Principles of Communication

Baseband Digital System (3)

LC 3-5

Lecture 14, 2008-10-31

2

Contents

Digital signal representationBinary and multilevel signalsLine codes and their SpectraSpectral efficiency

3

Digital SignalingMathematical Representation of the waveform

Estimation of Bandwidth

Voltage (or current) waveform for digital signals

( ) ( )

span timemessage :dimensions ofnumber :

functions orthogonal :data digital :

messageor wordPCM :

0

0

01

TN

(t)ww(t)

Tttwtw

k

k

N

kkk

ϕ

ϕ <<=∑=

4

Example: Message ‘X’ from a digital source - code word “0001101”

{ } { }7

0001101,,,,,,1 0 1 1 0 0 0 :

7654321

7654321

==

=======

Nwwwwwww

wwwwwwwwk

Baud (symbol rate)

sec in used s)(dimension symbols ofnumber : csymbols/se

0

0

TN N/ TD =

Bit rate (information rate)

sec in sent bits ofnumber : bits/sec

0

0

TN n/ TR =

5

Binary signal vs. multilevel signal

⎩⎨⎧

signal multilevel - values than twomoresignalbianry - values(two)binary

kw

How to detect the data at the receiver? (Matched filter detection)

( ) ( )

( )( ) functions orthogonal :

inputreceiver at the waveform:

21 ;1 *

0

0

ttw

, ..., N, kdtttwK

w

k

k

T

kk

ϕ

ϕ == ∫

6

Orthogonal Vector Space

( )

{ }1)()()(

if functions lorthonorma called be wouldand functions orthogonal : vectorldimensiona :

vectorldimensiona-:,...,,,

or

00

0

2

0

321

1

===

=

=

∫∫

∑

∗

=

dttdtttK

NN

wwww

w

T

j

T

jjj

j

N

N

jjj

ϕϕϕ

ϕ

ϕ

ww

w

7

ExampleThis 3-bit (binary) signal could be directly represented by

( )

)1,0,1(),,(otherwise,0

)1(,5)(

functions orthogonal :)(

)(21

321

3

1

3

1

==⎩⎨⎧ <<−

=

=⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −−= ∑∑

==

dddt

jTtTjtp

tp

tpdTjtpdts

j

j

jjj

jj

dThe orthonormal series representation

( )

1)()(

functions lorthonorma :)(

)(

0

0

3

1

=

=

∫

∑

∗

=

=

dttt

t

tsts

T

jj

j

M

jjj

ϕϕ

ϕ

ϕ

8

Example (con’t)

( ) ( )

( )( ) ( ) ( )TTTT

TdKdttp

ds

tpdtsts

t

jTtTjTT

tp

dttp

tp

dttptp

tpK

tpt

KA

AdtttAdttptpK

tAtp

jjjj

jjj

jjj

jjj

j

T

j

j

T

jj

j

j

jj

j

T

jj

T

jjj

jj

5,0,55)1,0,1(5

5)()(

)(

otherwise,0

)1(,1

5)(

)(

)(

)()(

)()()(

)()()()(

)()(Let

3

1

3

1

0

2

0

2

0

2

0

00

00

===

===

==

⎪⎩

⎪⎨⎧ <<−=====

=

===

=

∑∑

∫∫

∫∫

==

∗

∗∗

ds

ϕ

ϕ

ϕ

ϕϕ

ϕ

9

Bandwidth Estimation

The bandwidth of the waveform w(t) is

achieved. be willboundlower the type,sinc theare (t) theIf21

2

k

0

ϕ

DTNB =≥

10

Example

( ) ( )

0111001001001110 wordcode the toingcorrespond message particularA

signalingbinary for symbols, ofnumber :

0

ms8span timeThebits8log bits ofnumber The

256 messages ofnumber The

87654321

01

0

2

========

=

<<=

===

=

∑=

, w, w, w, w, w, w, ww

nNN

Tttwtw

TMn

M

N

kkkϕ

11

CASE 1. RECTANGULAR PULSE ORTHOGONAL FUNCTIONS

12ss

kT

)T(k-s

T

kk

k

s

kT

)T(k-

T

kkk

snull

bs

b

b

s

sk

kTTk

wdtw(t)T

dt(t)w(t)K

w

Tdtdt(t)(t)K

DT

B

RTT

nTN

TD

Tn

TR

nTT

TTktt

s

s

s

s

<<−

===

=×==

===

======

===

===

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−Π=

∫∫

∫∫∗

∗

ττ

τϕ

ϕϕ

ϕ

)1( time,sampling :

)(11

)11(

kHz11

kbaud111

kbits/s11

ms188

)()(

10

10

00

0

0

21

0

0

13

CASE 2. SINC PULSE ORTHOGONAL FUNCTIONS

14

12

0

0

0 0

12

1 1

12

( ( ) )( )

8 1 ms8

1 1 kbits/s

1 1 1 kbaud

500 Hz (Fig. 2.6)2

( ( ) )( ) ( )

[( ) ] ( ): sampling time,

sk

s

b

b

s b

abs

N Ns

k k kk k s

k s

t k Tt SaT

TTn

nRT T

N nD RT T T T

DB

t k Tw t w t w SaT

w w k T w

πϕ

πϕ

ττ τ

= =

⎛ ⎞− −= ⎜ ⎟

⎝ ⎠

= = =

= = =

= = = = = =

= =

⎛ ⎞− −= = ⎜ ⎟

⎝ ⎠= − =

=

∑ ∑

12( ) sk T−

15

Multilevel SignalingExample

( ) ( )

131301001110 wordcode the toingcorrespond message particularA

symboleach by carried bits :

44log

8log

signaling, multilevel 4For

symbols ofnumber :

0

ms8span timeThebits8log bits ofnumber The

256messagesofnumber The

4321

22

01

0

2

+=+=−=−=

=====

<<=

===

=

∑=

, w, w, ww

lL

nlnNL

N

Tttwtw

TMn

M

N

kkkϕ

16

Binary to Multilevel Conversion

Encoding SchemeBinary Input (l = 2 bits) Output Level (V)

11 +310 +100 -101 -3

17

CASE 1. RECTANGULAR PULSE ORTHOGONAL FUNCTIONS

18

0

0

12

0

0

0 0

0 1

0 1

( )( )

8 11 ms, 1 kbits/s8

1 /4, 500 baud

1 500 Hz

(1 1)

1 1 (

s

s

s

s

sk

s

bb

s

nulls

T kT

k k k s(k - )T

T kT

k k (k - )Tk s

t k TtT

T nT Rn T Tn N n l RN Dl T T T l

B DT

K (t) (t)dt dt T

w w(t) (t)dt w(t)dt wK T

ϕ

ϕ ϕ

ϕ

∗

∗

⎛ ⎞− −= Π⎜ ⎟

⎝ ⎠

= = = = = =

= = = = = = =

= = =

= = × =

= = =

∫ ∫

∫ ∫ )

: sampling time, ( 1) s sk T kT

τ

τ τ− < <

19

CASE 2. SINC PULSE ORTHOGONAL FUNCTIONS

20

12

0

0

0 0

12

1 1

12

12

( )( )

8 11 ms, 1 kbits/s8

1 /4, 500 baud

250 Hz2

( ( ) )( ) ( )

[( ) ] ( ): sampling time, (

sk

s

bb

s

abs

N Ns

k k kk k s

k s

t k TtT

T nT Rn T Tn N n l RN Dl T T T l

DB

t k Tw t w t w SaT

w w k T wk

ϕ

πϕ

ττ τ

= =

⎛ ⎞− −= Π⎜ ⎟

⎝ ⎠

= = = = = =

= = = = = = =

= =

⎛ ⎞− −= = ⎜ ⎟

⎝ ⎠= − =

= −

∑ ∑

) sT

21

Line CodesBinary 1’s and 0’s may be represented in various serial-bit signaling formats called line codes.The following are some of the desirable properties of a line code:

Self synchronizationLow probability of bit errorA spectrum that is suitable for the channelTransmission bandwidthError detection capabilityTransparency

22

Binary Line Codes

23

A digital signal (or line code) can be represented by

intervalbit : interval, symbol :shape pulse symbol :)(

)()(

b

bss

nsn

TlTTT

tf

nTtfats

=

−= ∑∞

−∞=

The general expression for the PSD of a digital signal is

product th thehaving ofy probabilit :

)()(

data theofation autocorrel :)()]([)(

)()(

)(

1

22

knni

I

iiiknn

k

kfTj

ss

aaiP

PaakR

kRtfFfF

ekRTfF

fP s

+

=+

∞

−∞=

∑

∑

=

=

= π

24

Unipolar NRZ

⎪⎪⎩

⎪⎪⎨

⎧

≠=⋅×+⋅×+⋅×+⋅×=

==⋅×+⋅×===

====

∑

∑∑

=+

=

=+

0,44

1)00(41)0(

41)0(

41)()(

0,22

1)00(21)()(

)()(

21)0()(

24

1

22

1

2

1 kAAAAAPaa

kAAAPaPaakR

aPAaP

iiiknn

iiinI

iiiknn

nn

25

Unipolar NRZ

)](11)[(4

])(11)[(4

])(11[4

)()(

)()(

)()()(

pulsesr rectangulaFor

])(11[4

)1(442

)(

)()(

)(

22

22

222

2

22

2

0

222

2

2200

20

20

0

00

fT

fTSaTATnf

TfTSaTA

Tnf

TA

TfTSaT

ekRTfF

fP

fTSaTfFTttf

Tnf

TAeAeAAekR

eTTnfefnTf

efecnTt

ss

s

n sss

s

n sss

ss

k

kfTj

s

sss

n ssk

kfTj

kk

kfTj

k

kfTj

k

kfTjs

n sk

kffj

n

k

tkfj

k

tkfjn

n

s

sss

s

δπδπ

δπ

π

δ

δδ

δ

π

πππ

ππ

ππ

+=−+=

−+⋅==

=↔⎟⎟⎠

⎞⎜⎜⎝

⎛Π=

−+=+=+=

=−⇒=−

⇒==−

∑

∑∑

∑∑∑∑

∑∑∑∑

∑∑∑

∞

−∞=

∞

−∞=

∞

−∞=

∞

−∞=

∞

−∞=

∞

≠−∞=

∞

−∞=

∞

−∞=

∞

−∞=

∞

−∞=

∞

−∞=

∞

−∞=

∞

−∞=

∞

−∞=

26

Unipolar NRZ

RDT

B

ffTSaT

ffTSaTfT

fTSaTfP

AAP

ATT

TA

TSadxxSa

TTA

dffT

fTSadffTSaTA

dffT

fTSaTAdffPP

snull

bb

ss

ss

s

ss

s

ss

s

sss

s

ss

s

===

+=

+=+=

=⇒=⇒=

=⎟⎟⎠

⎞⎜⎜⎝

⎛+⋅=

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

+==

∫

∫∫

∫∫

∞

∞−

∞

∞−

∞

∞−

∞

∞−

∞

∞−

1

)(21)(5.0

)(21)(

2)](11)[(

42)(

221

211

4

1)0()(14

)(1)()(4

)](11)[(4

)(

2

22

2

22

222

222

22

δπ

δπδπ

ππ

π

δππ

δπ

27

28

Sa Waveform NRZ

( )

)](11[14

)](11[14

])(11[14

])(11[4

1)(

)()(

1)()(

pulses sincFor 22

12

2

22

2

2

22

fTT

fTA

fTT

fTATnf

TTfTA

Tnf

TA

TTfT

ekRTfF

fP

TfTfF

TtSatf

Wf

WWtSa

ss

s

ss

s

n sss

s

n sss

ss

k

kfTj

s

ss

s

s

δ

δδ

δ

π

π

π

+⎟⎟⎠

⎞⎜⎜⎝

⎛Π=

+⎟⎟⎠

⎞⎜⎜⎝

⎛Π=−+⎟⎟

⎠

⎞⎜⎜⎝

⎛Π=

−+⋅⎟⎟⎠

⎞⎜⎜⎝

⎛Π

==

⎟⎟⎠

⎞⎜⎜⎝

⎛Π=↔⎟⎟

⎠

⎞⎜⎜⎝

⎛=

⎟⎠⎞

⎜⎝⎛Π↔

∑

∑∑∞

−∞=

∞

−∞=

∞

−∞=

29

2

2

2 1/ 2

1/ 2

2 2

2

1( ) ( )[1 ( )]4 1

1( ) ( ) ( )4 1 1

14

1 14 2

1 2 22 1 1( ) ( )[1 ( )] ( ) ( ) 04 1 2 1 2

s

s

s

s s

s

s s s

TsT

s

s

s s

s s

s s s

A T fP P f df f dfT T

A T f fdf f dfT T T

A T dfT

A T AT T

P A AT Tf fP f f f

T T T

δ

δ

δ δ

∞ ∞

−∞ −∞

∞ ∞

−∞ −∞

−

= = Π +

⎛ ⎞= Π + Π⎜ ⎟

⎝ ⎠⎛ ⎞

= +⎜ ⎟⎝ ⎠⎛ ⎞

= + =⎜ ⎟⎝ ⎠

= ⇒ = ⇒ =

= Π + = Π + =

∫ ∫

∫ ∫

∫

00

1.5 ( ) ( )1 2

1 1 2 (dimensionality theorem)2 2 2 2 2

bb

abs abss b

fT fT

N D RB N B TT T T

δΠ +

= = = = = ⇒ =

30

31

Polar NRZ

11

)()( 22

=⇒=

=

AP

fTSaTAfP bbNRZpolar π

32

Unipolar RZ

21

])(11)[2/(16

)( 22

21

=⇒=

−+=

=

∑∞

−∞=

APTnf

TfTSaTAfP

d

n bbb

bRZunipolar δπ

33

Bipolar RZ

PSD for bipolar (also known as pseudoternary or AMI) RZ signaling

21

)(sin)2/(8

)( 222

21

=⇒=

=

=

AP

fTfTSaTAfP

d

bbb

RZbipolar ππ

34

Manchester NRZ

11

)2/(sin)2/()(

24

24)(

222

=⇒=

=

⎟⎟⎠

⎞⎜⎜⎝

⎛ −Π+⎟⎟

⎠

⎞⎜⎜⎝

⎛ +Π=

AP

fTfTSaTAfP

TTt

TTttf

bbbNRZManchester

b

b

b

b

ππ

35

Multilevel NRZ

36

)emlity theordimensiona(222

1,1

21)( :pulses sincFor

1

)3(63)(2121)(

)()(

)(

)()()( pulsesr rectangulaFor

0,0

)]75311357)(7()75311357(5)75311357(7[641

)(641)()(

21])7()5()3()1(1357[81)(

81)()0(

8

222

22

8

11

222222228

1

2

1

2

lRD

TB

TfTfP

lRD

TB

fTSaTfTSaTT

fTSaTekR

TfF

fP

fTSaTfFTttf

k

aaPaakR

aPaR

sabs

ssNRZlevel

snull

bbsss

ss

k

kfTj

sNRZlevel

sss

I

iiknn

I

iiiknn

iin

I

iiin

s

===⎟⎟⎠

⎞⎜⎜⎝

⎛Π=

===

====

=↔⎟⎟⎠

⎞⎜⎜⎝

⎛Π=

≠=

−−−−+++−++−−−−++++−−−−+++=

==

=−+−+−+−++++===

−

∞

−∞=−

=+

=+

==

∑

∑∑

∑∑

πππ

π

π

L

37

In general, for the case of a multilevel polar NRZ signal with rectangular pulse shape

( )

lRB

fTlKSafP

null

bNRZmultilevel

=

= π2)(

Multilevel signaling is used to reduce the bandwidth of a digital signal compared with the bandwidth required for binary signaling.

38

Spectral EfficiencyThe spectral efficiency of a digital signal is given by the number of bits per second of data that can be supported by each hertz of bandwidth. That is,

(bits/s)/HzRB

η =

39

HomeworkLC 3-20, 3-23, 3-26, 3-32, 3-36