Bit-Interleaved Coded Modulation (BICM) - a Tutorial

Bit-Interleaved Coded Modulation(BICM) - a Tutorial

Vignesh Sethuraman and Bruce Hajek

University of Illinois

Urbana-Champaign

Bit-Interleaved Coded Modulation (BICM) - a Tutorial – p.1/23

http://www.ifp.uiuc.edu/~vsethura

http://tesla.csl.uiuc.edu/~hajek

Outline

• Outline

• Trellis-coded modulation (TCM) - an overview

• Channel model: Coded modulation (CM) andBICM

• Information-theoretic framework and results• Error probability analysis• Design guidelines and numerical results


Outline

• Outline• Trellis-coded modulation (TCM) - an overview

• Channel model: Coded modulation (CM) andBICM



Outline

• Outline• Trellis-coded modulation (TCM) - an overview• Channel model: Coded modulation (CM) and

BICM



Outline


BICM• Information-theoretic framework and results

• Error probability analysis• Design guidelines and numerical results


Outline


BICM• Information-theoretic framework and results• Error probability analysis

• Design guidelines and numerical results


Outline


BICM• Information-theoretic framework and results• Error probability analysis• Design guidelines and numerical results


Channel model

Consider a vector channel characteried by afamily of transition probability density functions(pdf)

{pθ(Y |X) : θ ∈ CM ; X, Y ∈ CN}

Channel state θ: stationary, finite memory random

process pθ(Y |X) =∏

k pθk(Yk|Xk)


Finite memory of channel stateprocess

There exists an integer ν > 0 such that, for all

r-tuples ν < k1 < . . . < kr and for all n-tuples

j1 < . . . < jn < 0, the sequences (θk1, . . . , θkr

) and

(θj1, . . . , θjn) are statistically independent.


Coded modulation (CM)

bn → ENC → cn → µ, χ→ π → pθ(Y |X)→π−1 → DEM, DECIdeal interleaver: For any K ⊂ Z with |K| <∞,

Eθ[∏

k

pθk(Yk|Xk)] =

∏

k

Eθk[pθk

(Yk|Xk)]


Detection for CM

• Full channel state information (CSI): Set ofML-symbol metrics is given by{log pθk

(Yk|z)}z∈χ

c = argmaxc∈C

∑

k

log pθk(Yk|µ(ck))}

• No CSI: Define p(Y |X) = Eθ[pθ(Y |X)]


BICM

Binary code C → ENC → π → µ, χ→ Channel

χ ⊂ CN : |χ| = 2m


Notation

• Codeword c→ π(c)→ Break intosub-sequences, m-bits each → µ

• Interleaver: π : k → (k‘, i)

• li(X): ith bit of label of X ∈ {0, 1}

• χib = {X ∈ χ : li(X) = b}


BICM contd.

Assume ideal interleaving

pθ(Y |li(x) = b) =

1

2m−1

∑

X∈χib

pθ(Y |X)

ML detection: For each signal time k‘, DEMproduces 2m such metrics:

λi(Yk‘, b) = log∑

X∈χib

pθk‘(Yk‘|X)


Simpli£ed Bit-metrics

log∑

j

Zj ≈ maxj

logZj

BICM branch metric:

λi(Yk‘, b) = maxX∈χi

b

log pθk‘(Yk‘|X)


Equivalent channel model

System can be seen as an equivalent parallelchannel.


Information-theoretic view ofBICM: Capacity

CM:

CCM = m− EX,Y

[

log2

∑

Z∈χ p(Y |Z)

p(Y |Z)

]

BICM:

CBICM = m I(b;Y |S)

= m−m∑

i=1

Eb,Y

[

log2

∑

Z∈χ p(Y |Z)∑

Z∈χibp(Y |Z)

]


Information-theoretic view ofBICM: Capacity (contd.)

Also, b→ XCM→ Y

Since, conditioned on X, Y and b are statistically

independent, CCM ≥ CBICM .


Information-theoretic view ofBICM: Cutoff rate

Cutoff rate:• Was important for comparing channels where

£nite complexity coding schemes were used.• Closed form expressions for cutoff rate of CM

and BICM are given.


Information-theoretic view ofBICM: Numerical results

Numerical results are presented for nonselectiveRician fading channels.

Y = gXejφ +N

• BICM is shown to be a more robust choicethan CM.

• No CSI: Choose χ to be N -ary orthogonal(N = 2m). eg PPM or Hadamard sequences.


Error Probability Analysis

Symmetrization:• In the parallel channel model, to make the

channel symmetric, de£ne µ = complementof µ.

• For each coded bit bi, let Ui be a binaryrandom variable determining whether µ or µis used.

• Assume U is known to the receiver.



CSI is assumed here.To calculate PEP(c, c) = PEP(c⊕ c) = f(d, µ, χ):

WlOG, assume c and c differ in d consecutive po-

sitions.



Union bound for linear binary codes:

Pb ≤

{

1

kc

∑∞d=1

WI(d)f(d, µ, χ) conv. codes kc

nc

1

Kc

∑Nc

d=1WI(d)f(d, µ, χ) Block codes Kc

Nc

where WI(d) is the total input weight of error

events at distance d.


Upper Bounds on f (d, µ, χ)

Bhattacharyya Union Bound:

• I{x1 ≤ x2} ≤√

x2

x1

•

I{pθ(Y |c, S) ≤ pθ(Y |c, S)} ≤

√

pθ(Y |c, S)

pθ(Y |c, S)

• f(d, µ, χ) ≤ Bd


Upper Bounds on f (d, µ, χ):Notation

χSc = χi1c1 × . . .× χikck × . . .× χidcd

χSc = χi1c1 × . . .× χikck × . . .× χidcd


Notation contd.

Path metric difference (corresponding tosequences):

δ =d∑

k=1

maxZk∈χ

ikck

log pθk(Yk|Zk)

−d∑

k=1

maxZk∈χ

ikck

log pθk(Yk|Zk)

= maxZ∈χ

Sc

log pθ(Y |Z)−maxZ∈χ

S

c

log pθ(Y |Z)


BICM Union Bound

P (c→ c|S, U) = P (δ ≤ 0|S, U)

= Ex

[

P

(

maxZ∈χ

Sc

pθ(Y |Z) ≤ maxZ∈χ

S

c

pθ(Y |Z)|S, U, x

)]

≤ Ex

[

P

(

pθ(Y |X) ≤ maxZ∈χ

S

c

pθ(Y |Z)|S, U, x

)]

≤ Ex

∑

Z∈χS

c

P (pθ(Y |X) ≤ pθ(Y |Z)|S, U, x)


Conclusion

• Presented an overview of TCM systems.

• Presented an introduction to BICM.• BICM was analyzed in an

information-theoretic framework.• Bounds on probability of error were

constructed and compared.• Design guidelines and numerical results were

studied.


Conclusion

• Presented an overview of TCM systems.• Presented an introduction to BICM.

• BICM was analyzed in aninformation-theoretic framework.

• Bounds on probability of error wereconstructed and compared.

• Design guidelines and numerical results werestudied.


Conclusion

• Presented an overview of TCM systems.• Presented an introduction to BICM.• BICM was analyzed in an

information-theoretic framework.

• Bounds on probability of error wereconstructed and compared.



Conclusion



constructed and compared.



Conclusion



constructed and compared.• Design guidelines and numerical results were

studied.


Documents

Bit-Interleaved Coded Modulation (BICM) - a Tutorial