INTERNATIONAL JOURNAL OF SATELLITE COMMUNICATIONS AND NETWORKING
`Int. J. Satell. Commun. Network. 2006; 24:261–281
`Published online 19 May 2006 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/sat.841
`
`Turbo-coded APSK modulations design for satellite
`broadband communications
`
`Riccardo De Gaudenzi1,*,y, Albert Guille´ n i Fa` bregas2, and Alfanso Martinez3
`
`1 European Space Agency (ESA-ESTEC), Noordwijk, The Netherlands
`2 Institute for Telecommunications Research, University of South Australia, Australia
`3 Technische Universitat Eindhoven, Eindhoven, The Netherlands
`
`SUMMARY
`
`This paper investigates the design of power and spectrally efficient coded modulations based on amplitude
`phase shift keying (APSK) modulation with application to satellite broadband communications. APSK
`represents an attractive modulation format for digital transmission over nonlinear satellite channels due to
`its power and spectral efficiency combined with its inherent robustness against nonlinear distortion. For
`these reasons APSK has been very recently introduced in the new standard for satellite Digital Video
`Broadcasting named DVB-S2. Assuming an ideal rectangular transmission pulse, for which no nonlinear
`inter-symbol interference is present and perfect pre-compensation of the nonlinearity, we optimize the
`APSK constellation. In addition to the minimum distance criterion, we introduce a new optimization based
`on the mutual
`information; this new method generates an optimum constellation for each spectral
`efficiency. To achieve power efficiency jointly with low bit error rate (BER) floor we adopt a powerful
`binary serially concatenated turbo-code coupled with optimal APSK modulations through bit-interleaved
`coded modulation. We derive tight approximations on the maximum-likelihood decoding error
`probability, and results are compared with computer simulations. The proposed coded modulation
`scheme is shown to provide a considerable performance advantage compared to current standards for
`satellite multimedia and broadcasting systems. Copyright # 2006 John Wiley & Sons, Ltd.
`
`Received 1 May 2005; Revised 1 February 2006; Accepted 29 March 2006
`
`KEY WORDS:
`
`turbo codes; amplitude-phase shift keying (APSK); modulation; bit-interleaved coded
`modulation (BICM); coded modulation; nonlinear channels; satellite communications
`
`1. INTRODUCTION
`
`A major strength of satellite communications systems lies on their ability to efficiently broadcast
`digital multi-media information over very large areas [1]. A notable example is the
`
`*Correspondence to: Riccardo De Gaudenzi, European Space Agency (ESA-ESTEC), RF Payload Systems division,
`Keplerlaan 1, P.O. Box 299, AG Noordwijk, The Netherlands.
`y E-mail: riccardo.de.gaudenzi@esa.int
`
`Copyright # 2006 John Wiley & Sons, Ltd.
`
`

`

`262
`
`R. DE GAUDENZI, A. GUILLE´ N I FA` BREGAS AND A. MARTINEZ
`
`so-called direct-to-home (DTH) digital television broadcasting. Satellite systems also provide a
`unique way to complement
`the terrestrial
`telecommunication infrastructure in scarcely
`populated regions. The introduction of multi-beam satellite antennas with adaptive coding
`and modulation (ACM) schemes will allow an important efficiency increase for satellite systems
`operating at Ku or Ka-band [2]. Those technical enhancements require the exploitation of
`power- and spectrally efficient modulation schemes conceived to operate over the satellite
`nonlinear channel. In this paper, we will design high-efficiency 16- and 32-ary coded modulation
`schemes suited for nonlinear satellite channels. The analysis presented here is complemented in
`[3] with the effects related to satellite nonlinear distortion, band-limited transmission pulse,
`demodulator timing, amplitude and phase estimation errors.
`To the authors’ knowledge there are few references in the literature dealing with 16-ary
`constellation optimization over nonlinear channels,
`the typical environment
`for satellite
`channels. Previous work showed that 16-QAM does not compare favourably with either
`trellis-coded (TC) 16-PSK or uncoded 8-PSK in satellite nonlinear channels [4]. The concept of
`circular APSK modulation was already proposed 30 years ago by Thomas et al. [5], where
`several nonband-limited APSK sets were analysed by means of uncoded bit error rate bounds;
`the suitability of APSK for nonlinear channels was also made explicit, but concluded that for
`single carrier operation over nonlinear channel APSK performs worse than PSK schemes. In the
`current paper, we will revert the conclusion. It should be remarked that Reference [5] mentioned
`the possibility of modulator pre-compensation but did not provide performance results related
`to this technique. Foschini et al. [6] optimized QAM constellations using asymptotic uncoded
`probability of error under average power constraints, deriving optimal 16-ary constellation
`made of an almost equilateral lattice of triangles. This result is not applicable to satellite
`channels. In Reference [7] some comparison between squared QAM and circular APSK over
`linear channels was performed based on the computation of the error bound parameter,
`showing some minor potential advantage of APSK. Further work on mutual information for
`modulations with average and peak power constraints is reported in Reference [8], which proves
`the advantages of circular APSK constellations under those power constraints. Mutual
`information performance loss for APSK in peak power limited Gaussian complex channels is
`reported in Reference [9] and compared to classical QAM modulations; it is shown that under
`this assumption APSK considerably outperforms QAM in terms of mutual information, the
`gain particularly remarkable for 16- and 64-ary constellations.
`Forward error correcting codes for our application must combine power efficiency and low
`BER floor with flexibility and simplicity to allow for high-speed implementation. The existence
`of practical, simple, and powerful such coding designs for binary modulations has been settled
`with the advent of turbo codes [10] and the recent re-discovery of low-density parity-check
`(LDPC) codes [11]. In parallel, the field of channel coding for nonbinary modulations has
`evolved significantly in the latest years. Starting with Ungerboeck’s work on TC modulation
`(TCM) [12], the approach had been to consider channel code and modulation as a single entity,
`to be jointly designed and demodulated/decoded. Schemes have been published in the literature,
`where turbo codes are successfully merged with TCM [13]. Nevertheless, the elegance and
`simplicity of Ungerboeck’s original approach gets somewhat lost in a series of ad hoc
`adaptations; in addition, the turbo-code should be jointly designed with a given modulation, a
`solution impractical for system supporting several constellations. A new pragmatic paradigm
`has crystallized under the name of bit-interleaved coded modulation (BICM) [13], where
`extremely good results are obtained with a standard nonoptimized, code. An additional
`
`Copyright # 2006 John Wiley & Sons, Ltd.
`
`Int. J. Satell. Commun. Network. 2006; 24:261–281
`DOI: 10.1002/sat
`
`

`

`TURBO-CODED APSK MODULATIONS DESIGN
`
`263
`
`advantage of BICM is its inherent flexibility, as a single mother code can be used for several
`modulations, an appealing feature for broadband satellite communication systems where a large
`set of spectral efficiencies is needed.
`This paper is organized as follows. Section 2 gives the system model under the ideal case of a
`rectangular transmission pulse.z Section 3 gives a formal description of APSK signal sets,
`describes the maximum mutual information and maximum minimum distance optimization
`criteria and discusses some of the properties of the optimized constellations. Section 4 deals with
`code design issues, describes the BICM approach, provides some analytical considerations based
`on approximate maximum-likelihood (ML) decoding error probability bounds, and provides
`some numerical results. The conclusions are finally drawn in Section 5.
`
`2. SYSTEM MODEL
`
`The baseband equivalent of the transmitted signal at time t; sT ðtÞ, is given by
`L1
`
`
`
`ffiffiffiffip X
`
`sT ðtÞ ¼
`
`P
`
`xðkÞpT ðt kTsÞ
`
`k¼0
`
`ð1Þ
`
`where P is the signal power, xðkÞ is the kth transmitted symbol, drawn from a complex-valued
`APSK signal constellation X; with jXj ¼ M; pT is the transmission filter impulse response, and
`Ts is the symbol duration (in seconds), corresponding to one channel use. Without loss of
`generality, we consider transmission of frames with L symbols. The spectral efficiency R is
`defined as the number of information bits conveyed at every channel use, and in measured in
`bits per second per Hertz (bps/Hz).
`The signal sT ðtÞ passes through a high-power amplifier (HPA) operated close to the saturation
`point. In this region, the HPA shows nonlinear characteristics that induce phase and amplitude
`distortions to the transmitted signal. The amplifier is modelled by a memoryless nonlinearity,
`with an output signal sAðtÞ at time t given by
`sAðtÞ ¼ FðjsT ðtÞjÞejðfðsT ðtÞÞþFðjsT ðtÞjÞÞ
`
`ð2Þ
`
`where we have implicitly defined FðAÞ and FðAÞ as the AM/AM and AM/PM characteristics of
`the amplifier for a signal with instantaneous signal amplitude A: The signal amplitude is the
`is decomposed as sT ðtÞ ¼
`instantaneous complex envelope, so that the baseband signal
`jsT ðtÞjejfðsT ðtÞÞ:
`In this paper, we assume an (ideal) signal modulating a train of rectangular pulses. These
`pulses do not create inter-symbol interference when passed through an amplifier operated in the
`nonlinear region. Under these conditions, the channel reduces to an AWGN, where the
`modulation symbols are distorted following (2). Let xA denote the distorted symbol
`corresponding to x ¼ jxjejfðxÞ 2 X; that is, xA ¼ FðjxjÞejðfðxÞþFðjxjÞÞ: After matched filtering and
`sampling at time kTs; the discrete-time received signal at time k; yðkÞ is then given by
`
`p
`
`ffiffiffiffiffi
`
`yðkÞ ¼
`
`Es
`
`xAðkÞ þ nðkÞ;
`
`k ¼ 0; . . . ; L 1
`
`ð3Þ
`
`z This assumption has been dropped in the paper [14].
`
`Copyright # 2006 John Wiley & Sons, Ltd.
`
`Int. J. Satell. Commun. Network. 2006; 24:261–281
`DOI: 10.1002/sat
`
`

`

`264
`
`R. DE GAUDENZI, A. GUILLE´ N I FA` BREGAS AND A. MARTINEZ
`
`with Es the symbol energy, given by Es ¼ PTs; xAðkÞ is the symbol at the kth time instant, as
`defined above, and nðkÞ  NCð0; N0Þ is the corresponding noise sample.
`This simplified model suffices to describe the nonlinearity up to the nonlinear ISI effect, and
`allows us to easily design constellation and codes. In the paper [14], the impact of nonlinear ISI
`has been considered, as well as other realistic demodulation effects such as timing and phase
`recovery.
`
`3. APSK CONSTELLATION DESIGN
`
`In this section, we define the generic multiple-ring APSK constellation family. We propose new
`criteria for the design of digital QAM constellations of 16 and 32 points, with special emphasis
`on the behaviour on nonlinear channels.
`
`3.1. Constellation description
`
`M-APSK constellations are composed of nR concentric rings, each with uniformly spaced PSK
`points. The signal constellation points x are complex numbers, drawn from a set X given by
`r1ejðð2p=n1Þiþy1Þ;
`i ¼ 0; . . . ; n1 1 ðring 1Þ
`r2ejðð2p=n2Þiþy2Þ;
`
`8>>>>>>><
`>>>>>>>:
`
`X ¼
`
`i ¼ 0; . . . ; n2 1 ðring 2Þ
`
`ð4Þ
`
`. r
`
`..
`
`P
`
`nR ejðð2p=nRÞiþynR Þ;
`i ¼ 0; . . . ; nnR 1 ðring nRÞ
`where we have defined n‘; r‘ and y‘ as the number of points, the radius and the relative phase
`shift for the ‘th ring. We will nickname such modulations as n1 þ þ nnR -APSK. Figure 1
`depicts the 4 þ 12- and 4 þ 12 þ 16-APSK modulations with quasi-Gray mapping. In
`particular, for next generation broadband systems [2, 15], the constellation sizes of interest
`are jXj ¼ 16 and 32; with nR ¼ 2 and 3 rings, respectively. In general, we consider that X is
`normalized in energy, i.e. E½jxj2Š ¼ 1; which implies that the radii r‘ are normalized such that
`nR
`‘ ¼ 1: Notice also that the radii r‘ are ordered, so that r15 5rnR:
`‘¼1 n‘r2
`Clearly, we can also define the phase shifts and the ring radii in relative terms rather than in
`absolute terms, as in (4); this removes one dimension in the optimization process, yielding a
`practical advantage. We let f‘ ¼ y‘ y1 for ‘ ¼ 1; . . . ; nR be the phase shift of the ‘th ring with
`respect to the inner ring. We also define r‘ ¼ r‘=r1 for ‘ ¼ 1; ; nR as the relative radii of the
`‘th ring with respect to r1: In particular, f1 ¼ 0 and r1 ¼ 1:
`
`3.2. Constellation optimization in AWGN
`Þ
`We are interested in finding an APSK constellation, defined by the parameters q ¼ ðr1; . . . ; rnR
`and / ¼ ðf1; . . . ; fnR
`Þ; such that a given cost function f ðXÞ reaches a minimum. The simplest,
`and probably most natural, cost function is the minimum Euclidean distance between any two
`points in the constellation. Section 3.2.1 shows the results under this criterion. These results are
`extended in Section 3.2.2, where the cost function is replaced by the mutual information of the
`AWGN channel; it also shown that significant gains may be achieved for low and moderate
`values of signal-to-noise ratio (SNR) by fine-tuning the constellation.
`
`Copyright # 2006 John Wiley & Sons, Ltd.
`
`Int. J. Satell. Commun. Network. 2006; 24:261–281
`DOI: 10.1002/sat
`
`

`

`TURBO-CODED APSK MODULATIONS DESIGN
`
`265
`
`Figure 1. Parametric description and pseudo-Gray mapping of 16 and 32-APSK constellations with
`n1 ¼ 4; n2 ¼ 12; f2 ¼ 0 and n1 ¼ 4; n2 ¼ 12; n3 ¼ 16; f2 ¼ 0; f3 ¼ p=16: For the first two rings: mapping
`below corresponds to 4 þ 12-APSK, mapping above to 4 þ 12 þ 16-APSK.
`
`ð5Þ
`
`1A
`
`ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
`
`Esjx x0j2
`2N0
`
`s0
`@
`
`Q
`
`3.2.1. Minimum Euclidean distance maximization. The union bound on the uncoded symbol
`error probability [16] yields,
`
`X x
`X x
`
`2X
`
`02X
`x0=x
`
`Pe4 1
`M
`
`
`
`ffiffiffiffiffiffip R
`
`1 x
`
`eðt2=2Þ dt is the Gaussian tail function. At high SNR Equation (5) is
`where QðxÞ ¼ 1=
`2p
`dominated by the pairwise term at minimum squared Euclidean distance d2
`min ¼ minx;x02X
`jx x0j2: Due to the monotonicity of the Q function, it is clear that maximizing this distance
`optimizes the error performance estimated with the union bound at high SNR.
`The minimum distance of the constellation depends on the number of rings nR; the number of
`points in each ring n1; . . . ; nnR ; the radii r1; . . . ; rnR ; and the offset among the rings f1; . . . ; fnR
`:
`The constellation geometry clearly indicates that the distances to consider are between points
`belonging to the same ring, or between points in adjacent rings. Simple calculations give the
`
`Copyright # 2006 John Wiley & Sons, Ltd.
`
`Int. J. Satell. Commun. Network. 2006; 24:261–281
`DOI: 10.1002/sat
`
`

`

`R. DE GAUDENZI, A. GUILLE´ N I FA` BREGAS AND A. MARTINEZ
`
`266
`
`following formula,
`
`
`
`
`
` 
`
`d2
`
`
`ring i ¼ 2r2i 1 cos
`
`2p
`ni
`
`ð6Þ
`
`for the distance between points in ith ring, with radius ri and ni points. For the adjacent rings the
`calculation is only slightly more complicated, and gives the following:
`d2
`
`
`rings i;iþ1 ¼ r2i þ r2iþ1 2ririþ1 cos y
`where y is the minimum relative offset between any pair of points of rings i and i þ 1;
`respectively. As the phase of point li in ring i is given by fi þ 2pli=ni; we easily obtain:
`li
`liþ1
`ni
`niþ1
`
`

`
`y ¼ min
`li;liþ1
`
`
`
`ðfi fiþ1Þ þ 2p
`
`
`
`
`
`

`
`ð7Þ
`
`ð8Þ
`
`The minimum distance of the constellation is given by taking the minimum of all these inter-ring
`and intra-ring values:
`
`d2
`min ¼ min
`i¼1;...;nR
`j¼1;...;nR1
`
`
`
`rings j; jþ1gfd2ring i; d2
`
`
`
`ð9Þ
`
`For the sake of space limitations, we concentrate on 16-ary constellations. Thanks to
`symmetry considerations, is clear that the best offset between rings happens when f2 ¼ p=n2:
`Figure 2 shows the minimum distance for several candidates: 4 þ 12-, 6 þ 10-, 5 þ 11 and
`1 þ 5 þ 10-APSK. It may be observed that the highest minimum distance is achieved for
`approximately r2 ¼ 2:0; except for 4 þ 12-APSK, where r2 ¼ 2:7: The results for f ¼ 0 are also
`plotted, and show that the corresponding minimum distance is smaller. We will see later in
`Section 3.2.2 how this effect translates into error rate performance.
`
`information maximization. The mutual
`3.2.2. Mutual
`information (assuming equiprobable
`symbols) for a given signal set X provides the maximum transmission rate (in bits/channel
`use) at which error-free transmission is possible with such signal set, and is given by (e.g.
`Reference [13]),
`
`f ðXÞ ¼ C ¼ log2 M Ex;n
`
`log2
`
`02X
`
`Interestingly, for a given SNR, or equivalently, for a given spectral efficiency R; an optimum
`constellation can be obtained, a procedure we apply in the following to 16- and 32-ary
`constellations.
`In general, closed-form optimization of this expression is a daunting task, so we resort to
`numerical techniques. Expression (10) can be easily evaluated by using the Gauss–Hermite
`quadrature rules, making numerical evaluation very simple. Note, however, that it is possible to
`calculate a closed-form expression for the asymptotic case Es=N0 ! þ1: First, note that the
`expectation in Equation (10) can be rewritten as
`
`
`
` ffiffiffiffiffi
`p
`
`
`
`
`
`#
`
`)
`
`
`exp 1
`N0
`
`X x
`
`02X
`
`"
`
`(
`
`lðXÞ ¼4 Ex;n
`
`log2
`
`Esjx x0j2 þ 2 Re
`
`ðx x0Þn
`
`Es
`
`ð11Þ
`
`Using the dominated convergence theorem [17], the influence of the noise term vanishes
`asymptotically, since the limit can be pushed inside the expectation. Furthermore, the only
`
`Copyright # 2006 John Wiley & Sons, Ltd.
`
`Int. J. Satell. Commun. Network. 2006; 24:261–281
`DOI: 10.1002/sat
`
`
`exp 1
`N0
`
`

`
`p
`
`ffiffiffiffiffi
`
`ðx x0Þ þ n
`
`Es
`
`

2
`
`

`
`
`
`n
`
`!
`
`
`
`#)
`
`

2
`
`ð10Þ
`
`"
`
`X x
`
`(
`
`

`

`TURBO-CODED APSK MODULATIONS DESIGN
`
`267
`
`4+12−APSK
`5+11−APSK
`6+10−APSK
`1+5+10−APSK
`
`0.7
`
`0.6
`
`0.5
`
`0.4
`
`0.3
`
`0.2
`
`0.1
`
`Minimum Euclidean Distance
`
`0
`
`0
`
`0.5
`
`1
`
`1.5
`Outer radius ρ
`
`2
`
`2.5
`
`3
`
`Figure 2. Minimum Euclidean distances for several 16-ary signal constellations. Solid lines correspond
`to f ¼ p=n2; dotted lines to f ¼ 0:
`
`remaining terms in the summation over x0 2 X are x0 ¼ x and those closest in Euclidean
`min ¼ minx02X x x0
`j2; of which there are nminðxÞ: Therefore the expectation
`distance d2
`j
`becomes
`
`
`
`
`
`
`
`
`
`
`
`log2 1 þ nminðxÞexp 1
`N0
`
`
`
`
`
`Noting that the exponential takes very small values, the approximation log2ð1 þ xÞ ’ x log2 e
`for jxj551 holds, thus by simplifying further the expectation we obtain:
`lðXÞ ’ Ex nminðxÞexp Es
`’ a exp Es
`N0
`N0
`
`
`where a is a constant that does not depend on the constellation minimum distance dmin nor on
`SNR. Then the capacity at large SNR becomes:
`f ðXÞ ¼ log2 M a exp Es
`N0
`
`lðXÞ ’ Ex
`
`Esd2
`min
`
`d2
`min
`
`log2 e
`
`d2
`min
`
`
`
`d2
`min
`
`ð12Þ
`
`ð13Þ
`
`ð14Þ
`
`It appears then clear that the procedure corresponds to the maximization of the minimum
`Euclidean distance, as in Section 3.2.1.
`Figure 3 shows the numerical evaluation of Equation (10) for a given range of values of r2
`and f ¼ f2 f1 for the 4 þ 12-APSK constellation at Es=N0 ¼ 12 dB: Surprisingly, there is no
`noticeable dependence on f: Therefore, the two-dimensional optimization can be done by
`simply finding the r2 that maximizes mutual information. This result was found to hold
`
`Copyright # 2006 John Wiley & Sons, Ltd.
`
`Int. J. Satell. Commun. Network. 2006; 24:261–281
`DOI: 10.1002/sat
`
`

`

`268
`
`R. DE GAUDENZI, A. GUILLE´ N I FA` BREGAS AND A. MARTINEZ
`
`30
`
`25
`
`20
`
`10
`
`15
`ϕ
`
`5
`
`2.5
`
`ρ
`
`2
`
`3.65
`
`3.6
`
`3.55
`
`3.5
`
`3.45
`
`3.4
`
`3.35
`
`3.3
`3
`
`C(bit/s/Hz)
`
`0
`Figure 3. Capacity surface for the 16-APSK (n1 ¼ 4; n2 ¼ 12), with Es=N0 ¼ 12 dB:
`
`1.5
`
`true also for the other constellations and hence, in the following, mutual information optimization
`results do not account for f: Figure 4 shows the union bound on the symbol error probability (5)
`for several 16-APSK modulations, and for the optimum value of r2 at R ¼ 3 bps=Hz (found with
`the mutual information analysis). Continuous lines indicate f ¼ 0 while dotted lines refer to the
`maximum value of the relative phase shift, i.e. f ¼ p=n2; showing no dependence on f at high
`SNR. This absence of dependency is justified by the fact that the optimum constellation separates
`the rings by a distance larger than the number of points in the ring itself, so that the relative phase
`f has no significant impact in the distance spectrum of the constellation.
`For 16-APSK it is also interesting to investigate the mutual information dependency on n1
`and n2: Figure 5(a) depicts the mutual
`information curves for several configurations of
`optimized 16-APSK constellations and compared with classical 16-QAM and 16-PSK signal
`sets. As we can observe, mutual information curves are very close to each other, showing a slight
`advantage of 6 þ 10-APSK over the rest. In particular, note that there is a small gain, of about
`0.1 dB, in using the optimized constellation for every R; rather than the calculated with the
`minimum distance (or high SNR). However, as discussed in Reference [14], 6 þ 10-APSK and
`1 þ 5 þ 10-APSK show other disadvantages compared to 4 þ 12-APSK for phase recovery and
`nonlinear channel behaviour.
`Similarly, Figure 5(b) reports capacity of optimized 4 þ 12 þ 16-APSK (with the correspond-
`ing optimal values of r2 and r3) compared to 32-QAM and 32-PSK. We observe slight
`capacity gain of 32-APSK over PSK and QAM constellations. Other 32-APSK constellations
`with different distribution of points in the three rings did not provide significantly better
`results.
`Finally Table I provides the optimized 16- and 32-APSK parameters for various coding rates,
`giving an optimum constellation for each given spectral efficiency.
`
`Copyright # 2006 John Wiley & Sons, Ltd.
`
`Int. J. Satell. Commun. Network. 2006; 24:261–281
`DOI: 10.1002/sat
`
`

`

`TURBO-CODED APSK MODULATIONS DESIGN
`
`269
`
`SER for different uncoded APSK modulations (ρ opt for R = 3 bit/s/Hz)
`
`16 QAM
`4+12 APSK φ=0
`4+12 APSK φ=15
`6+10 APSK φ=0
`6+10 APSK φ= 18
`5+11 APSK φ=0
`5+11 APSK φ=16.36
`1+5+10 APSK φ=0
`1+5+10 APSK φ=18
`
`7
`
`8
`
`9
`
`10
`
`11
`Eb/N0 (dB)
`
`12
`
`13
`
`14
`
`15
`
`16
`
`100
`
`10–1
`
`10–2
`
`10–3
`
`10–4
`
`SER
`
`10–5
`
`10–6
`
`10–7
`
`10–8
`
`6
`
`Figure 4. Union bound on the uncoded symbol error probability for several APSK modulations. Note that
`the continuous line and the dashed line are indistinguishable because they are superimposed.
`
`3.3. Constellation optimization for nonlinear channels
`
`3.3.1. Peak-to-envelope considerations. For nonlinear transmission over an amplifier, 4 þ 12-
`APSK is preferable to 6 þ 10-APSK because the presence of more points in the outer ring allows
`to maximize the HPA DC power conversion efficiency. It is better to reduce the number of inner
`points, as they are transmitted at a lower power, which corresponds a lower DC efficiency. It is
`known that the HPA power conversion efficiency is monotonic with the input power drive up to
`its saturation point. Figure 6 shows the distribution of the transmitted signal envelope for
`16-QAM, 4 þ 12-APSK, 6 þ 10-APSK, 5 þ 11-APSK, and 16-PSK. In this case, the shaping
`filter is a square-root raised cosine (SRRC) with a roll-off factor a ¼ 0:35 as for the DVB-S2
`standard [15]. As we observe, the 4 þ 12-APSK envelope is more concentrated around the outer
`ring amplitude than 16-QAM and 6 þ 10-PSK, being remarkably close to the 16-PSK case. This
`shows that the selected constellation represents a good trade-off between 16-QAM and 16-PSK,
`with error performance close to 16-QAM, and resilience to nonlinearity close to 16-PSK.
`Therefore, 4 þ 12þAPSK is preferable to the rest of 16-ary modulations considered. Similar
`advantages have been observed for 32-APSK compared to 32-QAM.
`
`3.3.2. Static distortion compensation. The simplest approach for counteracting the HPA
`nonlinear characteristic for the APSK signal, as already introduced in Section 2, is to modify the
`complex-valued constellation points at the modulator side. Thanks to the multiple-ring nature
`of the APSK constellation, pre-compensation is easily done by a simple modification of the
`
`Copyright # 2006 John Wiley & Sons, Ltd.
`
`Int. J. Satell. Commun. Network. 2006; 24:261–281
`DOI: 10.1002/sat
`
`

`

`270
`
`R. DE GAUDENZI, A. GUILLE´ N I FA` BREGAS AND A. MARTINEZ
`
`5
`
`10
`
`15
`
`Eb/N0 (dB)
`
`Capacity 4+12+16 APSK
`ρ
`pt
` 4+12+16 APSK
`ρ
`pt 4+12+16 APSK
`Capacity 32PSK
`Capacity 32QAM
`
`2o
`
`3o
`
`4+12-APSK
`5+11-APSK
`1+5+10-APSK
`6+10-APSK
`16-PSK
`16-QAM
`6+10−APSK dmin
`
`4
`
`3.5
`
`3
`
`2.5
`
`2
`
`1.5
`
`1
`
`Capacity (bit/s/Hz)
`
`0.5
`
`0
`
`(a)
`
`6
`
`5.5
`
`5
`
`4.5
`
`4
`
`3.5
`
`3
`
`2.5
`
`i
`
`Capacity (b/s/Hz),ρ
`
`2
`10
`
`(b)
`
`11
`
`12
`
`13
`
`14
`
`15
`Es/N0 (dB)
`
`16
`
`17
`
`18
`
`19
`
`20
`
`Figure 5. Capacity and ropt for the optimized APSK signal constellations vs QAM and PSK: (a) 16-ary
`constellations (zoom around 3 bps/Hz); and (b) 32-ary constellations.
`
`parameters r‘; and f‘: The objective is to exploit the known AM/AM and AM/PM HPA
`characteristics in order to obtain a good replica of the desired signal constellation geometry after
`the HPA, as if it had not suffered any distortion. This can be simply obtained by artificially
`increasing the relative radii r‘ and modifying the relative phases f‘ at the modulator side. This
`
`Copyright # 2006 John Wiley & Sons, Ltd.
`
`Int. J. Satell. Commun. Network. 2006; 24:261–281
`DOI: 10.1002/sat
`
`

`

`TURBO-CODED APSK MODULATIONS DESIGN
`
`271
`
`Table I. Optimized constellation parameters for 16- ary and 32-ary APSK.
`ropt
`1
`
`Coding rate, r
`
`Spectral eff. (bps/Hz)
`
`Modulation order
`
`ropt
`2
`
`N/A
`N/A
`N/A
`N/A
`N/A
`N/A
`
`5.27
`4.87
`4.64
`4.33
`4.30
`
`4 þ 12-APSK
`4 þ 12-APSK
`4 þ 12-APSK
`4 þ 12-APSK
`4 þ 12-APSK
`4 þ 12-APSK
`
`4 þ 12 þ 16-APSK
`4 þ 12 þ 16-APSK
`4 þ 12 þ 16-APSK
`4 þ 12 þ 16-APSK
`4 þ 12 þ 16-APSK
`
`2/3
`3/4
`4/5
`5/6
`8/9
`9/10
`
`3/4
`4/5
`5/6
`8/9
`9/10
`
`2.67
`3.00
`3.20
`3.33
`3.56
`3.60
`
`3.75
`4.00
`4.17
`4.44
`4.50
`
`3.15
`2.85
`2.75
`2.70
`2.60
`2.57
`
`2.84
`2.72
`2.64
`2.54
`2.53
`
`16 QAM
`4+12 APSK ρ=2.8
`5+11 APSK ρ=2.65
`6+10 APSK ρ=2.5
`16 PSK
`
`0.06
`
`0.05
`
`0.04
`
`0.03
`
`pdf
`
`0.02
`
`0.01
`
`0
`–20
`
`–15
`
`–10
`
`–5
`Signal power (dB)
`
`0
`
`5
`
`10
`
`Figure 6. Simulated histogram of the transmitted signal envelope power for 16-ary constellations.
`
`approach neglects nonlinear ISI effects at the matched filter output which are not present under
`the current assumption of rectangular symbols; ISI issues has been discussed in Reference [14].
`
`In the 16-ary APSK case, the new constellation points x0 follow (4), with new radii r01; r02; such
`
`
`
`2Þ ¼ r2. Concerning the phase, it is possible to pre-correct for the relativethat Fðr01Þ ¼ r1; and Fðr0
`phase offset introduced by the HPA between inner and outer ring by simply changing the
`relative phase shift by f0
`
`1Þ: These operations can be readily2 ¼ f2 þ Df; with Df ¼ fðr02Þ fðr0
`
`implemented in the digital modulator by simply modifying the reference constellation
`parameters r0; f0; with no hardware complexity impact or out-of-band emission increase at the
`
`Copyright # 2006 John Wiley & Sons, Ltd.
`
`Int. J. Satell. Commun. Network. 2006; 24:261–281
`DOI: 10.1002/sat
`
`

`

`272
`
`R. DE GAUDENZI, A. GUILLE´ N I FA` BREGAS AND A. MARTINEZ
`
`linear modulator output. On the other side, this allows to shift all the compensation effort into
`the modulator side allowing the use of an optimal demodulator/decoder for AWGN channels
`even when the amplifier is close to saturation. The pre-compensated signal expression at the
`modulator output is then
`
`
`
`ffiffiffiffip X
`
`L1
`
`spre
`T ¼
`
`P
`
`x0ðkÞpT ðt kTsÞ
`
`ð15Þ
`
`k¼0
`‘ and f0
`where now x0ðkÞ 2 X0 are the pre-distorted symbols with r0
`‘ for ‘ ¼ 1; . . . ; nR:
`
`4. FORWARD ERROR CORRECTION CODE DESIGN AND PERFORMANCE
`
`In this section, we describe the coupling of turbo-codes and the APSK signal constellations
`through BICM and we discuss some of the properties of this approach.} As already mentioned
`in Section 1, such approach is a good candidate for flexible constellation format transmission.
`The main drivers for the selection of the FEC code have been flexibility, i.e. use a single mother
`code, independently of the modulation and code rates; complexity, i.e. have a code as compact
`and simple as possible; and good performance, i.e. approach Shannon’s capacity bound as much
`as possible.
`We consider throughout a coded modulation scheme for which the transmitted symbols
`x ¼ ðx0; . . . ; xL1Þ are obtained as follows: (1) The information bits sequence a ¼ ða0; . . . ; aK1Þ
`is encoded with a binary code C 2 FN
`rate r ¼ K=N;
`(2)
`the encoded sequence
`2 of
`c ¼ ðc0; . . . ; cN1Þ 2 C is bit-interleaved, with an index permutation p ¼ ðp0; . . . ; pN1Þ; (3) the
`bit-interleaved sequence cp is mapped to a sequence of modulation symbols x with a labelling
`rule m : FM
`2 ! X; such that mða1; . . . ; aMÞ ¼ x: In addition to the description of the code, we also
`propose the use of some new heuristics to tune the final design of the BICM codes.
`
`4.1. Code design
`
`It was suggested in Reference [13] that the binary code C can be optimized for a binary channel
`(such as BPSK or QPSK with AWGN). We substantiate this claim with some further insights on
`the effect of the code minimum distance in the error performance. The Bhattachharyya union
`bound (BUB) on the frame error probability Pe for a BICM modulation assuming that no
`iterations are performed at the demapper side is given by [13]:
`AðdÞBðEs=N0Þd
`Pe4
`
`ð16Þ
`
`where AðdÞ is the number of codewords at a Hamming distance d; dmin is the minimum
`Hamming distance, with BðEs=N0Þ denoting the Bhattachharyya factor, which is given by
`
`X d
`
`X
`
`X1
`
`X
`
`log2 M
`
`8<
`:
`
`En
`
`ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
`vuut
`9=
`P
`P
`; ð17Þ
`
`z2Xi¼%b
`
`expðð1=N0Þjx z þ nj2Þ
`expðð1=N0Þjx z þ nj2Þ
`
`BðEs=N0Þ ¼
`
`1
`M log2 M
`
`i¼1
`
`b¼0
`
`x2Xi¼b
`
`z2Xi¼b
`
`} The optimization method based on the mutual information proposed in Section 3.2.2 can be easily extended to the case
`of the BICM mutual information [13] with almost identical results assuming Gray mapping. However, we use the
`proposed method in order to keep the discussion general and not dependent on the selected coding scheme.
`
`Copyright # 2006 John Wiley & Sons, Ltd.
`
`Int. J. Satell. Commun. Network. 2006; 24:261–281
`DOI: 10.1002/sat
`
`

`

`TURBO-CODED APSK MODULATIONS DESIGN
`
`273
`
`QPSK
`16QAM
`4+12APSK
`4+12+16APSK
`−7
`p = 10
`−4
`p = 10
`
`12
`
`10
`
`8
`
`6
`
`4
`
`2
`
`Bound on Minimum Distance d0
`
`0
`
`0
`
`1
`
`2
`
`3
`
`4
`5
`6
`Normalized Es/N0 (dB)
`Figure 7. Lower bound d0 vs normalized Es=N0 for target Pe ¼ 104; 107; QPSK and 16- and 32-ary
`modulations and Gray labelling.
`
`7
`
`8
`
`9
`
`10
`
`ðxÞ ¼ bg; where m1
`where Xi¼b ¼ fx 2 Xjm1
`ðxÞ ¼ b denotes that the ith position of binary label
`i
`i
`x is equal to b: Equation (17) can be evaluated very efficiently using the Gauss–Hermite
`quadrature rules. For sufficiently large Es=N0 the BUB in Equation (5) is dominated by the term
`at minimum distance, i.e. the error floor
`Pe ’ Admin BðEs=N0Þdmin
`
`ð18Þ
`
`From this equation, we can derive an easy lower bound on the d0 on the minimum distance of C
`for a given target error rate, modulation, and number of codewords at dmin:
`dmin5dd0e where d0 ¼ log Pe log Admin
`log B
`
`ð19Þ
`
`where dxe denotes the smallest integer greater or equal to x: Notice that the target error rate is
`fixed to be the error floor under ML decoding.} The lowest error probability floor is achieved by
`a code C with Admin ¼ 1: Figure 7 shows the lower bound d0 with Admin ¼ 1; as a function of
`Es=N0 for target Pe ¼ 104; 107; QPSK, 16-QAM, 16- and 32-APSK modulations and Gray
`labelling. In order to ease the comparison, a normalized SNR is used, defined as
`
`

norm
`
`Es
`N0
`
`¼
`
`Es
`N0
`
`1
`2R 1
`
`ð20Þ
`
`} Although this does not necessarily hold under iterative decoding, it does still provide a useful guideline into the
`performance.
`
`Copyright # 2006 John Wiley & Sons, Ltd.
`
`Int. J. Satell. Commun. Network. 2006; 24:261–281
`DOI: 10.1002/sat
`
`

`

`274
`
`R. DE GAUDENZI, A. GUILLE´ N I FA` BREGAS AND A. MARTINEZ
`
`where R is the spectral efficiency, and the normalization is thus to the channel capacity. The
`code rate has been taken r ¼ 3=4 for all cases. Note that a capacity-achieving pair modulation-
`code would work at a normalized Es=N0jnorm ¼ 1 or 0 dB.
`A remarkable conclusion is that BICM with Gray mapping preserves the properties of C
`regardless of the modulation used, since we observe that the requirements for nonbinary
`modulations are similar to those for binary modulations (in the error-floor region). In order to
`work at about 3 dB from capacity, that is, a normalized Es=N0jnorm ¼ 3 dB; the needed d0 is
`about 5 and 10 for a frame error rate of 104 and 107; respectively.
`We consider that C is a serial concatenatation of convolutional codes (SCCC) [18], with
`outer code CO of length LO and rate rO and inner code CI of length LI and rate rI:
`Obviously, LI ¼ N and rOrI ¼ r: The resulting spectral efficiency is R ¼ r log2 M:
`It
`provides two key advantages with respect
`to parallel
`turbo codes:
`lower error floor,
`possibly achieving the bit error rate requirements (BER 41010) without any external
`code; and simpler constituent codes simpler than in turbo codes or in