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`Case 2:22-md-03034-TGB ECF No. 106-1, PageID.8445 Filed 12/16/22 Page 2 of 9
`IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 8, AUGUST 2002
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`1250
`
`Bit-Interleaved Coded Modulation With Iterative
`Decoding and 8PSK Signaling
`
`Xiaodong Li, Aik Chindapol, Member, IEEE, and James A. Ritcey, Member, IEEE
`
`Abstract—We have suggested bit-interleaved coded modulation
`with soft decision iterative decoding (BICM-ID) for bandwidth-ef-
`ficient transmission over Gaussian and fading channels. Unlike
`trellis coded modulation, BICM-ID has a small free Euclidean
`distance but large diversity order due to bit interleaving. With
`iterative decoding, soft bit decisions can be employed to signif-
`icantly improve the conditional intersignal Euclidean distance.
`This leads to a large coding gain, comparable to that of turbo
`TCM, over both Gaussian and Rayleigh fading channels with
`much less system complexity. We address critical design issues
`to enhance the decoding performance and provide the analytical
`bounds on the performance with an ideal feedback assumption.
`We investigate the performance characteristics of BICM-ID
`through extensive simulations and show that at high signal to noise
`ratios, the performance of BICM-ID converges to the performance
`assuming error-free feedback.
`
`Index Terms—BICM, coded modulation, digital communica-
`tions, iterative decoding, turbo codes.
`
`I. INTRODUCTION
`
`A CCORDING to information theory, block code per-
`
`formance can be improved by increasing the codeword
`length. Yet, for a convolutional code or an equivalent block code
`formed from a convolutional code, the decoding performance
`is related to the constraint length of the code [1]. Typically, one
`can not benefit from using a long input data sequence, because
`the bits far apart on the trellis do not interact. Increasing the
`constraint length may bring significant improvement, but at
`the expense of exponentially increasing complexity in the
`maximum-likelihood (ML) decoder.
`One clever way to circumvent the above dilemma is the re-
`cently proposed turbo coding scheme [3], [4], where two or
`more short-memory convolutional codes are concatenated in
`
`Paper approved by R. D. Wesel, the Editor for Coding and Communication
`Theory of the IEEE Communications Society. Manuscript received December
`17, 1999; revised November 19, 2000 and July 20, 2001. This work was sup-
`ported by the National Science Foundation under Grant CCR-0073391 and a
`Royalty Research Fund Grant from the University of Washington. This paper
`was presented in part at the 33rd Annual Conference on Information Sciences
`and Systems (CISS’99), March 17–19, 1999, Baltimore, MD, and at the 1999
`IEEE International Conference on Communications (ICC’99), June 6–10. 1999,
`Vancouver, BC, Canada.
`X. Li was with the Department of Electrical Engineering, University of Wash-
`ington, Seattle, WA 98195 USA. He is now with Broadstorm Telecommunica-
`tions Inc., Bellevue, WA 98004 USA (e-mail: xli@broadstorm.com).
`A. Chindapol was with the Department of Electrical Engineering, University
`of Washington, Seattle, WA 98195 USA. He is now with the Networks Division,
`Siemens Information and Communication Mobile LLC, San Diego, CA 92127
`USA (e-mail: aik.chindapol@icm.siemens.com).
`J. A. Ritcey is with the Department of Electrical Engineering, University of
`Washington, Seattle, WA 98195 USA (e-mail: ritcey@ee.washington.edu).
`Publisher Item Identifier 10.1109/TCOMM.2002.801524.
`
`parallel or in serial. Due to the pseudorandom interleaving, a
`“global interaction” is introduced among the bits over an entire
`block. As a result, error protection is achieved not only through
`the constraints on the local trellis transitions, but also through
`the influence of other trellis sections. Although a true ML de-
`coder for such concatenated codes is hard to implement, itera-
`tive decoding methods which employ the maximum a posteriori
`probability (MAP) rule for each individual decoder have been
`shown to provide near-capacity performance [3]–[5]. Compared
`with convolutional codes, turbo codes effectively take advantage
`of the potential of large block length but with the reasonable de-
`coding complexity of simple constituent codes.
`Another simpler approach is to use iterative decoding with
`a serial concatenation of encoding, bit-by-bit
`interleaving
`and high-order modulation. Unlike turbo codes, this scheme
`requires only one set of encoder/decoder; therefore, the receiver
`complexity is significantly reduced. At a first glance, the block
`diagram is no different from that of conventional symbol-inter-
`leaved trellis-coded modulation (TCM), a bandwidth-efficient
`coding approach suggested by Ungerboeck [6]. Indeed, the
`scheme, called bit-interleaved coded modulation (BICM) [8],
`was first suggested by Zehavi [7] to increase the time diversity
`of coded modulation and therefore to improve the performance
`of TCM over fully interleaved Rayleigh fading channels.
`However, this improvement
`is achieved at
`the expense of
`reduced free (squared) Euclidean distance (FED), leading to a
`degradation over nonfading Gaussian channels [7], [8].
`In this paper, we show that BICM, a bandwidth-efficient ap-
`proach primarily considered for fading channels in the past,
`can in fact be used to provide excellent performance over both
`Gaussian and fading channels, with iterative decoding (ID). To
`maximize the gain of ID, we make critical changes to traditional
`Gray labeling used in Zehavi’s BICM transmitter design. We
`call our scheme BICM with iterative decoding (BICM-ID) [9],
`[10].
`The goal of this paper is to give a comprehensive set of per-
`formance analysis, simulation results and new labeling maps. In
`Section II, we first briefly review the scheme of BICM and its
`conventional decoding [7], [8]. There we expose the reasons for
`the performance degradation of BICM compared with conven-
`tional TCM over Gaussian channels. In Section III, we address
`system design issues critical to the performance of BICM-ID
`and give detailed information on our iterative decoding algo-
`rithm, signal labeling method and interleaver design. In Sec-
`tion IV, we provide performance analysis and show extensive
`simulation results for BICM-ID for both AWGN and Rayleigh
`fading channels. Section V concludes the paper.
`
`0090-6778/02$17.00 © 2002 IEEE
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`Case 2:22-md-03034-TGB ECF No. 106-1, PageID.8446 Filed 12/16/22 Page 3 of 9
`LI et al.: BICM-ID AND 8PSK SIGNALING
`1251
`
`Fig. 1. Block diagram BICM-ID with soft feedback.
`
`II. REVIEW OF BICM
`
`A. The BICM Transmitter
`
`The BICM transmitter is a serial concatenation of the en-
`and the memoryless modulator as
`coder, the bit interleaver
`per-
`shown in Fig. 1. Note that the pseudorandom interleaver
`mutes the encoding output binary bits, instead of coded sym-
`bols using a conventional symbol-interleaved system. To sim-
`plify our discussion, we assume the information transmission
`rate of 2 bits/s/Hz using a rate-2/3 convolutional code and 8PSK
`modulation. Extensions to other information rates, code rates or
`modulation schemes are possible. For example, BICM-ID with
`16QAM for fading channels is studied in [20].
`Denote the two input bits of a rate 2/3 encoder at time by
`and its corresponding three output bits (a code
`, where
`or
`is the th bit. After
`symbol) by
`permutation by a pseudorandom block interleaver, each three
`binary bits of the interleaver output are grouped together,
`and are mapped to a complex channel symbol
`chosen from -ary constellation
`by a signal label
`
`Fig. 2. For various labeling maps, the shaded regions correspond to the
`decision regions for each bit taking the value of 1. (a) Gray. (b) Set partitioning.
`(c) Semi set partitioning.
`
`, six bit metrics are generated, using the ML rule. For
`signal
`the three binary bits and 8PSK symbols
`
`(1)
`
`(3)
`
`.
`where the 8PSK signal set is
`With coherent detection, the received discrete-time baseband
`signal is
`
`(2)
`
`is the symbol energy,
`is the fading coefficient,
`where
`is complex additive white Gaussian noise with one-sided
`and
`. For the AWGN channel,
`. For a fre-
`spectral density
`is Rayleigh
`quency nonselective Rayleigh fading channel,
`. In this paper, we assume perfect
`distributed with
`is perfectly estimated
`channel state information (CSI); hence,
`and available to the receiver.
`
`B. Conventional Decoding for BICM
`Due to bit-based interleaving, true ML decoding of BICM
`requires joint demodulation and convolutional decoding and is
`therefore too complex to implement in practice. In [7], Zehavi
`suggested a suboptimal method using two separate steps: bit
`metric generation and Viterbi decoding. From each received
`
`. The
`where the signal subsets are
`indicates replacement by an equivalent statistic. For
`notation
`is 4.
`8PSK, the size of each subset
`In practice, the log-sum calculation in (3) is computed either
`by approximation
`
`or by table lookup for better accuracy. Finally,
`replaced by the squared Euclidean distance
`
`(4)
`
`is
`
`.
`
`C. Degradation of BICM Over Gaussian Channels
`Although BICM performs well over fading channels because
`of an increase in diversity order, one pitfall of BICM is the
`degradation over Gaussian channels due to the “random mod-
`ulation” caused by bit interleaving [7]. For example, referring
`to Fig. 2, where the shaded regions correspond to all received
`1, 2, 3 takes on the value “1”. With bit
`symbols for which bit
`interleaving and suboptimal decoding, the symbol may originate
`from any constellation point in the shaded region. As we will
`later see, iterative decoding resolves this random modulation.
`
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`Case 2:22-md-03034-TGB ECF No. 106-1, PageID.8447 Filed 12/16/22 Page 4 of 9
`IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 8, AUGUST 2002
`
`1252
`
`[7],
`It can be shown that the FED of BICM is
`is the free Hamming distance of a code and
`[11], where
`is the smallest Euclidean distance between the modulation con-
`,
`stellation points. For 8PSK modulation,
`is the energy of a channel symbol. In general, the FED
`where
`of BICM is a few dB below its counterpart TCM [7]. Therefore,
`conventional BICM is less efficient than TCM for Gaussian
`channels.
`
`III. BICM-ID
`
`Bit interleaving connects the coded bits, originally far apart
`in the sequence, to the same channel symbol. With ideal in-
`terleaving, coded bits forming a channel symbol are indepen-
`dent; therefore, the feedback from strong data sections (with
`less influence of channel noise) can remove the ambiguity in
`the high-order demodulation and enhance the decoding of weak
`data sections (those subject to undesirable noise patterns). With
`the perfect knowledge of the other two bits, which are provided
`by the decoding feedback, 8PSK modulation is effectively re-
`duced to binary modulation for each bit position. Hence, the in-
`terconstellation distance for the binary modulation can be sig-
`nificantly increased.
`Of course, if the feedback contains errors, we have picked
`a wrong binary constellation. Therefore, it is also important to
`reduce the effect of feedback errors and to control error propa-
`gation. These factors are considered in system design by using
`soft-decision feedback and well-designed interleavers. While
`more complex than our hard-decision feedback [10], soft feed-
`back is the key to realizing the inherent gains in BICM while
`mitigating error propagation.
`
`A. Iterative Decoding Using Soft Feedback
`The recent success of turbo codes has demonstrated the ad-
`vantages of iterative processing in the decoding of concatenated
`schemes. A good introduction by Hagenauer can be found in
`[12], where the method is called the “turbo principle.” Note that
`iterative decoding was also considered by Seshadri and Sunder-
`berg for multilevel coded modulation [13]. In [14], Woerz and
`Hagenauer also used the reliabilities of the decoding results to
`control the feedback.
`As shown in Fig. 1, our receiver uses a suboptimal, iterative
`method through individually optimal, but separate demodula-
`tion and convolutional decoding steps. The a posteriori proba-
`bilities for the coded bits can be calculated as
`
`(5)
`
`Note that, compared with (3), (5) considers the a priori proba-
`.
`bility
`At the initial demodulation, we assume the equally likely
`. Then, the soft-input–soft-output (SISO) module
`prior
`[5] is used for convolutional decoding and to generate the a
`posteriori bit probabilities for the information and coded bits.
`Following the notation of Benedetto et al. [5], we denote by
`the a priori probability for a random variable
`and
`the a posteriori probability. Note that
`is un-
`available and is not used in the entire decoding process. In ad-
`
`dition,
`are the extrinsic information, a
`and
`term well explained in the literature of turbo codes [3], [5].
`is interleaved and fedback, as
`On the second pass,
`, to the demodulator. Assuming
`,
`and
`are independent (a good interleaver assures near inde-
`,
`pendence), we obtain, for each
`
`(6)
`
`is the value of the th bit of the label
`where
`. Using (5) and (6), we derive the extrinsic a posteriori bit
`for
`probabilities for the second-pass demodulation
`
`(7)
`
`Therefore, when recalculating the bit metrics for one bit, we
`only need to use the a priori probabilities of the other bits in
`the same channel symbol. The regenerated bit metrics are put
`into the decoder and we iterate demodulation and decoding. The
`final decoded output is the hard decision on the extrinsic bit
`, which is also the total a posteriori prob-
`probability
`is unused.
`ability since
`In our implementation, the SISO decoder uses an additive
`“log-map” algorithm [5]. Also, the log-sum in (5) is approx-
`imated by max operations, aided by table lookups. These ap-
`proaches greatly reduce the system complexity.
`
`B. Signal Labeling
`In our design it is critical to note that different decoding
`methods are optimized with different signal constellation
`labels. In this paper, we consider Gray, set-partitioning (SP),
`and semi set-partitioning (SSP) as examples. A comparison
`of these labeling schemes for 8PSK is shown in Fig. 2. The
`are shown in the shaded areas
`decision regions for each bit in
`(only shown inside the unit circle) while the unshaded regions
`. It can be seen that all labeling schemes have
`correspond to
`the same minimum Euclidean distance between subsets of
`and
`but a different number of nearest neighbors. Therefore,
`for conventional BICM, Gray labeling has been considered
`to be optimal [7], [8] due to the smallest number of nearest
`neighbors.
`With perfect knowledge of all other bits, 8PSK modulation
`is translated to binary modulation selected from four possible
`sets of binary modulation. It can be seen that iterative decoding
`of BICM not only increases the intersubset Euclidean distance,
`but reduces a number of nearest neighbors to one as well. This
`leads to significant improvement over both AWGN and fading
`
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`Case 2:22-md-03034-TGB ECF No. 106-1, PageID.8448 Filed 12/16/22 Page 5 of 9
`LI et al.: BICM-ID AND 8PSK SIGNALING
`1253
`
`channels. Fig. 2 also illustrates the increase in the minimum
`Euclidean distance between subsets. It is obvious that Gray la-
`beling is not the preferred choice since the minimum distance
`between subsets is not increased. More detailed analysis on the
`effect of labeling schemes is given in the next section where the
`analytical bound for BICM-ID is derived.
`
`C. Interleaver Design
`The interleaver design is critical to the high performance
`of BICM-ID. We use pseudorandom interleavers with the
`following design objectives: 1) to increase FEDC and 2) to
`mitigate error propagation during iterative decoding. Readers
`familiar with turbo codes can see that some of our ideas are
`inspired by the spread-random interleavers suggested in [16].
`Here are our design rules.
`Rule 1: Modularity: The bit positions before and after in-
`value, i.e., for 8PSK
`terleaving must have the same modulo-
`, Bit 1 in the encoder output bit stream can only
`and
`be mapped to one of the positions at Bit 1, 4, 7,… in the inter-
`leaver output. Essentially, the entire interleaver is composed of
`subinterleavers. This ensures that the coded bits with different
`protection, due to their different positions at the channel-symbol
`labels, are distributed uniformly along the trellis.
`bits going to the same
`Rule 2: Reverse Spread: The
`trellis stages apart from
`channel symbol must be at least
`each other. This ensures feedback independence in bit metric
`recalculation and mitigates the error propagation through
`much larger than the code constraint
`iterative decoding.
`length is easily achievable. For a block containing
`information bits, a typical
`is 50.
`Rule 3: Forward Spread: The bits co-channel-symboled with
`stages should be spread at
`the bits from a trellis segment of
`stages far from each other. This ensures that a burst of
`least
`decoding errors spread evenly over the entire trellis and does not
`heavily affect, through bit-metric recalculation using the feed-
`back, another short trellis segment. It is usually difficult to en-
`force Rule 3 for a short block, even though the window sizes
`and
`are chosen very small. Therefore, we only try to mini-
`, the typical values
`mize the number of violations. For
`and
`we use are 3 and 6.
`of
`The design rules form a multicriterion objective function, of
`which each component can only be partially optimized in prac-
`tice. Our interleaver design algorithm uses these design rules as
`heuristics that guide iterative changes to an initial pseudoran-
`domly drawn permutation.
`
`IV. PERFORMANCE EVALUATION
`
`A. Performance Bound for AWGN Channels
`We first derive a BER upper bound for an idealized situation
`assuming error-free feedback (EFF). With ideal feedback, the
`8-PSK channel is transformed into 3 independent BPSK chan-
`, the minimum intersignal Euclidean
`nels. Normalizing
`,
`and
`distance for the three BPSK channels are
`for set-partitioning labeling while
`for Gray labeling.
`We first compute the pairwise error probability (PEP), the
`is transmitted but a code se-
`probability that a code sequence
`
`TABLE I
`COMPARISON BETWEEN FED OF TCM AND BICM AND FEDC OF BICM-ID.
`RATE-2/3 CODES AND 8PSK MODULATION WITH E = 1.
`PUNCTURED CODES FOR BICM AND BICM-ID
`
`1, 2, 3, the
`is selected at the decoder. Denote
`quence
`Hamming weight of the error pattern corresponding to the th
`bit position of the encoder output. The total Hamming weight
`. The squared Euclidean
`of the error pattern
`and
`is
`distance between the modulated sequences
`
`Therefore, the PEP is given by
`
`(8)
`
`(9)
`
`. Finally, we obtain the upper
`where
`(union) bound on the bit error probability for a rate-2/3 code as
`
`(10)
`
`is the total information weight corre-
`where
`sponding to all the error events with coded output weight
`). With ideal
`feedback, 8PSK modulation is
`(
`translated to binary modulation regardless of the labeling
`map. Therefore, from (10) it can be seen that only the FED
`conditioned on the ideal feedback (FEDC), which is defined as
`dominates the asymptotic
`
`performance of BICM-ID.
`In Table I, we compare the FED of TCM and BICM and the
`FEDC of BICM-ID. The large increase in FEDC over FED
`shows the potential of BICM-ID. Our extensive simulation
`results confirm that soft
`iterative decoding mitigates error
`propagation and practically realizes the potential of the con-
`ditional free Euclidean distance—FEDC. It can be seen that
`Gray labeling, a signal mapping optimized for conventional
`BICM [8], shows no improvement due to iterative decoding.
`Although SSP labeling has the largest FEDC, it has the largest
`number of nearest neighbors, which affect the first round
`performance, among all labeling maps considered as shown
`in Fig. 2. Therefore, an iterative decoding gain may not be
`evident at BER values of interest and SSP labeling is not further
`used for AWGN channels. This observation is confirmed by
`simulation results.
`
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`IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 8, AUGUST 2002
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`1254
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`B. Performance Bound for Rayleigh Fading Channels
`denotes the pairwise error probability (PEP)
`Let
`of BICM and depends only on Hamming distance , a labeling
`-ary signal constellation where
`. From
`map , and
`[8], the union bound of the PEP of BICM can be written in the
`form
`
`TABLE II
`HARMONIC MEAN OF THE MINIMUM SQUARED EUCLIDEAN DISTANCE d
`BEFORE AND AFTER FEEDBACK AND THE ASYMPTOTIC GAIN OVER GRAY
`LABELING. NOTE THAT THE ASYMPTOTIC GAIN DOES NOT DEPEND ON
`THE CODE STRUCTURE
`
`where
`
`(11)
`
`method [2, Sec. 13.3.2]. The union bound of probability of bit
`is given by
`error for convolutional codes of rate
`
`(15)
`
`are the sequences of label positions and labeling maps
`and
`is the Laplace
`and is the complement of . Note that
`transform of the probability density function of the metric dif-
`between
`and .
`ference
`When Gray labeling is used, irrelevant error events can be
`expurgated [8] from (11) and the PEP is rewritten as
`
`is the total input weight of error events at Ham-
`and
`is the minimum Hamming distance of
`
`where
`ming distance
`the code.
`Unlike AWGN channels, there is no dominating term in the
`performance bound for the Rayleigh channel. Using (12)-(15),
`the asymptotic performance of BICM over Rayleigh fading [8]
`is approximated by
`
`(12)
`
`where
`
`. However,
`denotes the nearest neighbor of
`and
`due to large gain introduced by iterative decoding, we are most
`interested in an analytical bound for the error free feedback per-
`formance, or error floor for short, to which the BICM-ID per-
`formance converges at low BER.
`contains only one
`,
`Given ideal feedback for each
`, whose label has the same binary bit values as
`term
`except at the th bit position. Note that
`is not
`those of
`depending on the labeling map .
`necessary the same as
`Therefore, by removing the innermost summation in (11), the
`PEP of the error floor of BICM-ID can be written as
`
`(13)
`
`where
`
`For the Rayleigh fading channel with perfect CSI, we have
`[2, Sec. 13.3.2]
`
`(14)
`
`Then, the PEP of the error floor of BICM-ID defined in (11) can
`be numerically evaluated by the Gauss–Chebyshev quadrature
`
`(16)
`is the minimum
`is the probability of bit error,
`where
`is the information rate and
`Hamming distance of the code,
`is the harmonic mean of the minimum squared Euclidean
`-ary constellation with a labeling map ,
`distance. For any
`can be calculated by
`
`(17)
`
`for
`for BICM and
`is
`. Note that
`where
`BICM-ID with ideal feedback as defined in (12) and (13), re-
`controls the
`spectively. From (16), it can be seen that
`curve while
`pro-
`slope of the probability of bit error
`vides the horizontal offset. Notice that a convolutional code
`and a labeling map
`have independent impacts on the perfor-
`mance of BICM-ID in Rayleigh fading whereas their effects on
`the BICM-ID performance in AWGN channels are intractable.
`As aforementioned, iterative decoding of BICM effectively
`increases the intersignal Euclidean distance among signal sets;
`therefore, the harmonic mean of the minimum squared Eu-
`clidean distance can also be increased. Hence, the error floor of
`BICM-ID is the horizontally shifted version of the performance
`curve of BICM without feedback. Numerical calculation of
`before and after feedback is shown in Table II. The offset gain
`after feedback and
`of Gray
`[20], the difference between
`labeled BICM without feedback, is also provided. This gives
`a quick comparison between BICM-ID with various labeling
`schemes and conventional BICM because the offset gain is the
`asymptotic iterative decoding improvement regardless of the
`
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`1255
`
`Fig. 3. Performance of BICM-ID with set-partitioning labeling in AWGN.
`16-state, rate-2/3 punctured code, 8PSK modulation, and 4000 information
`bits/block.
`
`Fig. 4. Effects of labeling methods on the performance of BICM-ID in
`AWGN. 16-state, rate-2/3 punctured code, and 8PSK modulation with 4000
`information bits/block.
`
`code structure. It is shown in Table II that Gray labeling yields
`the best first round performance while SSP labeling has the
`worst performance; however, SSP labeling has the largest
`after feedback and thus the largest asymptotic offset gain.
`For the same convolutional code, the asymptotic performance
`of BICM-ID in AWGN depends on the minimum of the min-
`imum Euclidean distance among signal sets whereas the per-
`formance over fading channels depends on the average (har-
`monic mean) of the minimum Euclidean distance. Therefore,
`it is interesting to see that BICM-ID with Gray labeling shows a
`slight asymptotic gain over conventional BICM in fading chan-
`nels while no improvement is expected in the AWGN channel.
`
`C. Simulation Results for AWGN Channels
`In this section, we provide the simulation results for BICM-
`ID over Gaussian channels. In particular, we show how close
`BICM-ID performance is to the EFF bound (10). We also
`show the effects of signal labeling and block length on the
`performance of BICM-ID. The performance of other coded
`modulation schemes including Ungerboeck’s TCM [6] and
`turbo-TCM suggested by Robertson and Worz [17] are also
`included for comparison. Unless stated otherwise, we focus
`on rate-2/3 coded 8PSK and use the 16-state, punctured code
`[18] for BICM-ID in our simulation. The bit interleavers are
`designed with the rules described in the previous section. For
`each BER data point, we simulate 10 information bits.
`1) Tightness of the EFF Bound: In Fig. 3, we show the per-
`formance of BICM-ID with set-partitioning labeling, soft deci-
`sion decoding and 4000 information bits in each block. Clearly,
`the gain through iterative decoding is significant and the actual
`decoding performance converges to the EFF bound.
`2) Effects of Signal Labeling: Fig. 4 shows the effects of
`signal labeling. Gray labeling, which is extensively used for
`conventional BICM [7], [8], offers the best first-pass perfor-
`mance, but yields almost no gain with iterative decoding. This is
`because its FEDC is the same as FED of BICM, which is quite
`small. In contrast, set-partitioning labeling gives the large gain
`through iterative decoding and the best overall performance, al-
`
`Fig. 5. Effects of block length on the performance of BICM-ID in AWGN.
`16-state, rate-2/3 punctured code, and 8PSK modulation with set-partitioning
`labeling.
`
`though its first-round performance is the worst. As expected,
`the performance using Mixed labeling [10], which is suitable
`for BICM-ID with hard-decision feedback [10], is between the
`aforementioned two labeling methods.
`3) Effects of the Block Length: In Fig. 5, we show the effects
`of block length. As in other schemes using iterative decoding, a
`large block length is desirable for BICM-ID. With 2000 infor-
`mation bits per block, the performance of BICM-ID converges
`to the EFF bound at the BER level of practical interest. The
`degradation resulted from using a 500-bit block is about 0.8 dB
`. On the other hand, a slight improvement can
`at BER
`be achieved by using 4000 bits.
`It should be noted that further increasing the block length
`leads to earlier convergence, in terms of SNR and the number
`of iterations, but does not improve the performance at high SNR
`as normally seen with standard turbo codes. BICM-ID can also
`be modeled as a serial concatenation of an encoder, a bit in-
`terleaver and a one-state (zero-memory) encoder (modulator).
`The one-state encoder is equivalent to a SISO decoder on the
`
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`Fig. 6. Performance comparison between 8-state BICM-ID, conventional
`64-state TCM and 8-state turbo-TCM in AWGN. Rate-2/3 code, 8PSK
`modulation, 2000 information bits/block and 8 iterations are used.
`
`Fig. 7. Performance of BICM-ID with SSP labeling in Rayleigh fading.
`8-state, rate 2/3 code, 8PSK and 4000 information bits/block.
`
`one-state trellis. Furthermore, none of the encoders is recursive.
`Therefore, there is no “interleaver gain” [4] from this configu-
`ration because iterative decoding does not decrease the multi-
`plicity of low weight error events.
`4) Comparison With Other Coded Modulation Schemes: In
`Fig. 6, we compare BICM-ID with conventional TCM and
`turbo-TCM. For TCM, the 64-state Ungerboeck code is used
`with Viterbi decoding, while two 8-state, nonpunctured convo-
`lutional codes are used for turbo-TCM [17]. For both BICM-ID
`and turbo-TCM, we use 2000 information bits/block and eight
`iterations. Reducing the iteration number to 4 only causes a
`small degradation in both cases. The gap between BICM-ID
`and turbo-TCM is about half a dB. Note that, to illustrate the
`performance of BICM-ID, we use a single 8-state rate 2/3
`[19], which has the same
`convolutional code with maximal
`complexity as one of the component codes used in turbo TCM.
`Regarding the complexity of the proposed BICM-ID, an ap-
`propriate measure of the maximum likelihood decoder from
`a convolutional code is the number of visited edges per de-
`convolutional code
`coded bits [2, Sec. 11.1.2]. For a rate
`with memory, a maximum likelihood decoding complexity is
`while the complexity of maximum a posteriori (MAP)
`decoding is roughly about three times that of maximum likeli-
`hood decoding [2, Appendix F]. Therefore, in this comparison,
`the complexity of 8-state BICM-ID is about half that of 8-state
`turbo TCM and about one third of that of 64-state TCM. The
`complexity of soft output demodulator shown in (7) is relatively
`small compared to the SISO decoder, which requires the for-
`ward and backward recursions.
`
`D. Simulation Results for Rayleigh Fading Channels
`In this section, we assume the Rayleigh fading channel. We
`compare our simulation results with the analytical bound on the
`error floor. The effects of labeling and block length on the per-
`formance of BICM-ID are also shown. For comparison, we in-
`clude the performance of Gray labeled 8PSK BICM using an
`8-state, rate 2/3 code [7]. To show the impact of iterative de-
`coding to the performance of BICM-ID in Rayleigh fading, we
`
`Fig. 8. Effect of block length on the performance of BICM-ID with SSP
`labeling in Rayleigh fading. 8-state, rate 2/3 and 8PSK modulation.
`
`use the same code as in [7] in place of the puncture code. The
`bit interleavers are designed using the the previously described
`rules.
`1) Tightness of the EF Bound: In Fig. 7, we show the
`performance of BICM-ID with the SSP labeling map. Each
`block contains 4000 information bits. With only two passes,
`BICM-ID shows the significant improvement over conventional
`BICM at all BER values of interest. At large signal-to-noise
`), the actual decoding performance converges to
`ratio (
`the EF bound. Notice the tightness of the bound. As previously
`shown, the simulation results show the asymptotic coding gains
`of about 5.7 dB and with the SSP labeling map. The simulations
`validate this analysis.
`2) Effects of the Block Length: We show the effects of
`block length in Fig. 8. Similar to the BICM-ID performance in
`AWGN, BICM-ID in Rayleigh fading also suffers from a short
`block length. Specifically, more than a dB loss can be observed
`at BER of 10 when the block size is reduced from 2000 to
`500 bits. However, a slight improvement is achieved when the
`block size of 4000 bits is used.
`
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`[10]
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`[11]
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`[9] X. Li and J. A. Ritcey, “Bit-interleaved coded modulation with iterative
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