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Digital Modulation 2– Lecture Notes –

Ingmar Land and Bernard H. Fleury

Department of Electronic Systems

Aalborg University

Version: November 15, 2006

i

Contents

1 Continuous-Phase Modulation 1

1.1 General Description . . . . . . . . . . . . . . . . . 1

1.2 Minimum-Shift Keying . . . . . . . . . . . . . . . . 6

1.3 Gaussian Minimum-Shift Keying . . . . . . . . . . 11

1.4 State and Trellis of CPM . . . . . . . . . . . . . . 19

1.5 Coherent Demodulation of CPM . . . . . . . . . . 26

2 Modern Representation of CPM 30

2.1 Initialization . . . . . . . . . . . . . . . . . . . . . 31

2.2 The Reference Phase Trajectory . . . . . . . . . . . 33

2.3 The Reference Frequency . . . . . . . . . . . . . . 37

2.4 The Tilted Phase and the New Symbols . . . . . . 38

2.5 Block Diagram of the CPM Transmitter . . . . . . 42

3 Laurent Representation of CPM 44

3.1 Minimum Shift Keying . . . . . . . . . . . . . . . . 44

3.2 Precoded MSK . . . . . . . . . . . . . . . . . . . . 51

3.3 Precoded Gaussian MSK . . . . . . . . . . . . . . 57

3.4 Modulation in EDGE . . . . . . . . . . . . . . . . 66

4 Trellis-Coded Modulation 72

4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . 72

4.2 Convolutional Codes . . . . . . . . . . . . . . . . . 81

4.3 Goodness of a TCM Scheme . . . . . . . . . . . . . 83

4.4 Set Partitioning . . . . . . . . . . . . . . . . . . . 87

4.5 Trellis Coded Modulation . . . . . . . . . . . . . . 88

4.6 Examples for TCM . . . . . . . . . . . . . . . . . . 92

Land, Fleury: Digital Modulation 2 SIPCom

ii

A ECB Representation of a Band-Pass Signal 102

A.1 The Block diagram . . . . . . . . . . . . . . . . . . 104

A.2 The Signal . . . . . . . . . . . . . . . . . . . . . . 106

A.3 Effect of QA Mod/Demod on Signal . . . . . . . . 107

A.4 Effect of QA Demodulation on Noise . . . . . . . . 110

A.5 Noise After LP Filtering and Sampling . . . . . . . 113


iii

References

[1] J. G. Proakis and M. Salehi, Communication Systems Engi-

neering, 2nd ed. Prentice-Hall, 2002.

[2] J. G. Proakis, Digital Communications. McGraw-Hill, 1995.

[3] P. Laurent, “Exact and approximate construction of digital

phase modulations by superposition of amplitude modulated

pulses (AMP),” IEEE Trans. Commun., vol. 34, no. 2, pp.

150–160, Feb. 1986.

[4] B. Rimoldi, “A decomposition approach to CPM,” IEEE

Trans. Inform. Theory, vol. 34, no. 2, pp. 260–270, Mar. 1988.

[5] C. E. Shannon, “A mathematical theory of communication,”

Bell System Technical Journal, vol. 27, pp. 379–423, 623–

656, July and Oct. 1948.


1

1 Continuous-Phase Modulation

The Classical Representation

Continuous-phase modulation allows for a bandwith-efficient trans-

mission. In this section, it is represented in the classical way. The

modern representation is discussed in the following section.

1.1 General Description

The general description of a CPM signal reads

X(t;α) =√

2P cos(2πf0t + ϕ(t;α)

)(1)

with the following notation:

The (semi-infinite) sequence of information-bearing symbols is

α = (α0, α1, . . .)

with the symbols

αn ∈ A = {±1,±3, . . . ,±M − 1} if M is even

αn ∈ A = {0,±2, . . . ,±M − 1} if M is odd

The mostly used case is that where M is even. (Or even a power

of 2. Why?)

Example: Binary CPM

M = 2: αn ∈ {−1,+1}. 3


2

The time-variant information-bearing phase

ϕ(t;α) = ϕ0 + 2πh∞∑

n=0

αnq(t− nT )

Remark: In some cases (see later), L−1 known symbols are trans-

mitted at the beginning. Then the summation starts with−(L−1).

The modulation index is h = p1p2

, where p1, p2 are natural numbers

that are relatively prime.

The function q(t) is the phase reponse. It satisfies

q(t) = 0 for t ≤ 0,

q(t) =1

2for t > LT .

LT..... tT

1

2

q(t)

The value L is the memory of the CPM system :

L = 1 : full response CPM

L > 1 : partial response CPM

The phase ϕ0 is arbitrary.

Without loss of generality, we can assume that ϕ0 is set such that

ϕ(0;α) = 0.

The carrier frequency is f0.

The power of the transmitted signal is P .

The symbol rate is 1T .


3

Instantaneous Frequency of a CPM signal

f (t;α) =1

2π

d

dt

[2πf0t + ϕ(t;α)

]

= f0 +1

2π

d

dt

[ϕ(t;α)

]

= f0 + hd

dt

[ ∞∑

n=0

αnq(t− nT )]

f (t;α) = f0 + h∞∑

n=0

αnq′(t− nT )

︸︷︷︸∆f(t;α) (frequency deviation)

The derivative

q′(t) =d

dtq(t)

of q(t) is called the frequency response of the CPM signal.

We can rewrite the CPM signal from (1) as a function of the in-

stantaneous frequency according to

x(t;α) =√

2P cos(

2π

t∫

0

f (t;α) dt)

=√

2P cos(

2πf0t + 2πh[ t∫

0

∞∑

n=0

αnq′(t− nT ) dt

])


4

Properties of the Frequency Response

q(t) = 0, t < 0 ⇒ q′(t) = 0, t < 0

q(t) =1

2, t > LT ⇒ q′(t) = 0, t > LT

q(LT ) =1

2⇒

LT∫

0

q′(t)dt =1

2

Example:

Sketch a valid frequency response for L = 4. 3


5

Block diagram of a CPM transmitter

(a) Using a phase modulator

..... t

q(t)

LT

(L = 2)

Phase

Modulator

f0

−1

+1

−1

+1 +1

−1

α(t) ψ(t;α)

T

ψ(t;α)

T

+2πhq(t− T )

−2πhq(t− T )

t t

+πk

−πk

x(t;α)

2πk

1

2

0

α(t) =

∞∑

n=0

αmδ(t− nT )

(b) Using a frequency modulator

t

q′(t)

LT

(L = 2)

f0

−1

+1

−1

+1 +1

−1

α(t)

TT t t

x(t; α)

0

h

V C D

∆f (t; α)

∆f (t; α)

1

2

α(t) =

∞∑

n=0

αmδ(t − nT )


6

1.2 Minimum-Shift Keying

MSK has the following parameters:

• Binary modulation symbols, i.e., M = 2

• Modulation index h = 12

• Phase response

qMSK(t) =

0 t < 012 · tT 0 ≤ t < T12 t ≥ T

tT

qMSK(t)

1

2

Example: MSK

Assume the transmitted symbol sequence

α = [−1 + 1 + 1− 1 + 1].

Sketch the phase and the signal. 3


7

Example: MSK, continued

Transmitted symbols αn:

-1 1 1 -1 1

T

+π2

−π2

ϕ(t; α)

x(t; α)

t

t

1

2πddtϕ(t; α) = 1

2ππ/2T = 1

4T

1

2πddtϕ(t; α) = −

1

4T

∆f = 1

2T

f1 = 1

T f2 = 3

2

1

T ⇒ ∆f = 1

2T

.= min. freq- sep for orthogonality

3


8

Phase tree of MSK

t

Phase

T 2T 3T 4T 5T 6T

π/2

π

3π/2

2π

5π/2

6π/2

−π/2

−π

−3π/2

−2π

−6π/2

−5π/2


9

Instantaneous frequency of MSK

q′MSK(t) =

0 for t < 01

2T for 0 ≤ t < T

0 for t ≥ T

T t

1

T

q′

MSK(t)

Then

f (t;α) = f0 +1

2

∞∑

n=0

αnq′MSK(t− nT )

︸︷︷︸∆f(t;α)

Example:

1

T 2T t

∆f (t; α)

−

1

4T

1

4T

-1 -111

3


10

In symbol interval [nT, (n + 1)T ), we have

f (t;α) = f0 +1

2· αn ·

1

2T

= f0 +αn4T

with αn ∈ {−1,+1}.Therefore the frequency offset is

∆f =(f0 +

1

4T

)−(f0 −

1

4T

)

=1

2T


11

1.3 Gaussian Minimum-Shift Keying

Problem

How to reduce the side lobes in the power spectrum of MSK while

keeping the essential features of this modulation scheme that co-

herent demodulation is possible?

Solution

1. Insert in the MSK VCO-based (non-causal) transmitter a

smoothing filter with impulse response gB(t) in front of the

MSK frequency response.

q′MSK(t) −→ Reduction of the side lobes!

VCO

tt−T T

h

T

1

2T

Smoothing filter MSK-transmitter

f0

α(t) =

∞∑

n=0

αnδ(t − nT )

q′(t)

x(t; α)gB(t) q′MSK(t)

2. Design gB(t) such that the overall frequency response

q′(t) = gB(t) ∗ q′MSK(t)

integrates to 12 as does q′MSK(t).

→ Coherent demodulation is possible!


12

The indexing parameter B is the 3 dB bandwidth of the impulse

response of the smoothing filter.

|F{gB(t)}|2

fB

3dB

Smoothing filter for GMSK

GMSK results by selecting the impulse response of the smoothing

filter to be a Gaussian pulse

gB(t) =1√

2πσTexp

− t2

2σ2T2 (2)

with

σ =

√ln 2

2πBT

Hence,

q′GMSK(t) = gB(t) ∗ q′MSK(t) (3)


13

We show now that q′GMSK(t) has the required property:

+∞∫

−∞

q′GMSK(t) dt =1

2(4)

Proof

We can view 2q′MSK(t) and gB(t) as the probability density func-

tions (pdfs) of two independent random variables, say U and V :

U ∼ 2q′MSK(t)

V ∼ gB(t)

Then from (3), 2q′GMSK(t) is the pdf of the sum

W = U + V

Equ. (4) follows then because the integral of pdfs equals unity.

Selected values of B

GSM : BT = 0.3 T =6

1.625= 3.69ms

DECT : BT = 0.5


14

Example

Gaussian Pulse

t/T4-4 -3 -2 -1 0 1 2 3

BT=0.2

BT=0.3

BT=0.5

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

g B(t

)

Frequency response

t/T4-4 -3 -2 -1 0 1 2 3

BT=0.2

BT=0.3

BT=0.5

0.5

0.45

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

g′ G

MSK

(t)


15

Phase response

qGMSK(t) =

t∫

−∞

q′GMSK(t ) dt

t/T4-4 -3 -2 -1 0 1 2 3

BT=0.2

BT=0.3

BT=0.5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

q GM

SK

(t)


16

Casual implementation of a GMSK transmitter

A casual implementation of a GMSK transmitter is obtained by

truncating the non-casual frequency response qGMSK(t) outside a

centered interval[− LT

2 , +LT2

]normalizing to 1/2, and time

delaying by LT2 :

q′GMSK(t) =

{12

1A q′GMSK(t− LT

2 ) for t ∈ [0, LT )

0 elsewhere

with

A =

+LT2∫

−LT2

q′GMSK(t) dt

A practical value is L = 3.

BT=0.2

BT=0.3

BT=0.5

0.5

0.45

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0 0.5 1 1.5 2 2.5 3 t/T

g′ G

MSK

(t)


17

Casual phase response

qGMSK(t) =

t∫

−∞

q′GMSK(t ) dt

t/T

BT=0.2

BT=0.3

BT=0.5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

q GM

SK

(t)

0 0.5 1.0 1.5 32.0 2.5


18

Phase tree of GMSK

Power spectrum of GMSK


19

1.4 State and Trellis of CPM

Let us consider the phase of a GMSK signal within some arbitrary

signaling interval say [nT, (n + 1)T ).

(n − 3)T (n − 2)T (n − 1)T nT (n + 1)T

qGMSK(t − (n − 3)T ) qGMSK(t − nT ) qGMSK(t − (n − 1)T ) qGMSK(t − (n − 2)T )

time

12

We consider first the special case L = 3 as depicted above. For

t ∈ [nT, (n + 1)T ), we can write:

ϕ(t;α) = ϕ0 + π

∞∑

i=0

αiqGMSK(t− iT )

= ϕ0 + π

n−3∑

i=0

αi1

2+

n∑

i=n−2

αiqGMSK(t− iT )

In the general case where L is arbitrary

ϕ(t;α) = ϕ0 +π

2

n−L∑

i=0

αi

︸︷︷︸θn

+

n−1∑

i=n−(L−1)

αiq(t− iT ) + αnq(t− nT )

= ϕ(t;αn, θn, αn−1, . . . , αn−(L−1)︸︷︷︸

Sn

)

= ϕ(t;αn,Sn)

for t ∈ [nT, (n + 1)T )


20

Thus, the behaviour of ϕ(t;α) in the signaling interval

[nT, (n + 1)T ) is entirely determined by the (L + 1)-dimensional

vector

(αn, θn, αn−1, . . . , αn−(L−1)︸︷︷︸

Sn∈S

)

with

αn : current transmitted symbol

Sn : state of ϕ(t;α) in n-th signaling interval

S : state space


21

State transitions (trellis) for GMSK with L = 3

Notice that

θn+1 = θn +π

2αn−(L−1)

(3π

2,−1, 1)

(3π

2, 1, 1)

(0, 1,−1)

(0, 1, 1)

(π

2,−1,−1)

(π

2,−1, 1)

(π

2,−1, 1)

(π

2, 1, 1)

(2π

2,−1, 1)

(2π

2,−1, 1)

(2π

2, 1, 1)

(3π

2,−1, 1)

(0,−1, 1)

(0,−1,−1)

(2π

2,−1,−1)

(3π

2,−1,−1)

(3π

2,−1, 1)

(3π

2, 1, 1)

(0, 1,−1)

(0, 1, 1)

(π

2,−1,−1)

(π

2,−1, 1)

(π

2,−1, 1)

(π

2, 1, 1)

(2π

2,−1, 1)

(2π

2,−1, 1)

(2π

2, 1, 1)

(3π

2,−1, 1)

(0,−1, 1)

(0,−1,−1)

(2π

2,−1,−1)

(3π

2,−1,−1)

αn = 1

αn = −1

Sn Sn+1

S


22

Phase behavior within one signaling interval as a function of

the state transitions for GMSK

(3π2,−1, 1)

(3π2, 1,−1)

(3π2,−1,−1)

(2π2,−1,−1)

(π2,−1,−1)

(0,−1,−1)

(0, 1,−1)

(2π2,−1, 1)

(π2,−1, 1)

(2π2, 1,−1)

(0,−1, 1)

(π2, 1,−1)

(3π2, 1, 1)

2π

3π2

2π2

π2

0

nT (n + 1)T t

τ

αm = +1

αm = −1

Sn ϕ(t; αn; Sn)

Initialization

ϕ0 = −−1∑

i=−(L−1)

αiq(t− iT )

θ0 = 0

where α−1, . . . , α−(L−1) are known symbols


23

Implementation of a CPM transmitterwith look-up tables

A. Using a phase modulator

Look-uptable

modulatorPhase

f0

αnαn

Computethe

phase state

Sn (αn, sn) 7→ ϕ(τ ; αn, Sn)

x(t; α)

ϕ(τ ; αn, Sn)

B. Using an inphase/quadrature modulator

Consider the inphase/quadrature representation of x(t;α),

where we assume ϕ0 = 0:

x(t;α) =

√2Es

Tscos(2πf0t + ϕ(t;α))

=

√2Es

Tscosϕ(t;α)

︸︷︷︸In-phase component

cos(2πf0t)

−√

2Es

Tssinϕ(t;α)

︸︷︷︸Quadrature component

sin(2πf0t)


24

αn αn

Look-uptable

Look-uptable

x(t; α)cos(2πf0t)

sin(2πf0t)

√

2Es

T

SnComputethe

phase state

cosϕ(t; αn, Sn)

−sinϕ(t; αn, Sn)

(αn, Sn) 7→ cosϕ(τ ; αn, Sn)

(αn, Sn) 7→ sinϕ(τ ; αn, Sn)

Lan

d,

Fleu

ry:

Digital

Mod

ulation

2S

IPC

om

25

Phase state computation

T T T T

Compute theαn

. . . .αn

αn−1

αn−2

αL−1

αL−1

(

n−L∑

i=0

αi

)

mod4

phase state

Sn

for GMSK (L = 3):

T T T

αn−2

αn−1

αn

αn−1 αn−2

(

n−3∑

i=0

αi

)

mod4


26

1.5 Coherent Demodulation of CPM

The initial state and the final state of the CPM transmitter are set

to known states using the following procedure:

• Before transmitting the data, L known symbols are trans-

mitted to drive the CPM transmitter into a known initial

state.

• After transmitting the data, L known symbols are transmit-

ted to to drive the CPM transmitter into a known final state.

Block of transmitted symbols

t = 0

. . . . α0

L − 1 known

symbols

. . . .αN−1

L + 1 known

symbols

N data symbols

α

ML decoding for the AWGN channel

Received signal:

Y (t) = x(t;α) + W (t)

White Gaussian noise

with spectral height N0

2

X(t; α)

W (t)

Y (t)


27

ML estimation

α = argmaxα∈AN

{ (N+L+1)T∫

0

y(t) x(t;α) dt

}

= argmaxα∈AN

{N+L∑

n=0

(n+1)T∫

nT

y(t) x(t;α) dt

}

= argmaxα∈AN

{N+L∑

n=0

[ (n+1)T∫

nT

y(t) cosϕ(t;αn,Sn) cos(2πf0t) dt

−(n+1)T∫

nT

y(t) sinϕ(t;αn,Sn) sin(2πf0t) dt

]}


28

ML receiver

cos(2πf0t)

−sin(2πf0t)

y(t)

Bank of correlators

αn ∈ Q

∫ (n+1)T

nT

(·)dt

Bank of correlators

∫ (n+1)T

nT

(·)dt

..

..

..

..

..

..

Viterbialgorithm

with

αn ∈ Qcosϕ(t; αn, Sn)

sinϕ(t; αn, Sn)Sn ∈ S

Sn ∈ S

states S

α


29

Performance of MSK and GMSK

Performace degradation of GMSK with respect to MSK


30

2 Modern Representation of CPM

In the modern representation of CPM [4], the CPM transmitter is

decomposed into a vector encoder with memory and a memoryless

modulator.

VectorEncoder

MemorylessModulator

{ {} } x(t) =

∞∑

j=0

xj(t − jT )uj xj

This modern representation of CPM has already been used in the

description of minimum-shift keying (MSK) in the lecture notes for

Digital Modulation I. (See therein.)


31

2.1 Initialization

We assume that the CPM transmitter is initialized with ”admissi-

ble” symbols. we select

α−1 = α−2 = . . . = α−(L−1) = −(M − 1)

Notice that −(M − 1) is a symbol common to both alphabets

corresponding to M odd and even.

The phase of CPM is then

ϕ(t;α) = ϕ0 + 2πh∞∑

j=−(L−1)

αjq(t− jT )

where α = (α0, α1, . . .) is the sequence of information-bearing

symbols.

The transmitted CPM signal reads:

x(t;α) =√

2P cos(2πf0t + ϕ(t;α))


32

Without loss of generality, we assume that ϕ0 is selected in such a

way that ϕ(0;α) = 0. Thus we obtain

ϕ0 = −2πh0∑

j=−(L−1)

αjq(−jT )

= −2πhL−1∑

j=0

α−jq(jT )

= −2πhL−1∑

j=1

α−jq(jT )

= 2πh(M − 1)

L−1∑

j=0

q(jT )


33

2.2 The Reference Phase Trajectory

We focus on the phase trajectory corresponding to the sequence

α− =(−(M − 1),−(M − 1), . . .

)

Inserting yields

ϕ(t;α−) = ϕ0 − 2πh(M − 1)

∞∑

j=−(L−1)

q(t− jT )

We investigate the behaviour of ϕ(t;α−) over each signaling inter-

val [nT, (n+ 1)T ). To this end, it is worth introducing the relative

time variable τ ∈ [0, T ). The absolute time variable t within this

signaling interval is related to τ according to

t = nT + τ

(i) Consider: n = 0, ⇒ t = τ

ϕ0(τ ) = ϕ(τ,α−)

= ϕ0 − 2πh(M − 1)

0∑

j=−(L−1)

q(τ − jT )

= ϕ0 − 2πh(M − 1)

L−1∑

j=0

q(τ + jT )

(The index in ϕ0(τ ) is the value of n.)

Inserting for ϕ0, we obtain

ϕ0(τ ) = −2πh(M − 1)

L−1∑

j=0

(q(τ + jT )− q(jT ))


34

ϕ0(τ ) = −2πh(M − 1)

L−1∑

j=0

(q(τ + jT )− q(jT ))

t

L = 3

τ T 2T 2T + τ 3TT + τ

1

2

q(t)

q(T + τ ) − q(T )

q(2T + τ ) − q(2T )

Notice that

• ϕ0(0) = 0

• ϕ0(T ) = −2πh(M − 1)q(LT ) = −πh(M − 1)

Behavior of ϕ0(τ ):

τ

T

ψ0(τ )

−πh(M − 1)


35

(ii) Consider: n = 1, ⇒ t = T + τ

ϕ1(τ ) = ϕ(T + τ ;α−)

= ϕ0 − 2πh(M − 1)

1∑

j=−(L−1)

q(τ + T − jT )

= ϕ0 − 2πh(M − 1)

L−1∑

j=−1

q(τ + (j + 1)T )

= ϕ0 − 2πh(M − 1)

L∑

j=0

q(τ + jT )

= −2πh(M − 1)

L−1∑

j=0

q(τ + jT )−L−1∑

j=0

q(jT )

− πh(M − 1)

= −πh(M − 1) + ϕ0(τ )

(iii) Consider: n ≥ 1, ⇒ t = nT + τ

ϕn(τ ) = ϕ(nT + τ,α−)

= ϕ0 − 2πh(M − 1)

n∑

j=−(L−1)

q(τ + nT − jT )

= ϕ0 − 2πh(M − 1)

(L−1)+n∑

j=0

q(τ + jT )

= ϕ0 − 2πh(M − 1)

L−1∑

j=0

q(τ + jT ) + nπh(M − 1)

= nπh(M − 1) + ϕ0(τ )


36

t

T 2T 3T nT

−πh(M − 1)

−2πh(M − 1)

−3πh(M − 1)

−nπh(M − 1)

−πh(M − 1) tT

(n + 1)T

ϕ(t; α)

ϕ(t; α−)

ϕ0 (τ

)−

nπh(M

−1)

ϕ0 (τ

)−

2πh(M

−1)

ϕ0 (τ

)−

πh(M

−1)

ϕ0 (τ

)


37

2.3 The Reference Frequency

For the sake of simplifying the presentation we assume that

ϕ(t;α−) = −πh(M − 1)t

Tor equivalently that

ϕ0(τ ) = −πh(M − 1)τ

T

In order for ϕ0(τ ) to exhibit this behavior, q(t) must satisfy the

condition

−2πh(M − 1)

L−1∑

j=0

(q(τ + jT )− q(jT )) = −πh(M − 1)τ

T

which holds ifL−1∑

j=0

(q(τ + jT )− q(jT )) =τ

2T(5)

Thus for α = α−, the CPM signal transmitted reads

x(t;α−) =√

2P cos(2πf0t− πh(M − 1)

t

T

)

=√

2P cos(2πf1t)

with

f1 = f0 −h(M − 1)

2T

This frequency f1 is the retained reference frequency with respect

to which the phase of CPM is expressed.

x(α) =√

2P cos(

2πf1t + ϕ(t;α) + πh(M − 1)t

T︸︷︷︸φ(t;α)

)


38

2.4 The Tilted Phase and the New Symbols

The so called tilted phase φ(t;α) reads

φ(t;α) = ϕ(t;α) + πh(M − 1)t

T

= ϕ0 + 2πh∞∑

j=−(L−1)

αjq(t− jT ) + πh(M − 1)t

T

We investigate the behavior of φ(t;α) within each signaling inter-

val.

(i) Consider: n = 0, ⇒ t = τ ∈ [0, T )

φ0(τ ;α) = φ(0 · T + τ ;α)

= ϕ0 + 2πh

L−1∑

j=0

α−jq(τ + jT ) + πh(M − 1)τ

T

As we have (5) due to the assumption regarding ϕ0(τ ), we conclude

that

τ

T= 2

L−1∑

j=0

q(τ + jT )− 2

L−1∑

j=0

q(jT )

= 2

L−1∑

j=0

q(τ + jT )−[πh(M − 1)

]−1

ϕ0

Inserting yields

φ0(τ ;α) = 2πh

L−1∑

j=0

α−jq(τ + jT ) + 2πh(M − 1)

L−1∑

j=0

q(τ + jT )

= 2πh

L−1∑

j=0

(α−j + (M − 1)

)q(τ + jT )


39

We introduce the new information-bearing symbols

Uj =1

2

(αj + (M − 1)

)∈{

0, 1, . . . ,M − 1}

Notice that the new symbol alphabet is similar regardless of

whether M is odd or even.

Making use of these ”new” symbols, we can recast φ0(τ ;α) ac-

cording to

φ0(τ ;α) = φ(τ ;α)

= 4πh

L−1∑

j=0

U−jq(τ + jT )


40

(ii) Consider: n ≥ 1, ⇒ t = nT + τ

φn(τ ;α) = φ(nT + τ ;α)

= ϕ0 + 2πhn∑

j=−(L−1)

αj q(τ + nT − jT )

+ πh(M − 1)nT + τ

T

= ϕ0 + 2πh

n+(L−1)∑

j=0

αn−jq(τ + jT )

+ πh(M − 1)n + πh(M − 1)τ

T

= ϕ0 + 2πhL−1∑

j=0

αn−jq(τ + jT ) + πh(M − 1)τ

T

+ 2πh

n+(L−1)∑

j=L

αn−j1

2+ πh(M − 1)n

= 4πh

L−1∑

j=0

Un−jq(τ + jT ) + πh( n−L∑

j=−(L−1)

[αj + (M − 1)

])

= 4πhL−1∑

j=0

Un−jq(τ + jT ) + 2πhn−L∑

j=−(L−1)

Uj

= 4πhL−1∑

j=0

Un−jq(τ + jT ) + 2πh

( n−L∑

j=−(L−1)

Uj

)

mod p2︸︷︷︸Vn

Remember the definition of the modulation index: h = p1/p2.


41

Let us define

Vn =

( n−L∑

j=−(L−1)

Uj

)

mod p2

Then

φn(τ ;α) = 4πh

L−1∑

j=0

Un−j q(τ + jT ) + 2πhVn

= φ(τ ;Un, Un−1, . . . Un−(L−1), Vn

)

The vector

Sn :=(Un−1, . . . Un−(L−1), Vn

)

defines the state in the new description (modern description).

(Cf. definition pf state in the traditional description.)


42

2.5

Blo

ckD

iagra

mofth

eC

PM

Tra

nsm

itter

Th

ep

revious

discu

ssions

leadto

the

followin

gb

lockd

iagram.

T T T Tpa

. . .

..

..

..

Encoder Memoryless modulator

un

un−1

un−(L−2)

un−(L−1)

vn

vn

un−(L−1)

un−(L−2)

un−2

un−1

un

un−2

−sin(2πf1τ )

√

2P

cos(2πf1τ )

x(τ ; un; sn)

Look-uptable

Look-uptable

un

cos(φ(τ ; un; Sn))

sin(φ(τ ; un; Sn))

Lan

d,

Fleu

ry:

Digital

Mod

ulation

2S

IPC

om

43

Example: GMSK with L = 3

Tilted phase tree:

0 1 2 5−1

3 4 6 7

0

1

2

3

4

5

6

t/T

φ(t

;α)/

π

Trajectories of φ(τ ;Un, Sn

):

0.2 0.3 0.5 0.6 0.7 0.8 0.9 1.0

0.4

0.6

0.8

1.2

1.6

0.2

0.1 0.4 τ/T

1.0

1.4

1.8

2.0

0

φ(τ

;un,s

n)/

π

3


44

3 Laurent Representation of CPM

CPM signals can be equivalently represented by a superposition of

PAM signals [3]. In this section we use only the first term of this

representation as a (fairly) good approximation of the CPM signal.

3.1 Minimum Shift Keying

The MSK Signal

The MSK signal is given by

x(t;α) =

√2Eb

Tcos(

2πf0t + ϕ(t;α))

with the information-bearing phase

ϕ(t;α) = π

N−1∑

n=0

αn qMSK(t− nT ),

t ∈ [0, NT ).

The information symbols are

αn ∈{−1,+1

}, n = 0, . . . , N − 1.

The phase pulse qMSK(t) has the following shape:

tT

qMSK(t)

1

2


45

In the nth signaling interval, t ∈ [nT, nT + T ), t = nT + τ ,

τ ∈ [0, T ), we have

ϕn(τ ;α) =π

2

n−1∑

i=0

αi

︸︷︷︸θn

+π

2

τ

Tαn.

Remember that θn is the phase at the beginning of the nth signaling

interval.

Notice that

θn ∈{{

0, π}

for n even{π2 ,

3π2

}for n odd

The initialization is θ0 = 0.


46

In-phase and quadrature components of MSK

Consider t ∈ [0, NT ). The MSK signal can be decomposed into

its inphase and its quadrature components:

x(t;α) =

√Eb

Tcos(ϕ(t;α)

)

︸︷︷︸xI(t;α)

·√

2 cos(2πf0t

)

−√Eb

Tsin(ϕ(t;α)

)

︸︷︷︸xQ(t;α)

·√

2 sin(2πf0t

)

Consider now the pulse

pMSK(t) =

{1√T

cos( πt2T ) for t ∈ [−T,+T ],

0 elsewhere.

It is normalized such that it has energy one, i.e.,

+T∫

−T

(pMSK(t)

)2

dt = 1.

−T T t0

1/√

T

pMSK(t)


47

Using this pulse, the inphase component xI(t;α) and the quadra-

ture component xQ(t;α) can be written as follows:

xI(t;α) =

√Eb

Tcos(ϕ(t;α)

)

=

N∑

n=0n even

x1,n · pMSK(t− nT )

x1,n =√Eb cos (θn) ∈

{−√Eb,+

√Eb

}

xQ(t;α) =

√Eb

Tsin(ϕ(t;α)

)

=

N∑

n=1n odd

x2,n · pMSK(t− nT )

x2,n =√Eb sin (θn) ∈

{−√Eb,+

√Eb

}

The symbol αN is a known suffix symbol.

The modulation symbols can thus be considered as two-

dimensional symbols

Xn =

(X1,n

X2,n

)∈ S

out of the signal constellation

S =

{(+√Eb

0

)

︸︷︷︸s1

,

(−√Eb

0

)

︸︷︷︸s2︸︷︷︸

n even

,

(0

+√Eb

)

︸︷︷︸s3

,

(0

−√Eb

)

︸︷︷︸s4︸︷︷︸

n odd

}


48

Illustration of the signal constellation:

s4

s3

s2 s1

n even

n odd

Example: Phase Trajectories

Draw the phase and the trajectories of the inphase and quadrature

components for the following example.

n αn θn cos θn sin θn0 +1 θ0 = 0(init) 1 0

1 −1 π2 0 1

2 +1 0 1 0

3 +1 π2 0 1

N − 1 = 4 +1 π −1 0

Suffix N = 5 (+1) 3π2 0 −1

(1)

3


49

Complex Representation

The Vector Xn = (X1,n , X2,n)T can be conceived as the complex

number

Xn = X1,n + jX2,n

=√Eb exp(jθn) θn =

π

2

n−1∑

i=0

αi

The (complex-valued) signal constellation is thus

s3

s4

s1

1

−j

−1

j

s2

The values of the complex-valued modulation symbol can be re-

cursively computed:

Xn =√Eb · exp

(jπ

2

n−1∑

i=0

αi

)

= exp(jπ

2αn−1

)·√Eb exp

(jπ

2

n−2∑

i=0

αi

)

= jαn−1 ·Xn−1


50

Notice that

exp(jπ

2αn−1

)= (j)αn−1 = jαn−1

as αn−1 ∈ {−1,+1}.

Thus the MSK signal can be written as

X(t;α) = <{[

N∑

n=0

Xn pMSK(t− nT )

]√

2 exp(j2πf0t)

}

for t ∈ [0, NT ).

The initialization is

X0 = j α−1 X−1 =√Eb.

This can be achieved by

X−1 = j√Eb,

α−1 = −1.

Block Diagram

T

T

αn αn−1

j

xnpMSK(t)

jαn−1

xn−1N∑

n=0

δ(t − nT )

Re{·}x(t; α)

√2 exp(j2πf0t)


51

3.2 Precoded MSK

Precoding

The information symbols are mapped from the {0, 1} representa-

tion to the {+1,−1} representation: un 7→ αn

un

{0, 1} {−1, +1}

αn0

1 −1

+1

The precoding can be done in both representations:

T

0

1 −1

+1

0

1 −1

+1

T

un−1

un

un−1un

un

αn

αn−1

αnαn−1

αnαn−1

Notice that un−1 ⊕ un (addition in F2 ≡ GF (2)) corresponds

to αn−1 · αn (multiplication in R). This justifies the particular

mapping.


52

Complex Representation

Transmitted symbols:

Xn = j(αn−1 αn−2)Xn−1, n = 0, . . . , N − 1

Initialization:

α−2 = α−1 = −1

X−1 = j√Eb

The symbols are generated as follows:

n Xn

0 j j√Eb =

√Eb = j0 α−1

√Eb

1 j α0 α−1 j0 α−1

√Eb = j1 α0

√Eb

2 j α1 α0 j1 α0

√Eb = j2 α1

√Eb

3 j α2 α1 j2α1

√Eb = j3 α2

√Eb

... ...

n jn αn−1

√Eb

This gives the block diagram of precoded MSK using the complex

representation of the symbols:

T

αn

jn

αn−1

pMSK(t)

N∑

n=0

δ(t − nT )

Re{·}xn x(t; α)

√2 exp(j2πf0t)


53

Vector Representation

n even: Xn = (j)n√Eb αn−1

= (j2)n2√Eb αn−1

= (−1)n2√Eb αn−1 (real-valued)

Xn =(

(−1)n2

√Eb αn−1︸︷︷︸

Xn,1

, 0︸︷︷︸Xn,2

)T

n odd: Xn = (j)n√Eb αn−1

= j(j)n−1√Eb αn−1

= j(−1)n−1

2√Eb αn−1 (imaginary-valued)

Xn =(

0︸︷︷︸Xn,1

, (−1)n−1

2

√Eb αn−1︸︷︷︸

Xn,2

)T

This results in the following block diagram:

T

pMSK(t)

pMSK(t)

αn αn−1

t ∈ [0, 1]√

Eb

nT

n even

n odd

N∑

n=0

δ(t − nT )

(−1)bn

2c Xn,2

X(t; α)

Xn,1

√2 cos(2πf0t)

−√

2 sin(2πf0t)


54

Dem

od

ula

tion

inth

eA

WG

NC

han

nel

pMSK(t)

pMSK(t)

T

pMSK(t)

pMSK(t)

T

t ∈ [0, 1]

N∑n=0

δ(t − nT )

W (t)

Y (t)

nT

Vector

decoder

non-causal filter

−T

αn αn−1

√Eb

nT

n even

n odd

αn αn−1

√Eb

rn

rn

Vectordecoder

xn,1

αn−1

pMSK(−t) = pMSK(t)

tT0

Vn,1

Vn,2

xn,2

nT

αn−1

E[Vn,iVm,j] = δijN0

2rm−n

Rpp(τ ) =

∫+∞

−∞pMSK(t)pMSK(t + τ )dt

rn = Rpp(nT )

(−1)bn

2c

(−1)bn

2c

Xn, 1

Xn,2

√2 cos(2πf0t)

−√

2 sin(2πf0t)

√2 cos(2πf0t)

−√

2 sin(2πf0t)

X(t; α)

Yn,1

Yn,2

Lan

d,

Fleu

ry:

Digital

Mod

ulation

2S

IPC

om

55

The effective discrete-time impulse response is

rn = Rpp(nT ),

Rpp(τ ) =

∫pMSK(t) pMSK(t + τ ) dt.

The noise after sampling has the covariance (mean is zero)

E[Vn,iVM,j

]= δi,j

N0

2rm−n.

The optimum vector decoder for MSK can be simplified. To see

this, consider the impulse response:

rn = Rpp

rn =

1 for n = 0,

r1 for |n| = 1,

0 elsewhere.

−T T 2T−2T 0 n

rn

Thus, in the upper (lower) branch, symbols are effectively trans-

mitted at even (odd) time indices.


56

Example for the upper branch:

n

x0,1

T

2T

3T

4T

5T n

x0,1

x2,1 x4,1

x4,1x2,1

r1x2,1r1x2,1

r1x4,1 r1x4,1

r1x0,1

xn,1

Sampling time

xn,1 ∗ rn

← upper branch

The optimum vector decoder for the AWGN channel can thus be

described as follows.

T

αn αn−1

√Eb

rn

rn

Vn,1

Vn,2

nT

Sign(·)αn−1

(−1)bn

2c (−1)b

n

2c

Xn,1

Xn,2

Yn,1

Yn,2

Remark: More information about the discrete-time representation

of QAM-like signals is provided in the appendix.


57

3.3 Precoded Gaussian MSK

The lenght of the phase pulse is set to L = 3, as before.

Precoded GMSK Signal

The signal reads

x(t;α) =

√2Eb

Tcos(

2πf0t + ϕ(t;α))

with

ϕ(t;α) = π

N−1∑

n=−2

(αnαn−1) qGMSK(t− nT ) + ϕ0

where

• αn ∈ {−1,+1};prefix symbols (GSM): αn = −1 for n = −3,−2,−1,

sufffix symbols (GSM): αn = −1 for n = N,N + 1, N + 2

• GMSK phase pulse qGMSK(t):

t/T0 0.5 1.5 2 31 2.5

0.1

0.2

0.3

0.4

0.5

0.6

qGMSK(t)


58

• ϕ0 is set so that ϕ(0;α) = 0. Hence,

ϕ0 = −π2∑

n=1

qGMSK(nT )

Within the n-th signaling interval, i.e., for

t ∈ [nT, nT + T ), n = 0, . . . , N − 1.

ϕn(τ ;α) = ϕ0

+π

2

n−3∑

i=−2

(αi αi−1)

+

n∑

i=n−2

(αi αi−1) qGMSK(τ )

In-phase and Quadrature Components

The signal can be decomposed as

X(t;α) =

√Eb

Tcosϕ(t;α)

︸︷︷︸XI(t;α)

·√

2 cos(2πf0t)

−√Eb

Tsinϕ(t;α)

︸︷︷︸XQ(t;α)

·√

2 sin(2πf0t).


59

The inphase component and the quadrature component can be

approximated as follows:

XI(t;α) ≈N+2∑

n=−2n even

X1,n · pGMSK(t− nT )

X1,n = (−1)bn2c√Eb αn−1

XQ(t;α) ≈N+1∑

n=−1n odd

X2,n pGMSK(t− nT )

X2,n = (−1)bn2c√Eb αn−1

The pulse pGMSK(t) is often denoted as c0(t) pulse, referring to [3],

and it is defined as follows.

Let LT , L = 3, be the length of the truncated GMSK pulse

qGMSK(t). Then

c0(t) = s0(t) · s1(t) · s2(t)

(for larger L, more pulses have to be multiplied) with

sl(t) = sin(φ(t + lT )

)

φ(t) =

{π · qGMSK(t) for t < LTπ2 − π · qGMSK(t) for t ≥ LT

The resulting pulse c0(t) has length (L + 1)T = 4T .


60

The resulting pulse is depicted for L = 3:

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

t

pGMSK(t)

2T 3T 4T0 T


61

Block diagram of precoded GMSK

Using the vector representation:

T

αn αn−1

√Eb

nT

n even

n odd

(−1)bn

2c

N+2∑

n=−2

δ(t − nT )

pGMSK(t)

pGMSK(t)

Xn,1

Xn,2

X(t; α)

√2 cos(2πf0t)

−√

2 sin(2πf0t)

Using the complex representation:

T Re{·}

N+2∑

n=−2

δ(t − nT )√

2exp(j2πf0t)

αn αn−1

jn√

Eb

pGMSK(t)x(t; α)Xn

Initialization (prefix symbols):

α−3 = α−2 = α−1 = −1,

X−2 = j−2α−3

√Eb =

√Eb.

Termination (suffix symbols):

αN = αN+1 = αN+2 = +1.


62

Dem

od

ula

tion

ofp

reco

ded

GM

SK

inth

eA

WG

Nch

an

-

nel

T

T

t ∈ [0, 1]

W (t)

nT

Vector

decoderαn αn−1

√Eb

nT

n even

n odd

αn αn−1

√Eb

rn

Vectordecoder

Vn,1

Vn,2

nT

rn = Rpp(nT )

(−1)bn

2c

(−1)bn

2c

pGMSK(t)

pGMSK(t)

N+2∑n=−2

δ(t − nT )

pGMSK(t)

pGMSK(t)

{αn}N−1

n=0

{αn}N−1

n=0

Rpp(τ ) =

∫ LT

2

−LT

2

pGMSK(t)pGMSK(t + τ )dt

rn

Xn,1

Xn,2

Yn,1

Yn,2

Xn,1

Xn,2

√2 cos(2πf0t)

−√

2 sin(2πf0t) −√

2 sin(2πf0t)

√2 cos(2πf0t)

Lan

d,

Fleu

ry:

Digital

Mod

ulation

2S

IPC

om

63

The effective impulse response has the (deterministic) autocorre-

lation function:

Rpp(τ ) =

∫pGMSK(t) pGMSK(t + τ ) dt

0.4

−5 −4 −3 −2 −1

0.1

0

0.2

0.3

0.5

0.6

0.7

0.8

0.9

1.0

0 1 2 3 4 5 τ/T

Rpp(τ )

We have approximately

rn =

1 for n = 0

0.05 for |n| = 2

0 for |n| > 2

Hence, ISI occurs in each branch. For optimum decoding, the

Viterbi algorithm (VA) may be applied.


64

(Close-to-optimum) Vector Decoder

T

αn αn−1

√Eb

rn

Vn,1

Vn,2

nT

(−1)bn

2c

nT

nT

n even

n odd

0 ≤ n ≤ N + 2

0 ≤ n ≤ N + 2

rn

VA

VA

{αn}N−1

n=0,n even

{αn}N−1

n=1,n odd

Two-state

Two-state

Xn,1

Xn,2

nT

−2 ≤ n ≤ N + 2

α

Metrics to be used in the upper-branch VA (even n):

λ(αe) =

N−1∑

n=0n even

(−1)bn2c√Eb αn−1·

·[yn − r1(−1)bn2−1c √Eb αn−3

]

Metrics to be used in the lower-branch VA (odd n):

λ(αo) =

N−1∑

n=1n odd

(−1)bn2c√Eb αn−1·

·[yn − r1(−1)bn2−1c √Eb αn−3

]


65

GSM Burst Structure

Normal burst

3 Tail bits

Synchronization burst

Information bits

8,25guardbits

8,25guardbits

1 control bitTime slot

3 Tail bits

(57)

Trainingsequence Information bits

Information bits

Information bits

(57)

(39)(64)(39)

(26)

1 control bit

Training sequence


66

3.4 Modulation in EDGE

Objective

Modify the GSM format in order to increase the bit rate while

keeping the spectrum requirement of this modulation scheme.

Selected Solution

• Extend the signal space in the Laurent representation of

GMSK

−→ 8PSK, i.e., 3 bits / symbol

• Include an additional rotation of the symbols

−→ rotation of 3π/8 in each signaling interval

Signal Constellation and Mapping

Im

(1, 1, 1)

(U1, U2, U3)

(0, 1, 0)

(0, 1, 1)

(0, 0, 1)

(1, 0, 0)

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

(0, 0, 0)

Re


67

Phase Rotation

Rotation by 3π8 in each signaling interval

Im

Re

4 5

6

0

7

3π

8

1

2

3

Im

Re

4 5

6

0

7

3π

8

1

2

3

By comparison, for 8 PSK without such phase rotation:

Im

Re

0

2

6

3

7

1

5

4

Im

Re

0

2

6

3

7

1

5

4


68

Pulse Shaping

p(t) = pGMSK(t)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

t

pGMSK(t)

2T 3T 4T0 T

EDGE Transmitter

pGMSK(t)

√Eb

Re{·}Xn

X(t; u)[Un,1, Un,2, Un,3]

exp(jn3π8

)N−1∑

n=0

δ(t − nT )

√2 exp(j2πf0t)


69

EDGE Receiver

pGMSK(t)Vector

decoder

nT

[Un,1, Un,2, Un,3]Y (t) Yn

√

2 exp(−j2πf0t)

Discrete-time Representation of the EDGE System

Xnrn

Vector

decoder

Vn

[Un,1, Un,2, Un,3]

exp(jn3π8

)

[Un,1, Un,2, Un,3]

The effective impulse response is

rn = Rpp(nT )

Rpp(τ ) =

LT∫

0

pGMSK(t) pGMSK(t + τ ) dt

The noise has the covariance

E[VnVn+l

]= N0rl.


70

0.4

−5 −4 −3 −2 −1

0.1

0

0.2

0.3

0.5

0.6

0.7

0.8

0.9

1.0

0 1 2 3 4 5 τ/T

Rpp(τ )

Optimum Vector Decoder for EDGE

For optimum vector decoding of EDGE in the AWGN channel, the

Viterbi algorithm may be applied. The corresponding trellis has

• J = 82 = 64 states and

• 8 transitions per state.

The branch metric is

λ(u) =

N−1∑

i=0

<{x∗n exp

(−jn3π

8

)·

·[yn − r1xn−1 exp

(j(n− 1)

3π

8

)

− r2Xn−2 exp(j(n− 2)

3π

8

)]}


71

Discrete-time Model

Using the complex-valued representation, the discrete-time model

of EDGE is give by the following block diagram:

Xn

VA

64-statern

[Un,1, Un,2, Un,3]

Vn

[Un,1, Un,2, Un,3]

exp(jn3π8

)

The vector with information bits looks like

U =[[U0,1 , U0,2 , U0,3

], . . . ,

[UN−1,1 , UN−1,2 , UN−1,3

]]

Burst format of EDGE

Data symbols

(57)

Trainingsequence

(26)

Data symbols

(57)

Tai

lSym

(3)

Tai

lSym

(3)

Guard

(8,25)

Time slot

Con

trol

sym

(1)

symbols

Normal burst

Con

trol

sym

(1)


72

4 Trellis-Coded Modulation

4.1 Motivation

We consider the vector representation of a digital communication

system transmitting across the AWGN channel.

Encoder Decoder

{0, 1}K {0, 1}K

[U0, ..., UK−1]

[W 0, ...., W N−1]

[U0, ..., UK−1][X0, ..., XN−1] [Y 0, ..., Y N−1]

We restrict our attention to the case where the dimension of the

signal constellation X = {S0 . . . ,SM−1} is one or two.

Example:

MPAM: dim X = 1 MPSK (M > 2) and QPAM : dim X = 2.

2-AM

4-PSK 8-PSK

16-QASK8-AMPM

32-AMPM 64-QASK

8-AM

4-AM

16-AM

3


73

For these two cases, the vector channel is either of the form.

• dimX = 1

independent

Xj Yj

Wj ∼ N (0, N0

2), W0, ..., WN−1

• dimX = 2

Xj1

Xj2 Yj2

Wj1

Wj2

Yj1

XjY jWj ∼ N

((

0

0

)

,

(

N0/2 0

0 N0/2

))

In both cases, the noise samples are independent.


74

Rate of the encoder

The Encoder maps bit-blocks onto modulation symbols

x ∈ X ={s0, . . . , sM−1

}.

Its rate is

R =K

Nbits/symbol

Two equivalent necessary conditions for the encoder to be one-to-

one are

2K ≤MN ⇔ R =K

N≤ log2(M)

In the case of uncoded modulation, we have

N = 1, K = log2(M) ⇒ R = log2(M)

Bit Error Probability

The BER is defined as

Pb =1

K

K−1∑

j=0

P[Uj 6= Uj

]


75

Capacity of the AWGN Channel

With the constraint that the transmitted vectors belong to a finite

signal constellation X = {s1 . . . , sM}:

CX = maxp(s1),...,p(sM )

M∑

m=1

p(sm) ·

·(∫

p(y|sm

)log2

( p(y|sm

)p(sm)

∑Mm′=1 p

(y|sm′

)p(sm′)

)dy

)

Remark

The capacity CX depends on the signal constellation X.

Dropping the restriction that X is finite while assuming a given

average power Es = E[|Xj|2

],

C := log2

(1 + SNR

), SNR =

Es

N0

The dimension of C is bit / transmitted symbol.


76

Tight lower bounds for the capacities of the signal constellations

given on page 72:

Channel Coding Theorem

Provided R < C, Pb can be made arbitrarily small by selecting

appropriate encoders.

In other words, reliable communication is possible at ratesR < C.

Converse of the Channel Coding Theorem

If R > C, reliable communication is impossible.


77

Example: Some comparisons

• Uncoded 4−PSK:

◦ R = 2 bits/T

◦ Pb = 10−5 at SNR = 12.9 dB

• Coded 8−PSK:

Remark: C∗ = 2 bits/symbols at 5.9 dB


78

Thus, with coded 8PSK it is theoretically possible to transmit

reliably 2 bits/symbol at SNR = 5.9 dB.

• Coding gain of coded 8PSK transmitting 2 bits/symbol ver-

sus uncoded 4PSK at BER = 10−5.

12.9− 5.9 = 7 dB

• Coding gain of coded digital transmission across the AWGN

channel

◦ with no restriction on the signal constellation except the

average transmitted power,

◦ transmitting (on average) 2 bits/symbol

versus coded 4PSK at BER = 10−5:

12.9− 4.7 = 8.2 dB

• Similar figures for the above coding gains hold when other

signal constellations are considered.

3

Remark

By doubling the number of vectors in a signal constellation X,

containing 2K vectors, almost all is achieved in terms of coding gain

for reliable transmission of K bits/T versus uncoded transmission

using the signal constellation X. (The transmission rates are the

same!)


79

A way to realize the above coding gains by doubling the number

of vectors in the signal constellation is as follows:

with rate mapping

Vector

Uj,1

Vj,1

Xj

Encoder

Vj,K

Vj,K+1

{0, 1}K+1{0, 1}K X

|X| = 2K+1

Uj,K

.

.

.

.

.

.{0, 1}K+1 → X

r = KK+1

Open issues:

1. how to select the encoder, and

2. how to design the vector mapping,

to obtain a scheme for which Pb is small.

For the encoder, block coding or convolutional coding may be used.

In many schemes, convolutional codes are applied due to their low

decoding complexity. (The Viterbi algorithm is applicable.)

A schemes that consists of the combination of a convolutional en-

coder with a vector mapping device is called Trellis-Coded Modu-

lation (TCM).


80

Block Diagram of a TCM Scheme

.

.

.

mapping

Vector

Uj,1

Vj,1

Convolutional

Encoder

Xj

{0, 1}K

r = KK+1

{0, 1}K+1

|X| = 2K+1Vj,K+1

Uj,K memory

X

Vj,K{0, 1}K+1 → X

length ν

The overall rate is given by

R = r · log2(K + 1) = K

Objective

Find TCM schemes that achieve a good bit-error-rate performance.

Example: Coded 8PSK

TT

011

010

001

000

111

100

Vj,1

Vj,2

Vj,3

Uj,1

Uj,2

101

110

Xj

(V3, V2, V1)

The parameters are r = 2/3, ν = 2, R = 2/3 · log2 8 = 2. 3


81

4.2 Convolutional Codes

Two examples of convolutional encoders. (See e.g. Proakis or

Morelos-Zaragoza for a general description.)

Example: 1

TT

Vj,1

Vj,2

Vj,3

Uj,1

Uj,2

Code rate:

r =number of input bits

number of output bits=

2

3

Memory length:

ν = number of shift registers = 2

Trellis:

00

10

01

11 11

01

01

00000

001

011 011

labels: (v3, v2, v1)

100

010

110

101

101

111001

States

3


82

Example: 2

T

TUj,2

Uj,1

Vj,1

Vj,2

Vj,3

Uj−1,1

Uj−2,2

Code rate:

r =2

3

Memory length:

ν = 2

Trellis:

00

01

10

11

00

01

10

11

3


83

4.3 Goodness of a TCM Scheme

Admissible sequences

Let A∞ denote the set of sequences (x0,x1, . . .) generated by a

given TCM scheme. These sequences are called admissible for

the considered TCM scheme.

Example: 1 continued

A∞ ∈

{xj} =(s0 , s0 , s0 , s0 , s0, . . .

)

{x′j} =(s0 , s4 , s0 , s0 , s0, . . .

)

{x′′j} =(s2 , s1 , s2 , s0 , s0, . . .

)

A∞ /∈{{x′′′j } =

(s0 , s5 , s3 , s2 , s0, . . .

)

0

4

6

2

2

4

1

5

7

3

5

17

3

0 0

21

1

2

3

4

5

6

7

0

60

4 4√

(2 −

√

2)Es

√

Es

√

2E

s

√

(2+√

2)E

s

3


84

Euclidean Distance

Let {xj} and{x′j}

denote two sequences in A∞.

The Euclidean distance between {xj} and {x′j} is

d({xj}, {x′j}

)=

( ∞∑

j=0

∣∣xj − x′j∣∣2)1/2


d({xj}, {x′j}

)=√

4Es = 2√Es

d({xj}, {x′′j}

)=

√(2 + (2−

√2) + 2

)Es = 2.1414

√Es

3

Free Euclidean Distance

The minimum Euclidean distance between pairs of admissible se-

quences of a TCM scheme is called the free Euclidean distance of

the TCM scheme:

df = min{xj},{x′j} ∈ A∞

xj 6=x′j

d({xj}, {x′j}

)


It can be shown that

df = d({xj}, {x′j}

)= 2

√Es

3


85

Probability of a Sequence Error

Probability that the Viterbi decoder makes a wrong decision on

the path in the trellis:

Pe ≥ N(df) · Q( df√

2N0

)

where

(i) Q(z) is the Gaussian error function

Q(z) =

∞∫

z

1√2π

exp

{−1

2u2

}du

(ii) N(df) is the average number of sequence pairs in A∞ that

have the Euclidean distance df .

Remark

The above lower bound for Pb is tight for high SNR. The reason is

that in this case an erroneously detected path has a sequence likely

to have an Euclidean distance to the true transmitted sequence

equal to df , when ML decoding is applied.

Remember that given a finite observed sequence, {yj} =(y0,y1, . . . ,yL−1

), the Viterbi algorithm (which realizes the ML

decoder) searches for the finite sequence generated by the trellis

(with initial state 0) which minimizes

L−1∑

j−0

∣∣yj − xj∣∣2


86

Optimality Criterion

Optimality criterion for finding a vector mapping for a given con-

volutional code:

“Maximize the free Euclidean distance !”

TCM schemes maximizing the free Euclidean Distance among all

TCM with a specified memory length are called optimal.

The method proposed by Ungerboeck to identify ”good” vector

mapping given a certain convolutional encoder (or equivalently,

given a certain trellis) relies on the technique of set partitioning.


87

4.4 Set Partitioning

Example: 8PSK

√

Es

d0 =√

(2 −√

2)Es

0 1B0B1

sqrtEs

d1 =√

2Es

C00 1 0 1C2

C1C3

d2 = 2√

Es

000 100 010 110 001 101 011 111

D7D3D1 D5D6D2D41000

v2

v3

v1

D01 1 0 1

3

Goals of set partitioning:

1. The subsets should be similar .

2. The points inside each subset should be maximally separated.


88

4.5 Trellis Coded Modulation

General form of the encoding process:

Convolutional

Encoder

..

..

..

..

..

..

..

..

Selectsubset

Vector Mapping

{1, 2, . . . , 2N1}

Select vector

from subset

{1, 2, . . . , 2K−K1}

r′ = K1

N1

Uj,1

Uj,K1

Uj,K1+1

Uj,K

Vj,1

Vj,N1−1

Vj,N1

Vj,N1+1

Vj,N

Xj

Remarks

1. Number of states in the trellis : 2ν

2. Number of branches leaving each state : 2K1

3. Number of parallel branches in each transition : 2K−K1


89


Parameters: K = 2, K1 = 1, N1 = 2, N = 3

TT

Vj,1

Vj,2

Vj,3

Uj,1

Uj,2

xj

0 1 0 1 0 1 0 1

0 0 1 1 0 0 1 1

0 0 0 0 1 1 1 1

0 1 2 3 4 5 6 7

04

62

2

4

1

5

7

3

5

173

0 0

21

60

4 4C0

C1

C2

C3

C2

C3

C0

C1

(0, 4) (2, 6)

(3, 7)(1, 5)

(2, 6) (0, 4)

(1, 5)(3, 7)

3


90

Rules for the Assignments

Experimentally inferred rules for assigning the signal vectors to the

transitions:

1. All vectors in X are used with the same frequency.

2. The vectors assigned to branches originating from or merging

to the same state belong to the same subset at level 1.

3. Vectors with maximum Euclidean distances are assigned to

parallel branches.

Remarks

• Rule 1 guarantees that the trellis code has a regular structure.

• Rule 2 and Rule 3 guarantee that the minimum distance

between paths in the trellis which diverge from any state and

re-emerge later exceeds the free Euclidean distance of the

uncoded modulation scheme.


(df)2coded8PSK

= 4Es ≥ 2Es = (df)2uncoded

4PSK

3


91


Consider the following trellis code:

T

TUj,2

Uj,1 Uj−1,1

Uj−2,2

Vj,1

Vj,2

Vj,3

0 1 2 3 4 5 6 7

0 0 0 0 1 1 1 1

0 0 1 1 0 0 1 1

0 1 0 1 0 1 0 1

Xj

Despite the fact that this trellis code has no parallel branches, its

free Euclidean distance is smaller than that of the trellis code of

Example 1. (See Exercise 4.1). 3


92

4.6 Examples for TCM

In the following, some examples of specific TCM schemes are given

and their error probabililites are discussed.


93

TCM with 8PSK and 2 bits/T


94

Implementation


95

Probability of error-event and of bit-error


96


97

TCM with 16QAM and 3 bits/T

Set Partitioning


98


99

Error event probability


100

CCITT V.32 standard

CCITT V.32 standard for 9.6 Kbit/s and 14.4 bit/s modems.

Convolutional Coder

The code is designed to guarantee invariance to phase rotations

of ±π/2 and π. These values are the phase ambiguities which

result when a phase-locked loop (PLL) is employed for carrier-

phase estimation.


101

Mapping


102

A ECB Representation of a Band-Pass Signal

Band-pass signals can equivalently be represented as complex-

valued base-band signals. This is often called the equivalent com-

plex base-band (ECB) representation.

This concept is discussed in more detail in the following.


103


104

A.1

Th

eB

lock

dia

gra

m

rn

rn

Vn,1

Vn,2

Xn,1

Xn,2 Yn,2

Yn,1

1

T sitT

1

T sitT

X(t)

W (t)

Y (t)

Xn,1

Xn,2

1

T sitT

1

T sitT

nT

p(t)

p(t)

Yn,1

Yn,2

N∑

n=0

δ(t − nT )

p(f )Assumption:

f0 ≥ 1

2T

B ≤ 1

T

1/T

− 1

2T1

2T

Yn,1 = rn ∗ Xn,1 + Vn,1

Yn,2 = rn ∗ Xn,2 + Vn,2

p(−t)

p(−t)

X2(t)

X1(t)

√2 cos(2πf0t)

−√

2 sin(2πf0t)

√2 cos(2πf0t)

−√

2 sin(2πf0t)

Lan

d,

Fleu

ry:

Digital

Mod

ulation

2S

IPC

om

105

Complex discrete-time representation:

Xn YnYn = Yn,1 + jYn,2

Vn

Yn = rn ∗ Xn + Vn

Xn = Xn,1 + jXn,2

Vn = Vn,1 + jVn,2

rn


106

A.2 The Signal

Consider a real-valued BP signal:

x(t) = x1(t)√

2 cos(2πf0t)− x2(t)√

2 sin(2πf0t)

= <{[x1(t) + jx2(t)︸︷︷︸

= x(t)

][√2 cos(2πf0t) + j

√2 sin(2πf0t)︸︷︷︸√

2 exp(j2πf0t)

]}

= <{x(t)√

2 exp(j2πf0t)}

The signal x(t) is a complex-valued LP signal, and it is called the

ECB representation of x(t).

The spectra

X(f ) = F{x(t)

}X(f ) = F

{x(t)

}

are related by

X(f ) =

√2

2

[S(f − f0) + S(−f − f0)

]

The energies of the signals are

Es = Es


107

A.3 Effect of QA Mod/Demod on Signal

x(t) = x1(t)√

2 cos(2πf0t)− x2(t)√

2 sin(2πf0t)

x1(t) =∑

i

xi,1 δ(t− iT ) ∗ p(t)

x2(t) =∑

i

xi,2 δ(t− iT ) ∗ p(t)

Assumption: p(t) is a LP signal, i.e., its Fourier transform P (f ) =

F{p(t)} has the property

P (f ) = 0 for |f | > f0

Tricks:

2 cos2 x = 1 + cos 2x

2 sin2 x = 1− cos 2x

2 sinx cosx = sin 2x

In-phase component:

z1(t) = x(t)√

2 cos(2πf0t) ∗ p(−t)=[x1(t) 2 cos2(2πf0t)︸︷︷︸

1 + cos(4πf0t)

−x2(t) 2 sin(2πf0t) cos(2πf0t)︸︷︷︸sin(4πf0t)

]∗ p(−t)

= x1(t) ∗ p(−t) +[x1(t) cos(4πf0t)− x2(t) sin(4πf0t)

]

︸︷︷︸BP

∗ p(−t)︸︷︷︸LP

︸︷︷︸= 0


108

z1(t) = x1(t) ∗ p(−t)=∑

i

xi,1 δ(t− iT ) ∗ p(t) ∗ p(−t)︸︷︷︸= Rp(t)

=∑

i

xi,1 Rp(t− iT )

After sampling:

z1,n = z1(nT )

=∑

i

xi,1 Rp(nT − iT )︸︷︷︸Rp

((n− i)T

)= rn−i

=∑

i

xi,1 · rn−i = x1,n ∗ rn

rn = Rp(nT )

Quadrature component:

(...similar calculations...)

zn,2 = z2(nT ) = Xn,2 ∗ rn


109

A

x1(t)

A

A/2

A

A/2

A A

A

A/2

A√

2

2

x1(t)√

2cos(2πf0t) ...cos(2πf0t)

A

pLP (t)

A

A/2

x2(t)

x2(t)(−√

2)sin(2πf0t)

jA√

2

2

A

A/2

...pLP (t)

x1(t)√

2cos(2πf0t) + x2(t)(−√

2)sin(2πf0t)

−2f0 −f0 f0 2f0

...(−√

2)sin(2πf0t)

Lan

d,

Fleu

ry:

Digital

Mod

ulation

2S

IPC

om

110

A.4 Effect of QA Demodulation on Noise

Block Diagram

√

2cos(2πf0t)

√

2sin(2πf0t)

Z2(t)

Z1(t) Y1(t)

Y2(t)

W (t)

p(−t)

p(−t)

Y1(nT ) = Yn,1

Y2(nT ) = Yn,2

w(t): WGN with Sw(f ) = N02

Noise before LP filtering

Z1(t) = W (t)√

2 cos(2πf0t)

E[Z1(t) Z1(t + τ )

]=

= E[W (t)W (t + τ )

]︸︷︷︸

=N0

2δ(τ )

(√

2)2 cos(2πf0t

)cos(2πf0(t + τ )

)

=N0

2δ(τ ) 2 cos2(2πf0t)︸︷︷︸

1 + cos(4πf0t)

=N0

2δ(τ ) +

N0

2δ(τ ) cos(4πf0t)

︸︷︷︸time-variant past

= RZ1(τ ; t)

→ Time-variant (periodic) auto-correlation function


111

Noise after LP filtering

Assumption: p(t) is a LP signal, i.e., its Fourier transform P (f ) =

F{p(t)} has the property

P (f ) = 0 for |f | > f0

Autocorrelation function of Y1(t):

Y1(t) = W (t)√

2 cos(2πf0t) ∗ p(t)

E[Y1(t)Y1(t + τ )

]=

= E[∫

W (t− s)√

2 cos(2πf0(t− s)

)p(s)ds ·

·∫W (t + τ − s′)

√2 cos

(2πf0(t + τ − s′)

)p(s′)ds′

]

=

∫ ∫E[w(t− s) w(t + τ − s′)

]

︸︷︷︸N0

2δ(τ − s′ + s)

·

· 2 · cos(2πf0(t− s)

)· cos

(2πf0(t + τ − s′)

)·

· p(s) p(s′) ds ds′

Integrate w.r.t. s′ ⇒ 6= 0 for τ − s′ = −s ⇔ s′ = τ + s

=N0

2

∫2 cos2

(2πf0(t− s)

)︸︷︷︸

1 + cos(4πf0(t− s)

)p(s) p(τ + s) ds

=N0

2Rp(τ ) +

N0

2

∫cos(4πf0(t− s)

)p(s) p(τ + s) ds

︸︷︷︸= cos

(4πf0(t− τ )

)p(τ ) ∗ p(−τ )


112

E[Y1(t) Y1(t + τ )

]=N0

2Rp(τ )

+N0

2

∫cos(4πf0(t− s)

)p(s) p(τ + s) ds

︸︷︷︸(1)

Rp(τ ) = p(τ ) ∗ p(−τ )

(1) =[cos(4πf0(t− τ )

)p(τ )

]∗ p(−τ ) (6)

The Fourier transform w.r.t. τ of

cos(4πf0(t− τ )

)= cos

(4πf0(τ − t)

)= cos

(4πf0τ

)∗ δ(τ − t)

is

1

2

[δ(f − 2f0) + δ(f + 2f0)

]exp(−j2πft) =

=1

2

[δ(f − 2f0) exp(−j4πf0t) + δ(f + 2f0) exp(j4πf0t)

]= (3)

Consider the Fourier transform w.r.t. τ of (6):

(2) =[(3) ∗ P (f )

]· P (−f )

=1

2

[P (f − 2f0) exp(−j4πf0t)

+ P (f + 2f0) exp(j4πf0t)]· P (−f )

= 0

because P (f ) = 0 for |f | > f0 according to the assumption.


113

⇒ E[Y1(t) Y1(t + τ )

]

︸︷︷︸RY1(τ )

=N0

2Rp(τ )

⇒ RY1(τ ) =N0

2Rp(τ )

A.5 Noise After LP Filtering and Sampling

RY1(kT ) =N0

2Rp(kT )︸︷︷︸

= rk

=N0

2rk


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