1744
`
`IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 19, NO. 9, SEPTEMBER 2001
`
`Turbo Equalization: Adaptive Equalization and
`Channel Decoding Jointly Optimized
`
`Christophe Laot, Alain Glavieux, and Joël Labat
`
`Abstract—This paper deals with a receiver scheme where
`adaptive equalization and channel decoding are jointly optimized
`in an iterative process. This receiver scheme is well suited for
`transmissions over a frequency-selective channel with large
`delay spread and for high spectral efficiency modulations. A
`-ary channel decoder is
`low-complexity soft-input soft-output
`proposed. Turbo equalization allows intersymbol interference to
`be reduced drastically. For most time-invariant discrete channels,
`the turbo-equalizer performance is close to the coded Gaussian
`channel performance, even for low signal-to-noise ratios. Finally,
`results over time-varying frequency-selective channel proves the
`excellent behavior of the turbo equalizer.
`Index Terms—Channel coding, equalizers, fading channel, inter-
`symbol interference.
`
`I. INTRODUCTION
`
`T HE DEVELOPMENT of digital communications systems
`
`over multipath channels has seen a considerable number
`of works during the last decade. Indeed, the increasing demand
`for high spectral efficiency modulation requires regular system
`evolution in order to improve performance. The increasing data
`rates through bandlimited channels introduce intersymbol inter-
`ference (ISI) which drastically deteriorates the received signal.
`As a consequence, it is necessary for the optimal receiver to
`deal with this phenomenon in order to achieve acceptable per-
`formance.
`Conventional solutions generally involve both equalization
`and channel coding which are done separately. In what follows,
`we introduce a new receiver scheme, called a turbo equalizer,
`where adaptive equalization and channel decoding are jointly
`optimized in order to improve the global performance. Equaliza-
`tion is achieved by means of an ISI canceller which completely
`removes ISI when transmitted data are a priori known. This as-
`sumption is meaningless in practice. Nevertheless, it is possible
`to obtain a reliable estimate of these data by using information
`provided by a previous processing involving both equalization
`and channel decoding. A turbo equalizer [1]–[3] allows the re-
`ceiver to benefit from channel decoder gain thanks to an iterative
`process applied to the same data block.
`In fact, the turbo-equalizer performance depends on channel
`selectivity and/or its time variation. For a large number of
`time-invariant channels, the turbo equalizer succeeds in com-
`pletely removing the ISI and exhibits the same performance as
`
`Manuscript received May 1, 2000; revised December 1, 2000, and March 1,
`2001. This work was supported in part by FTR&D, DMR/DDH.
`The authors are with ENST Bretagne, BP 832, 29285 Brest Cedex, France
`(e-mail:
`christophe.laot@enst-bretagne.fr;
`alain.glavieux@enst-bretagne.fr;
`joel.labat@enst-bretagne.fr).
`Publisher Item Identifier S 0733-8716(01)04188-9.
`
`the coded additive white Gaussian noise channels (AWGN).
`For time-varying channels, the turbo equalizer eliminates ISI
`and leads to a diversity gain.
`Although the turbo equalizer is an original approach to
`combat ISI, many authors have already proposed solutions
`using an ISI canceller, a maximum likelihood sequence esti-
`mator (MLSE), and a channel decoder.
`In 1981, Gersho and Lim [4] and Mueller and Salz [5] pro-
`posed an equalizer including a matched filter followed by a
`linear ISI canceller which uses past and/or future transmitted
`data. When data are known, this equalizer totally overcomes ISI.
`In other cases, a linear equalizer estimates data previously. Due
`to this, the receiver performance is strongly dependent on the bit
`error rate (BER) at the linear equalizer output. Furthermore, this
`receiver cannot benefit from information provided by a channel
`decoder.
`Many years later, Eyuboglu [6] proposed a receiver com-
`bining a decision feedback equalizer (DFE), a channel decoder,
`and a periodic interleaver. With this approach, the equalizer can
`use hard decisions provided by the channel decoder. For a low
`BER at the decoder output, this receiver can reach optimum
`DFE and coding performance. This receiver is thus very sen-
`sitive to decoding errors. Later, Zhou and Proakis [7] added
`an iterative process to this system in order to improve adap-
`tive parameter estimation. The performance of such a receiver is
`largely suboptimum when the channel is strongly frequency-se-
`lective.
`Finally in 1995, a receiver called a turbo-detector [8], [9],
`whose principle is borrowed from turbo-codes [10], combined
`a maximum a posteriori (MAP) detector with a MAP decoder
`through an iterative process. The performance has proved to
`be near optimum for many transmission channels. Neverthe-
`less, the turbo-detector is essentially dedicated to weak spec-
`tral efficiency modulations and a channel exhibiting a weak
`delay spread owing to prohibitive computational complexity.
`In order to reduce the turbo-detector complexity, the MAP de-
`tector can be advantageously replaced by an ISI canceller. This
`new receiver, called a turbo-equalizer [1]–[3], makes it pos-
`sible to almost completely overcome ISI over time-invariant
`and/or time-varying Rayleigh channels for high spectral effi-
`ciency modulations.
`This paper presents the turbo-equalizer scheme and its
`performance. It is organized as follows. Section II describes
`the transmission model where the information data are coded,
`interleaved, and then transmitted over a frequency-selective
`channel, using
`-QAM signaling. Section III introduces the
`turbo-equalizer structure which combines an adaptive equalizer
`called an interference canceller, a deinterleaver, a soft-input
`
`0733–8716/01$10.00 © 2001 IEEE
`
`Exhibit 1013
`U.S. Patent No. 6,108,388
`
`

`

`LAOT et al.: TURBO EQUALIZATION: ADAPTIVE EQUALIZATION AND CHANNEL DECODING JOINTLY OPTIMIZED
`
`1745
`
`Fig. 1. Principle of the transmission scheme.
`
`-ary symbols, and an interleaver, the
`soft-output decoder for
`whole process being iteratively repeated. Section IV provides
`simulation results for 4-QAM, 16-QAM, and 64-QAM over
`frequency-selective channels. Section V presents our conclu-
`sions.
`
`II. PRINCIPLE OF THE TRANSMISSION SCHEME
`Let us consider the transmission scheme depicted in Fig. 1. A
`convolutional code is fed in by independent binary data
`rate
`taking the values 0 or 1 with the same probability. Each set
`;
`is associated with
`of 2-m encoded data
`-ary complex symbol
`where symbols
`and
`with variance
`take equiprobable values in the
`with
`. Symbols
`and unitary variance
`are then
`
`with symbol duration
`interleaved and called
`.
`The signal at the output of the equivalent discrete channel
`with variance
`. The observed
`is corrupted by an AWGN
`channel noisy output
`can be written
`
`set
`
`(1)
`
`are the coefficients of the discrete channel impulse
`where
`response, which produces ISI. In this approach, the channel is
`considered time-invariant.
`The transfer function of this channel is given by
`
`(2)
`
`The signal-to-noise ratio (SNR) at the turbo-equalizer input is
`equal to
`
`SNR
`
`(3)
`
`where
`
`the mean energy received by information data;
`the monolateral noise power spectral density at the
`input of the receiver;
`the autocorrelation channel function defined by
`
`(4)
`
`III. TURBO-EQUALIZER STRUCTURE
`
`For a turbo equalizer, equalization and channel decoding are
`jointly performed in an iterative way as for a turbo-decoder [10].
`is carried out by a module fed
`Each iteration
`and decoded data
`originating
`in by both samples
`. The turbo-equalizer scheme is de-
`from the module
`picted in Fig. 2 where the delays are equal to the latency of each
`module.
`
`Fig. 2. Turbo-equalizer principle.
`
`Each module consists of an equalizer, a deinterleaver, a
`symbol-to-binary converter (SBC), a soft-input soft-output
`(SISO) binary decoder, a binary to symbol converter (BSC),
`and an interleaver as depicted in Fig. 3. Each module provides
`, called
`. This information
`an estimation of the symbol
`will be used by the adaptive equalizer of the next module.
`Note that a module uses the same SISO binary decoder for all
`-ary modulations. In this principle scheme, the combination
`of the three functions SBC, SISO binary decoder, and BSC
`constitutes an approximation of a SISO -ary decoder.
`
`A. Equalizer Structure
`The equalizer is close to an intersymbol interference canceller
`(IC) which allows ISI to be completely removed, providing that
`are known. Generally these symbols are unknown
`symbols
`by the receiver. As a consequence, the equalizer is a subop-
`symbols
`timum IC which replaces transmitted symbols by
`and adjusts its filters co-
`estimated by the previous module
`efficients in an adaptive way. When the SNR is sufficient, the
`iterative process gradually increases the reliability of the esti-
`mated symbols and the adaptive equalizer reaches the perfor-
`mance of the optimum IC. Fig. 4 depicts the equalizer structure,
`consisting of two transversal filters fed by the received samples
`and the data
`, respectively, estimated from the previous
`module.
`For each stage, the equalizer is updated according to the
`mean-square-error (MSE) criterion defined as
`MSE
`has been dropped for convenience.
`where the superscript
`In order to determine the optimum IC, it is necessary to as-
`are known. With the con-
`sume that the symbols
`having to be
`straint for the central coefficient of the filter
`, it can be shown [4], [5] that the IC op-
`equal to zero
`timum filters have the following transfer function:
`
`(5)
`
`(6)
`
`(7)
`
`

`

`1746
`
`IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 19, NO. 9, SEPTEMBER 2001
`
`Once convergence is established, the algorithms are decision-di-
`rected and minimize the estimated MSE given by
`
`MSE
`
`(12)
`
`where
`equalizer output.
`be the normalized phase and quadrature com-
`and
`Let
`ponents of the output equalizer, respectively,
`
`is a tentative decision taken at the
`
`and the decision rules for a
`
`-QAM are
`
`if
`
`if
`if
`
`if
`
`(13)
`
`(14)
`
`Fig. 3. Principle scheme of module p 1.
`
`Fig. 4. Equalizer structure.
`
`where the weighting coefficient
`
`is equal to
`
`are the coefficients of filters
`
`and
`and
`spectively.
`Thus, the IC output is ISI-free and equal to
`
`and
`
`with the output MSE given by
`
`MSE
`
`and the SNR at the IC output is equal to
`
`SNR
`
`(8)
`
`, re-
`
`(9)
`
`(10)
`
`(11)
`
`As a consequence of the comparison of (11) with (3), it clearly
`appears that ISI is completely removed by the equalizer, without
`noise enhancement.
`are generally unknown by the re-
`Transmitted symbols
`is fed in
`ceiver and the equalizer is suboptimum because
`by estimated symbols instead of transmitted symbols. This sub-
`optimality is taken into account in an adaptive way to adjust the
`, the esti-
`equalizer coefficients. So, at the first iteration
`mated symbols are equal to zero and the equalizer approximates
`, the
`the MMSE linear equalizer. After several iterations
`likelihood of the estimated symbols is expected to be right and
`the adaptive equalizer is close to the optimum IC defined by the
`transfer function (6) and (7).
`
`B. Adaptive Equalization
`Adaptive algorithms [11], [12] such as stochastics gradient
`least mean square (SGLMS) or recursive least square (RLS)
`can be used for updating equalizer parameters. These algorithms
`minimize the MSE defined by (5). In general, they require an ini-
`tial or even periodic data sequence (learning sequence) known
`by the receiver to ensure the convergence of the algorithms.
`
`(
`
`) into the relation (14) gives
`
`(
`
`) by
`
`Substitution of
`value.
`the
`For each iteration, the equalizer structure is depicted in Fig. 4.
`and estimated symbols sequence
`Output channel sequence
`provided by the channel decoder output of the previous
`module feed the equalizer. The equalizer output is given by
`
`(15)
`
`where
`
`and
`are the received samples vector and the
`estimated mean values vector, respectively.
`and
`the equalizer parameters vector corresponding to filters
`and
`, respectively.
`and
`are appropriate values
`greater or equal to . denotes the transposition.
`For a time-invariant channel, the SGLMS algorithm is used
`and initialized by a learning sequence
`for all iterations
`at the beginning of the transmission. Corresponding update
`equations are given by
`
`are
`
`(16)
`
`(17)
`
`is an appropriate step size.
`where
`For a time-varying channel, the RLS algorithm is used for
`and data aided by a periodic learning se-
`all iterations
`
`

`

`LAOT et al.: TURBO EQUALIZATION: ADAPTIVE EQUALIZATION AND CHANNEL DECODING JOINTLY OPTIMIZED
`
`1747
`
`, the estimated symbols
`quence. At the first iteration
`are unknown and the equalizer is a purely adaptive transversal
`and from relations (6) and
`filter. For the other iterations
`can be calculated with
`(7), the equalizer coefficients
`
`and
`
`For the nonuniform rule, the relations between the input
`couple
`and the output couple
`are given by
`
`if
`
`if
`
`is even
`
`is odd
`
`and
`
`(18)
`
`(19)
`
`Generally, channel coefficients are unknown and can be esti-
`mated by an RLS algorithm.
`in (18)
`with its estimated value
`The substitution of
`and (19) allows the equalizer coefficient to be approximated.
`The advantage of this approach is to have a smaller number of
`taps to be adjusted in order to enable the algorithm to follow
`channel fluctuation rapidly. However, this approach does not
`take into account the suboptimality of IC during the very first
`iterations.
`Some modifications can improve the performance of the
`adaptive equalizer. To increase the speed of the convergence,
`and
`can be substituted with
`in (12), (15), (16), and
`,
`(17) during the learning sequence. For the iteration
`are more accurate than
`decisions on estimated symbols
`in
`tentative decisions at the equalizer output and replace
`(12), (16), and (17). Furthermore, when a frequency offset
`exists between the transmitter and receiver oscillators, the
`equalizer can integrate a phase-locked loop (PLL) [13].
`
`C. Interleaving and Deinterleaving Functions
`
`The interleaving function allows temporal error sequence dis-
`tribution to be modified and splits the error series. Used gener-
`ally with time-varying channels, the interleaver is an essential
`function of the turbo-equalizer even if the channel is time-in-
`variant. Over a severe frequency-selective channel, the likeli-
`hood of the estimated data is weak and the equalizer output
`presents series of errors with large values which perturbates the
`channel decoder. Due to this, the interleaving dimension may
`be sufficient in comparison with the error sequence length but
`also in comparison with the error value. Some results related
`to interleaving performance versus interleaver size are given in
`Section IV.
`As presented in Fig. 1, the turbo-equalizer interleaves sym-
`. An alternative approach is to replace the symbol-inter-
`bols
`leaver by a bit-interleaver located between the channel encoder
`and the mapper. This approach is often used and gives excel-
`lent performance. Nevertheless, it can be demonstrated [1] that
`theoretical bounds for high-order modulation give better perfor-
`mance with a symbol-interleaver than a bit-interleaver.
`It is for this reason that our turbo-equalizer uses a matrix
`interleaver on symbols. Symbols
`are written line by line
`in a matrix and read following a given rule.
`For the uniform rule, the relations between a coordinate input
`associated with the symbol
`and the output
`couple
`associated with the symbol
`are given by
`coordinates
`
`.
`
`where
`Parameters
`
`.
`means modulo
`and
`are, respectively, equal to
`
`if
`
`if
`
`is even
`
`is odd
`
`if
`if
`if
`if
`
`and
`
`if
`if
`if
`if
`
`.
`is equal to
`where
`Nonuniform interleaving in comparison with uniform inter-
`leaving allows the matrix size to be reduced for equivalent per-
`formance.
`
`D. Symbol to Binary Converter (SBC)
`This function enables the same channel decoder to be used
`-QAM modulation.
`regardless of the state number of the
`, representative of
`The SBC associates values
`binary coded data
`, at each sample
`provided by the equalizer
`. Values
`are defined as the logarithm of the likelihood ratio
`(LLR) of binary coded data conditionally to the observation
`(
`) representative of symbol
`(
`)
`
`(20)
`
`is a constant. Its value will be defined later.
`where
`is a representative form of a binary
`A symbol
`, such that
`coded data vector, with dimension
`. Let us denote
`the symbol
`associated with one among
`possible realizations of
`. By fixing
`, we define a new vector
`that has
`possible realizations. By applying
`Bayes’ rule, the LLR given by (20) may be written as
`
`(21)
`
`

`

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`
`IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 19, NO. 9, SEPTEMBER 2001
`
`is the probability density function (pdf) of ob-
`where
`conditionally to the transmitted symbol
`. This
`servation
`pdf follows a Gaussian law according to (9) and (13) and the
`LLR may be expressed as
`
`Fig. 5. Example of Gray mapping for 16-QAM and 64-QAM modulations.
`
`The complexity of the previous relation can be reduced by using
`the Logarithm Jacobian defined by
`
`(22)
`
`E. Soft-Input Soft-Output (SISO) Channel Decoder
`The channel decoder is a SISO which is an approximate ver-
`sion of the MAP algorithm [15]. These outputs are given by the
`LLR of coded data
`
`(23)
`
`When the distance between
`possible to write
`
`and
`
`is sufficiently large, it is
`
`(24)
`
`In this case, for a sufficiently large SNR, the LLR may be ap-
`proximated by
`
`, the estimation of coded data provided to
`For
`the SBC is only a function of the equalizer output and is given
`by
`
`(25)
`
`(26)
`
`to the previous value simplifies the receiver com-
`Fixing
`plexity because the noise variance disappears and its estimation
`is not necessary.
`Consideration of a particular mapping for a coded bit can sim-
`-bit vector
`plify this expression. The mapping associates an
`-bit symbols differ by only
`with a symbol. When two adjacent
`a single bit, as depicted in Fig. 5, the encoding (Gray mapping)
`allows the BER to be minimized.
`With Gray mapping, (26) can be approximated by
`
`(29)
`
`, are the samples
`
`where the observations, called
`vided by the SBC.
`Equation (29) has been determined from the Berrou–Adde
`algorithm [16]. This algorithm is less optimum than other algo-
`rithms but gives good performance with reasonable complexity
`requirements [15].
`
`pro-
`
`F. Binary-to-Symbol Converter (BSC)
`it is necessary for the trans-
`To feed the equalizer filter
`mitted symbols to be known or estimated. When transmitted
`symbols are unknown, it is possible to get an estimated value
`from coded data LLRs provided by the channel
`decoder of the previous module. This paragraph proposes a so-
`-ary symbol.
`lution to convert the binary decoder output to an
`denotes the symbol
`associated with
`As previously,
`possible realizations of
`. Then, estimated
`one among
`of
`may be approximated by its mean value
`
`(30)
`
`By using the fact that coded data are decorrelated, this expres-
`sion may be written as
`
`(31)
`
`Let us consider the case of a 4-QAM modulation. Symbol
`is associated with one coded piece of data
`and has two possible values. The estimated symbol is equal to
`
`and in the same manner
`
`(27)
`
`(32)
`
`From (29), when coded data are independent and identically
`distributed (i.i.d), the LLR of coded data is equal to
`
`The difference between the exact LLR (22) and its approxima-
`tion (27) has been evaluated and the loss is weak with respect to
`the calculation gain [14].
`
`(28)
`
`the probability of having
`
`may be expressed by
`
`(33)
`
`(34)
`
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`

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`
`1749
`
`Consequently, by substituting (34) into (32), the estimated
`symbol is
`
`(35)
`
`may be obtained in the same way. For high-order modulation,
`(32) must be changed according to the Gray mapping described
`in Fig. 5.
`
`IV. SIMULATION RESULTS
`
`A. Time-Invariant Channels
`For simulations, the information data were coded by a 1/2 rate
`convolutional code with a free distance equal to 7 and generator
`polynomials equal to 23, 35 (expressed in octals). Turbo-equal-
`-QAM signaling schemes
`izer performance was evaluated for
`4, 16, and 64) with several discrete equivalent channel re-
`(
`sponses. Time-invariant channels may be characterized by their
`. Simulations use equiv-
`coefficients vector
`alent discrete channels proposed by Porat and Friedlander [17]
`and Proakis [11]:
`
`The output channel power was normalized to unity. Consid-
`, it is necessary to normalize
`ering data with variance
`. For each simulation,
`the coefficients vector by fixing
`we have represented by a dashed line the theoretical bound
`of the turbo-equalizer, which corresponds to an IC fed by
`transmitted symbols. At
`the IC output, ISI is completely
`cancelled and transmitted symbols are only corrupted by a cor-
`related noise. This noise is whitened by interleaving, allowing
`the channel decoder to run under optimal conditions. For a
`multipath time-invariant channel, the theoretical bound of the
`turbo-equalizer corresponds to the performance of Gaussian
`channel with coding. The turbo-equalization goal is to reach
`this limit.
`From theoretical results, the turbo-equalizer needs parameter
`in order to run. This parameter is given by (10) and can be
`with
`estimated by
`corresponds to the decision symbol
`, where
`from equalizer output at the first iteration and at the decision
`symbol from BSC output for the other iterations, respectively.
`is a positive constant denoting the forgetting factor equal to
`0.995.
`To allow equalizer convergence a training sequence made up
`of 2048 symbols is transmitted. At the first iteration, the number
`is equal to 31, the central coefficient of
`of tap weights of
`the filter is initialized to 1 and adaptation step sizes for SGLMS
`algorithms are equal to 0.003 for the training period and 0.0005
`for the tracking period. For the other iterations, the number of
`and 41 for
`, respectively, and
`taps is equal to 21 for
`filter is initialized to 1. The adaptation
`the central tap of
`
`Fig. 6. Turbo-equalization performance over the Porat and Friedlander
`channel for a 4-QAM modulation.
`
`Fig. 7. Turbo-equalization performance over several channels for a 4-QAM
`modulation at the fifth iteration.
`
`step size for SGLMS algorithms is equal to 0.003 for the training
`period and 0.000 25 for the tracking period.
`was evaluated in the tracking period
`The BER versus
`over coded data using the Monte Carlo method (for at least 100
`transmission errors. For results presented in Figs. 6–9, a
`matrix following a uniform law as defined in Section III-C
`performed the interleaving.
`Fig. 6 depicts the BER at the output of the turbo-equalizer as a
`function of the number of iterations , for a 4-QAM modulation
`over the Porat and Friedlander channel. For this channel, only
`three iterations and an SNR greater than 3 dB are necessary for
`the performance of a coded Gaussian channel without ISI to be
`reached. Thus, for a sufficiently large SNR, the turbo-equalizer
`
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`

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`IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 19, NO. 9, SEPTEMBER 2001
`
`Fig. 8. Turbo-equalization performance over several channels for a 16-QAM
`modulation at the fifth iteration.
`
`Fig. 9. Turbo-equalization performance over several channels for a 64-QAM
`modulation at the fifth iteration.
`
`improves its global performance at each iteration and reaches
`the theoretical bound, provided that a sufficient number of iter-
`ations is processed. In fact, the “turbo effect” is primed when
`the estimated data feeding the equalizer reach a BER threshold
`sufficiently low allowing the equalizer to cancel a large amount
`of ISI. This BER threshold depends on the first iteration perfor-
`mance which is mainly a function of the transversal equalizer
`performance and channel coding gain.
`The performance comparison with linear MMSE equalizer
`(LE-MMSE) and MAP detector is shown in Fig. 6. The per-
`formance of the LE-MMSE equalizer corresponds to the first
`of the turbo-equalizer. It worth noting that for
`iteration
`a low SNR the adaptive Decision Feedback Equalizer (DFE-
`MMSE) presents approximately the same performance as the
`LE-MMSE. Table I presents some performance comparisons
`.
`between different receivers for a BER equal to
`Fig. 7 presents the turbo-equalizer performance for a 4-QAM
`modulation over the three previously mentioned channels, at the
`fifth iteration. The turbo-equalizer over Proakis A and Porat et
`al. channels totally overcomes the ISI, even for weak SNRs.
`However, for the Proakis B channel, the turbo-equalizer does not
`enable the theoretical bound to be reached. For this channel, an
`input filter creates an impulse noise
`error introduced at the
`at the output of the equalizer that strongly affects the channel
`decoder and decreases the channel coding gain. This phenom-
`enon appears for highly frequency-selective channels. To im-
`prove the turbo-equalizer performance over these channels one
`can increase the interleaver size as described below.
`Fig. 8 presents the turbo-equalizer performance for a
`16-QAM modulation over the three previous channels at
`the fifth iteration. Like in Fig. 7, the performance of the
`turbo-equalizer over Proakis A and Porat et al. channels
`roughly reaches the theoretical bound, whereas, for the Proakis
`B channel performance is poor. From a general point of
`view, using large spectral efficiency modulation increases the
`
`TABLE I
`PERFORMANCE COMPARISON BETWEEN DIFFERENT RECEIVERS OVER THE
`PORAT AND FRIEDLANDER CHANNEL FOR A 4-QAM MODULATION
`
`pathological behavior over severe frequency selective channels.
`Note that, in terms of complexity, a MAP detector over the
`Porat et al. channel for MAQ16 needs a trellis with 65 536
`states whereas the turbo-equalizer uses simple filters for better
`performance.
`The performance achieved at the fifth iteration for a 64-QAM
`modulation over the three previously described channels is rep-
`resented in Fig. 9. The turbo-equalizer reaches the theoretical
`bound when the SNR is greater than 8 dB for the Proakis A
`channel and 11 dB for the Porat et al. channel, respectively. The
`poor performance of the transversal equalizer at the first itera-
`tion over the Proakis B channel for a 64-QAM does not allow
`the turbo-effect to be started and the performance is worse.
`Fig. 10 presents the turbo-equalizer performance at the fifth
`iteration as a function of the interleaver type and size over
`the Proakis B channel for a 4-QAM modulation. When the
`interleaver size is sufficient, the turbo-equalizer performance
`reaches the theoretical bound. If the interleaver size decreases
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`LAOT et al.: TURBO EQUALIZATION: ADAPTIVE EQUALIZATION AND CHANNEL DECODING JOINTLY OPTIMIZED
`
`1751
`
`Fig. 10. Turbo-equalization performance over the Proakis B channel for a
`4-QAM modulation at the fifth iteration as a function of the interleaver type
`and size.
`
`Fig. 11. Turbo-equalization performance over a multipath Rayleigh channel
`for a 4-QAM modulation.
`
`turbo-equalizer performance is suboptimum because the equal-
`izer outputs are correlated and there is a loss of the channel
`decoding gain. Simulations show that the interleaver must very
`large when the channel is highly selective.
`
`B. Time-Varying Channels
`Radio communication propagation is generally subject to
`multipaths corrupted by Doppler effects, which depend on the
`relative speed between the emitter–receiver and the carrier
`frequency. The channel coefficents may be modeled by a
`complex-valued independent process which may be expressed
`as
`
`correspond to the maximum Doppler and at the
`and
`where
`mean power associated with the path, respectively. Parameters
`and
`are uniform random variables over
`. For
`was fixed to 10. Generally the Doppler effect is
`simulations
`by .
`characterized by the product of a Doppler band
`For simulations, the information data were coded by a 1/2 rate
`convolutional code with generator polynomials equal to 23, 35.
`The turbo-equalizer performance was evaluated for 4-QAM sig-
`parameter
`nalling schemes over a Rayleigh channel. The
`is fixed to 0.001.
`are assumed to be emitted by
`Data with variance
`slots of 125 symbols whose first 25 symbols are known, from
`the receiver. Thus, the training sequence represents 20% of the
`transmission flow. All the slots are sequentially transmitted over
`the continually varying Rayleigh multipath channel. The loss of
`due to the use of a periodic training
`1 dB in the ratio
`sequence was not taken into account for the plotted curves.
`The algorithm used with the Rayleigh channel is described
`has nine
`in Section III-B. The transversal equalizer
`
`coefficients. This value is weak and can involve some perfor-
`mances loss. Nevertheless, it necessary to have a weak number
`of coefficients in order to enable the RLS algorithm to follow
`the time-varying channel. The central coefficient is initialized
`to one and the weighting factor equal to 0.965. For the other
`, the equalizer coefficients are calculated
`iterations
`from the estimated coefficients channels, which were previously
`obtained from an RLS algorithm. The number of taps for the
`channel estimator is equal to 5 and the weighting factor 0.965.
`The BER was evaluated for at least 500 errors at the last itera-
`tion.
`Results presented in Fig. 11 are given for a Rayleigh channel
`which has three paths with the same power mean and each path
`. A
`matrix fol-
`is separated by a symbol duration
`lowing a nonuniform law performed the interleaving.
`We have represented by a dashed line the theoretical bound
`of the turbo-equalizer, which corresponds to an optimum IC
`fed by transmitted symbols. The turbo-equalization goal is to
`reach the theoretical bound. At the optimum IC output, ISI is
`completely cancelled and the energy from the different channel
`taps is collected. This explains the diversity gain in comparison
`with the Rayleigh non frequency selective channel plotted by
`a dashed–dotted line. After five iterations, performance of the
`turbo-equalizer is close to the theoretical bound, which indicates
`a good behavior of this new receiver for a large class of chan-
`nels.
`
`V. CONCLUSION
`
`This paper describes a reduced complexity receiver for
`-QAM, called a turbo-equalizer, that uses equalization
`a
`and coding jointly optimized in an iterative process. For a
`large class of frequency-selective channels, turbo-equalization
`allows ISI to be totally overcome and reaches the coded
`Gaussian channel performance that actually seems to be a
`theoretical bound. Moreover, using an adaptive equalizer
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`IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 19, NO. 9, SEPTEMBER 2001
`
`allows high-order modulation over channels with a large delay
`spread to be processed, which is impossible with a conventional
`Viterbi detector (MLSE). To achieve the turbo-equalizer, a
`-ary channel decoder has been used, the complexity
`SISO
`of which is practically independent of the modulation order.
`Finally, simulations over a radiomobile-type channel show that
`turbo-equalization could be used for all digital receivers when
`information data are coded and interleaved.
`
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