Optimisation of 16APSK Constellation with a New Iterative Decoder on the Nonlinear Channel Nghia H. Ngo, Steven S. Pietrobon, S. Adrian Barbulescu and Craig Burnet
Abstract— Amplitude phase shift keying (APSK) modulation presents an attractive scheme to overcome nonlinearity effects on nonlinear channels compared to conventional square quadrature amplitude modulation (QAM). In this paper, we investigate the performance of a bit interleaved coded modulation with iterative decoding (BICM-ID) system in which APSK modulation is used to achieve a spectral efficiency of 3 bit/s/Hz. A new iterative receiver is proposed where an extra soft-demapper is used between the two soft input soft output (SISOs) decoders. Extrinsic information exchange for the data and parity bits is optimised according to an appropriate iteration scheduling. Simulation results show a gain of 0.15 dB at a bit error rate (BER) of 10 4 with this new iterative decoding scheme and the optimised constellation over a nonlinear satellite channel. Index Terms— Non-uniform constellation, high order modulation, nonlinear channel, spectral efficiency
I. I NTRODUCTION Power and bandwidth are the two main constraints in satellite communications. High demands for these scarce resources result in the need to use higher order modulation. In general, higher order modulations such as M-ary QAM, phase shift keying (PSK) or APSK are used to achieve a greater bandwidth efficiency, at the cost of an increase in the transmitted power. In a nonlinear satellite channel where constellation warping, spectral spreading and intersymbol interference (ISI) are the three main effects, there are several issues related to the use of higher order modulation The minimum Euclidean distance between signal points is decreased and the number of nearest neighbor signal points is increased. This results in a significant degradation in performance. Higher modulations suffer more from nonlinear distortion caused by the high power amplifier (HPA) in the satellite channel, which results in the loss of signal power and a higher error floor. Carrier phase recovery, synchronisation and tracking become harder and difficult issues. Recent research results in [1] and [2] show that performance of APSK modulation is slightly worse than a conventional square QAM signal set on a linear additive Gaussian white noise (AWGN) channel. However, on a nonlinear satellite channel where high energy signal points are greatly distorted by the HPA, APSK modulation outperforms QAM. It is widely known that a system which employs iterative demapping and decoding, The authors are with the Institute for Telecommunications Research, University of South Australia, Mawson Lakes SA 5095, Australia. Steven S. Pietrobon is also with Small World Communications, 6 First Ave., Payneham South SA 5070, Australia. E-mail:
[email protected],
[email protected] fAdrian.Barbulescu,Craig.Burnet
[email protected].
e.g., BICM-ID, in which error control techniques are combined with coded modulation, offers more flexibility and robustness compared to a conventional system without iterative demapping and decoding, e.g., bit interleaved coded modulation (BICM) [3]. The results in [3–5] show that when extrinsic information is exchanged between the soft-demapper and the decoder, a significant gain is observed. This gain is strongly dependent on how the signal constellation is labelled. In this paper, we propose a new iterative turbo receiver that differs from a conventional turbo decoder. An extra softdemapper is used between the two SISO decoders to update and improve the channel estimation coming into the next decoder. This allows improved extrinsic information exchange for the data and parity bits. In addition, we also investigate the performance of 16APSK modulation with this new iterative receiver on a nonlinear channel and optimise its design parameters. Simulations over a nonlinear channel for a turbo code in a parallel concatenated (PCCC) scheme and the new iterative receiver show a 0.15 dB improvement at a bit error rate (BER) of 10 4 , compared to conventional binary turbo decoding. In the next section, the nonlinear satellite channel is described. Section III introduces the optimisation for the constellation over a nonlinear channel. Section IV presents the new iterative receiver scheme. Simulation results and discussion are shown in Section V. Finally, Section VI concludes the paper. II. N ONLINEAR S ATELLITE C HANNEL
At simplified block diagram of a nonlinear satellite channel is given in Figure 1. The HPA in Figure 1 is modelled by two Uplink
y(t)
hT (t)
t
Satellite Channel
hI (t)
s(t) r(t)
HPA
hO (t)
Downlink
rt
nt hR (t)
Fig. 1. The Nonlinear Channel
functions given in [6] as follows
A(r) =
a r
1 + a r 2
(1)
(r) = 1 + r r2
(2)
IBOdB = 10log PPsat
(3)
2
where r is the amplitude, A(r) is the AM/AM conversion and (r) is the AM/PM conversion in radians. Parameters for these equations are a = 2, a = 1, = =3 and = 1. The transmit and receive filters hT (t) and hR (t) are root-Nyquitst with = 0:2. The satellite channel input and output filter hI (t) and hO (t) are assumed to have a bandwidth much greater than the transmitted signal and so these filters are assumed to have no effect. The amplifier input backoff (IBO) is defined as the ratio between the amplifier input saturation power and the input signal power. in
It is desirable to operate the system at low IBOs to maximise signal power output from the HPA. However, this results in a significant distortion to the high energy signal points, especially for higher order modulations. III. C ONSTELLATION C ONSTRUCTION
AND
O PTIMISATION
This section describes the construction and optimisation of a 16APSK constellation over a nonlinear channel. Performance of a BICM-ID system is dependent on the constellation structure and the signal mapping. Therefore, we optimise the parameters to design this particular constellation and investigate its performance for several mapping schemes. In general, a
t 16APSK constellation can be constructed in many different
t ways. For example, we can layout these 16 points into two rings hT (t) in which four points are located in the inner ring and twelve hI (as t) others are located in the outer ring. We denote this structure 4-12. Another design is to have a three ring signal set in which hOthe (t) 1-5-10 is the number of signal points in the most inner ring, y ( t) middle ring and the most outer ring, respectively. Recent work nt in [1] and [2] showed that the former scheme is preferable since s(t) its modulation symbols have only two power levels, which minhRhas (t) imises the envelop fluctuations in the transmitted signal and r ( t) a better DC power conversion characteristic. Therefore, in this r paper, we focus on the construction and optimisation of thist Uplink constellation structure. Satellite Channel The inner and outer rings of the 4-12APSK constellation are Downlink the represented by radius r1 and r2 , respectively. To simplify design, we fix the value of r1 to one and vary r2 . An important point drawn from [1] is that the performance of a BICM system is affected by , which is defined as the ratio between r1 and r2 as follows r = 2: (4)
r1
Another interesting observation is that the system performance is not greatly influenced by the phase rotation between the signal points in the two rings. Thus, we selected a value for = =4 as given in [1]. The complex 4-12 APSK constellation points can be described as
X=
(
2
ej( n1 i) i = 0; 1; n1 1 j ( n22 i+=4) e i = 0; 1; n2 1
(5)
Where n1 and n2 are the number of signal points in the inner and outer rings, respectively. It is widely known that a HPA in a satellite causes severe distortion to the input signals at low IBO, especially the high energy input signals. In other words, when the average input signal power is close to the saturation region in the HPA, high energy signal points are compressed together. This results in a significant degradation in performance and a higher error floor. In addition, higher order modulation also suffers from losses due to power spectral spreading and ISI. From the first observation described above, we can fine tune to reduce these nonlinear effects. In a memoryless AWGN channel, this optimisation process can be done analytically by maximising the average mutual information, or the constrained capacity of a given constellation. We have [7]
d2r; m # ) 2 C = Eb;r m log2 Qm P (6) d2r; exp( ) 2 i=1 2Si;bi m where S is a uniform distributed 2 -ary signal set, Eb;r is the expectation with respect to b = b0 ; b1 :::bm 1 and r, dr; is the Euclidean distance between received signal r and , Si;bi P
"
2S exp(
f
g
is the subset of all 2 S in which bit position i in has a value equal to bi 2 f0; 1g. The expectation Eb;r in (6) involves complicated numerical integrations which can be simplified by using a Gaussian Hermite approximation [8]. Results in [1] for code rate 3=4 showed the optimal is 2.85. It was shown in [5] that Gray mapping is the best mapping for BICM, but no improvement was observed for performing iterations in an AWGN channel for a BICM-ID system. We
t investigated several other mapping schemes such as anti Gray
t mapping and natural mapping. hT (t)We observed that Gray mapping gave the best performance.h Figure 2 shows the Gray mapping I (t) for 4-12APSK and 16QAM.
hO (t) y(t) nt s(t) hR (t) r(t) rt
1.5 0010
1010 1
0110
1110
0.5 1111
0 1101
1011
0011
1001
0001
0111
0101
−0.5 0100
1100 −1
−1.5 −1.5
−1
−0.5
Uplink
0000
1000
0
Satellite Channel 0.5
1
1.5
1.5
1
0.5
0010
0110
1110
1010
0011
0111
1111
1011
0001
0101
1101
1001
0000
0100
1100
1000
0
−0.5
−1
−1.5 −1.5
−1
−0.5
0
0.5
1
1.5
Downlink
Fig. 2. Gray mapping for 4-12APSK and 16QAM (bt;0 ; bt;1 ; bt;2 ; bt;3 )
On a nonlinear channel with memory, it is much harder to analyse the constrained capacity in (6). We therefore used Monte Carlo simulations to do an extensive search to find the optimum at different IBOs, based on the optimal iteration scheduling described as follows. The PCCC encoder in Figure 3 uses two identical convolutional codes with feed forward and feedback polynomials of [37/23] in octal notation. The encoder memory p is four and the interleavers are of S -random type with S N=2 [9]. The code rate is 3=4 and 4-12APSK modulation with = 4 is used. For a fixed size block length of 8196 bits and at 1 dB IBO, the complexity of a soft-demapper is negligible compared to the complexity of a SISO decoder [10].
ite Channel
0;1;2
3d
Downlink
d PCCC
p
bt;0 ; bt;1; bt;2 4−12 APSK
3p
q 3q
Puncture
3
bt;3
or 16QAM
Fig. 3. The Encoder
t
−1
10
1 outer−12 inner
t
t hT (t) hI (t)
2−6 3−4 4−3 6−2
hO (t) y(t) To make a fair comparison that is feasible in practice, we nt consider a system complexity of 24 SISO decoders and ignores(t) the soft-demapper’s complexity. This is equivalent to iterating hR (t) 12 times between these decoders. We denote the iterations be-r(t) tween the two SISO decoders and between the SISO decoder r t
BER
12−1
and the soft-demapper as the inner and outer iterations, respecUplink tively. An appropriate iteration scheduling was found based on Satellite Channel several possible combinations of the inner and outer loops. This Downlink means the best iteration scheduling for the old iterative decoder 0;1;2 and the new iterative decoder are selected to make the compar- 3 ison. Figures 4 and 5 show the performance of a receiver b t;0 ; bt;for 1 ; bt;2 different iteration schedules at a BER down to 10 3 . In the oldbt;3 iterative receiver, it is optimal if we do one outer iteration and twelve inner iterations. This is the same as BICM. In the new it- t Fig. 5. erative decoder, it is better do do four outer iterations and three t hT (t) inner iterations.
hI (t)
1 outer−12 inner
t
t hT (t) hI (t)
2−6 3−4 4−3 6−2 12−1
hO (t) y(t) nt s(t) hR (t) r(t) rt
−3
10
5.6
5.8
6
6.2 EbN0(dB)
6.4
6.6
Iteration scheduling of 24 SISO decoder for the new iterative decoder
9
hO (t) y(t) nt s(t) hR (t) r(t) rt
1 dB IBO 3 dB IBO 6 dB IBO AWGN
8.5
8
EbN0 (dB)
−1
10
−2
10
7.5
7
Uplink 6.5 BER
Satellite Channel −2
Downlink
10
0;1;2 3
bt;0 ; bt;1; bt;2 bt;3
Uplink
6
5.5
2
3
4
5 6 7 Ratio between the inner and outer rings
8
9
Fig. 6. Performance of 4-12APSK rate 3/4 for different on the linear and nonlinear channel at a BER of 10 4
ite Channel Downlink
0;1;2 3
;0 ; bt;1 ; bt;2 bt;3
−3
10
5.6
5.8
6
6.2 EbN0(dB)
6.4
6.6
Fig. 4. Iteration scheduling of 24 SISO decoder for the old iterative decoder
We used the optimal iteration scheduling from above to simulate for a wide range of at different IBOs and a BER of 10 4 . Figure 6 depicts the performance of these 4-12APSK constellations. The optimal is determined from the graph at which the minimum Eb N0 is achieved for the same level of BER. When the system is operated at a lower IBO, we need to increase the ratio between r1 and r2 to get the optimal .
IV. T HE R ECEIVER A traditional BICM-ID for a PCCC turbo receiver is replaced by a new iterative receiver shown in Figure 7. The transmitted symbol xt is transmitted through a nonlinear channel. The received signal is
rt = Nt ( t ) + nt
(7)
where Nt is a nonlinear function which represents the effects of the HPA and the filters and nt is AWGN with zero mean and variance 2 . At the receiver, the log likelihood ratio (LLRs) is estimated
SISO2
from rt as follows
=
j
t :bt;i =0 ft ( t )P ( t bt;i
= 0)
4−12APSK old−ID 4−12APSK new−ID 16QAM−oldID 16QAM−newID
−2
10
0
1 1 m 2 1 where t is one of 2 signal points and the summation is over all points t conditioned on bit bt;i being either 0 or 1. ft ( t )= 0;1;2 j rt t j2 exp[ N0 ℄. For a nonlinear channel, we replace t with ~t ,0 the centroid estimation point, that was averaged over the block.1 The a priori values P ( t jbt;i = b) in the soft-demapper are2 13 calculated as 3 X mY1 p P (bt;k ) P ( t jbt;i = b) = (9) q bt :bt;i =b k=0 g(bt )= t k6=i
−1
10
pq
De
BER
L(bt;i ) =
0;1;2
P :b =1 P (rt j t )P ( t jbt;i = 1) ln P t t;i P (r j )P ( jb = 0) 3 t t t t;i
t :bt;i =0 1 1 P :b =1 ft ( t )P ( t jbt;i = 1) 0;1;2 (8) ln P t t;i 1
−3
10
−4
10
−5
10
5.5
6
6.5
7 EbN0 (dB)
7.5
8
where P (bt;k ) is the a priori probability of bit position Pun ture k in a symbol at time t, which is equal to 0.5 in the first iteration Fig. 9. Performance of 4-12APSK and 16QAM for old-ID and new-ID on a and is updated through extrinsic values output from the turbo nonlinear channel at 1 dB IBO decoder for the next iteration. The function g (bt ) maps the binary bt into the complex t . To take into account the transmit As can be seen in Figure 9, for a spectral efficiency of 3 and receive filters, we upsampled four times and normalised the signal energy to one before adding Gaussian noise. The noise bit/s/Hz in a nonlinear satellite channel, the new iterative revariance 2 is estimated based on the new centroid points ~t ceiver is about 0.15 dB better than the old iterative decoder at a BER of 10 4 when the same modulation is used. Performance and the noisy symbols. In a multiple concatenated coded system, it is desirable to of the 4-12APSK scheme is about 1 dB better than 16QAM with use the most updated information to pass among decoders. In Gray mapping, at the same BER of 10 4 . The improvement Figure 7 the soft-demapper M1 1 calculates the extrinsic log is due to the fact that the channel estimation coming into the likelihood ratio (LLR) for the data and parity bit sequences d1 , second SISO decoder is improved by the extra soft-demapper p1 and q 1 based on the received symbol rt , as in (8). A priori between the two SISOs. Compared to the result in [1] in which information for d~2 and q~2 are assumed to be zero in the first it- a serial concatenated convolutional code was considered, this eration. These soft output estimation sequences are multiplexed PCCC scheme with the new receiver shows a gain of 1.28 dB at and de-interleaved using S-random interleavers. 0;1;2 and 3 a BER of 10 4 . are the interleavers connecting the encoder to the signal mapper. is the interleaver used in the PCCC encoder. To simplify VI. C ONCLUSION the diagram in Figure 7, Figure 8 depicts the input and output notation for these interleavers. The de-interleaved sequences In this paper, optimisation of a 16APSK constellation on 1 d1 and p1 1 are fed into SISO1 and the output extrinsic infor- a nonlinear channel was investigated. We found the optimal mation of d~1 and p~1 are estimated based on the SISO algorithm parameters to construct a 4-12APSK constellation at different described in [10]. The difference between a conventional PCCC IBOs through Monte Carlo simulation. A new iterative receiver turbo decoder and the decoder in this paper is that an extra soft- with negligible added complexity is proposed for a BICM-ID demapper M2 1 is placed between the two SISO decoders. The system and an appropriate scheduling was simulated. Simulasoft-demapper M2 1 is identical to M1 1 , but it uses d~1 , p~1 and tion results showed 0.15 dB improvement at a BER of 10 4 for q~2 as the a priori information in which q~2 is set to zero in the this approach. This is better than the result shown in [1] where first iteration. The updated LLR extrinsic information d2 and q 2 no extrinsic information is considered between the decoder and are estimated from the second soft-demapper M2 1 in a similar the soft-demapper. Investigations using EXIT charts to provide fashion to that for M1 1 and fed into SISO2. The outputs d~2 and insight of the system behavior is a subject for future research. q~2 are interleaved and fed back to M1 1 in the next iteration. The use of non-uniform probability distributions for 4-12APSK will also be investigated. V. S IMULATION R ESULTS In this section, we present some of the simulation results for R EFERENCES the nonlinear channel. Simulations for this channel are at a spectral efficiency of 3 bit/s/Hz. We use the same encoder as [1] R. De Gaudenzi, A. Guillen i Fabregas and A. Martinez, “Turbo-coded APSK modulations for satellite broadband communications - Part II: Endfor previous simulations. The Gray mapping for 16QAM is to-end performance,” submitted to IEEE Trans. on Wireless Comm., Feb. 2004. shown in Figure 2.
2 2 1
d2 r t p2 1 q2 1 d~2 q~2 SISO1 SISO2
M1
1
d~1 d 1 1 d1 1 0;1;2 0;1;2 1 p1 q1 1 p1 SISO1 p~1 3 3 q1 1
0;1;2 3 1
M2
d2
1
d2
0;11;2 1 p2 q2 1 p2 3
d~2
1
q2 1 SISO2 q~2
1
0;1;2
Fig. 7. New iterative demapping and decoding receiver
0 1 1 1 2 1
0;11;2
pq
3 1
p q
pq
3 1
0 1 2
0;1;2
De Pun ture
p q
p q
Fig. 8. Input and output symbols for the bit interleavers
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p q
3
pq
p q
p q
Pun ture
3
pq
