fading channels is only inversely proportional to the signal-to- noise ratio, which is not very reliable for digital communica- tion applications. Improving their ...
An Improved PSK Scheme for Fading Channels Slimane Ben Slimane
Radio Communication Systems Department of Signals, Sensors and Systems Royal Institute of Technology 100 44 Stockholm, Sweden Abstract | The performance of uncoded PSK schemes over fading channels is only inversely proportional to the signal-tonoise ratio, which is not very reliable for digital communication applications. Improving their performance has been approached by means of coded modulation schemes where code redundancy combined with interleaving introduces some degree of diversity that depends on the complexity of the code. This paper proposes an alternative way in improving the performance of PSK schemes over fading channels by looking at the reference scheme rst. It is shown that by using interleaving combined with a proper signal constellations and still using symbol-by-symbol detection, a higher diversity is obtained and the performance of uncoded PSK schemes over fading channels is considerably improved. With this simple modi cation, the obtained performance is comparable to that of the best 4-state Trellis Coded Modulation (TCM) 8PSK schemes reported in the literature. By optimizing the reference scheme rst, it is shown that this technique optimizes the performance of Ungerboeck's Trellis Coded 8PSK schemes over fading channels. Infact, without altering their performance over the additive white Gaussian noise channel, significant coding gain over fading channels is achieved.
of diversity in fading channel, suggesting that for at fading, the standard trellis codes may not be the correct approach [2]. Following these suggestions, we decided to look at the reference uncoded scheme and in particular QPSK schemes. Our investigation showed that the performance of the reference scheme over fading channels can in fact be improved without increasing the complexity of the system. In this paper, an improved PSK scheme for fading channel applications is introduced. Combining interleaving with a proper signal constellations and still using symbol-by-symbol detection, the performance of PSK schemes over fading channels is considerably improved. This performance improvement is due to a diversity gain obtained from the quadrature components of the PSK scheme. Results showed that the modi ed scheme performs as well as the best designed standard 4-state 8PSK TCM schemes. However, to assure this performance channel state information is required.
I. Introduction
II. System Model
F
ADING causes signi cant degradation to the performance of digital communication systems. For frequency-nonselective, slowly fading channel, coded modulation/interleaving showed good performance improvement. Interleaving destroys the fading correlation making the coded scheme acts as a kind of diversity technique [1-4]. The order of diversity for these coded schemes is de ned by the length of the shortest error event path through the trellis of the code, higher diversity is obtained at the expense of more complexity. Since the minimum squared Euclidean is secondary in fading channels, an optimum scheme for AWGN channels may not be optimum for fading channels. Recent results [2, 4] raised some questions concerning the design of optimum coded schemes for fading channels. In [2], the author showed that by interleaving each bit independently, the diversity order of trellis coded 8PSK schemes is increased by the number of coded bit positions. However, the Euclidean distance properties of the coded scheme are destroyed causing some degradation in the AWGN channel compared to the original scheme. Also the bit separations before demodulation at the receiver seems to be a big task. In [4], the author used a more straightforward way by interleaving the signal coordinates of trellis coded QAM schemes, where each coordinate is interleaved independently. The diversity order in this case could be increased by at most a factor of two. These results clearly indicate the importance
The block diagram of the PSK modulator is shown in Figure 1. The baseband modulator generates the quadrature components, xn and yn, of the transmitted signal. Each quadrature component is then independently interleaved. The signal interleavers are chosed such that after deinterleaving the two components will be independent. The transmitted signal can written as follows
s(t) =
where
+1 X
xn p(t ? nTs) cos(2fct) n=?1 +1 X + yn p(t ? nTs ) sin(2fct); n=?1
(1)
xn = A cos + 24 (l ? 1) ; l = 0; 1; 2; 3; yk = A sin + 24 (m ? 1) ; m = 0; 1; 2; 3; (2) with l and m independent. fc represents the carrier frequency of the system and the initial phase, , is selected in such a way that the squared Euclidean distance between QPSK signal constellations is maximized for both components, inphase and quadrature, as shown in Figure 2. For
the two closest constellation points of the QPSK scheme, we can write d2I = 1 + cos(2); d2Q = 1 ? cos(2); where d2I and d2Q are the normalized squared Euclidean distances taken along the inphase and quadrature directions, respectively. The sum of these two distances is obviously independent of and represents the minimum squared Euclidean distance of conventional QPSK which guarantees the regular performance of coherent QPSK over Additive White Gaussian Noise (AWGN) channels. The use of this initial phase is e ective only when the signal is transmitted over fading channels. xn
an
cn
S/P
Interleaver 1
Baseband
cos (wc t)
modulator
sin (wc t)
bn
yn
s(t)
At the receiver (Figure 3), the signal is rst demodulated and then each quadrature component is deinterleaved independently. Thus, at the kth symbol interval, the two dimensional received vector takes the following form
rk = ak sk + nk ; where sk is the vector representation of the transmitted signal at time kTs , the elements of the vector nk are i.i.d Gaussian random variables with zero-mean and variance N0 =2, and that of the vector ak are also i.i.d (normalized) random fading amplitudes. The received vector is processed by the demodulator using symbol-by-symbol detection. Thus, the optimum demodulator computes the squared Euclidean distance between the received vector and each of the four signal vectors of the QPSK scheme then decides in favor of the one closest to rk . Assuming a perfect channel state information, the average bit error probability conditioned on the fading amplitude vector can be written as follows:
P (s ! s^ja) = 21 erfc
Interleaver 2
Fig. 1. Block diagram of modulator for the 4-phase PSK scheme.
The digital communication channel is modeled by a multiplicative factor representing the e ect of fading and an additive term representing the AWGN channel. The received signal is then written as r(t) = a(t)s(t) + n(t): (3) Assuming a suciently slow fading channel, the received signal can be rewritten as: r(t) = [ja(t)jsI (t) + nc (t)] cos(wc t) + [ja(t)jsQ (t) + nc (t)] sin(wc t) (4)
θ
d2 (s; s^) = a2I d2I + a2Q d2Q : The average bit error probability is then obtained by averaging over the probability density function of the fading amplitudes. Note that when there is no fading the squared Euclidean distance of the QPSK scheme stays the same and the performance over AWGN channels is unchanged. In this investigation, we assume that the fading amplitude is modeled as Rayleigh at fading. That is, aI and aQ are i.i.d. random variables each having a probability density function ?x2 x 0 pa (x) = 20xe elsewhere with E fa2 g = 1. Averaging over the probability density function of both aI and aQ , we get
rrn
cos (wc t) Baseband
Decision
sin (wc t) demodulator Deinterleaver 2
"
s
#
2 Pb = 2(d2 ? d2 ) 1 ? 1 +dIdE2bE=2=N20N 0 I Q I b s " # d2Q Eb =2N0 d2Q ? 2(d2 ? d2 ) 1 ? 1 + d2 E =2N 0 I Q Q b
Fig. 2. Signal constellations of the 4-phase PSK scheme.
r(t)
!
Eb 2 2N0 d (s; s^) ;
where we have dropped the subscript k. d2 (s; s^) represents the weighted squared Euclidean distance between s and s^ and is given by
d2I
Deinterleaver 1
r
cn
rin
Fig. 3. Block diagram of demodulator for the 4-phase PSK scheme.
where
(5)
d2I = 1 + cos(2); d2Q = 1 ? cos(2); 0 =4
It is observed from (5) that the bit error probability is a function of the phase rotation . When = 0, the QPSK
signal constellations are pairwise symmetric with respect to both axes. In this case the Euclidean distance between any two signal points along one of the quadrature components is zero, and thus the performance of the system reduces to conventional QPSK s
#
(6)
The above result (Eq. (6)) represents the worst case performance. In fact when is increased, the Euclidean distance is split between the two quadrature components and a diversity of degree two is obtained. As a result the performance of the system is improved. As for the optimum performance, it is obtained when the Euclidean distance is optimally divided between the quadrature components, which is obtained for a phase rotation of = =8. Figure 4 illustrates the performance of the modi ed QPSK for the optimum case. We notice that considerable performance improvement is obtained compared to conventional QPSK scheme, about 8 dB gain at a bit error probability of 10?3. This performance improvement is obtained without increasing the complexity of the system, that is, still using symbol-by-symbol detection. Also shown in Figure 4 are simulation results for this modi ed QPSK scheme, giving the same results as obtained analytically. Figure 5 shows simulation results for the performance of the modi ed QPSK scheme together with that of three di erent rate 2/3, 4-state TCM 8PSK schemes [5-7] over Rayleigh fading channels. It is observed that the performance of this modi ed scheme is comparable to that of the coded schemes. This suggests that these coded schemes may not be optimized for the fading channel under consideration [2]. Since the performance of uncoded PSK schemes can be improved by just using combined interleaving with a proper signal constellations, we can apply the same technique to the expanded signal set of a trellis coded PSK scheme. When properly used an extra degree of diversity can be obtained and the performance of the overall coded scheme will be improved. III. Application to TCM Schemes
Consider the set partitioning of an 8PSK scheme given in Figure 6. We notice that by using this particular signal constellations for the 8PSK scheme, the signal constellations of the two subsets A0 and A1 now have signal constellations similar to that obtained in the previous section for the modi ed QPSK scheme (Figure 2). That is, a squared Euclidean distance between signal points split between the two quadrature components. Therefore, using this set partitioning, applying Ungerboeack's design rules and interleaving each quadrature component independently we obtain optimum coded schemes for both AWGN and fading channels. Since Ungerboeck's TCM schemes are optimum in AWGN channels, it is only required to modify the signal constellations of the 8PSK scheme. To illustrate this, we have considered the case of 4-state TCM 8PSK schemes with three
−1
10
−2
10 Bit Error Probability
"
0 : Pb = 21 1 ? 1 +EbE=N=N b 0
0
10
−3
10
−4
10
regular QPSK modified QPSK simulation
−5
10
−6
10
0
5
10
15 Eb/No, dB
20
25
30
Fig. 4. Performance of uncoded QPSK scheme (regular and modi ed) over Rayleigh fading channels with perfect channel state information.
di erent codes. The rst code known as Ungerboeck code [5] is optimum for AWGN channels but its trellis contains parallel paths and therefore does not get any diversity gain over fading channels. The two other codes [6, 7] (designed for fading channels, i.e., their trellises do not contain parellel paths) perform better in fading channels however their performance in AWGN channels is inferior to that of Ungerboeck's code. Figure 7 illustrates the performance of those three codes over Rayleigh fading channels. The dashed line indicates the performance of the standard TCM scheme and the solid line indicates the performance of the corresponding modi ed TCM scheme. The rst observation is that by using the modi ed signal constellations combined with quadrature interleaving improves the performance of all these three codes. The second observation is that this technique closes the gap between the performance of all three codes. Originally, the performance of Ungerboeck's code is inferior (about 5 dB away at a bit error rate of 10?3) to that of the two others. As for now, the performance of the three codes is comparable for most ranges of SNRs. Infact, for bit error rates of 10?3 or higher, Ungerboeck code performs better than the other two. The results of Figure 7 show the importance of the minimum squared Euclidean distance even for fading channels applications. The squared Euclidean distance controls the system performance at low signal-to-noise ratio. Thus, for coded schemes with maximized Euclidean distances such as TCM schemes only a reasonable number of diversity is needed to achieve good performance over fading channels. Figure 8 illustrates the performance of Ungerboecks 8state TCM 8PSK scheme over Rayleigh fading channels. It is observed that this modi ed technique makes the performance of the TCM scheme about 1 dB better than the case when bit-interleaving [2] is used. This improvement is due
A: 8PSK −1
10
−2
Bit Error Probability
10
−3
10
A0
A1
−4
10
4−state TCM of [5] modified QPSK 4−state TCM of [7]
−5
10
4−state TCM of [6]
Fig. 6. Set partitioning of an 8PSK signal set. −6
10
5
10
15
20
25
30
Eb/No, dB
−1
10
Fig. 5. Performance comparison of the modi ed uncoded QPSK scheme and 4-state TCM 8PSK schemes over Rayleigh fading channels, perfect channel state information.
where
and
0
1
0
1
Eb ; d20 = 1 ? p1 N 2 0 Eb ; d21 = 1 + p1 N 2 0 s
2
xm = 1 +d1d2 1 which at high signal-to-noise ratio can be approximated as P 2 b
Eb 4 N0
The above equation indicates that the diversity order of 8state TCM 8PSK scheme increased from two to four. This is shown in the performance improvement obtained in Figure 8.
−2
10 Bit Error Probability
to the fact this system takes full advantage of both diversity and Euclidean distance. As mentioned earlier bit interleaving does not take advantage of the distance properties of the TCM scheme. Also given in Figure 8 is lower bound on the bit error probability where only the shortest error event path through the scheme trellis diagram has been considered. Following the procedure given in [8], this bound can be written as: 5 + 5 4 2 1 + + Pb 32 1 + xm (1 + xm )2 (1 + xm )3 (1 + xm )4 1 (7) 2 (1 + 0:5d )(1 + 0:5d2)(1 + d2 )(1 + d2 )
coded scheme of [5] coded scheme of [7] coded scheme of [6]
−3
10
−4
10
−5
10
5
10
15
20
25
30
Eb/No, dB
Fig. 7. Performance of 4-state, TCM 8PSK schemes over Rayleigh fading channels, perfect channel state information.
IV. Conclusions
An improved PSK scheme for fading channels applications has been introduced. It has been shown that by modifying the signal constellations of the regular QPSK scheme combined with independent interleaving for the quadrature components of the system, and still using symbol-by-symbol detection detection, the performance of the QPSK scheme is considerably improved. Performance improvement of about 8 dB at a bit error probability of 10?3 is obtained. As this modi cation does not a ect the performance of the original scheme over the AWGN channel, its application to trellis coded modulation schemes was very bene cial. In fact, by applying the same procedure to the reference 8PSK scheme the performance of the equivalent TCM scheme improved considerably compared to the original scheme. When applied to 4-state TCM 8PSK schemes for example, the per-
−1
10
standard Ung. bit inter. Ung. [2] modified Ung. −2
Bit Error Probability
10
−3
lower bound
10
−4
10
−5
10
0
2
4
6
8 10 Eb/No, dB
12
14
16
18
Fig. 8. Performance of 8-state, TCM 8PSK scheme over Rayleigh fading channels, perfect channel state information.
formance of codes with parallel paths becomes comparable to that of codes with no parallel paths, and all the codes perform better than the original code. It has also been shown that when applied to 8-state TCM 8PSK scheme, the equivalent scheme outperforms the same scheme with bit-interleaving by about 1 dB. References [1] D. Divsalar and M. K. Simon, \The design of trellis coded MPSK for fading channels: Performance criteria,", IEEE Trans. Commun., vol. COM-36, NO. 9, pp. 1004-1012, September 1988. [2] E. Zehavi, \8-PSK trellis codes for a Rayleigh channel,", IEEE Trans. Commun., vol. 40, pp. 873-884, May 1992. [3] C. Schlegel, D. J. Costello, Jr., \Bandwidth ecient coding for fading channels: code construction and performance analysis," IEEE J. Select. Areas, Commun., vol. 7, pp. 1356-1368, December 1989. [4] B. D. Jelicic and S. Roy, \Design of trellis coded QAM for at fading and AWGN channels,", IEEE Trans. Veh. Techn.,, vol. 44, pp. 192-201, February 1995. [5] G. Ungerboeck, \Channel coding with multilevel/phase signals," IEEE Trans. Inform. Theory, vol. IT-28, No. 1, pp. 55-67, January 1982. [6] S. H. Jamali and T. Le-Ngoc, \A new 4-state 8PSK TCM scheme for fast fading, shadowed mobile radio channels,", IEEE Trans. on Veh. Tech., vol. VT-40, pp. 216-223, Feb. 1991. [7] S. G. Wilson and Y. S. Leung, \Trellis-coded phase modulation on Rayleigh channels," in Proc. Int. Conf. Commun., Seattle, WA, June 7-10, 1987. [8] S. Ben Slimane and T. Le-Ngoc, \Tight bounds on the error probability of coded modulation schemes in Rayleigh fading channels," IEEE Trans. Veh. Techn., vol. 44, pp. 121-130, February 1995.
