IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 44, NO. 3, MAY 1998
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`Bit-Interleaved Coded Modulation
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`Giuseppe Caire, Member, IEEE, Giorgio Taricco, Member, IEEE, and Ezio Biglieri, Fellow, IEEE
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`Abstract—It has been recently recognized by Zehavi that the
`performance of coded modulation over a Rayleigh fading channel
`can be improved by bit-wise interleaving at the encoder output,
`and by using an appropriate soft-decision metric as an input
`to a Viterbi decoder. The goal of this paper is to present in
`a comprehensive fashion the theory underlying bit-interleaved
`coded modulation, to provide tools for evaluating its performance,
`and to give guidelines for its design.
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`Index Terms—Bit-interleaving, channel capacity, coded modu-
`lation, cutoff rate, fading channel.
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`I. INTRODUCTION AND MOTIVATIONS
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`EVER since 1982, when Ungerboeck published his land-
`
`mark paper on trellis-coded modulation [19], it has been
`generally accepted that modulation and coding should be
`combined in a single entity for improved performance. Of
`late, the increasing interest for mobile-radio channels has led
`to the consideration of coded modulation for fading channels.
`Thus at first blush it seemed quite natural to apply the same
`“Ungerboeck’s paradigm” of keeping coding combined with
`modulation even in a situation (the Rayleigh fading channel)
`where the code performance depends strongly, rather than
`on the minimum Euclidean distance of the code, on its
`minimum Hamming distance (the “code diversity”). Several
`results followed this line of thought, as documented by a
`considerable body of work aptly summarized and referenced
`in [14] (see also [5, Ch. 10]). Under the assumption that
`the symbols were interleaved with a depth exceeding the
`coherence time of the fading process, new codes were designed
`for the fading channel so as to maximize their diversity.
`This implied in particular that parallel transitions should be
`avoided in the code, and that any increase in diversity would
`be obtained by increasing the constraint length of the code.
`A notable departure from Ungerboeck’s paradigm was the
`core of [24]. Schemes were designed aimed at keeping as
`their basic engine an off-the-shelf Viterbi decoder for the
`de facto standard, 64-state rate-
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`Fig. 1. Block diagram of transmission with coded modulation (CM) and bit-interleaved coded modulation (BICM). In the case of CM,  denotes interleaving
`at the symbol level. In the case of BICM,  denotes interleaving at the bit level.
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`a branch metric deinterleaver
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`(a)
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`(b)
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`Fig. 2. 16QAM signal set with (a) a suitable set partitioning labeling and
`(b) Gray labeling.
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`binary labeling map
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`IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 44, NO. 3, MAY 1998
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`Fig. 3. Equivalent parallel channel model for BICM in the case of ideal interleaving.
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`the transmission of
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`Similarly, with no CSI we obtain
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`IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 44, NO. 3, MAY 1998
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`Fig. 4. BICM and CM capacity versus. SNR for 4PSK, 8PSK, and 16QAM over AWGN with coherent detection (SP denotesset-partitioning labeling).
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`Fig. 5. BICM and CM capacity versus SNR for 4PSK, 8PSK, and 16QAM over Rayleigh fading with coherent detection and perfect CSI (SP denotes
`set-partitioning labeling).
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`we conjecture that Gray labeling maximizes BICM capacity)
`while the performance of BICM with SP labeling is several
`decibels worse. Similar differences between Gray and SP
`labelings can also be observed from the cutoff rate of the
`Rayleigh fading channel.
`Our next results are based on cutoff rate. This parameter
`appears to be more suitable than capacity to compare BICM
`and CM, possibly because there is no fixed relation between
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`Fig. 6. BICM and CM cutoff rate versus SNR for QAM signal sets with Gray (or quasi-Gray) labeling over AWGN with coherent detection.
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`Fig. 7. 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.
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`dB over BICM 256QAM at
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`Fig. 8. BICM and CM cutoff rate versus. SNR for orthogonal signal sets over AWGN with noncoherent detection. The cutoff rate is expressed in information
`bit per channel use, where a channel use corresponds to N complex dimensions.
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`Fig. 9. BICM and CM cutoff rate versus SNR for orthogonal signal sets over Rayleigh fading with noncoherent detection and no CSI. The cutoff rate is
`expressed in information bit per channel use, where a channel use corresponds to N complex dimensions.
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`introduced by BICM is marginal (thus leaving room for
`“pragmatic” approaches [24]). On the contrary, BICM is
`much more appropriate for the Rayleigh fading channel. As a
`consequence, if the channel model—as is the case for example
`for mobile radio—fluctuates in time between the extremes of
`Rayleigh and AWGN, BICM proves to be a more robust choice
`than CM.
`2)
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`very low rate CM schemes.3 At
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`Convergence of the Union Bound: Under the assumptions
`of Section II-A, with ideal interleaving the channel is mem-
`oryless and stationary. Hence,
`the random coding bound
`based on the Bhattacharyya union bound for (time-varying)
`convolutional codes given in [23, Ch. 5] can be applied to
`our case. In particular, for
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`and where
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`does not belong to the pairwise error region of the BICM
`decoder, and can be neglected. We conclude that
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`Example 3 (Continued): If we apply the BICM EX to the
`case of uncoded 4PSK with Gray labeling over AWGN we
`obtain
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`IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 44, NO. 3, MAY 1998
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`For any other labeling, the asymptotic ratio between the
`PEP upper and lower bounds is
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`Fig. 10. BER of BICM 8PSK code obtained from the optimal 8-state, rate-2=3 code and Gray labeling. Rayleigh fading with perfect CSI.
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`TABLE I
`VALUES OF Nmin(1) AND d2
`h FOR SOME PSK AND QAM SIGNAL SETS.
`(THE SIGNAL SET AVERAGE ENERGY IS NORMALIZED TO 1. LABELS “ G”
`and “W ” DENOTE QUASI-G RAY AND THE LABELING OF 32QAM PROPOSED
`BY WEI IN [25] FOR DESIGNING UNEQUAL ERROR PROTECTION CM)
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`the BICM expurgated bound (or approximation), and SIM
`computer simulation. All
`the simulation results presented
`hereafter were obtained by using the suboptimal branch metric
`(9).
`1) Effect of Finite-Depth Interleaving: Here we prove that
`interleaving is indeed necessary, although it need not be very
`deep if the channel has a short memory. Fig. 10 shows the
`BER of BICM over independent Rayleigh fading with perfect
`CSI, where
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`class of binary parallel concatenated codes [15] (also known
`as “turbo codes”).
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`A. Numerical Results
`In this subsection we show a selection of numerical re-
`sults aimed at
`illustrating some features and applications
`of BICM. In the figures, curves marked by BUB denote
`union-Bhattacharyya bound, UB the BICM union bound, EX
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`Fig. 11. BER of BICM 16QAM code obtained from the optimal 64-states rate-1=2 code. AWGN channel.
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`(very tight) bound only for Gray labeling. With SP labeling,
`the BICM EX curve underestimates the actual BER. This is
`because, without Gray labeling, there exist many more nearest
`neighbors for each transmitted signal sequence, while the
`BICM EX counts just one neighbor. Hence, if the labeling
`is not Gray, the BICM EX curve yields too optimistic values.
`However, the Bhattacharyya union bound and the BICM UB
`always provides an upper bound (also for the SP labeling),
`although the latter is rather loose in the range of BER values
`of practical interest.
`3) BICM PSK/QAM Codes for the Fading Channel:
`Fig. 12 shows the BER over Rayleigh fading channels of
`BICM codes obtained by concatenating the same rate-
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`Fig. 12. BER of BICM obtained from the optimal 64-state rate-1=2 code and PSK/QAM signal sets. Rayleigh fading with perfect CSI.
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`Fig. 13. BER of BICM for R = 2 and R = 1 with  = 128 (64 states). The codes are obtained by concatenating the best rate-2=3 punctured code with
`8PSK, the best rate-1=2 code with 16QAM, the best rate-1=2 code with 4PSK and the best rate-1=4 code with 16QAM. Rayleigh fading with perfect CSI.
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`set expansion yields a coding gain only at very low BER
`values. The actual BER curves intersect at
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`Fig. 14. OBICM and OCC over AWGN with noncoherent detection. Simulation results are shown for N = 256 only.
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`TABLE III
`UPPER BOUNDS TO MINIMUM EUCLIDEAN DISTANCE AND
`CODE DIVERSITY FOR TCM AND BICM CODES FOR 16QAM
`(AVERAGE ENERGY NORMALIZED TO 1) WITH R = 3 bit/dim
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`where
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`Fig. 15. OBICM and OCC over Rayleigh fading with noncoherent detection and no CSI. Simulation results are shown for N = 256 only.
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`TABLE IV
`CODE DIVERSITY OF OBICM AND OCC FOR
`THE SAME DECODER COMPLEXITY AND RATE.
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`OBICM compares favorably with OCC only for low
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