (NRZ).

With RZ coding, the waveform returns to a zero-volt level for a portion (usually half) of the bit interval.


Line coding is used to match the digital signal to the physical characteristics of the channel (e.g. zero dc content) and facilitate synchronization at the receiver.

• a communication channel should not transmit dc components, as it is the waste of signal power.

• In order to maintain synchronization, there should be enough timing information built into the code so that the clock signal can be easily exacted. A long series of binary 1s and 0s should not cause a problem in time recovery.


Unipolar code (on-off code)A “1” is transmitted by a pulse and a “0” is transmitted by nopulse.

Polar code“1” is transmitted by a positive pulse p(t) and “0” is transmittedby a negative pulse – p(t).

Bipolar code“0” is encoded by no pulse and “1” is encoded by a pulse p(t) or – p(t), depending on whether the previous “1” is encoded by p(t) or – p(t). In short, pulse representing consecutive 1’s alternate in sign. This format has zero dc level (assuming an equal number of 1s and 0s).


The nonreturn-to-zero (NRZ) format is disadvantages becauseit has a dc component and does not allow for self-clocking.

The NRZ-bipolar (NRZ-B) format eliminates the dc component (assuming that an equal number of 1s and 0s occur) but it is also not self-clocking.

The polar return-to-zero (RZ) format includes clock information, has zero dc level (again, for equal numbers of 1s and 0s), but has three voltage levels (for bipolar).


Manchester codeA “1” is represented by a “1” level during the first half-bit

interval, then shift to “0” level for the latter half-bit interval;

a “0” is indicated by the reverse representation.

Manchester code uses a level transition of one direction or theother in the middle of every bit interval. This provides the receiver with clocking information and produces a zero dc level even if the 1s and 0s are not equal.


The output of a PCM system is a string of 1’s and 0’s.

If they are transmitted over the transmission medium

such as the copper wire, they can be directly

transmitted as two voltage levels + V and – V, if a

polar code is adopted.

This is baseband digital transmission, the signals are

transmitted at their baseband frequency.


If digital signals are to be transmitted over a

bandpass channel by use of high frequency carrier

wave such as signal transmission through space using

antenna, some form of modulation has to be used.

This is referred to bandpass digital transmission.


Digital carrier modulation

There are three basic forms of digital modulation,

corresponding to AM, FM and PM are known as

amplitude-shift keying (ASK), frequency-shift

keying (FSK), and phase-shift keying (PSK), in which

the amplitude, frequency or phase of the sinusoidal

carrier is switched between either of two values

corresponding to a 1 or a 0.


Amplitude Shift Keying (ASK)

In ASK, the modulated signal can be expressed as

Note that the modulated signal is still an on-off signal.

Thus, ASK is also known as on-off keying (OOK).

cos for a 1 bit( ) 0 for a 0 bit{ c

c

A tx t


Frequency Shift Keying (FSK)

In FSK, the modulated signal can be expressed as

1

2

1 2

cos for a 1 bit( )

cos for a 0 bit

where

,

{c

c c

A tx t

A t


Phase Shift Keying (PSK)

In PSK, the modulated signal can be expressed as

cos for a 1 bit( )

cos( ) for a 0 bit

cos for a 1 bit( )

cos for a 0 bit

{

{

cc

c

cc

c

A tx t

A t

or

A tx t

A t


Digital carrier demodulation

The detection of digital carrier modulation systems is virtually identical to that of analog modulation systems, i.e. the coherent demodulation method is used.

OOK signals can be demodulated also by using an envelope detector.

Questions (Digital Questions (Digital Communications)Communications)

Questions1. What is intersymbol interference? How is it generated? 2. What is eye diagram? What is its application?3. What pulse shape has zero intersymbol interference?4. What is the condition for zero intersymbol interference?5. How are the two levels, “1” and “0”, represented in unipolar bi

nary signal and in polar binary signal respectively?6. What is the main advantage of source coding?7. How bit streams are electrically represented in communication

systems?8. What are suitable methods to transmit PCM signals through sp

ace using antenna?

Exercise Problems (Digital Exercise Problems (Digital Communications)Communications)

1. A digital transmission system employs a signal element waveform at the input to the decision circuit given by

p(t) = (1 – cos2t/T) / 2(t/T)2 Show that this provides zero ISI for a binary signalling rate of 1/T

bits/s.

Exercise Problems (Digital Exercise Problems (Digital Communications)Communications)

2. Consider a source which produces message based on three symbols: A, B, C. The symbols are found to occur, on average, with relative frequencies A = 0.5, B = 0.25, C = 0.25. It is required to encode messages in a binary format; two coding schemes are proposed:

Scheme1: A = 01, B = 10, C = 11Scheme2: A = 1, B = 01, C = 00Show that, on average, a smaller number of bits per message are

required for scheme 2 than for scheme 1.

.

10

102sin2)(

6

6

t

ttha

3. (a) The overall transfer function of a transmission medium is given as Ha(w). The inverse Fourier transform of this function is

If the sequence of pulses, shown below, is to be transmitted through themedium given above, discuss whether there will be any possibility oferror due to intersymbol interference at the receiver. Explain. (Hint:there is an inverse time-bandwidth relationship, i.e. B ~ 1 /τ, whereτispulse duration)

(b) If we now send the same sequence of pulses through another transmission medium which has the transfer function Hb(w) shown below,

will there be intersymbol interference? Explain.

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