Digital Modulation Schemes Proakis Slides

Chapter 3

Digital Modulation Schemes

Wireless Information Transmission System Lab.Wireless Information Transmission System Lab.Institute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat--sensen UniversityUniversity

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ContentsContents

◊ 3.1 Representation of Digitally Modulated Signals

◊ 3.2 Memoryless Modulation Methods

◊ 3.3 Signaling Schemes with Memoryg g y

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Chapter 3.1: Representation of Digitally Modulated SignalsWireless Information Transmission System Lab.Wireless Information Transmission System Lab.Institute of Communications EngineeringInstitute of Communications Engineering

Digitally Modulated Signals

g gg gNational Sun National Sun YatYat--sensen UniversityUniversity

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3.1 Digitally Modulated Signals3.1 Digitally Modulated Signalsg y gg y g

◊ Digital Modulation (or digital signaling):◊ The process of mapping a digital sequence to signals for

transmission over a communication channel.

◊ In the process of modulation, usually the transmitted signals are bandpass signals suitable for transmission in the bandwidth providedbandpass signals suitable for transmission in the bandwidth providedby the communication channel.

◊ The transmitted signal can be memoryless or with memory.

4


◊ Memoryless Modulation◊ Each of length k sequence is mapped into one of the sm(t),

1 ≤ m≤ 2k, signals regardless of the previous transmitted signals

◊ Equivalent to map a k-bit sequence to one of M=2k signals.

5


◊ Modulation with memory◊ Mapping from the set of current k bits and the past (L−1)k bits to the

set of M=2k possible signals◊ The modulation scheme can be viewed as a mapping from the

current state and the current input of the modulator to the set of output signals resulting in a new state of the modulator.output signals resulting in a new state of the modulator.

◊ This defines a finite-state machine with 2(L−1)k states : at t = l−1, the modulation is in state ( ){ }1

1 1, 2, , 2 L klS −∈ …,

at t =l, the input sequence is , then the modulation moves to the next state

{ }1 , , ,l −

( )1,l s l lS f S I−={ }1,2, , 2k

lI ∈ …

◊ L is the constraint length of modulation.◊ Can be effectively represented by a Markov Chain.

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◊ Digital modulation is linear if

otherwise, it is nonlinear.1 1 2 2 1 2 1 2( ), ( ) ( ) ( )b m t b m t b b m t m t→ → ⇒ + → +

◊ Assume that the signal waveforms sm(t), 1 ≤ m ≤ M=2k

are transmitted at every seconds◊ Signal interval : sT

sT

◊ Signaling rate or Symbol rate :◊ Bit interval :

1/s sR T=

2/ / logb s sT T k T M= =◊ Bit rate : 21/ / logb s s sR T k T kR R M= = = =

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◊ Let the energy of sm(t) is Em, the average signal energy is

1

M

avg m mm

E p E=

= ∑◊ pm is the probability of the m-th signal◊ If equiprobable messages, i.e. pm = 1/M, then

1

1 M

avg mE EM

= ∑◊ If Em = E, then Eavg = E 1mM =

◊ The average energy per bit when pm = 1/M is

lavg avg

bavg

E EE

k M= =

◊ The average power is 2logbavg k M

avg

bavgbavg

EP RE

T= =

8

bT

Ch 3 2 M l M d l iChapter 3.2 : Memoryless Modulation Methods


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3.2 Memoryless Modulation Methodsy

◊ Pulse Amplitude Modulation (PAM)◊ Phase Modulation◊ Quadrature Amplitude Modulation

◊ π/4-DQPSK◊ Dual-Carrier Modulation (DCM)

◊ Multidimensional Signaling◊ Orthogonal Signaling◊ Frequency-Shift Keying (FSK)◊ Hadamard Signals◊ Biorthogonal Signaling◊ Simplex Signaling

Si l W f f Bi C d

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◊ Signal Waveforms from Binary Codes

3.23.2--1 Pulse Amplitude Modulation (PAM)1 Pulse Amplitude Modulation (PAM)p ( )p ( )

◊ In PAM, the (baseband) signal waveforms are

◊ p(t) : pulse of duration T( ) ( ), 1m ms t A p t m M= ≤ ≤

◊ {Am}: amplitudes corresponding to M=2k possible k-bit symbols ◊ The amplitudes of sm(t) are ( )1, 3, 5, , 1 .mA M= ± ± ± ± −

◊ The energy in signal sm(t) is2 1 , 1, 2,mA m M m M= − − = …

◊ Average bit energy is

2 2 2( )m m m pE A p t dt A E∞

−∞= =∫ ( ) ( )2

22 2 21

1 3 5 16

M MM

−+ + + + − =

22 2 2 2 2

1

2 ( 1)(1 3 5 ( 1) )

3

Mp p p

avg mm

E E M EE A M

M M=

−= = + + + + − =∑

11

2p 2/ ( 1) / 3logbavg avgE E k M E M= = −


◊ In bandpass PAM signals with lowpass equivalents of the Amg(t),

◊ fc : carrier frequency

2( ) Re[ ( ) ] ( ) cos(2 )cj f tm m m cs t A g t e A g t f tπ π= =

2A

Am and g(t) are real.

◊ Energy of sm(t) :2

2m

m gAE E= From 2.1-21

◊ Compared with generic form of PAM signaling( ) ( ) cos(2 )cp t g t f tπ= / 2p gE E=

◊ Average Signal Energy :2 2

p( 1) ( 1)3 6

gavg

M E M EE

− −= =

◊ Average Bit Energy :3 6avg

2( 1)avg gE M EE

−= =

12

26 logbavgEk M

= =


◊ {sm(t), m=1,…,M} is one-dimensional with basis signal:

◊ Baseband PAM :

◊ Bandpass PAM :

p( ) ( )t p t Eφ =

( ) 2 / ( )cos(2 )g ct E g t f tφ π=◊ p◊ Using these basis signals, sm(t) is written as:

◊ Baseband PAM:

( ) ( ) ( )g cg fφ

( ) ( ) ( )s t A p t A E tφ◊ Baseband PAM:

◊ Bandpass PAM:Th di i l i f ( )

p( ) ( ) ( )m m ms t A p t A E tφ= =

g( ) ( ) cos(2 ) / 2 ( )m m C ms t A g t f t A E tπ φ= =

◊ The one-dimentional vector representation of sm(t) are

◊ Baseband PAM: p , 1, 3, , ( 1)m m mA E A M= = ± ± … ± −s

◊ Bandpass PAM:◊ Bandpass PAM is also called amplitude-shift keying (ASK)

g / 2, 1, 3, , ( 1)m m mA E A M= = ± ± … ± −s

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p p f y g ( )


◊ The mapping from k information bits to M=2k amplitudes is not unique

◊ Gray coding: the adjacent signal amplitudes differ by one digit only one-bit error with erroneous selection of an adjacent amplitude

◊ Euclidean distance between (sm, sn)2

p|| || | |

| | / 2

mn m n m nd A A E

A A E

= − = −s s

◊ The minimum distance PAM is

g | | / 2m nA A E= −( )2m nA A− =

2min p g 2

12log2 21 bavgMd E E E

M= = =

−2

14

2

2

( 1)6 log

gbavg

M EE

M−

=


◊ The following bandpass signal is double-sideband (DSB)

◊ Requires twice the channel bandwidth

2( ) Re[ ( ) ] ( ) cos(2 )cj f tm m m cs t A g t e A g t f tπ π= =

◊ We may use single-sideband (SSB) PAM:

◊ is the Hilbert transform of

2ˆ( ) Re[ ( ( ) ( )) ]cj f tm ms t A g t jg t e π= ±

ˆ ( )g t ( )g t◊ Bandwidth is half of DSB

◊ When M = 2, s1(t) = -s2(t)◊ E1= E2 =1◊ Cross-correlation coefficient = -1

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◊ Binary antipodal signaling


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3.23.2--2 Phase Modulation2 Phase Modulation

◊ Phase modulation is also called phase-shift keying (PSK)22 ( 1)/( ) Re ( ) , 1, 2, ,

( ) cos[2 2 ( 1) / ]

cj f tj m Mm

c

s t g t e e m M

g t f t m M

ππ

π π

−⎡ ⎤= = …⎣ ⎦= + −

◊ g(t) : signal pulse shape( ) ( ) ( ) ( )

( ) ( ) ( ) cos cos 2 ( )sin sin 2

c

m C m C

g fg t f t g t f tθ π θ π= −

◊ : phase of the M=2k transmitted signals2 ( 1) /m m Mθ π= −

◊ Signal waveforms have equal energy/ 2avg m gE E E E= = ≡

◊ Average bit energy : g g

g gE EE E= = ≡ (3 2-26)

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22 2logbavg bE Ek M

= = ≡ (3.2 26)


◊ g(t)cos(2πfc t) and g(t)sin(2πfc t) are orthogonal◊ Two basis functions from DSB signal

1( ) 2 / ( ) cos(2 )t E g t f tφ π=1( ) 2 / ( ) cos(2 )g ct E g t f tφ π

2 ( ) 2 / ( )sin(2 )g ct E g t f tφ π= −

◊ These can be used for expansion of {sm(t)}

2 2( ) ( 1) ( ) i ( 1) ( )g gE Eπ π⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟

◊ The signal space has dimension N=2 and vector representations

1 2( ) cos ( 1) ( ) sin ( 1) ( )2 2

g gms t m t m t

M Mπ πϕ ϕ⎛ ⎞ ⎛ ⎞= − + −⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

2 2cos ( 1) , sin ( 1)2 2

g gm

E Em m

M Mπ π⎛ ⎞⎛ ⎞ ⎛ ⎞= − −⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠

s

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⎝ ⎠ ⎝ ⎠⎝ ⎠


◊ M=2, BPSK M=4, QPSK M=8, 8-PSK

◊ BPSK=Binary PAMy◊ The mapping is not unique and Gray coding is preferred◊ A variant of four-phase PSK (QPSK), called π/4-QPSK, is

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◊ A variant of four phase PSK (QPSK), called π/4 QPSK, is obtained by introducing an additional π/4 phase shift.


◊ The Euclidean distance between signal points

2g

2|| || 1 cos ( )mn m nd E m nMπ⎡ ⎤⎛ ⎞= − = − −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

s s

◊ The minimal distance (|m-n|=1) :

221 2 id E Eπ π⎡ ⎤ 2 1 cos 2sin θθ −=2

min 1 cos 2 sing gd E EM M

⎡ ⎤= − =⎢ ⎥⎣ ⎦

22 l iM Eπ⎛ ⎞⎜ ⎟

2

sin2

2sin 1 cos 2

θ

θ θ

=

→ = −

◊ For large M

222 log sin bM E

M⎛ ⎞= ×⎜ ⎟⎝ ⎠

sin( / ) ( / )M Mπ π≈ ( ) 3.2-262log

gb

EE

M≡g ( ) ( )

22

min 2

log2 bMd E

Mπ

≈

22 log M

20

2M

3.23.2--3 3 QuadratureQuadrature Amplitude ModulationAmplitude ModulationQQ pp

◊ The bandwidth efficiency of PAM/SSB can be obtained by impressing two k-bit symbols on two quadrature carriers⎯ cos(2πfc t) and sin(2πfc t).◊ Quadrature PAM or QAM

◊ In QAM, signal waveforms are2( ) Re[( ) ( ) ]cj f t

m mi mqs t A jA g t e π= +

◊ Ami and Amq : information-bearing signal amplitudesmi A ( )cos(2 ) ( )sin(2 ), m 1,2, ,Mc mq cg t f t A g t f tπ π= − = …

◊ Alternatively, it may be expressed as( ) ( )2( ) Re cos(2 )m cj j f t

m m m c ms t r e g t e r g t f tθ π π θ⎡ ⎤= = +⎣ ⎦2 2 ( )

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◊ and 2 2m mi mqr A A= + ( )1tanm mq miA Aθ −=


◊ QAM: combined amplitude (rm) and phase (θm) modulation◊ Similar to PSK, two basis functions:

1( ) 2 / ( ) cos(2 )g ct E g t f tφ π=

◊ Using this basis2 ( ) 2 / ( )sin(2 )g ct E g t f tφ π= −

g

◊ Vector representation of sm(t):1 2( ) / 2 ( ) / 2 ( ), 1, 2,...,m mi g mq gs t A E t A E t m Mφ φ= + =

p m( )

◊ Signal energy:( )1 2( , ) / 2, / 2m m m mi g mq gs s A E A E= =s

2 2 2( / 2)( )m m g mi mqE s E A A= = + (3.2-39)g gy◊ Euclidean distance between signal points

( )( )m m g mi mq

2 2 2gE⎡ ⎤

( )

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2 2 2( ) ( ) ( )2

gmn m n mi ni mq nqd A A A A⎡ ⎤= − = − + −⎣ ⎦s s

. . 2; 0;mi ni mq nqe g A A A A− = − =


◊ May combine M1-level PAM and M2-phase PSK to construct M=M1M2 combined PAM-PSK

◊ If M1=2n and M2=2m , the combined PAM-PSK simultaneouslytransmit m + n = log2 M1M2 binary digits

◊ (ex) M=8 (M1=2, M2=4) M=16 (M1=4, M2=4)

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◊ When the signal amplitudes (Ami , Amq) take value on (2m−1−M), qm=1,…,M, signal space is rectangular◊ Minimal distance is

min 2 gd E= E.g. 2; 0.mi ni mq nqA A A A− = − =

◊ If M=22k, (M=4,16,64,256,..) and ( ), 1, 3, , 1i qm mA A M= ± ± ± −…

2 21 ( )M MgEE A A+∑ ∑1 1

( )2

2 ( 1) 1

gavg m nm n

g

E A AME M M M E

= == +

− −= × =

∑ ∑

◊

2 3 3 gE

M×

13lbavg g

ME EM

−=

◊

23logbavg gM

2min

6 log1 bavgMd E

M=

2d E

24

min 1 bavgM − min 2 gd E=

2

1

M

nn

A=

∑

( ) ( )( )22

1 12 1 2 1

M M

n n

M M M

n M n M= =

= − − = − +∑ ∑( )

( ) ( )1

2

1

12

1 2 16

N

i

N

i

N Ni

N N Ni

=

=

⋅ +=

⋅ + ⋅ +=

∑

∑

( ) ( )( ) ( ) ( ) ( ) ( )

22

1 1 1

2

4 4 1 1

1 2 1 1

M M M

n n nn M n M

M M M M M= = =

= − + + +

⋅ + ⋅ + ⋅ +

∑ ∑ ∑1i=

( ) ( ) ( ) ( ) ( )( ) ( )

21 2 1 14 4 1 1

6 21 1 1

M M M M MM M M

M M M

+ + += ⋅ − ⋅ + + ⋅ +

= ⋅ ⋅ + ⋅ −( ) ( )3

( ) ( ) ( )2 2 2 1 1 13

M M M

m n mA A M A M M M⎛ ⎞+ = ⋅ + ⋅ ⋅ − ⋅ +⎜ ⎟⎝ ⎠

∑∑ ∑( ) ( ) ( )

( ) ( )1 1 1

2

1

3

1 1 13

m n m

M

mm

M A M M M M

= = =

=

⎝ ⎠

⎛ ⎞= ⋅ + ⋅ ⋅ ⋅ − ⋅ +⎜ ⎟⎝ ⎠

∑

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( )2 13

M M= ⋅ ⋅ −


◊ Bandpass PAM, PSK and QAM are of the general form2( ) Re[ ( ) ], 1, 2,...,cj f t

m ms t A g t e m Mπ= =

◊ PAM : Am is real, and ; N=1◊ PSK : Am is complex, and ; N=2( )2 1 /j m M

mA e π −=( )1, 3, , 1mA M= ± ± ± −…

◊ QAM: Am is complex, and ; N=2both amplitude and phase carry information

m mi mqA A jA= +

(bit) (symbol)

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Comparison of PAM, PSK, and QAMComparison of PAM, PSK, and QAMp Qp Q

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MemorylessMemoryless Modulation MethodsModulation Methods

◊ Phase-modulated signals (π/4-Shifted QPSK, π/4-DQPSK)

yy

◊ The carrier phase used for the transmission of successive symbols is alternately picked from one of the two QPSK constellations in the following figure and then the other.

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yy

◊ It follows that a π/4-shifted QPSK signal may reside in any one of eight possible phase states:

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yy

◊ Attractive features of the π/4-shifted QPSK scheme◊ The phase transitions from one symbol to the next are

restricted to ±π/4 and ±3π/4.◊ Envelope variations due to filtering are significantly

reduced.◊ π/4-shifted QPSK signals can be noncoherently detected,

h b id bl i lif i h i d ithereby considerably simplifying the receiver design.◊ Like QPSK signals, π/4-shifted QPSK can be differently

encoded in which case we should really speak of π/4encoded, in which case we should really speak of π/4-shifted DQPSK .

◊ π/4 DQPSK is adopted in IS 54/136

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◊ π/4-DQPSK is adopted in IS-54/136.


◊ Dual-Carrier Modulation (DCM)

yy

◊ Multi-band OFDM (IEEE 802.15.3 a Ultra Wideband)◊ The coded and interleaved binary serial input data, b[i] where

i = 0, 1, 2, …, shall be divided into groups of 200 bits and converted into 100 complex numbers using a technique called d l i d l tidual-carrier modulation.

◊ The conversion shall be performed as follows:1 Th 200 d d bi d i 50 f 4 bi1.The 200 coded bits are grouped into 50 groups of 4 bits.

Each group is represented as (b[g(k)], b[g(k)+1], b[g(k) + 50)] b[g(k) + 51]) where k [0 49] and50)], b[g(k) + 51]), where k [0, 49] and

( ) [ ][ ]

0, 24225 492 50

kkg k

kk⎧ ∈⎪= ⎨⎪⎩

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( ) [ ]25, 492 50 kk⎨ ∈+⎪⎩


◊ Dual-Carrier Modulation (DCM)

yy

2. Each group of 4 bits (b[g(k)], b[g(k)+1], b[g(k) + 50)], b[g(k) + 51]) shall be mapped onto a four-dimensional constellation, and converted into two complex numbers (d[k], d[k + 50]). p ( [ ], [ ])

3. The complex numbers shall be normalized using a normalization factor KMOD.

◊ The normalization factor KMOD = 10-1/2 is used for the dual-carrier modulationcarrier modulation.

◊ An approximate value of the normalization factor may be used, as long as the device conforms to the modulation accuracy requirements.

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Dual-Carrier Modulation (DCM)

yy

33


Dual-Carrier Modulation (DCM) Encoding Table

yy

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3.23.2--4 Multidimensional Signaling4 Multidimensional Signalingg gg g

◊ To construct signal waveforms corresponding to higher dimensions, we may use time or frequency domain or both

◊ Time domain: ◊ Divide a time interval into N subintervals with length T=T1/N,◊ Use binary PAM to transmit N-dimensional vector◊ If N is even, simultaneously transmits two N-dimensional vectors by

modulating the amplitude of quadrature carriers◊ Frequency Domain:◊ Frequency Domain:

◊ Divide a frequency band to have N sub-band with width Δf◊ Frequency separation Δf must be large to avoid interference◊ Frequency separation Δf must be large to avoid interference

◊ May use both time and frequency domains to jointly transmit N-dimensional vector

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dimensional vector

3.23.2--4 Multidimensional Signaling4 Multidimensional Signalingg gg g

◊ Subdivision of time and frequency axes into distinct slots.

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Orthogonal SignalingOrthogonal Signalingg g gg g g

◊ Define a set of orthogonal signals {sm(t)} with equal energy,

( ), ( ) 1 ,0, m n

E m ns t s t m n M

m n=⎧

< >= ≤ ≤⎨ ≠⎩◊ {sm(t)} are linearly independent (N = M), define orthonormal set

( ) ( ) / , 1j jt s t E j Nφ = ≤ ≤

◊ Vector representations of {sm(t)}( ) ( ) ,j j jφ

( ) = EE1

2

( ,0,0,...,0)

(0, ,0,...,0)

E

E

=

=

s

s2

=logbE

M

= 2d E

(0,0,...,0, )M E

=

=s

2

= 2

2log

mn

min

d E

d E

ME

37

2 = 2log bME

FrequencyFrequency--Shift Keying (FSK)Shift Keying (FSK)q yq y y g ( )y g ( )

◊ A special case of orthogonal signaling :2( )= Re[ ( ) ], 1 ,0

= 2 / cos(2 2 )

cj f tm mls t s t e m M t T

E T f t m ft

π

π π

≤ ≤ ≤ ≤

+ Δ

◊

= 2 / cos(2 2 )cE T f t m ftπ π+ Δ

2 (t) = 2 / , 1 , 0j m ft

mls E T e m M t Tπ Δ ≤ ≤ ≤ ≤◊ is to guarantee that each signal has an energy equal to E.◊ Messages are transmitted by signals that differ in frequency

2 /E T

◊ ASK, PSK, QAM are linear modulationsFSK is non-linear modulationsk h h li i l f

The sum of two QAM signalsis another QAM signal.

◊ To keep the orthogonality among signals, for m≠n*

0 0( ) ( ) 0 Re[ ( ) ( ) ] 0

T T

m n ml nls t s t dt s t s t dt= ⇒ =∫ ∫fΔ

38

◊ has to satisfy the above conditionfΔ ( ) ( ) ( ), ,

,(2.1 26) Re

l lx y x yx y

x t y t

E Eρ ρ− = =

FrequencyFrequency--Shift Keying (FSK)Shift Keying (FSK)q yq y y g ( )y g ( )

◊ 2 ( )

0

2( ), ( )T j m n ft

ml nlEs t s t e dt

Tπ − Δ= ∫0

( )2 sin( ( ) ) ( )

ml nl

j T m n f

TE T m n f e

T m n fππ

π− Δ− Δ

=− Δ

∫

◊

( )T m n fπ Δ

2 sin( ( ) )Re ( ), ( ) cos( ( ) )( )ml nl

E T m n fs t s t T m n ff

π π− Δ⎡ ⎤ = − Δ⎣ ⎦( ), ( ) ( ( ) )

( )2 sin(2 ( ) )

2 ( )

ml nl fT m n f

E T m n fT f

ππ

⎣ ⎦ − Δ− Δ

=Δ

sinsinc θθ =

◊ {sm(t)} is orthogonal ⇔ sinc(2Tπ(m−n)Δf )=0, for all m ≠ n

2 ( ) 2 sinc(2 ( ) )

T m n fE T m n f

π − Δ= − Δ

sincθθ

{ m( )} g ( ( ) f ) ,◊ If , k is a positive integer, sinc(2T π(m−n)Δf )=0◊ Minimum frequency separation to guarantee orthogonality

/ 2f k TΔ =

39

◊ Minimum frequency separation to guarantee orthogonality1/ 2f TΔ =

◊ 2 ( )

0

2( ), ( )T j m n ft

ml nlEs t s t e dt

Tπ − Δ= ∫

( )

( )

0

222

ml nl

Tj m n ft

T

E eT j f

π − Δ

= ⋅Δ

∫

( )( )

0

2

2

2 1j m n fT

T j m n f

E e π

π− Δ

− Δ

−= ⋅

( )( ) ( ) ( )( )

2

2j m n fT j m n fT j m n fT

T j m n f

e e eEπ π π

π− Δ − Δ − − Δ

− Δ

−( )( )

( )

22 sin( ( ) ) j T m n f

T j m n fE T m n f π

π

π − Δ

= ⋅− Δ

− Δ sinj je eθ θ

θ−−( )( ( ) )

( )j T m n ff e

T m n fπ

πΔ=

− Δsin

2 jθ =

40

HadamardHadamard SignalsSignalsgg

◊ The orthogonal signals constructed from Hadamard matrices◊ Hadamard matrices are 2n× 2n and

[1] n n⎛ ⎞⎜ ⎟

H HH H0 1[1]; n n

nn n

+= = ⎜ ⎟−⎝ ⎠H H

H H1 1⎡ ⎤

⎢ ⎥H ⎡ ⎤1 1 1= ⎢ ⎥−⎣ ⎦

H

1 1 1 1⎡ ⎤

1 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 1

⎡ ⎤⎢ ⎥− − − −⎢ ⎥⎢ ⎥

2

1 1 1 11 1 1 11 1 1 1

⎡ ⎤⎢ ⎥− −⎢ ⎥=⎢ ⎥

H 3

1 1 1 1 1 1 1 11 1 1 1 1 1 1 1

=1 1 1 1 1 1 1 1

⎢ ⎥− − − −⎢ ⎥− − − −⎢ ⎥⎢ ⎥H1 1 1 1

1 1 1 1⎢ ⎥− −⎢ ⎥− −⎣ ⎦

1 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 1

⎢ ⎥− − − −⎢ ⎥

− − − −⎢ ⎥⎢ ⎥

41

1 1 1 1 1 1 1 11 1 1 1 1 1 1 1

⎢ ⎥− − − −⎢ ⎥

− − − −⎢ ⎥⎣ ⎦

HadamardHadamard SignalsSignalsgg

◊ Hadamard matrices:◊ Symmetric◊ All rows are orthogonal

◊ Using Hadamard matrices to generate orthogonal signals◊ (ex) using H2: 1

2

[ ]

[ ]

E E E E

E E E E

=

= − −

s

s

3

4

[ ]

[ ]

E E E E

E E E E

= − −

= − −

s

s◊ The set of signals may be used to modulate 4-D orthonormal basis

4 [ ]E E E Es

( ) ( )4

, 1 4j js t s t mφ= ≤ ≤∑

42◊ Energy in each signal is 4E and Eb=2E

( ) ( )1

, 1 4m mj jj

s t s t mφ=

≤ ≤∑

BiorthogonalBiorthogonal SignalingSignalinggg g gg g

◊ A set of M biorthogonal signals is constructed from M/2 orthogonal signals

◊ Simply including the negatives of orthogonal signals

◊ Only require M/2 orthogonal signalsy q g g◊ Correlation between signals: ρ = 0 or -1◊ Distance between signals: 2 or 2d E E=g◊ Minimal distance:

2 or 2d E E

min 2d E=

43

Simplex SignalingSimplex Signalingp g gp g g

◊ Let {sm(t)} be a set of M orthogonal waveforms with energy Eand vector representation {sm}

◊ The mean of {sm} is

◊ Construct M simplex signals :1

1 M

mmM =

= ∑s s

◊ Construct M simplex signals :

◊ Translate the origin of M orthogonal signals to the point, 1, 2,...,m m m M′ = − =s s s

s◊ Translate the origin of M orthogonal signals to the point◊ Energy per waveform:

22 2 1 1|| || (1 )m m E E E EM M M

′ = − = − + = −s s s

s

◊ Cross-correlation between signals:m m M M M

1/ 1Re[ ] =m n Mρ′ ′⋅ −

= = −s s

44

Re[ ] =1 1/ 1mn

m n M Mρ = = −

′ ′⋅ − −s s

Simplex SignalingSimplex Signalingp g gp g g

◊ Simplex signals require less energy◊ Simplex signals are equally correlated◊ Distances between signal points remain the same: = 2d E◊ Signal dimensionality: N = M − 1

4M

45

4M =

ComparisonsComparisonspp

◊ The signal space dimensionality of the class of orthogonal, biorthogonal, and simplex signals is highly dependent on the constellation size.

hi i i A S d QA◊ This is in contrast to PAM, PSK, and QAM systems.

F fi d E th i i di t i th t i◊ For fixed Eb, the minimum distance in these systems increases with increasing M.◊ This is in sharp contrast to PAM PSK and QAM signaling◊ This is in sharp contrast to PAM, PSK, and QAM signaling.

46

Signal Waveforms from Binary CodesSignal Waveforms from Binary Codesg yg y

◊ Signaling wave forms generated from M binary code words

◊ , for all m and j1 2[ ], 1, 2,...,m m m mNc c c m M= =c

{0,1}mjc ∈

◊ Each component of a code word is mapped to a BPSK waveform:1 2 / cos 2 , 0mj c c c cc E T f t t Tπ= ⇒ ≤ ≤

◊ Tc = T/N0 2 / cos 2 , 0mj c c c cc E T f t t Tπ= ⇒ − ≤ ≤

◊ Ec = E/N◊ The M code words {cm} are mapped to a set of M waveforms

{ (t)} hi h h t f{sm(t)}, which have vector forms

f ll d j1 2[ ], 1, 2,...,m m m mNs s s m M= =s

47

◊ , for all m and j/mjs E N= ±

Signal Waveforms from Binary CodesSignal Waveforms from Binary Codesg yg y

◊ Block length = N◊ Dimension of M waveforms = N◊ 2N possible waveforms from 2N possible binary code words◊ 2N signal points ⇔ vertices of N-dim. hypercube w. center at origin◊ May select M < 2N signal waveforms for transmission◊ Cross-correlation depends on how to select M waveforms◊ Cross-correlation coefficient of adjacent points:

(1 2 / ) 2E N N− −

◊ Minimal distance:

(1 2 / ) 2E N NE N

ρ = =

min 2 (1 ) 4 /d E E Nρ= − =

48

Chapter 3.3: Signaling Schemes with Memory

Wireless Information Transmission System Lab.Wireless Information Transmission System Lab.Institute of Communications EngineeringInstitute of Communications Engineering

Memory

g gg gNational Sun National Sun YatYat--sensen UniversityUniversity

49

Signaling Schemes with MemorySignaling Schemes with Memoryg g yg g y

◊ Three different baseband signals:◊ NRZ: (Non-return-to-zero)

◊ “1”: A“0” A◊ “0”: -A

◊ Equivalent to a binary PAM or s BPSK signal.◊ NRZI: (Non-return-to-zero inverted)◊ NRZI: (Non return to zero, inverted)

◊ “0”: amplitude unchanged◊ “1”: amplitude transit to another

◊ Delay modulation:◊ “0”: No delay

“1” Si l d l d b T/2◊ “1”: Signal delayed by T/2

50

Signaling Schemes with MemorySignaling Schemes with Memoryg g yg g y

◊ NRZI: (Non-return-to-zero, inverted)◊ Differential encoding: ◊ Modulation with memory

T M k h i

1k k kb a b −= ⊕

◊ Two-state Markov chain◊ If P[ak=1] = P[ak=0] = 1/2,

1/ 2 1/ 2⎡ ⎤

◊ Steady state probability

1/ 2 1/ 21/ 2 1/ 2

⎡ ⎤= ⎢ ⎥

⎣ ⎦P

[ ]1/ 2 1/ 2=p◊ If P[ak=1] = 1−P[ak=0] = p,

11

p pp p

−⎡ ⎤= ⎢ ⎥

⎣ ⎦P

◊ NRZI signaling can be representedusing trellis diagram

1p p⎢ ⎥−⎣ ⎦

51

g g

CPFSKCPFSK

◊ Continuous-Phase Frequency-Shift Keying (CPFSK)◊ The phase of signal is constrained to be continuous◊ Conventional FSK:

◊ Shifting the carrier by◊ Switching one oscillators to another results in relative large spectral

, 1m f m MΔ ≤ ≤

side lobes ⇒ requires large frequency band◊ To avoid large side lobe, the information-bearing frequency modulates

a single carrier whose frequency is changed continuouslya single carrier whose frequency is changed continuously⇒ The signal is phase-continuous⇒ Continuous-phase FSK (CPFSK)⇒ Continuous phase FSK (CPFSK)⇒ The modulation has memory because the phase of the carrier is

constrained to be continuous

52

CPFSKCPFSK

◊ Begin with a PAM signal: ( ) ( )nd t I g t nT= −∑◊ {In} : sequence of amplitudes ◊ g(t): rectangular pulse with duration T and amplitude 1/2T

n

◊ d(t) is used to frequency-modulate the carrier◊ The equivalent lowpass waveform is

⎡ ⎤

◊ fd : peak frequency deviation

04 ( )( ) 2 /

tdj Tf d d

t E T eπ τ τ φ

ν −∞

⎡ ⎤+⎢ ⎥⎣ ⎦∫=d

◊ φ0: initial phase of the carrier◊ The carrier modulate signal s(t) for v(t) is

[ ]0( ) 2 / cos 2 ( ; )cs t E T f t tπ φ φ= + +I

( ; ) 4 ( ) 4 ( )t t

d dt Tf d d Tf I g nT dφ π τ τ π τ τ⎡ ⎤= = −⎢ ⎥∑∫ ∫I

(bandpass signal)(3.3-8)

53

( ; ) 4 ( ) 4 ( )d d nn

t Tf d d Tf I g nT dφ π τ τ π τ τ−∞ −∞ ⎢ ⎥⎣ ⎦

∑∫ ∫I

CPFSKCPFSK

◊ Although d(t) is discrete, the integral of d(t) is continuous⇒ s(t) is continuous-phase signal

◊ The phase of carrier in nT ≤ t ≤ (n+1)T11

( ; ) 2 4 ( )

2 ( )

n

d k d nk

t f T I f Tq t nT I

hI t T

φ π π

θ

−

=−∞

= + −

+

∑I

2 ( )n n hI q t nTθ π= + −

0 0 t <⎧⎪

◊

l ti ( ) f b l t t T

( ) / (2 ) 0 1/ 2 t T

q t t T t T⎪= ≤ ≤⎨⎪ >⎩

1nh Iθ −∑◊ : accumulation (memory) of symbols up to t = nT◊ : modulation index

1nn kk

h Iθ π=−∞

= ∑2 dh f T= (n-1)th symbol

54

ContinuousContinuous--Phase ModulationPhase Modulation

◊ CPFSK is a special case of continuous-phase modulation (CPM)◊ The carrier phase of CPM signal is

( ; ) 2 ( ), ( 1)n

k kt I h q t kT nT t n Tϕ π= − ≤ ≤ +∑I

◊ {Ik}: sequence of M-ary symbols , Ik ∈{±1,±3,…,±(M−1)}◊ {hk}: sequence of modulation indices

( ; ) ( ), ( )k kk

qϕ=−∞∑

{ k} q◊ When hk = h, for all k, modulation index is fixed for all symbols◊ If hk varies from one symbol to another ⇒ multi-h CPM

◊ q(t): normalized waveform shape, could be represented by

0( ) ( )

tq t g dτ τ= ∫

◊ If g(t) = 0, for t > T ⇒ full-response CPM◊ If g(t) ≠ 0, for t > T ⇒ partial-response CPM

◊ CPM signal has memory introduced through the phase continuity

55

◊ CPM signal has memory introduced through the phase continuity


◊ Full-response CPMLREC, L=1, results in CPFSK

LRC, L=1

◊ Partial-response CPM

CPFSK

p

LREC, L=2 LRC, L=2

◊ LREC: Rectangular pulse with duration LT: ( ) 1 , 02

g t t LTLT

= ≤ ≤

◊ LRC: Raise-cosine pulse with duration LT:◊ For L > 1, additional memory is introduced by g(t)

( ) 1 21 cos , 02

tg t t LTLT LT

π⎛ ⎞= − ≤ ≤⎜ ⎟⎝ ⎠

2LT

56


◊ Gaussian minimum-shift keying (GMSK) pulse:

( )( ) ( )2 / 2 2 / 2

ln 2

Q B t T Q B t Tg t

π π⎡ ⎤− − +⎡ ⎤⎣ ⎦⎣ ⎦=

◊ Bandwidth B : represents the -3 dBbandwidth of the Gaussian pulsep

◊ Pulse duration ⇑ ,bandwidth ⇓◊ In GSM system, 0.3BT =

57


◊ Phase trajectories φ (t;I) Phase trajectory binary CPFSK:

generated by all possible {In} canbe sketched as a phase tree

◊ Right figure: Phase tree of CPFSK with In = ±1n ◊ Phase tree is piecewise linear

because g(t) is rectangular◊ Phase trajectory is not smooth

58


◊ Phase trajectory of {In }= (+1,-1,-1,-1,+1,+1,-1,+1)◊ : CPFSK◊ : partial-

CPMresponse CPM with LRC, L=3

◊ Phase trellis: the phase trajectory is plotted in modulo 2p j y p◊ When h=m/p, m and p are relative prime, for full response CPM signal:

2 ( 1)0 when is evenm m p m mπ π π⎧ ⎫−Θ = ⎨ ⎬0, , ,..., , when is evens m

p P pΘ = ⎨ ⎬

⎩ ⎭2 (2 1)0, , ,..., , when is odds

m m p m mπ π π⎧ ⎫−Θ = ⎨ ⎬

59

, , , , ,s p P p⎨ ⎬⎩ ⎭


◊ For partial-response CPM, the number of phase states may increase up to

1 even mLpMS

−⎧= ⎨

◊ Up-Right : phase trellis of

12 odd m t LS

pM −= ⎨

⎩

CPFSK with h = 1/2◊ St = 4◊ Not the real phase trajectory,

only represents phase at t = nT◊ Down-Right: state diagram of

CPFSK with h = 1/2

60

MinimumMinimum--Shift Keying (MSK)Shift Keying (MSK)y g ( )y g ( )

◊ MSK is a special case of binary CPFSK◊ h =1/2◊ g(t) is rectangular pulse with duration T 1

( ; ) 2 4 ( )n

d k d nt f T I f Tq t nT Iφ π π−

= + −∑I11( ; ) ( )

2

n

k nk

t I I q t nTϕ π π−

=−∞

= + −∑I 1

( ) ( )

2 ( )

d k d nk

nk nk

f f q

h I hI q t nT

φ

π π=−∞

−

=−∞= + −

∑

∑

Th d l t d i i l i ( 3 3 8)

1 ( ), ( 1)2n n

t nTI nT t n TT

θ π −= + ≤ ≤ +

◊ The modulated carrier signal is (see 3.3-8)1( ) cos 22c n n

t nTs t A f t IT

π θ π⎡ − ⎤⎛ ⎞= + + ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦21 1 cos 2 ( ) , ( 1)

4 2c n n n

T

A f I t n I nT t n TT

π π θ

⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎡ ⎤= + − + ≤ ≤ +⎢ ⎥⎣ ⎦

61

4 2T⎢ ⎥⎣ ⎦

MinimumMinimum--Shift Keying (MSK)Shift Keying (MSK)y g ( )y g ( )

◊ The binary CPFSK signal can be expressed as a sinusoid having two possible frequencies in nT ≤ t ≤ (n+1)T

11

4cf fT

= −

Th bi CPFSK i l b i b2

41

4c

T

f fT

= +

◊ The binary CPFSK signal can be written by

11( ) cos 2 ( 1) , 1, 22

ii i ns t A f t n iπ θ π −⎡ ⎤= + + − =⎢ ⎥⎣ ⎦

◊ Equivalent to a FSK signal withI FSK i i f i h li f

( ) ( ) , ,2i i nf⎢ ⎥⎣ ⎦

2 11

2f f f

TΔ = − =

◊ In FSK, minimum frequency separation to ensure orthogonality of the signals is

◊ Thus binary CPFSK with h=1/2 is called minimum-shift keying

2T1

2f

TΔ =

62

◊ Thus, binary CPFSK with h 1/2 is called minimum-shift keying

Offset QPSK (OQPSK)Offset QPSK (OQPSK)Q ( Q )Q ( Q )

◊ In QPSK, each 2 information bits is mapped into one of the constellation points

◊ Assume the information bits are 11000111

Mapping for QPSK signal:

◊ Split into 11,00,01,11◊ In each of 2Tb period,

11: / 2(1 )

00 : / 2( 1 )

E j

E j

+

− −( )

01: / 2( 1 )

11: / 2(1 )

j

E j

E j

− +

+

Phase transition:

◊ Phase transition occurs at 2nTb

11: / 2(1 )E j+

63

occurs at 2nTb


◊ 180° phase changes causes abrupt change inthe signal, resulting large spectral side lobe.

◊ To prevent 180° phase changeoffset QPSK (OQPSK)

◊ In OQPSK, the I-phase and Q-phase Q p Q pcomponents are misaligned by Tb

◊ At t = nTb, one of I-phase and Q-phaseb, p Q pcomponent changes

prevent 180° phase changep p g◊ But, 90° phase change occurs more frequently◊ QPSK and OQPSK have the same PSD

64

◊ QPSK and OQPSK have the same PSD


◊ The OQPSK signal is

2 2 1( ) ( 2 ) cos 2 ( 2 ) sin 2n c n cn n

s t A I g t nT f t I g t nT T f tπ π∞ ∞

+=−∞ =−∞

⎡ ⎤⎛ ⎞ ⎛ ⎞= − + − −⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎣ ⎦∑ ∑

◊ Lowpass equivalent of s(t) is

n n=−∞ =−∞⎝ ⎠ ⎝ ⎠⎣ ⎦

2 2 1( ) ( 2 ) ( 2 )l n nn n

s t A I g t nT j I g t nT T∞ ∞

+=−∞ =−∞

⎡ ⎤ ⎡ ⎤= − − − −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

∑ ∑◊ MSK can be expressed as a form of OQPSK with

sin 0 t 2Ttπ⎧ ≤ ≤⎪sin 0 t 2T( ) 2

0 otherwise g t T

≤ ≤⎪= ⎨⎪⎩

(See Problem 3.26)

65


◊ With the sum of I-phase and Q-

( ) sin , 0 t 2T2

tg tT

π= ≤ ≤

p Qphase has constant amplitude and is frequency-modulated.

66


◊ MSK: continuous phase but jump in frequency

◊ OQPSK: constant frequency but 90° jump in phase everyj p p yT seconds.

◊ QPSK: constant frequency but 180° or 90° jump in j pphase every 2T seconds.

67

Linear Representation of CPM SignalsLinear Representation of CPM Signalsp gp g

◊ CPM is a nonlinear modulation with memory, but it can be represented as linear superposition of waveforms

◊ A lowpass equivalent of a CPM signal is

◊

( ; )( ) 2 / , ( 1)j t Iv t E T e nT t n Tφ= ≤ ≤ +

( ; ) 2 ( ), ( 1)n

kt h I q t kT nT t n Tφ π= − ≤ ≤ +∑Ik =−∞

1 2 ( )

n L n

k kk k n L

h I h I q t kTπ π−

=−∞ = − +

= + −∑ ∑t

∫◊ , the duration of g(t) is LT◊

0( ) ( )

tq t g dτ τ= ∫ 1

0

exp[ ( ; )] exp exp{ 2 [ ( ) ]}Ln L

k n kk k

j t j h I j hI q t n k Tϕ π π−−

−⎛ ⎞

= − −⎜ ⎟⎝ ⎠

∑ ∏I0k k=−∞ =⎝ ⎠

Cumulated phase up to

Product of L phase termsn LI

68

p p n L−


◊ Assume that h is not an integer and symbols Ik∈{±1}◊ The k-th phase term sinexp{ 2 [ ( ) ]} exp{ 2 [ ( ) ]}

sinn k n khj hI q t n k T j hI q t n k Th

ππ ππ− −− − = − −

sinsin{ 2 [ ( )] } sin{2 [ ( ) ]}exp( )

sin sinn k

hh hq t n k T hq t n k Tj hI

h h

ππ π ππ

π π−

− − − − −= +

◊ Define the signal pulsesin 2 ( ) / sin , 0 hq t h t LTπ π ≤ ≤⎧

⎪0 ( ) sin[ 2 ( )] / sin , 2

0 , otherwise s t h hq t LT h LT t LTπ π π⎪= − − ≤ ≤⎨

⎪⎩

◊

1

exp[ ( ; )]

exp { [ ( ) )] exp( ) [ ( ) ]}Ln L

j t

j h I s t k L n T j hI s t k n T

φ

π π−−⎛ ⎞= + + − + − −⎜ ⎟∑ ∏

I

69

0 00

exp { [ ( ) )] exp( ) [ ( ) ]}k n kk k

j h I s t k L n T j hI s t k n Tπ π −=−∞ =

= + + − + − −⎜ ⎟⎝ ⎠

∑ ∏


◊1

0 0exp[ ( ; )] exp { [ ( ) )] exp( ) [ ( ) ]}Ln L

k n kj t j h I s t k L n T j hI s t k n Tφ π π−−

−⎛ ⎞

= + + − + − −⎜ ⎟⎝ ⎠

∑ ∏I

◊ By multiplying over L terms, we obtain 2L terms

0k k=−∞ =⎜ ⎟⎝ ⎠

∑ ∏

◊ 2L-1 terms are distinct◊ 2L-1 terms are time-shifted version of 2L-1 distinct terms

70


◊ The final result is 1

,2 1

0

exp[ ( ; )] ( )L

k nj hAk

n k

j t e c t nTπϕ− −

=

= −∑ ∑I1L−

◊

◊

1

0 0 , ,1

( ) ( ) [ ( ) ], 0 min[ (2 ) ]L

k k n k nnn

c t s t s t n La T t T L a n−

=

= + + ≤ ≤ × − −∏1n L

k n m n m k mA I I a−

= −∑ ∑◊

◊ ak,n = 0 or 1 are coefficients in binary representation of k, ,

1k n m n m k m

m m−

=−∞ =∑ ∑

11 12 0 1 2 1

Lm Lk a k

−− −= =∑

◊ Binary CPM is expressed as a weighed sum of 2L-1 real pulses {ck(t)}(t) i th t i t t t

, ,12 0,1,..., 2 1k m

mk a k

=

= = −∑

◊ c0(t) is the most important component:◊ Has longest duration◊ Contains the most signal energy

71

g gy

Pulse Code ModulationPulse Code Modulation(Line Code)


72

PCM Waveform RepresentationsPCM Waveform Representationspp

73

PCM Waveform : NRZPCM Waveform : NRZ--LL

1 0 1 1 0 0 0 1 1 0 1

+E

0

-E

◊ NRZ Level (or NRZ Change)“One” is represented by one level◊ One is represented by one level.

◊ “Zero” is represented by the other level.

74

PCM Waveform : NRZPCM Waveform : NRZ--MM

1 0 1 1 0 0 0 1 1 0 1

+E

0

-E

◊ NRZ Mark (Differential Encoding)“One” is represented by a change in level◊ One is represented by a change in level.

◊ “Zero” is represented by a no change in level.

75

PCM Waveform : NRZPCM Waveform : NRZ--SS

1 0 1 1 0 0 0 1 1 0 1

+E

0

-E

◊ NRZ Space (Differential Encoding)“One” is represented by a no change in level◊ One is represented by a no change in level.

◊ “Zero” is represented by a change in level.

76

PCM Waveform : UnipolarPCM Waveform : Unipolar--RZRZpp

1 0 1 1 0 0 0 1 1 0 1

+E

0

-E

◊ Unipolar - RZ“One” is represented by a half bit width pulse◊ One is represented by a half-bit width pulse.

◊ “Zero” is represented by a no pulse condition.

77

PCM Waveform : PolarPCM Waveform : Polar--RZRZ

1 0 1 1 0 0 0 1 1 0 1

+E

0

-E

◊ Polar - RZ“One” and “Zero” are represented by opposite◊ One and Zero are represented by opposite level polar pulses that are one half-bit in width.

78

PCM Waveform : BiPCM Waveform : Bi--φφ--LLφφ

1 0 1 1 0 0 0 1 1 0 1

+E

0

-E

◊ Bi-φ-L (Biphase Level or Split Phase Manchester 11 + 180o)180 )◊ “One” is represented by a 10.◊ “Zero” is represented by a 01.

79

◊ Zero is represented by a 01.

PCM Waveform : BiPCM Waveform : Bi--φφ--MMφφ

1 0 1 1 0 0 0 1 1 0 1

+E

0

i ( i h k h 1)

-E

◊ Bi-φ-M ( Biphase Mark or Manchester 1)◊ A transition occurs at the beginning of every bit period.

“O ” i t d b d t iti h lf bit i d◊ “One” is represented by a second transition one half bit period later.

◊ “Zero” is represented by no second transition

80

◊ Zero is represented by no second transition.

PCM Waveform : BiPCM Waveform : Bi--φφ--SSφφ

1 0 1 1 0 0 0 1 1 0 1

+E

0

-E

◊ Bi-φ-S ( Biphase Space)◊ A transition occurs at the beginning of every bit period.◊ “One” is represented by no second transition.◊ “Zero” is represented by a second transition one-half bit period

later

81

later.

PCM Waveform : Dicode PCM Waveform : Dicode -- NRZNRZ

1 0 1 1 0 0 0 1 1 0 1

+E

0

-E

◊ Dicode Non-Return-to-Zero◊ A “One” to “Zero” or “Zero” to “One” changes polarity.◊ Otherwise, a “Zero” is sent.

82

PCM Waveform : Dicode PCM Waveform : Dicode -- RZRZ

1 0 1 1 0 0 0 1 1 0 1

+E

0

-E

◊ Dicode Return-to-Zero◊ A “One” to “Zero” or “Zero” to “One” transition produces a half

d ti l it hduration polarity change.◊ Otherwise, a “Zero” is sent.

83

PCM Waveform : Delay ModePCM Waveform : Delay Modeyy

1 0 1 1 0 0 0 1 1 0 1

+E

0

-E

◊ Dicode Non-Return-to-Zero◊ A “One” is represented by a transition at the midpoint of the bit

intervalinterval.◊ A “Zero” is represented by a no transition unless it is followed

by another zero. In this case, a transition is placed at the end of bi i d f h fi

84

bit period of the first zero.

Line Code: 4B3TLine Code: 4B3T

O - -

85

Line Code: 4B3TLine Code: 4B3T

◊ Ternary words in the middle column are balanced in their DC content.

◊ Code words from the first and third columns are selected alternately to maintain DC balance.

◊ If more positive pulses than negative pulses have been transmitted, l 1 i l t dcolumn 1 is selected.

◊ Notice that the all-zeros code word is not used.

86

Line Code: Multilevel TransmissionLine Code: Multilevel TransmissionLine Code Multilevel TransmissionLine Code Multilevel Transmission

◊ Multilevel transmission with 3 bits per signal interval◊ Multilevel transmission with 3 bits per signal interval.

87

Criteria for Selecting PCM WaveformCriteria for Selecting PCM WaveformCriteria for Selecting PCM WaveformCriteria for Selecting PCM Waveform

◊ DC component: eliminating the dc energy from the signal’s power spectrum.

◊ Self-Clocking: Symbol or bit synchronization is required for any digital communication systemdigital communication system.

◊ Error detection: some schemes provide error detection without introducing additional error-detection bits.introducing additional error detection bits.

◊ Bandwidth compression: some schemes increase bandwidth utilization by allowing a reduction in required bandwidth for a i dgiven data rate.

◊ Noise immunity.C t d l it◊ Cost and complexity.

88

Spectral Densities of Various PCM WaveformsSpectral Densities of Various PCM Waveformspp

89

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