Lecture 5

Trellis Coded ModulationReferences :1. G. Ungerboeck, “Trellis coded Modulation With Redundant signals sets. Part I, Part II” IEEE Commun. Mag. vol.25, pp.5-12, pp.12-22, Feb.1987.2. G. Ungerboeck, “Channel Coding With Multi-level/Phase signals” IEEE Tran. Inform. Theory vol.IT-28, pp.55-67, Jan.1982.3. Book : “Introduction to Trellis-Coded Modulation With Applications”.

Motivation of TCM

Generally, there exist two possibilities to compensate for the rate loss caused by using error-correcting codes :

<l> increasing the modulation rate if the channel permits bandwidthexpansion.

<ll> enlarging the signal set of the modulation system if the channel is band-limited.

→ If modulation and error-correction coding are performed in theclassical independent manner, disappointing results are obtained.

Example:To use an error-correcting code with the same rate as uncoded 4-PSK, we can use either <l>or <II>.

<l> a convolutional code with rate 2/3 and 4-PSK modulation.Comment: The duration T 2/3 T, therefore bandwidth W 2/3 W!

<ll> a convolutional code with rate 2/3 and 8-PSK modulation.Comment: If the 4-PSK system operates at an error of , at the same SNR, the “raw” error rate at the 8-PSK demodulator exceeds because of the smaller spacing between the 8-PSK signals.

510−

Patterns of at least 3 bits errors must be corrected to reduce the error rate to that of the uncoded 4-PSK system.→ A rate- , 64-state binary convolutional code has

After all this effort, error performance only breaks even with that of uncoded4-PSK!

∴

23 m in 7 .Hd =

Two problems contribute to this unsatisfactory situation :

<l>The independent “hard” signal decisions made prior to decoding→ cause an irreversible loss of information in the receiver.

Let unquantized “soft” output samples of the channel be

where : the discrete signals sent by the modulator.: samples of an additive white Gaussian noise process

The decision rule of the optimum sequence decoder :Among the set C of all coded signal sequences, determine with minimum squared Euclidean distance from . That is,

The minimum squared Euclidean distance (MSED) is called the squared “free distance”.

n n nr a w= +

nanw

{ }ˆna{ }nr

{ }{ }

2ˆ arg minn

n n na C

a r a∈

= −∑

{ } { }{ } { }

22

,

minn n

n n

free n na ba b C

d a b≠

∈

= −∑

<ll> Mapping code symbols of a code optimized for Hamming distance into nonbinary modulation signals.→ does not guarantee that a good Euclidean distance structure is

obtained.

Squared Euclidean and Hamming distances are equivalent only in the case of binary modulation or four-phase modulation.

Binary modulation systems with codes optimized for Hamming distance and soft decision decoding have been well established since the late 1960s for power-efficient transmission at spectral efficiencies of less than 2 bit/sec/Hz.

Consider a transmission of 2 bit/T by uncoded 4-PSK modulation, where occurs at SNR=12.9 dB. If thenumber of channel signals is doubled, (e.g. 8-PSK), an error-free transmission of 2 bit/T is theoretically possible at SNR = 5.9dB. But only 1.2dB can further be gained if there is no constraint on the number of signals.

By doubling the number of channel signals, almost all is gained in terms of channel capacity that is achievable by signal-set expansion.

( ) 510rP e −=

Examples of TCM

Asymptotic coding gain (ACG)

where is the minimum Euclidean distance for uncoded

transmission, is the free distance for coded transmission,

and and E are the average energies spent to transmit with uncoded and coded transmission respectively.

2

2m in

/A C G

/freed E

d E=

′

mind

mind

E′

Uncoded QPSK

0 00

2 22

4

6

4 4

6 6

2min 2d =

2 0 2Δ =

4 0

6

Four-state trellis coded 8-PSK

220

2212 22

2sin 0.5868

2 2

2 4

π⎛ ⎞Δ = =⎜ ⎟⎝ ⎠

Δ = =

Δ = =

(a)Parallel transitions are associated with signals with maximum distance , the signals in the subsets (0,4), (1,5), (2,6) or (3,7).

(b) Four transitions originating from or merging in one state are labeled with at least distance between them, the signals in the subsets (0,4,2,6) or (1,5,3,7).

(c) All 8-PSK signals are used in the trellis diagram with equal frequency.

2Δ

1Δ

The 8-PSK signals are assigned to the transitions in the four state trellis in accordance with the following rules :

Any two signal paths that diverge in one state and remerge in another after more than one transition have at least squared Euclidean distance between them.

← parallel transitions

ACG over uncoded QPSK : (dB)

any state transition along any coded 8-PSK sequence transmitted, there exists only one nearest-neighbor signal at free distance, which is the rotated version of the transmitted signal.

2 2 21 0 1Δ + Δ + Δ

( )2 2 2 21 0 1 2Δ +Δ + Δ > Δ

2 4f r e ed∴ =

41 0 lo g 32=

180

Realization

1

0

2

1

32

0

3

D D

0 0 0 0 1 1 1 1

0 0 1 1 0 0 1 1

0 1 0 1 0 1 0 1

signal : 0 1 2 3 4 5 6 7

1u0u

The code is invariant to a signal rotation by , but to no other rotations.

180

At high SNR, the error event probability is generally well approximated by

where

is the variance of Gaussian noise samples in each signal dimension.

is the (average) number of nearest-neighbor signal sequences with distance .

( )2

freer free

dP e N Q

σ⎛ ⎞

≈ ⋅ ⎜ ⎟⎜ ⎟⎝ ⎠

( )2

21 ,2

y

xQ x e dy

π∞ −

= ∫2σ

freeNfreed

In TCM schemes with more trellis states and other signal sets , is not necessarily found between parallel transitions, and will generally be an average number larger than one.

freed

freeN

2 2 21 0 2.586freed = Δ +Δ =

2.58610 log 1.12

=

Two-state trellis coded 8-PSK

ACG over uncoded QPSK: (dB)

Eight-state trellis code for amplitude/phase modulation

0Δ

{ }0 4 2 6, , ,D D D D { }1 5 3 7, , ,D D D D 1 02Δ = Δ

{ }0 4,D D { }2 6,D D { }1 5,D D { }3 7,D D2 04Δ = Δ

iD3 08Δ = Δ

Distance :A0 :

or :or or or :

, i = 0~7 :

8-state trellis→ 4 transitions diverge from and merge into each state

To each transition, one of the subset is assigned.If A0 contains signals, each of its subsets will comprise signals→ parallel transitions→ signals can be sent from each state, as required to encode m

bits per modulation interval.

The assignment of signal subsets to transitions satisfies the same three rules of coded 8PSK.

0 7, ,D D12m+ 22m−

22m−

2m

To compute :The four transitions from or to the same state are always assigned either the subsets or .→ A squared Euclidean distance of at least when sequences diverge and when the remerge (1) If paths remerge after two transitions, SED between the

diverging transitions total SED

(2) If paths remerge after three or more transitions, at least one intermediate contributes an additional SED total SED .

(3) Parallel transition SED == ACG = (dB).

2freed

{ }0 4 2 6, , ,D D D D { }1 5 3 7, , ,D D D D

≥ 206Δ≥ 2

04Δ

0D 0D

4D6D

2Δ 1Δ

20Δ

205Δ≥

208Δ

∴ 2freed 2

05Δ= ACG =2020

510 log 42Δ

⋅ ≈Δ

(dB).

Roughly speaking, we can get the coding gain with :4 states : 3dB8 states : 4dB16 states : 5dB128 or more states : up to 6dB

(1) Doubling the number of states does not always yield a code with better error performance. → The distance growth is limited.

(2) Numbers of nearest neighbors, and neighbors with next-largest distance are increased.→ prevent real coding gains exceeding the ultimate limit set by channel capacity.

Design of TCM Schemes

General structure of encoder for TCM

m bits are transmitted per operation.

bits ( ) are expanded by a rate - binary convolutional encoder into coded bits.

The bits are used to select one of subsets of a redundant

-ary signal set.The remaining uncoded bits determine which of the signals is to be transmitted.

12m+

12m+

m m−

2m m−

m m m≤ ( )/ 1m m+1m+

m

Set partitioning :

If the labels of two subsets

agree in the last q positions,

but not in the bit , then the

signals of the two subsets

are elements of the same

subset at level q in the

partition tree; thus they have

at least distance .

The free Euclidean distance of a TCM code , where

is the minimum distance between parallel transitions and is the

minimum distance between nonparallel paths in the TCM trellis diagram.

( )1min ,free m freed d m+⎡ ⎤= Δ⎣ ⎦

1m+Δ ( )free md

( ){ } { } ( )

2 2minnn

free qn

d m≠

≥ Δ∑ ee 0

Minimization has to be carried out over all non-zero code sequences that deviate at, say, time 0 from the all-zero sequence and remerge with it at a later time.

{ }ne{ }0

is the number of trailing zeros in . For example, =2,

if . not !

( )nq e ( )nq ene3, , ,1,0,0m

n nn e e⎡ ⎤⎣ ⎦=e ( ) 0,qΔ =0 1m+Δ

Let and be two code sequences. Since the convolutionalcode is linear, is also a code sequence. The distance between signals in the subsets selected by and is lower-bounded by .

{ }nZ { } { }n n n′ = ⊕Z Z e

{ }ne

n′ZnZ ( )nqΔ e

Usually, this smallest distance equals for all . Only when the signal subsets contain very few signals may the bound not be satisfied with equality. For example, for 8PSK, . But for all possible or or or, the squared Euclidean distance is larger then !

This equation is of key importance in the compute distance between every pair of TCM signal sequences.

( )nqΔ e ne

[ ]1 0 1n = →e ( ) ( ) 00 0.586nn qq = →Δ = Δ =ee

{ },n n n n′ = ⊕Z Z Z e { }000,111= { }001,100 { }010,111 { }011,11020Δ

Two encoder realizationsfeedback-free encoder.systematic encoder with feedbackA systematic encoder cannot generate a catastrophic code.

Parity-check equation : A linear convolutional code of rate is most compactly defined by a parity-check equation which puts a constraint on the code bits in a sliding time window of length :

/( 1)m m+

1υ +

( )1 1 00

0m

i i i i i in nn

ih Z h Z h Zυ υ υ υ− − − +

=⊕ ⊕ ⊕ =∑

where : constraint length: modulo-2 addition: the i-th code bit at time j

: binary parity-check coefficients of the code

υ⊕

ijZ

{ }0 ,1ilh ∈

Valid code sequences satisfy this equation at all times n. Note that the equation defines only the code sequences, not the input/output relation of an encoder.

A minimal encoder is realized with binary storage elements.That is, this code has trellis states.2υ

υ

For example2 1 0 0

2 1 3 0nn n nZ Z Z Z− − −⊕ ⊕ ⊕ =

0 2 1 02 1 3 n n n nZ Z Z Z− − −⇒ = ⊕ ⊕

Another 8-state TCM

Effects of Carrier-Phase OffsetReference: L. F. Wei, “Rotationally invariant convolutional channel coding with expanded signal space-Part II : nonlinear codes,” IEEE J. Select. Area Commun., pp.672-686, 1984.

The soft-decision decoder operates on a sequence of complex-valued signals,where : transmitted TCM signals

: additive Gaussian noise: phase offset

{ } { }jn n nr a e wφΔ= ⋅ +

nanwφΔ

The coded 8PSK fail at = 22.5°. In contrast, uncoded4PSK requires a higher signal-to- noise ratio at small phase offset, but has an operating range up to = 45° in the absence of noise. These results are typical for TCM schemes.

φΔ

φΔ

In the trellis diagrams of TCM schemes, there exist long distinct paths with low growth of signal distance between them, that is, paths which have either the same signals or signals with smallest distance assigned to concurrent transitions.

0Δ

In most digital carrier- modulation systems, decision-directed loops are employed for carrier-phase tracking. The figure at left side illustrates, for 4-state 8PSK, the mean estimate of and its variance as a function of the actual value of the phase offset.

φΔ

Conventionally, the problem of phase ambiguity is solved by using a differential encoding technique.

For 2D signal constellations, it is impossible to achieve 90° invariance with linear codes (the best that can be done is 180° invariance ), and hence nonlinear codes are needed for full rotational invariance.

In 1984, Wei introduced nonlinear elements into the convolutional encoder of the 8-state. This made the code invariant to 90° rotations while maintaining its coding gain of 4dB. This TCM was adopted in the CCITT (now ITV-T ) V.32、V.33 Recommendation (for voice-band modem ).

Multi-dimensional TCM schemes with 90° phase invariance can be obtained with linear codes.

Multi-Dimensional Trellis Coded Modulation

References: 1. S.S. Pietrobon et.al “Trellis-Coded Multidimentional Phase Modulation,”

IEEE Tran. Inform Theory, pp.63-89, Jan.19902. S.S. Pietrobon and D.J. Costello, Jr. “Trellis Coding with Multidimentional

QAM Signal Sets,” IEEE Tran. Inform. Theory, pp.325-336, March 1993

2×8PSK set partition0 0u = 0 1u =

1 0u = 1 1u =

2 0u = 2 1u =

0 0.586Δ =

1 02Δ = ⋅Δ

2 12Δ > ⋅Δ

3 22Δ = ⋅Δ

Consider ( 2, 2, 1 ) block code. Information bits are .0 1,u u

Consider ( 4, 4, 1 ) block code. Information bits are . 0 1 3 4, ,,u u u u

.

L=2

( )

[ ][ ]

m

----------------------------------------0 1 2 0 1

1 2 1 1 1

Tmm md N T

2×1y { }1 2, 0,1, ,7y y ∈

2y

Consider 8PSK: the first 8PSK point: the second 8PSK point

{ }2 1 004 2 , for 1,2 with 0,1i

j j j jy y y y j y= + + = ∈

5 4 31

2

2 1 0

4 0 24 4 2

0 1 0 + 2 1 1

y Z Z Zy

Z Z Z

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

= + +

+ +

for i =0,1,2( 2) 1 2

2ii iy Z Z× + ×= ⊕

( 2) 11

iiy Z × +=

With modulo-2 addition

With modulo-8 addition

( mod 8)

Consider 3 X 8PSK

Consider 4× 8PSK

For QAM