Bit interleaved coded modulation

transcript

Seminar on Signal Processing in Wireless Communications 2011

Bit Interleaved Coded Modulation

Mridula Sharma

February 28, 2011

Outline• Introduction

• System Model: CM(Coded Modulation) and BICM (Bit

Interleaved Coded Modulation)

• Information-theoritical Framework and Results

• Error Probability Analysis

• BICM-ID

•BICM-OFDM

•Summary

Introduction• 1982: Ungerboeck: landmark paper on Trellis Coded Modulation (TCM)

• highly efficient transmission of information over band-limited channels such as telephone lines

• 1992: Zehavi: performance of coded modulation over Rayleigh fading channel can be improved

• Bit-wise interleaving at the encoder output• Appropriate soft-decision metric as an input to Viterbi decoder

• Modulation + Coding: Single entity for improved performance

• Bit Interleaved Coded Modulation (BICM)

• 1998: Caire: Information-theoritical view on BICM

Introduction•Wireless Fading Channels

• Non-recursive non-Systematic Convolutional (NSC) code

• Type of Serial Concatenated Code (SCC)

• Coded bits are interleaved prior to modulation

• increase the diversity order of TCM schemes

• uses bit-interleavers for all the bits of a symbol

• number of bit-interleavers equals to the number of bits assigned

to one non-binary codeword

• interleaved bits are collected into Gray labeled non-binary

symbols

Introduction•Purpose of the bit-interleaver:

• Disperse the burst errors and maximize the diversity order of the system• Uncorrelate the bits associated with the given transmitted symbol

Binary Encoder

Bit -Interleaver

Symbol Mapper

Flat fading Channel

m-bits define a symbol

Due to the interleaving the input bits to the mapper are approx. independent

Introduction

Fig : BICM Overview

The combination of binary encoding, bitwise interleaving, and M-ary modulationactually yields better performance in fading than symbol-wise interleaving andtrellis-coded modulation (Caire 1998)

BinaryEncoder

BitwiseInterleaver

Binaryto M-arymapping

M-ary-modulator

Soft-InBinary

Decoder

BitwiseDeinterleaver

LLRBit Metric

Calculation

Receiverfrontend

Complex flat-fading

ku kc kc' ms )(ts

)(trrk'kku

Gray Mapping• Let χ denote a signal set of size M=2m with a minimum Euclidean distance dmin

• A binary map µ= {0, 1} m χ is a Gray labeling for χ if for all i= 1……..m and bϵ {0, 1}, each x ϵ χ b

i has at most one z ϵ χ bi at

distance dmin

Fig : 16QAM Symbol arrangement chart with Gray labeling

Gray Mapping• Key component of a BICM system

• Main Function: to produce an equivalent channel that has ʋ parallel, independent, memoryless binary channels

• Each channel corresponds to a position in the label of a signal x ϵ χ

• For each codeword at the output of the binary encoder, the interleaver assigns at random a position in the label of the signals to transmit the coded bits

Set Partitioning

Set Partitioning• As proposed by Ungerboeck:

• Errors for bit a1 can easily occur, because adjacent symbols of 8PSK will necessarily have different a1s

• If a1 is assumed to be correct, then a2 changes every other symbol of 8PSK and a symbol distance the same as that of QPSK will be obtained

• If a1 and a2 are assumed to be correct, then a3 can be determined if a decision can be made as to which diagonal symbol has been received, and a symbol distance the same as that of BPSK will be obtained

Fig : 16QAM Symbol arrangement chart with Set Partitioning

Building Blocks• Encoder (ENC)

• Interleaver π

• Modulator, modeled by a labeling map μ and a signal set χ, i.e., a finite set of points in the complex N-dimensional Euclidean space CN

• A stationary finite-memory vector channel whose transition probability density function pƟ(y|x), x,y ϵ CN may depend on a vector parameter Ɵ

• Demodulator (DEM)

• Branch Metric Deinterleaver π -1

• Decoder (DEC)

System Model

Fig: Block diagram of transmission with coded modulation (CM) and bit-interleaved coded modulation (BICM). In the case of CM, π denotes interleavingat the symbol level. In the case of BICM, π denotes interleaving at the bit level.

ENC π µ, χ pƟ(y|x) DEM π -1 DEC

Vector Channel Model• Consider a vector channel characterized by a family of transition probability density functions (pdf)

{ pƟ(Y|X) : Ɵ ϵ CM; X,Y ϵ CN }

• Channel state Ɵ: stationary, finite memory random processpƟ(Y|X) = ∏k pƟk (Yk|Xk)

• Finite Memory of Channel State Process : There exists an integer ʋ > 0 such that, for all r-tuples ʋ < k1 < . . . < kr and for all n-tuples j1 < . . . < jn < 0, the sequences (Ɵk1 ; . . . ; Ɵkr) and (Ɵj1 ; . . . ; Ɵjn) are statistically independent

Vector Channel Model• Large number of typical communication channels can be represented

• Additive White Gaussian Noise (AWGN) channel (Ɵ = constant)• AWGN channel with random phase (Ɵ is the residual phase due to

imperfect carrier phase recovery) • Frequency nonselective slow-fading channels (Ɵ describes the

multiplicative fading process)

• But Inter-symbol Interference (ISI), or frequency selectivity infading channels cannot be accounted for

• Channel state depends on the input sequence

Coded Modulation

•Non-uniform error correction to non-uniform symbol distances for multiphase/ multi-level modulation

• Digital modulation• Error correction

Coded Modulation

• Detection for CM: (assuming ideal interleaver)• Full channel state information (CSI): rule for the transmitted code

sequence

• No CSI: channel is not memoryless, • Also, assuming ideal interleaver: For any K ϲ Ƶ with |K|<∞,

• new average transition pdf: p(Y|X)= EƟ[pƟ(Y|X)]

Coded Modulation

Fig : Configuration of an 8PSK modulator using coded modulation

Fig : Configuration of an Ungerboeck coded modulator

Coded Modulation

Fig : Performance of coded modulation using convolutional code

Binary Code Ĉ ENC μπ

χ Channel

Bit Interleaved Coded Modulation(Notation)

• ĉ π (ĉ) Break into sub-sequences, m-bits each μ

• Interleaver, π : k (k‘, i)

• li(x): ith bit of label X ϵ {0, 1}

• χib = { X ϵ χ: li(X) = b}

• Assuming Ideal Interleaving,

• ML detection: For each signal time k’, DEM produces 2m such metrics:

Bit Interleaved Coded Modulation(Simplified Bit Metrics)

• BICM Branch Metric:

• Sub-optimal branch metrics can be obtained by the log sum approximation which is good as long as the sum in the LHS is dominated by a single term as typically occurs in channels with high SNR

Equivalent Channel Model• System can be seen as an equivalent parallel channel model

Fig: Equivalent parallel channel model for BICM in the case of ideal interleaving

Information-theoritic view of BICM: Capacity

Unlike CM, the capacity of BICM depends on how bits are mapped to symbols

• As with CM, BICM Capacity can be computed using a Monte Carlo integration

• CM

• BICM

• Also, b X CM Y

• Since, conditioned on X, Y and b are statistically independent,

CCM ≥ CBICM

Fig: CM and BICM capacity for 16QAM in AWGN

Information-theoritic view of BICM: Cut-off Rate

• Cut-off Rate: • Was important for comparing channels where finite

complexity coding schemes were used• Cut off Rate Ȓ o of the discrete-input continuous-output

channel generated by a CM scheme

, perfect CSI

, no CSI

• The cutoff rate of a BICM scheme can be obtained from theBhattacharyya bound on the average bit-error probability Pb

of the parallel channel model in the absence of coding •By considering the ML bit metrics with perfect CSI

where B denotes the average Bhattacharyya factor of the BICM channel, with perfect CSI

• Perfect CSI

• No CSI

• Note: A single channel use of the BICM channel is equivalent to m-channel uses of a binary-input channel with average Bhattacharyya factor B

• Hence, cut-off rate Ȓ o for BICM: (resorting to Monte Carlo numerical integration for calculation)

Ȓ o= m(1-log2(B+1))

Information-theoretic view of BICM: Numerical results

• Numerical results are presented for nonselective Rician fading channels (which encompass Rayleigh and AWGN as special cases)

Y = g X ejɸ + N

• N is a complex zero-mean Gaussian i.i.d. random vector with covariance

•g is a scalar complex fading gain• ɸ is the carrier phase, independent of X and g and uniformly distributed over [-π, π]

Fig: BICM and CM cutoff rate versus SNR for QAM signal sets with Gray (or quasi-Gray) labeling over AWGN with coherent detection

Fig: BICM and CM cutoff rate versus SNR for QAM signal sets with Gray (or quasi-Gray) labeling over Rayleigh fading with coherent detection and perfect CSI

Information-theoretic view of BICM: Numerical results

• BICM is shown to be a more robust choice than CM

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

Error Probability Analysis

• Symmetrization:

• Time-varying labeling map: In the parallel channel model, to make the channel symmetric

• μ’ = compliment of μ

• For each coded bit bi, let Ui be a binary random variable determining whether μ’ or μ is used

• Assume U is known to the receiver

• Assuming CSI

• c and ĉ denoting two distinct sequences stemming from

the same state and merging after ɭ ≥ 1 trellis steps

• Assume c and ĉ differ in d consecutive positions

• Pairwise Error Event: {c ĉ }

• Pairwise Error Probability (PEP): P(c ĉ)

• P(c ĉ)= f (d, µ, χ)

• Union Bound for linear binary codes:

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

Upper Bound on f (d, µ, χ)• Given earlier by Bhattacharyya Union Bound

• f (d, µ, χ) ≤ Bd

• BICM Union Bound derived free of Bhattacharyya and Chernoff upper bounds

• loose but provided basis for tight upper bounds

• Tight upper bound to the PEP of BICM for Rician fading channels with perfect CSI

BICM with Iterative Decoding (BICM-ID)

BinaryEncoder

BitwiseInterleaver

Binaryto M-arymapping

M-ary-modulator

Soft-InBinary

Decoder

BitwiseDeinterleaver

LLRBit Metric

Calculation

Receiverfrontend

Complex flat-fading

ku kc kc' ms )(ts

)(trrk'kku

BitwiseInterleaverSoft-Output Estimates

of Coded Bits

BICM-ID• Converts a 2m ary signaling scheme into m independent parallel binary schemes

•First iteration - Gray labeling optimal here• Gray labeling has a lower number of nearest neighbors compared to SP -

based labeling.• The higher the number of nearest neighbor the higher the chances for a

bit to be decoded into wrong region

• Second iteration• The soft information allows to confine the decision region into a pair of

constellation points• We want to maximize the minimum Euclidean distance between any two

points in the possible phasor pairs for all the bits

BICM-ID

• Feeding back from decoder to demod can improve the performance of noncoherent M-FSK

•For M=16 and r=⅓ coding, the improvement is 0.7 dB in Rayleigh flat fading

•The additional complexity is negligible• No extra iterations needed• Only need to update demod metrics during each iteration

BICM in Orthogonal Frequency Division Mutiplexing (OFDM)

• Employed in the WLAN standard IEEE 802.11a • channel can be considered quasi-static and frequency-

selective

• Powerful, yet easily implementable scheme • channel coherence bandwidth is about the same like the

Fourier transmission bandwidth

• Random position of coded bits in the subcarrier symbols (assuming an ideal interleaver)

• good performance of BICM in OFDM schemes

BICM-OFDM

Fig: System Model of an adaptive BICM-OFDM

FEC π Mod IFFT GI

DEC π-1 Mod-1 FFT GI-1

bc x d

802.11a Transmitter

channelencoder

andpuncturer

QAMmapper iFFT

add cyclicextension(guard)

addtrainingsymbols

interpol.and filter,

limiter

bit interleaver

add pilotsymbols

D/A up-converter

amplifier

binary source

• Channel encoder (error correcting coding) and QAM symbol mapper are connected through a bit interleaver

802.11a Receiver

decimateandfilter

synchr.frequencycorrection FFT QAM

demapper

bit deinterleaver

de-punct.and

channeldecoder

down-converter

amplifier A/D

frequ.offset

estimator

channelestimator

andtracker

binary sink

pilotremoval

Summary

• BICM

• system model

• analyzed in information-theoritical framework

• error probability analysis

• BICM-ID

• BICM-OFDM

References[1] G. Caire, G. Taricco, E. Biglieri, “Bit-interleaved coded modulation”, IEEE Trans. Inf. Theory, vol. 44, no. 3, pp. 927-946, May 1998

[2] Martinez_et_al,“Bit-Interleaved Coded Modulation in the Wideband Regime“,2008

[3] E. Zehavi, “8-PSK trellis codes for a Rayleigh channel”, IEEE Trans. Commun., vol. 40, pp. 873-884, May 1992

[4] Bockelmann et al, “Efficient Coded Bit and Power Loading for BICM-OFDM“, IEEE 2009

[5] Samahi et al, “Comparative Study for Bit-Interleaved Coded Modulation with Iterative Decoding”, 2009 Fifth Advanced International Conference on Telecommunications

Thank You!!!

Bit interleaved coded modulation

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