Signal Shaping by
`QAM for
`AWGN Channels and Applications Using
`Turbo Coding
`Dirk Sommer and Gerhard P. Fettweis
`Dresden University of Technology, Germany
`
`Abstract: A non-uniform signal set constella-
`tion is used in order to provide shaping gain.
`An approximately Gaussian distribution of the
`transmitter output signal is achieved by using
`equally likely signal points with unequal spac-
`ing. In this way it is possible to obtain shaping
`gain without any additional computational ef-
`fort. The method is also applicable when bit in-
`terleaved coded modulation in conjunctionwith
`parallel decoding of the individual bit levels is
`used.
`- Channel
`capacity, entropy,
`quadrature amplitude modulation, error cor-
`rection coding.
`
`I. INTRODUCTION
`
`From information theory it is known that, in addi-
`tion to coding gain, shaping gain can be obtained if the
`amplitude of the transmitter output follows a Gaussian
`distribution.
`In most cases, a uniform
`constellation together
`with different probabilities for each signal point is used
`to provide shaping gain. The idea has been investi-
`gated in
`Other examples are trellis shaping
`the use of prefix codes
`or similar, subdivision of
`the signal constellation into variable-size regions
`Recently, shaping also has been used in conjunction
`decoding
`While
`with multilevel
`trellis shaping requires an additional Viterbi-algorithm
`at the transmitter but maintains a fixed data rate, the
`approaches in [3;
`are conceptually simpler but pos-
`sibly impractical since the non-constant data rate may
`involve buffer over- or underrun. Also,
`has to be carried out frequently in this case to
`avoid catastrophic error propagation.
`Non-uniform (NU) signal point constellations have
`a long tradition,
`in order to create a signal set for
`, some-
`hierarchical (rate adaptive) transmission [6;
`times also called multiresolution modulation. An ap-
`proach to use an asymmetric coded-modulation scheme
`in order to optimize trellis coding has been presented
`in
`
`In this paper we make use of a non-uniform signal
`point constellation in order to obtain shaping gain.
`Contrary to other approaches to signal shaping, each
`assumed to be chosen with the same
`signal point
`
`probability. Therefore we arrange the points in a way
`that a Gaussian distribution of the output signal at
`the transmitter results,
`using more points at the
`lower power levels and less at higher.
`It is shown that for higher order modulation schemes
`using these constellations, shaping gains in the or-
`der of 1
`are possible. Furthermore, it is demon-
`strated that the method provides similar gain when
`bit-interleaved coded modulation (BICM), recently in-
`troduced by Caire et.
`in conjunction with Gray
`labeling and simple parallel decoding of the different
`bit levels is used. Finally, some examples using turbo
`coding over the AWGN channel are given in order to
`verify that the principle works when used with com-
`mon coding schemes.
`In the following, all considerations regarding the
`channel capacity are made with respect to amplitude
`shift keying (ASK). This can of course simply be ex-
`tended to
`by adding a second dimension.
`The transmission model is depicted in Fig. 1. The
`info bits
`are first coded and
`interleaved.
`Then m interleaved coded bits
`are
`A of the signal space using
`mapped onto a symbol
`uniform or non-uniform ASK with Gray labeling. The
`passes a memoryless white Gaussian noise
`symbol
`channel, and y is received.
`
`Figure 1.
`
`model
`
`SHAPING PRINCIPLE
`A. Signal set constellation
`In order to find a signal set with the desired prop-
`erties, an empirical approach can be made as follows.
`Each point of the signal constellation is assumed to be
`chosen with the same probability, and we want the out-
`put of the transmitter to give an approximately Gaus-
`=
`sian distribution. Therefore, using N-ASK,
`we split the Gaussian cumulative distribution function
`(CDF) into N sections of equal probability,
`the or-
`dinate of the CDF is partitioned into N equal parts.
`Since each signal point shall represent exactly one of
`the sections, each signal point (placed on the abscissa)
`
`1
`
`LGE 1006
`
`

`

`will be chosen such that the corresponding probability
`is exactly in the middle of a section.
`In other words, the points
`on the abscissa of the
`=
`Gaussian PDF
`(to be approximated)
`will be chosen such that
`
`for
`illuminates the
`Fig.
`are normalized by a factor
`procedure. Finally the
`, such that the average power of the trans-
`h , =
`mitted signal is one
`
`by
`
`Alternatively, it has been tried to find the points
`calculating the mean of each probability section,
`=
`where
`and
`are the
`of the current section, but the results did not change
`significantly.
`Fig.
`illustrates how the Gaussian CDF is ap-
`proximated. The corresponding PDF consists of
`at locations
`functions of height
`Signal points
`values for various ASK constellations are listed in the
`Appendix.
`B. Peak-to-average power ratio expansion
`
`4-ASK
`
`32-ASK
`64-ASK
`128-ASK
`
`2.3
`3.1
`3.8
`
`Table 1. PAPR expansion (NU-ASK vs. uniform ASK)
`
`One of the drawbacks of the presented shaping prin-
`ciple is the expansion of the peak-to-average power ra-
`tio (PAPR) compared to uniform ASK. If for some
`reason not
`the average but the maximum transmis-
`sion power is limited it might be better to dispense
`with that sort of signal shaping and exploit the higher
`signal to noise ratio of a uniformly distributed signal
`set instead, or to use other methods. Table 1 lists the
`PAPR expansion for the different modulation schemes.
`In all cases the PAPR expansion exceeds the shaping
`gain (see
`so the kind of shaping presented
`here is generally not useful when the power constraint
`is only on the maximum power.
`111. CAPACITY FOR GAUSSIAN NU-ASK
`In the following, the capacity of the non-uniform
`signal constellation is calculated under the assumption
`of equal probability for each signal point. Two cases
`are
`
`U
`
`(a) Signal points u , for 8-NU-ASK
`
`U
`
`(b) Approx. of the Gaussian CDF by 8-NU-ASK
`
`Figure 2. Signal set constellation
`
`A . Signal set capacity
`First, the capacity of the signal set that can
`ciply be reached at a certain SNR is calculated,
`when all information contained in received signal y is
`exploited. This could possibly be done using multi-
`level-coding (MLC) multi-stage-decoding (MSD) as
`proposed in
`or by BICM with additional feed-
`back decoding (AFD) as presented in section IV (see
`This capacity will be referred to as
`also Fig.
`(signal set capacity). Under the assumption of
`equally likely source signals,
`can be calculated as
`follows
`
`Y )
`
`-
`
`the
`
`where
`
`X
`variable
`random
`the
`represents
`the bits of
`..) denotes the mutual
`I(
`constellation,
`..)
`information between two random variables, and
`the entropy.
`
`2
`
`

`

`32-NU-ASK vs. 32-ASK
`
`-5
`
`0
`
`5
`
`10
`
`15
`SNR
`
`20
`
`25
`
`30
`
`35
`
`40
`
`(a) Signal set capacity
`
`32-NU-ASK vs. 32-ASK
`
`6 5
`
`1 0
`
`-10
`
`6
`
`5
`
`B. Parallel decoding capacity
`Second, each bit-level is considered separately,
`the assumption is made that each bit level is decoded
`without prior knowledge of the decoding results of
`some other bit
`levels. The corresponding decoding
`scenario is depicted in Fig.
`From the received
`signal y , a metric is calculated for each bit. Subse-
`quent decoding is performed without the attempt to
`improve that metric using the decoding results. Caire
`et.
`noticed
`that in this case the capacity strongly
`depends on the labeling scheme and conjectured that
`Gray labeling maximizes the capacity. Therefore no
`other labeling schemes will be considered in the sequel.
`As remarked in
`this is equivalent to the capacity
`when MLC is used with parallel independent decoding
`of the different bit levels. Adopting the term of
`the
`(parallel decoding
`capacity will be referred to as
`of individual levels).
`can be calculated as
`
`1
`
`which means that different from (3) one has to sum
`up the mutual information between Y and the indi-
`vidual bits
`while averaging the conditional entropy
`= over the possibilities
`= k , k
`In the sequel, we use the term PD for BICM, too, in
`order to distinguish from BICM with AFD, although
`the bit levels are not really decoded in parallel.
`C. Shaping gain results
`The shaping gain is usually defined as the reduction
`in average power of the signal set at afixed data rate [2]
`compared to a rectangular (hypercube) constellation.
`With the approach taken here this definition is not
`applicable since there would be no shaping gain at all.
`Consequently, we need a different definition and use
`the term shaping gain for the gain in SNR of a
`compared to a u n i f o r m ASK a t a certain capacity.
`In order to determine the shaping gain, (3) and (4)
`have been evaluated for the constellations presented in
`section
`Fig.
`and Fig.
`give an example of
`the capacity of the 32-NU-ASK constellation for
`and
`respectively. As can be seen from Fig.
`is very close to the optimum if less than
`3 [bits
`channel use] are transmitted. In both cases the highest
`use],
`if a
`gain occurs at about 2.5
`code rate of R
`0.5 is used. Evaluating other
`ASK constellations confirms that this generally seems
`to be the case. With turbo codes a powerful class of
`codes operating at this rate exists that can exploit this
`gain. However, contrary to common coded modulation
`or multilevel'coding schemes, the number of bits of the
`
`-5
`
`0
`
`5
`
`10
`
`15
`SNR
`
`20
`
`25
`
`30
`
`35
`
`40
`
`(b) Parallel decoding capacity
`
`Figure 3. Capacities for NU-ASK vs. ASK
`
`signal set has to be doubled instead being expanded by
`only 1bit, since there is no shaping gain at code rates
`close to 1. This is a further drawback if the PAPR is
`of some importance.
`
`Gain
`4-ASK
`
`0.07
`
`0.01
`
`16-ASK 0.59
`32-ASK
`0.82
`
`I
`
`0.50
`0.81
`
`128-ASK 1.14 1.18
`
`Table 2. Shaping gain at R
`
`0.5
`
`and
`Table 2 lists the capacity gains of
`at R = 0.5.
`the use of NU-ASK for shap-
`ing offers advantages only for higher order modulation
`schemes. However, it can provide shaping gains up to
`independent of whether BICM with PD or a
`1
`scheme exploiting
`is used.
`
`3
`
`

`

`Bitwise
`
`metric calc.
`
`(a) BICM with P D
`
`Bitwise
`Inter-
`leaver
`
`I
`
`I
`
`(b) BICM with AFD
`
`Figure 4. Decoder structure
`
`IV. APPLICATIONS USING TURBO CODES
`Turbo codes, first introduced in
`are known as
`a class of powerful codes that can approach the chan-
`nel capacity relatively close. The BCJR-algorithm
`= for the
`used for turbo decoding requires
`A are as-
`branch metrics. When all signal points
`sumed to be equally likely,
`= k ) can be simply
`determined by summing up the transition probabili-
`= k
`1},
`ties over all signal points
`with
`for each bit of the ASK symbol,
`
`is the (one dimensional) noise variance, as-
`where
`sumed to be known at the receiver.
`If a-priori infor-
`=
`mation of the bits
`is provided, then
`can be calculated as
`
`which means that additionally, the probability of the
`current signal point
`is taken into account, not using
`the input probability of the bit, for which the output
`soft value has to be determined. The expression in
`eq. (6) is a variant of the method presented in
`for GSM speech frame decoding, and has also recently
`been used in
`for feedback demodulation of BICM.
`in (5) and (6) can be ne-
`Constant factors
`glected since they cancel out when likelihood ratios
`=
`= 1) are computed for subse-
`quent decoding.
`In the following, the theoretical results have been
`tested using turbo codes over the AWGN channel. Two
`cases have been considered. First, BICM with simple
`
`In
`subsequent turbo decoding was used, see Fig.
`this case, the maximum transmission rate is bounded
`
`Second, the extrinsic soft values of the turbo decoder
`for info and parity bits were fed back into the metric
`calculator in order to serve as a-priori information ac-
`cording to
`see Fig.
`(AFD, additional feedback
`decoding). The corresponding capacity is
`The
`soft values were calculated according to the method
`presented in
`8000 bits (slightly
`The interleaver length was
`constel-
`adapted to give a multiple of the actual
`lation size) for the turbo coding and twice that much
`(one complete coded burst) between coding and modu-
`lation. Both interleavers performed random permuta-
`tion of the
`values. The turbo code was the one
`a 16-state code with polynomi-
`presented in
`als 21 and 37 (octal representation) and every second
`parity bit punctured.
`V. CODING RESULTS
`and Fig.
`compare turbo coding results
`Fig.
`for the AWGN channel using different modulation or-
`ders without and with AFD, respectively.
`As can be seen, the difference between
`and
`NU-&AM matches the shaping gain results of table 2
`relatively close at
`independent of whether
`AFD was used or not. Lower BER have not been con-
`sidered since the typical turbo code error floor starts
`already in that region. If additional feedback for the
`metric calculation is provided, the results improve by
`about 0.5 - 1
`The necessary SNR for
`ASK with
`and
`
`(NU-ASK),
`respectively, are listed in table 3. The
`
`16-ASK (256-&AM)
`32-ASK
`64-ASK (4096-QAM j
`
`16-ASK (256-&AM)
`32-ASK
`
`11.94
`15.05
`18.10
`
`12.75
`15.96
`
`12.53
`15.87
`19.10
`(ASK)
`13.25
`16.77
`
`Table 3. Capacity at R 0.5
`
`are in all cases
`SNR necessary to reach
`within 1- 2
`from the capacity result.
`For all simulation, 12 iterations of inner turbo de-
`coding have been used. With AFD, 5 outer iterations
`have been made additionally resulting in a total of 60
`iterations. This is impractical, however, the number
`could be dramatically lowered, if some suitable crite-
`rion on when to stop an inner (turbo code) iteration
`was applied. Simulations show that for the first outer
`iterations only a few inner iterations are necessary.
`
`4
`
`

`

`1
`
`1
`
`1
`
`1
`
`024-14096-QAM.
`
`Compared to uniform ASK with R
`0.5, shaping by
`NU-ASK does not entail any additional computational
`effort. Only the lookup tables for signal transmission
`and metric calculation have to be replaced, so signal
`shaping with gains up to about 1
`has been demon-
`strated without changing the
`structure.
`
`SNR
`
`(a) With PD (without AFD)
`
`25611
`
`SNR
`
`(b) With AFD
`
`VII. APPENDIX
`Signal point constellations for NU-ASK
`
`4-NU-ASK
`f0.377525
`
`8-NU-ASK
`f0.170563
`
`16-NU-ASK
`f0.081626
`f0.807881
`
`41.362892
`
`f0.961643
`
`f1.662971
`
`f1.050833
`
`f0.418533
`51.371404
`
`f0.039934
`f0.367358
`f0.739052
`f1.254465
`
`50.120107
`f0.453957
`f0.847173
`fl.446126
`
`f0.200994
`f0.544127
`
`f 1.709493
`
`f0.283207
`f0.638580
`f1.099117
`f2.196958
`
`64-NU-ASK
`
`f0.059337
`
`410.099030
`
`f1.201035
`f1.617051
`
`f1.285369
`f1.779256
`
`f1.379905
`f2.007313
`
`f1.488176
`f2.441711
`
`Figure 5.
`BICM
`
`Comparison between NU-$AM and uniform QAM for
`
`128-NU-ASK
`
`f0.0295222
`
`f0.0492204
`
`f0.0689388
`
`VI. CONCLUSIONS
`A simple method to obtain shaping gain for higher
`order ASK modulation on AWGN channels was pre-
`sented.
`Using non-uniform ASK in conjunction with BICM,
`shaping gains in the order of the theoretically pre-
`dicted capacity gains can be achieved with a real cod-
`ing system, independent of whether parallel decoding
`(no AFD) or additional feedback decoding is used. Im-
`provements due to AFD are within 0.5 - 1
`Compared to other shaping methods, the achieved
`gain is lower, and even for higher order modulation
`scheme it does not seem approach the ultimate gain
`of 1.53
`Also, the fact that shaping gain is exhib-
`ited only when the constellations are used with code
`rates R = 0.5 might lead to a higher decoding effort
`in comparison to higher rate coding schemes. Further-
`more, the enlarged signal point constellation, entailing
`a considerably higher peak-to-average power ratio, can
`be prohibitive in some applications.
`The main Advantage is the simplicity of the method.
`
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