DVB-S2 Spectrum Efficiency Improvement with Hierarchical Modulation
DDVB-S2 Spectrum Efficiency Improvement withHierarchical Modulation
Hugo M´eric ∗† and Jos´e Miguel Piquer †∗ INRIA Chile, Santiago, Chile † NIC Chile Research Labs, Santiago, ChileEmail: [email protected], [email protected]
Abstract —We study the design of a DVB-S2 system in order tomaximise spectrum efficiency. This task is usually challenging dueto channel variability. Modern satellite communications systemssuch as DVB-SH and DVB-S2 rely mainly on a time sharing strat-egy to optimise the spectrum efficiency. Recently, we showed thatcombining time sharing with hierarchical modulation can providesignificant gains (in terms of spectrum efficiency) compared tothe best time sharing strategy. However, our previous design doesnot improve the DVB-S2 performance when all the receiversexperience low or large signal-to-noise ratios. In this article, weintroduce and study a hierarchical QPSK and a hierarchical 32-APSK to overcome the previous limitations. We show in a realisticcase based on DVB-S2 that the hierarchical QPSK providesan improvement when the receivers experience poor channelcondition, while the 32-APSK increases the spectrum efficiencywhen the receivers experience good channel condition.
I. I
NTRODUCTION
In most broadcast applications, the Signal-to-Noise Ratio(SNR) experienced by each receiver can be quite different.For instance, in satellite communications the channel qualitydecreases with the presence of clouds in Ku or Ka band, orwith shadowing effects of the environment in lower bands.The first solution for broadcasting was to design the systemfor the worst-case reception, but this leads to poor performanceas many receivers do not exploit their full potential. Two otherschemes were proposed in [1] and [2]: time division multiplex-ing with variable coding and modulation, and superpositioncoding. Time division multiplexing, or time sharing, allocates aproportion of time to communicating with each receiver usingany modulation and error protection level. This functionality,called Variable Coding and Modulation (VCM) [3], is inpractice the most used in standards today. If a return channel isavailable, VCM may be combined with Adaptive Coding andModulation (ACM) to optimise the transmission parameters[3]. In superposition coding, the available energy is sharedamong several service flows which are sent simultaneously inthe same band. This scheme was introduced by Cover in [1]in order to improve the transmission rate from a single sourceto several receivers. When communicating with two receivers,the principle is to superimpose information for the user withthe best SNR. This superposition can be done directly at theForward Error Correction (FEC) level or at the modulationlevel as shown in Fig. 1 with a 16 Quadrature AmplitudeModulation (16-QAM).Hierarchical modulation is a practical implementation of Hierarchical IQ Fig. 1: Hierarchical modulation using a 16-QAM. Each con-stellation symbol carries data from 2 streams.superposition coding. Although hierarchical modulation hasbeen introduced to improve throughput, it has many othersapplications such as providing unequal protection [4], broad-casting local content [5], improving the performance of relaycommunication system [6] or backward compatibility [7],[8]. Note that none of the previous works use hierarchicalmodulation to improve the spectrum efficiency.Our work focuses on using hierarchical modulation inmodern broadcast systems to increase the transmission rate.For instance, even if the Low-Density Parity-Check (LDPC)codes of DVB-S2 approach the Shannon limit for the AdditiveWhite Gaussian Noise (AWGN) channel with one receiver[9], the throughput can be greatly increased for the broadcastcase. Indeed, we recently showed that combining ACM withhierarchical modulation improves the spectrum efficiency ofa DVB-S2 system [10]. To that end, we used the hierarchical16 Amplitude and Phase-Shift Keying (16-APSK) modulation.The performance improvement is significant, however thereis no gain when all the receivers experience low or largeSNRs. In this paper, we introduce and study a hierarchicalQuadrature Phase-Shift Keying (QPSK) and a hierarchical 32-APSK to tackle these limitations. The main contribution is thepresentation of two novel hierarchical modulations to improvethe spectrum efficiency of an AWGN broadcast channel.The paper is organised as follows: Section II introducesand studies two new hierarchical modulations. In Section III,we show how hierarchical modulation can improve the perfor-mance of satellite communication systems. Section IV studieson a DVB-S2 use case the spectrum efficiency improvementwith the previous modulations. Finally, Section V concludesthe paper by summarising the results and presenting the futurework. Digital Video Broadcasting - Satellite - Second Generation a r X i v : . [ c s . N I] O c t I. H
IERARCHICAL MODULATION
This part first introduces the principle of hierarchical mod-ulation. Then we introduce the hierarchical QPSK and 32-APSK modulations. Finally, we evaluate with simulations theperformance of both modulations on an AWGN channel.
A. Hierarchical modulation
As already mentioned, hierarchical modulations merge sev-eral streams in a same symbol. The available energy isshared between each stream. In this paper, two streams areconsidered. When hierarchical modulation is used for unequalprotection purposes, these flows are called High Priority (HP)and Low Priority (LP) streams. However, unequal protectionis not the goal of our work, so we will now refer to HighEnergy (HE) and Low Energy (LE) streams for the streamscontaining the most and the least energy, respectively.As each stream usually does not use the same energy, hier-archical modulations often rely on non-uniform constellations where the symbols are not uniformly distributed in the space.The geometry of non-uniform constellations is described usingthe constellation parameter(s). For instance, the hierarchical16-QAM in Fig. 1 uses a non-uniform 16-QAM describedby the constellation parameter α = d h /d l , where d h is theminimum distance between two constellation points carryingdifferent HE bits and d l is the minimum distance betweenany constellation point (see Fig. 1). The energy ratio betweenthe two streams is E he E le = (1 + α ) , (1)where E he and E le correspond to the amount of energyallocated to the HE and LE streams, respectively [5].In the literature, the most common hierarchical modulationsare the hierarchical 16-QAM and 8-PSK. We now introducethe hierarchical QPSK and 32-APSK modulations. For bothmodulations, we use an energy argument to choose the con-stellation parameter(s). B. Hierarchical QPSK and 32-APSK1) Hierarchical QPSK:
Fig. 2 shows the hierarchicalQPSK. To describe the constellation geometry, it requires oneparameter, θ . Without loss of generality, we can assume that (cid:54) θ (cid:54) π/ . The uniform QPSK corresponds to θ = π/ .When θ < π/ , the Least Significant Bit (LSB) in the mappingof each symbol is more protected than the Most Significant Bit(MSB) in the sense that its Bit Error Rate (BER) is smaller.The HE stream is transmitted with the LSB (see Fig. 2). θ Fig. 2: Hierarchical QPSK. Constellation parameter: θ For a given θ and energy per symbol ( E s ), we have E he = ρ he E s where ρ he = cos( θ ) . (2)The parameter ρ he corresponds to the amount of energyallocated to the HE stream. Remark that E le = (1 − ρ he ) E s .Even if the design of the hierarchical QPSK is easy, weshow in Section IV that it improves the performance of aDVB-S2 system when the receivers experience poor channelconditions. To the best of our knowledge, the hierarchicalQPSK has not been studied or used before.
2) Hierarchical 32-APSK:
Fig. 3 presents the hierarchical32-APSK and the mapping used in this paper. The mapping isbased on the Gray mapping of a 32-QAM. In each quadrant,the two MSB are identical (underlined bits in Fig. 3). Thesebits serve to transmit the HE stream. The constellation param-eters are the ratio between the radius of the middle ( R ) andinner ( R ) rings γ = R /R , the ratio between the radius ofthe outer ( R ) and inner ( R ) rings γ = R /R and the halfangle between the points on the outer ring in each quadrant θ (see Fig. 3). HE streamBits assigned to the R R R
110 001000 101010 111 011100110001 000101010111011 100110000101010111011 100 110 001000 101010 111 011100001 Fig. 3: Hierarchical 32-APSK. Constellation parameters: θ , γ = R /R and γ = R /R We consider the energy of the HE stream, given by theenergy of a QPSK modulation where the constellation pointsare located at the barycenter of the eight points in eachquadrant. Using the polar coordinates, the barycenter in theupper right quadrant is e iπ/ R + R (1+ e iθ + e − iθ )+ R ( e iθ + e − iθ + e iθ/ + e − iθ/ )8 . (3)Moreover, the symbol energy E s is expressed as E s = 4 R + 12 R + 16 R
32 = 1 + 3 γ + 4 γ R . (4)Combining (3) and (4), the distance d of the barycenter tothe origin is d = 1 + γ (1 + 2 cos θ ) + 2 γ (cos θ + cos θ/ (cid:112) γ + 4 γ ) (cid:112) E s . (5)Finally, the energy of the HE stream is equals to d whichis the energy of a QPSK modulation with each constellationpoint located at a distance d from the origin. Thus we canwrite E he = ρ he E s where ρ he = (1 + γ (1 + 2 cos θ ) + 2 γ (cos θ + cos θ/ γ + 4 γ ) . (6) ) Remark: Equations (2) and (6) introduce ρ he as the ratiobetween the energy of the HE stream E he and the symbolenergy E s . As the HE stream contains more energy than theLE stream, we verify that ρ he (cid:62) . . C. Performance of the hierarchical QPSK and 32-APSK
In practical systems, several values of ρ he have to be chosen.For the hierarchical QPSK, a given ρ he corresponds to onevalue of θ as shown in (2). Table I resumes the adoptedconstellation parameters ( θ is expressed in degree).TABLE I: Constellation parameters for the hierarchical QPSK ρ he θ
45 42 39 36 33 30 27 24 18
For the hierarchical 32-APSK, once the value of ρ he isknown, there remains to pick one ( γ , γ , θ ) triple. We solve(6) with Matlab and choose the triple that minimises thedecoding threshold of the HE stream averaged over all theDVB-S2 code rates. To obtain a fast evaluation of the decodingthresholds in function of the constellation parameters, we usethe method described in [11]. Table II presents the adoptedtriples.TABLE II: Hierarchical 32-APSK constellation parameters ρ he γ γ θ Finally, the performance in terms of BER of the hierarchicalQPSK and 32-APSK are evaluated with simulations. We usethe Coded Modulation Library [12] that already implementsthe DVB-S2 LDPC. The LDPC codewords are 64 800 bits long(normal FEC frame) and the iterative decoding stops after 50iterations if no valid codeword has been decoded. Moreover,in our simulations, we wait until 10 decoding failures beforecomputing the BER. If the BER is less than − , then we stopthe simulation. Our stopping criterion is less restrictive thanin [9] (i.e., a packet error rate of − ) because simulationsare time consuming. However, our simulations are sufficientto detect the waterfall region of the LDPC and then theperformance of the code. For instance, Fig. 4 presents the BERof the HE stream for the hierarchical QPSK with ρ he = 0 . .III. H IERARCHICAL MODULATION FOR S ATCOM SYSTEMS
In [10], we compared the two following schemes: timesharing with or without hierarchical modulation, referred toas hierarchical modulation and classical time sharing, re-spectively. This part reviews how the combination of ACMand hierarchical modulation (i.e., hierarchical modulation timesharing) improves the performance of a DVB-S2 system basedon classical time sharing. A more detailed description can befound in [10]. In this section and the rest of the paper, weconsider an AWGN channel. We assume that the transmitterhas knowledge of the SNR at the receivers. A concreteexample is a DVB-S2 system that implements ACM. B i t E rr o r R a t e Es/No (dB) R=1/4R=1/3R=2/5R=1/2R=3/5R=2/3R=3/4R=4/5R=5/6R=8/9R=9/10
Fig. 4: BER for the hierarchical QPSK HE stream ( ρ he = 0 . ) A. Case with two receivers
We first consider one source communicating with two re-ceivers, each one with a particular signal-to-noise ratio
SN R i ( i = 1 , ). Given SN R i and the transmission parameters,receiver i has a spectrum efficiency R i which is the amountof useful data transmitted on the link over a given bandwidth.For instance, if the receiver successfully decodes a signalmodulated with a QPSK and a coding rate of 1/3, then itsspectrum efficiency equals × / bit/s/Hz. In our study, thephysical layer is based on the DVB-S2 standard [13].When communicating with two receivers, non-hierarchicalmodulations achieve spectrum efficiencies of the form ( R , or (0 , R ) . Hierarchical modulation allows spectrum efficien-cies of the form ( R , R ) as the transmitted symbols carryinformation to both receivers. When two spectrum efficienciespairs ( R , R ) and ( R ∗ , R ∗ ) are available, the time sharingstrategy achieves any spectrum efficiency pair ( τ R + (1 − τ ) R ∗ , τ R + (1 − τ ) R ∗ ) , (7)where (cid:54) τ (cid:54) is the fraction of time allocated to ( R , R ) .Thus, if a set χ = (cid:8) ( R i , R i ) | i = 1 , .., k (cid:9) of spectrum efficien-cies pairs is available, any pair ( R , R ) inside the convex hullof χ can be achieved.In this paper, we are interested in offering the same (time-averaged) spectrum efficiency to all the receivers . We note R hm and R ts as the spectrum efficiencies offered to bothreceivers by the hierarchical modulation and classical timesharing strategy, respectively. In [10], the receiver with thebest SNR decodes the LE stream, while the receiver with theworst SNR decodes the HE stream. However, when the SNRsof both receivers are close, it is also interesting that the receiverwith the best SNR decodes the HE stream. We consider bothcases thereafter.Fig. 5 shows on an example the spectrum efficiency im-provement when the receivers experience a SNR of 7 and 10dB (to lighten Fig. 5, only the hierarchical 16-APSK is used). B. Case with n receivers We consider a broadcast system with n receivers. Whentransmissions do not involve hierarchical modulation , weassume that receiver i has a spectrum efficiency R i which R e c e i v e r s pe c t r u m e ff i c i en cy Receiver 2 spectrum efficiency(R hm , R hm )(R ts , R ts ) Hier. modulation time sharingClassical time sharing8−PSK with code rate 2/316−APSK with code rate 3/4hier. 16−APSK ( ρ he =0.75)hier. 16−APSK ( ρ he =0.8)hier. 16−APSK ( ρ he =0.85)hier. 16−APSK ( ρ he =0.9) Fig. 5: Spectrum efficiency for 2 receivers (
SN R = 7 dB, SN R =
10 dB)corresponds to the best spectrum efficiency it can manage.Then the classical time sharing offers the following spectrumefficiency to all the receivers R ts = n (cid:88) j =1 R j − . (8)For the hierarchical modulation time sharing, the first stepis to group the receivers in pairs in order to use hierarchicalmodulation. Many possibilities are available. We adopt thefollowing strategy: from any set of receivers, we pick the tworeceivers with the largest SNR difference, group them andrepeat this operation. We showed in [10] that this strategygenerally leads to significant performance improvement.Once the pairs have been chosen, we compute for eachpair the spectrum efficiency as previously described with tworeceivers. We note R hm,i the spectrum efficiency for each pair( (cid:54) i (cid:54) n/ ). As the terms R hm,i are different, we need toequalise the spectrum efficiency between each receiver. Thisis done using time sharing. Finally, the spectrum efficiency is R hm = n/ (cid:88) j =1 R hm,j − . (9)In [10], we compare the values of R hm and R ts for variousscenarios. Using the grouping strategy presented above, weachieved significant gains (up to 10%) with the hierarchicalmodulation time sharing. However, there is no gain when allthe receivers experience low or large SNRs.IV. P ERFORMANCE EVALUATION
To evaluate the performance of the two hierarchical modu-lations proposed in Section II for real systems, we first presenta model to estimate the SNR distribution of the receiversfor an AWGN channel. Then we evaluate the performance ofhierarchical modulation time sharing and discuss the spectrumefficiency improvement of each modulation.
A. Channel model
We consider the set of receivers located in a given spot beamof a geostationary satellite broadcasting in the Ka band. Themodel takes into account two main sources of attenuation: the relative location of the terminal with respect to the center of(beam) coverage and the weather. Concerning the attenuationdue to the location, the idea is to set the SNR at the centerof the spot beam SNR max and use the radiation pattern of aparabolic antenna to model the attenuation. An approximationof the radiation pattern is G ( θ ) = G max (cid:32) J (cid:0) sin( θ ) πDλ (cid:1) sin( θ ) πDλ (cid:33) , (10)where J is the first order Bessel function, D is the antennadiameter and λ = c/f is the wavelength [14]. Our simulationsuse D = 1 . m and f = 20 GHz. Moreover, we consider atypical multispot system where the edge of each spot beamis 4 dB below the center of coverage. Assuming a uniformrepartition of the population, the proportion of the receiversexperiencing an attenuation between two given values is theratio of the ring area over the disk as shown in Fig. 6. Thering area is computed knowing the satellite is geostationaryand using (10). (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)
Fig. 6: Satellite broadcasting areaFig. 7, provided by the Centre National d’Etudes Spatiales(CNES), presents the attenuation distribution of the Broad-casting Satellite Service (BSS) band. More precisely, it is atemporal distribution for a given location in Toulouse, France.In this paper, we assume that the SNR distribution for thereceivers in the beam coverage at a given time is equivalentto the temporal distribution at a given location. C u l m u l a t i v e D en s i t y F un c t i on Attenuation (in dB)
Fig. 7: Attenuation distribution (due to weather)Finally, our model combines the two attenuations previouslydescribed to estimate the SNR distribution. From a set ofreceivers, we first compute the attenuation due to the location.Then, for each receiver we draw the attenuation caused by theweather according to the distribution in Fig. 7. G a i n ( pe r c en t age ) SNR max (dB) Hier. QPSKHier. 8−PSKHier. 16−APSKHier. 32−APSK
Fig. 8: Spectrum efficiency gains of the hierarchical modulation time sharing with 500 receivers. Each curve is an averageover 100 simulations. The gains of the hierarchical 16-APSK are consistent with [10, Fig. 9].
B. Transmission parameters for the simulations
Our simulations involve the following modulations: QPSK,8-PSK, 16-APSK, 32-APSK and their hierarchical versions.The error performance of the non-hierarchical modulationsare summarised in [13, Table 13]. The hierarchical QPSKand 32-APSK have been introduced in Section II and theirperformance are given in the Appendix. We use the fourhierarchical 16-APSK modulations studied in [10]. For thehierarchical 8-PSK, the constellation parameter θ is definedas the half angle between two points in the same quadrant.We chose θ = 30 ◦ , ◦ , ◦ , ◦ for the simulations. C. Simulations results
Fig. 8 presents the gains (in terms of spectrum efficiency)of hierarchical modulation time sharing compared to classicaltime sharing for a broadcasting area with 500 receivers. Foreach simulation, the SNR value of each receiver is drawnaccording to the distribution presented above. Note that thisSNR is fixed over all times for a given simulation. We alsoassume that the transmitter has knowledge of the SNR atthe receivers. In practice, this corresponds to a system thatimplements ACM. For one system configuration (i.e., theparameter
SN R max is set), we present the average gains over100 simulations. Moreover, we show the results independentlyfor each hierarchical modulation in order to visualise how itaffects performance.For low SNR configuration, the hierarchical QPSK is theonly modulation that improves the performance. For instance,there is a gain of 9% for
SN R max = 1 dB and we observe animprovement up to 12% for
SN R max ≈ dB. However, when SN R max (cid:62) dB, the hierarchical QPSK does not increasethe spectrum efficiency anymore and we need to increase themodulation order. For large SNR configuration, both the hierarchical 16-APSK and 32-APSK improve the performance. When dB (cid:54) SN R max (cid:54) dB, the hierarchical 16-APSK obtains a slightadvantage. However, the results point out that the hierarchical32-APSK performs better than the hierarchical 16-APSK for SN R max (cid:62) dB. Indeed, the hierarchical 32-APSK stilloffers some gains, e.g., around 3% for SN R max = 16 dB.Unlike the others modulations, the hierarchical 8-PSK doesnot provide any significant gain. Moreover, either the hierar-chical QPSK (
SN R max (cid:54) . dB) or the hierarchical 16-APSK ( SN R max (cid:62) . dB) outperforms the hierarchical 8-PSK. When all the modulations are used (the curve is notpresented to make Fig. 8 easier to read), the gains follow thecurve of the QPSK for SN R max (cid:54) dB, then the curve ofthe 16-APSK until SN R max = 13 dB and finally the curveof the 32-APSK. We just observe a slight improvement when
SN R max is around 6 dB.In practical systems, the transmission parameters (modu-lation and coding rates) have to be signaled. In this paper,the simulations involve 22 hierarchical modulations and thereare 11 code rates. Thus it requires 12 bits of signalisation( log (11 × × ≈ . ). However, it is possible to reducethat number. For instance, the hierarchical 8-PSK is not usefuland can be removed. To avoid completely the signaling ofthe modulation, a solution is to use modulation recognitionthat consists in identifying at the receiver the modulation usedby the transmitter [15]. However, it is necessary to verifythat this does not decrease too much the performance andthat the complexity at the receiver is still acceptable. Anothersolution to reduce the number of transmission parameters isinvestigated in [16].Recently, a new method called Bit Division Multiplexing(BDM) has also been proposed to increase the throughput ofABLE III: Hierarchical QPSK decoding thresholds (in dB) Code ρ he = 0 . ρ he = 0 . ρ he = 0 . ρ he = 0 . ρ he = 0 . ρ he = 0 . ρ he = 0 . ρ he = 0 . ρ he = 0 . rate HE/LE HE LE HE LE HE LE HE LE HE LE HE LE HE LE HE LE1/4 -2.6 -3.1 -2.2 -3.5 -1.7 -3.8 -1 -4.2 -0.4 -4.4 0.4 -4.7 1.2 -4.9 2.1 -5.2 4.51/3 -1.4 -1.8 -0.9 -2.2 -0.4 -2.6 0.2 -2.9 0.9 -3.2 1.6 -3.4 2.5 -3.6 3.4 -4 5.82/5 -0.5 -1 0 -1.3 0.5 -1.7 1.1 -2 1.8 -2.3 2.5 -2.5 3.3 -2.8 4.3 -3.1 6.71/2 0.9 0.5 1.4 0.1 1.2 -0.3 2.5 -0.6 3.2 -0.9 3.9 -1.1 4.8 -1.3 5.7 -1.7 8.13/5 2.1 1.7 2.6 1.3 3.1 1 3.7 0.7 4.4 0.4 5.1 0.1 6 -0.1 6.9 -0.4 9.32/3 3 2.6 3.5 2.2 4 1.8 4.6 1.5 5.3 1.3 6 1 6.9 0.8 7.8 0.4 10.23/4 4 3.5 4.5 3.1 5 2.8 5.6 2.5 6.3 2.2 7 2 7.8 1.8 8.8 1.4 11.24/5 4.6 4.2 5.1 3.8 5.6 3.4 6.2 3.1 6.9 2.8 7.6 2.6 8.5 2.4 9.4 2 11.85/6 5.1 4.7 5.6 4.3 6.1 3.9 6.7 3.6 7.4 3.3 8.1 3.1 9 2.9 9.9 2.5 12.38/9 6.1 5.7 6.6 5.3 7.1 5 7.7 4.7 8.4 4.4 9.1 4.1 10 3.9 10.9 3.6 13.39/10 6.3 5.9 6.8 5.5 7.4 5.2 7.9 4.9 8.6 4.6 9.4 4.3 10.2 4.1 11.1 3.8 13.5 broadcasting systems [17]. In fact, hierarchical modulation isa special case of BDM. In [17], the authors study BDM froma theoretical point of view and use uniform constellations. Westrongly believe that BDM combined with non-uniform con-stellations could further improve the performance of satellitecommunication systems.V. C ONCLUSION
We introduce and study the hierarchical QPSK and thehierarchical 32-APSK. We propose these two modulationsto increase the spectrum efficiency offered to the receiversin a DVB-S2 system. To the best of our knowledge, thesemodulations have not been studied before. Here, we chosethe constellation parameters according to an energy argument.Then we showed that hierarchical QPSK achieves significantgains for low SNRs (up to 12%). The hierarchical 32-APSKis efficient for large SNRs but the gains are less important.Future work will explore a better computation of the 32-APSK constellation parameters to obtain higher gains for largeSNRs. We also plan to investigate the combination of non-uniform constellations with bit division multiplexing to fur-ther increase the spectrum efficiency of satellite broadcastingsystems. A
PPENDIX
Table III and IV summarise the decoding thresholds (ob-tained by simulations) to achieve a BER of − for thehierarchical QPSK and 32-APSK over an AWGN channel.TABLE IV: Hierarchical 32-APSK decoding thresholds (indB) Code ρ he = 0 . ρ he = 0 . ρ he = 0 . ρ he = 0 . ρ he = 0 . rate HE LE HE LE HE LE HE LE HE LE1/4 0 6 -0.6 6.6 -1.1 7.6 -1.6 9 -1.9 10.51/3 1.7 7.3 0.9 8 0.4 9 -0.1 10.6 -0.6 122/5 3 8.5 2.2 9.2 1.5 10.2 0.9 11.7 0.4 131/2 5.2 10.1 4.2 10.8 3.4 11.8 2.7 13.3 2 14.73/5 7.5 11.5 6.2 12.3 5.1 13.2 4.3 14.6 3.4 16.22/3 9 12.5 7.5 13.3 6.4 14.2 5.4 15.6 4.5 17.23/4 11 13.6 9.3 14.4 8 15.2 6.8 16.6 5.8 18.24/5 12.3 14.4 10.4 15.2 9 15.9 7.8 17.2 6.6 18.95/6 13.3 15.1 11.3 15.9 9.9 16.5 8.5 17.7 7.2 19.58/9 15.4 16.4 13.2 17.2 11.6 17.6 10.2 18.7 8.7 20.69/10 15.9 16.6 13.6 17.4 12 17.9 10.5 18.9 9 20.8 R EFERENCES[1] T. Cover, “Broadcast channels,”
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