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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
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
Bit-Interleaved Coded Modulation (BICM) - a Tutorial – p.2/23
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
Bit-Interleaved Coded Modulation (BICM) - a Tutorial – p.2/23
Outline
• Outline• Trellis-coded modulation (TCM) - an overview• Channel model: Coded modulation (CM) and
BICM
• Information-theoretic framework and results• Error probability analysis• Design guidelines and numerical results
Bit-Interleaved Coded Modulation (BICM) - a Tutorial – p.2/23
Outline
• Outline• Trellis-coded modulation (TCM) - an overview• Channel model: Coded modulation (CM) and
BICM• Information-theoretic framework and results
• Error probability analysis• Design guidelines and numerical results
Bit-Interleaved Coded Modulation (BICM) - a Tutorial – p.2/23
Outline
• Outline• Trellis-coded modulation (TCM) - an overview• Channel model: Coded modulation (CM) and
BICM• Information-theoretic framework and results• Error probability analysis
• Design guidelines and numerical results
Bit-Interleaved Coded Modulation (BICM) - a Tutorial – p.2/23
Outline
• Outline• Trellis-coded modulation (TCM) - an overview• Channel model: Coded modulation (CM) and
BICM• Information-theoretic framework and results• Error probability analysis• Design guidelines and numerical results
Bit-Interleaved Coded Modulation (BICM) - a Tutorial – p.2/23
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)
Bit-Interleaved Coded Modulation (BICM) - a Tutorial – p.3/23
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.
Bit-Interleaved Coded Modulation (BICM) - a Tutorial – p.4/23
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)]
Bit-Interleaved Coded Modulation (BICM) - a Tutorial – p.5/23
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)]
Bit-Interleaved Coded Modulation (BICM) - a Tutorial – p.6/23
BICM
Binary code C → ENC → π → µ, χ→ Channel
χ ⊂ CN : |χ| = 2m
Bit-Interleaved Coded Modulation (BICM) - a Tutorial – p.7/23
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}
Bit-Interleaved Coded Modulation (BICM) - a Tutorial – p.8/23
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)
Bit-Interleaved Coded Modulation (BICM) - a Tutorial – p.9/23
Simpli£ed Bit-metrics
log∑
j
Zj ≈ maxj
logZj
BICM branch metric:
λi(Yk‘, b) = maxX∈χi
b
log pθk‘(Yk‘|X)
Bit-Interleaved Coded Modulation (BICM) - a Tutorial – p.10/23
Equivalent channel model
System can be seen as an equivalent parallelchannel.
Bit-Interleaved Coded Modulation (BICM) - a Tutorial – p.11/23
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)
]
Bit-Interleaved Coded Modulation (BICM) - a Tutorial – p.12/23
Information-theoretic view ofBICM: Capacity (contd.)
Also, b→ XCM→ Y
Since, conditioned on X, Y and b are statistically
independent, CCM ≥ CBICM .
Bit-Interleaved Coded Modulation (BICM) - a Tutorial – p.13/23
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.
Bit-Interleaved Coded Modulation (BICM) - a Tutorial – p.14/23
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.
Bit-Interleaved Coded Modulation (BICM) - a Tutorial – p.15/23
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.
Bit-Interleaved Coded Modulation (BICM) - a Tutorial – p.16/23
Error Probability Analysis
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.
Bit-Interleaved Coded Modulation (BICM) - a Tutorial – p.17/23
Error Probability Analysis
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.
Bit-Interleaved Coded Modulation (BICM) - a Tutorial – p.18/23
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
Bit-Interleaved Coded Modulation (BICM) - a Tutorial – p.19/23
Upper Bounds on f (d, µ, χ):Notation
χSc = χi1c1 × . . .× χikck × . . .× χidcd
χSc = χi1c1 × . . .× χikck × . . .× χidcd
Bit-Interleaved Coded Modulation (BICM) - a Tutorial – p.20/23
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)
Bit-Interleaved Coded Modulation (BICM) - a Tutorial – p.21/23
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)
Bit-Interleaved Coded Modulation (BICM) - a Tutorial – p.22/23
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.
Bit-Interleaved Coded Modulation (BICM) - a Tutorial – p.23/23
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.
Bit-Interleaved Coded Modulation (BICM) - a Tutorial – p.23/23
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.
• Design guidelines and numerical results werestudied.
Bit-Interleaved Coded Modulation (BICM) - a Tutorial – p.23/23
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 werestudied.
Bit-Interleaved Coded Modulation (BICM) - a Tutorial – p.23/23
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.
Bit-Interleaved Coded Modulation (BICM) - a Tutorial – p.23/23
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