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Digital Communications

Baseband demodulation (Detection) :

ISIE-mail: [email protected]

Sanaa University

Faulty of Engineering

Department of Electrical Engineering

Communications Sections

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Digital Communications 2

Today we are going to talk about:

Another source of error:

Inter-symbol interference (ISI)

Nyquist theorem

The techniques to reduce ISI

Pulse shaping

Equalization

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3

Generalized One Dimensional Signals

One Dimensional Signal Constellation

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4

Binary Baseband Orthogonal Signals

Binary Antipodal Signals

Binary orthogonal Signals

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5

Constellation Diagram

Is a method of representing the symbol states of modulatedbandpass signals in terms of their amplitude and phase

In other words, it is a geometric representation of signals There are three types of binary signals:

Antipodal

Two signals are said to be antipodal if one signal is thenegative of the other

The signal have equal energy with signal point on the realline

ON-OFF Are one dimensional signals either ON or OFF with

signaling points falling

on the real line

)()( 01 tsts

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6

With OOK, there are just 2 symbol states to map onto theconstellation space

a(t) = 0 (no carrier amplitude, giving a point at the origin)

a(t) =A cos wct (giving a point on the positive horizontalaxis at a distanceA from the origin)

Orthogonal

Requires a two dimensional geometric representation sincethere are two linearly independent functions s1(t) and s0(t)

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Error Probability of Binary Signals

Recall:

Where we have replaced a2by a0.

18.2 0

01 Bequationaa

QPB

3.2.4 Error performance

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8

To minimize PB, we need to maximize:

or

We have

Therefore,

02

2

01 )(

aa

0

01

aa

2

1 0

2

0 0 0

( ) 2

/ 2

d da a E E

N N

2

1 0 1 0

2

0 0 0 0

( ) 21 1

2 2 2 2

d da a a a E E

N N

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)1(22

)()(2)()(

)()(

0 01

2

0 0

2

0 1

2

0 01

bbbb

TTT

T

d

EEEE

tstsdttsdtts

dttstsE

)63.3(2 0

N

EQP d

B

The probability of bit error is given by:

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10

The probability of bit error for antipodal signals:

The probability of bit error for orthogonal signals:

The probability of bit error for unipolar signals:

0

2

N

EQP bB

02N

EQP bB

0N

E

QP b

B

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Bipolar signals require a factor of 2 increase in energy comparedto orthogonal signals

Since 10log102 = 3 dB, we say that bipolar signaling offers a 3 dB

better performance than orthogonal

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Comparing BER Performance

For the same received signal to noise ratio, antipodal provides lowerbit error rate than orthogonal

4,

2

,

0

10x8.7

10x2.9

10/

antipodalB

orthogonalB

b

P

P

dBNEFor

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13

3.1.4 Relation Between SNR S/N) and E

b

/N

0

In analog communication the figure of merit used is the average

signal power to average noise power ration or SNR.

In the previous few slides we have used the term Eb/N0in the bit

error calculations. How are the two related?

Eb can be written as STband N0is N/W. So we have:

Thus Eb/N0can be thought of as normalized SNR.

Makes more sense when we have multi-level signaling.

Reading: Page 117 and 118.

0 /

b b

b

E ST S W

N N W N R

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3.3 Inter-Symbol Interference (ISI)

ISI in the detection process due to thefiltering effects of the system

Overall equivalent system transfer function

creates echoes and hence time dispersion

causes ISI at sampling time

)()()()( fHfHfHfH rct

i

ki

ikkk snsz

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How is ISI created?

There are a variety of reasons(bandlimited channels, multipath,fading). The end result is something likethis

1 0 0 0

1 0 0 0

1 0 1 0

1 0 0 0ISI causes detection error

0s may look like 1s and

vice versa

transmitted

received

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Inter-symbol interference

Baseband system model

Equivalent model

Tx filter Channel

)(tn

)(tr Rx. filterDetector

kz

kTt

kx kx1x 2x

3xT

T)(

)(

fH

th

t

t

)(

)(

fH

th

r

r

)(

)(

fH

th

c

c

Equivalent system

)( tn

)(tz

Detector

kz

kTt

kx kx1x 2x

3xT

T)()(fHth

filtered noise

)()()()( fHfHfHfH rct

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Nyquist bandwidth constraint

Nyquist bandwidth constraint: The theoretical minimum required system bandwidth to

detect Rs[symbols/s] without ISI is Rs/2[Hz].

Equivalently, a system with bandwidth W=1/2T=Rs/2[Hz] can support a maximum transmission rate of

2W=1/T=Rs[symbols/s] without ISI.

Bandwidth efficiency, R/W[bits/s/Hz] :An important measure in DCs representing data

throughput per hertz of bandwidth.

Showing how efficiently the bandwidth resources areused by signaling techniques.

Hz][symbol/s/222

1

W

RW

R

T

ss

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Ideal Nyquist pulse (filter)

T2

1

T2

1

T

)(fH

f t

)/sinc()( Ttth

1

0 T T2TT20

T

W

2

1

Ideal Nyquist filter Ideal Nyquist pulse

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Output in the frequency domain

In the frequency domain, input is a sinc.

RbRb

transmit

receive

Rb/2

Clearly, the received pulse

is severely bandlimited

compared to its original

shape. How does it look like

in time?

Di t t d l i ti l

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Distorted pulse in time=pulsedispersion

As a result of bandlimiting, we witnesspulse dispersion, i.e. spreading

transmitted

received

Tb/2

Tb

time

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Ideal Nyquist pulse (filter)

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Nyquist pulses (filters)

Nyquist pulses (filters):

Pulses (filters) which results in no ISI at thesampling time.

Nyquist filter:

Its transfer function in frequency domain isobtained by convolving a rectangular function withany real even-symmetric frequency function

Nyquist pulse:

Its shape can be represented by a sinc(t/T)function multiply by another time function.

Example of Nyquist filters: Raised-Cosine filter

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Pulse shaping to reduce ISI

Goals and trade-off in pulse-shaping

Reduce ISI

Efficient bandwidth utilization

Robustness to timing error (small side

lobes)

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The raised cosine filter

Raised-Cosine Filter

A Nyquist pulse (No ISI at the sampling time)

Wf

WfWWWW

WWf

WWf

fH

||for0

||2for2||

4cos

2||for1

)( 00

02

0

Excess bandwidth:0WW

Roll-off factor0

0

W

WWr

10 r

20

0

00 ])(4[1

])(2cos[))2(sinc(2)(

tWW

tWWtWWth

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The Raised cosine filtercontd

2)1(Baseband sSB

sRrW

|)(||)(| fHfH RC

0r5.0r

1r

1r

5.0r

0r

)()( thth RC

T2

1

T4

3

T

1T4

3

T2

1

T

1

1

0.5

0

1

0.5

0 T T2 T3TT2T3

sRrW )1(Passband DSB

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Equalization

ISI due to filtering effect of the

communications channel (e.g. wirelesschannels) Channels behave like band-limited filters

Equalization is a technique to remove ISI caused bythe channel

)(

)()( fj

ccc

efHfH

Non-constant amplitude

Amplitude distortion

Non-linear phase

Phase distortion

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Equalizationcontd

Frequencydown-conversion

Receivingfilter

Equalizingfilter

Thresholdcomparison

For bandpass signals Compensation for

channel induced ISI

Baseband pulse

(possibly distored)Sample

(test statistic)Baseband pulse

Received waveform

Step 1waveform to sample transformation Step 2decision making

)(tr

)(Tzi

m

Demodulate & Sample Detect

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Equalizationcontd

Equalization using

MLSE (Maximum likelihood sequenceestimation)

Filtering

Transversal filtering Zero-forcing equalizer

Minimum mean square error (MSE) equalizer

Decision feedback

Using the past decisions to remove the ISI contributedby them

Adaptive equalizer

Pulse shaping and equalization to

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Pulse shaping and equalization toremove ISI

Square-Root Raised Cosine (SRRC) filter and Equalizer

)()()()()(RC fHfHfHfHfH erct

No ISI at the sampling time

)()()()(

)()()(

SRRCRC

RC

fHfHfHfH

fHfHfH

tr

rt

Taking care of ISI

caused by tr. filter

)(

1)(

fHfH

ce

Taking care of ISIcaused by channel

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