Qualcomm Inc., 10555 Sorrento Valley Rd., San Diego, CA 92121. ABSTRACT. Recently, there has been much progress in improving the reliability of digital ...
8-PSK TRELLIS CODES ON RAYLEIGH CHANNEL bY Ephraim Zehavi * Department of Elecmcal Engineering, Technion Israel Instituteof Technology and Qualcomm Inc., 10555 Sorrento Valley Rd., San Diego, CA 92121
I n this paper trellis-coding techniques. [ 1-31 are ABSTRACT Recently, there has been much progress in improving the reliability of digital communication over a fading channel by means of the trellis-coding technique, whereby an M-ary channel signal constellation is used in conjunction with convolutional codes. In this paper, an alternative approach is described, which increases the diversity. This approach, yields a better coding gain over a Rayleigh channel than some previous works.
I. INTRODUCTION In radio communication systems operating over a multipath environment, such as a mobile radio or indoor wireless system, the communication channels can be modeled as a Rayleigh fading channel. Previous contributions [2, 31 considered the performance of a trellis coded system in a fading environment, with the system based on a symbol-by symbol interleaver. We will call it the baseline system and it is shown in Figure 1. For the baseline system the order of diversity is governed by the minimum number of distinct symbols between two codewords. Thus, diversity can be increased by avoiding parallel transitions and increasing the constraint length of the code.
*
considered as a means of improving the reliability of 8-PSK. It is shown that the combination of a rate 2/3 convolutional code, and 3 bit interleavers provides a good modulation/coding method for the Rayleigh channel. In Sections I1 and I11 a suboptimal coded system is proposed which combines a r e g u l a r convolutional code, bit interleaving, and an appropriate decoding technique. The bounds on the error performance of this approach indicate that the best convolutional code in conjunction with 8-PSK modulation and an appropriate soft decision memc is an attractive mean of improving the reliability. In particular over a Rayleigh channel this system requires E d N , of about 14.5 dB to provide a bit error rate of 10-5 (for a code with 64 states) which is 1.2 dB less than the requirement for the baseline system which uses the optimal trellis codes for Rayleigh channel.
11. SUBOPTIMAL CODED SYSTEM The diversity of a coded system can be increased by using a regular convolutional code followed by three bit interleavers. The diversity achieved by this approach is proportional to dfree of the code, and the error performance is governed by some product of dfr, terms. The analysis model of the proposed system is shown in Figure 2. In the following, ideal interleavers and deinterleavers are assumed, so that the combined interleavers, mapping and modulation can be viewed as three statistically independent modulators and
This work was partially performed while E. Zehavi was a Visiting Researcher at Qualcomm Inc., San Diego, CA 92121. The research was supported by Qualcomm Inc. and the J . S . Frankel Research Fund.
channels. The outputs of the encoder C = {id, i = l , 2 , 3 , ire transmitted by three random modulators. 1
2
3
Let us denote by (cp,cp, cp) the 3-tuple output of the encoder at time p. There are three channel symbols, i
i
xp, i = l , 2, 3, carrying the encoder output bits cp. With ideal random interleavers, the mapping from binary 3 tuple to 8-PSK channel signal is as shown in Figure 3, we define the following subsets,
29.2.1. .0536
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0 1989 IEEE
is suggested. The i-th metric computation unit produces two metiics corresponding to the two
Sy ={x.x( k) =a e x p S:=
S ! =
i
possible values of the bit c,
F),k=0,1,6,7} { x : x ( k ) = a e x p ( j 7),k=O,l,2,3} {x:x(k) =a e x p ( j
at time p. The decoder
input unit computes the branch memcs corresponding 1 2 3 to all possible values of €,,= (c,, c,, c,). For each such value the decoder input unit computes the sum i
i
C
of mi(ypSi; p,) denoted by m(Yp, C,;
and I 0 Si = 0 -Si ,i = l , 2 , 3 .
where Q is the signal set ,
y),
{x: x(k) = a e x p ( j k=0,..,7}. The transmitted channel symbol associated with the Q=
i
output bit cp = c at time p, can be any signal from the subset Si (c=O,l) with equal probability equal to 1/4. Thus, the i-th bit cb, induces a partition of the signal 0
1
set into two subsets S i and S i , i=1,2,3, as shown in Figure 3. This model of random modulation is due to the random interleavers, and to the suboptimal nature of the proposed decoding system which does not use the side information associated with the ordering of the transmitted symbols. The channel produces a noisy discrete-time sequence i
i
i
[y,), written as y, = p, "p
+
i
n,. Here, JJ,, is the
i received signal at the i-th channel at time p , and "p is a complex noise process, where Re{n> and Im{n) are uncorrelated, zero-mean Gaussian r.v.'s each with variance c? = No. p i is a random variable
these metrics are passed to the decoder, which employs the Viterbi algorithm to find the binary data sequence I(; (or the path with the highest cumulative sum of metrics). We note that the random modulation technique has the advantage of higher diversity while preserving reasonable minimum squared Euclidean distance, which is an important feature for transmission over a fading channel. The number of distinct symbols between two codewords is lower-bounded by dfre of the code. Thus, one should expect that this approach will be superior to the baseline system over a Rayleigh fading channel. 111. ERROR PERFORMANCE OF THE PROPOSED SYSTEM The error performance of the proposed system is evaluated by using the Chernoff bound technique 13, 41 and the generating function approach [5-71. To find the upper bound on the error performance of a trellis coded system we have to compute the A
A
A
A
A
A
coded sequence C = ( C l , C2, ., C, .., CN),instead of € , when the latter is transmitted. It can be shown that for three independent, random interleavers, the above inequality is upper bounded by
where e, = cb@ c i ,and
and that the phase variations of the channel are tracked by the receiver, assuming that the receiver can perform coherent detection.
D l ( a ) = 1 @e;
XE
,a,
pairwise error probability, namely P(€+ € I € which represents the probability of choosing the
representing the random amplitude fluctuation of the received signal. We assume that the receiver has perfect side information on the fade amplitude { pi J
At the receiver, the faded, noisy version of the transmitted channel signal is passed through three metric computation units. For the proposed system a receiver which uses the suboptimal metric mi(yL,sf; p ) = - min ~ liy -pd(/12, , i C=O,I, i = l N 1
e).Finally,
.
A.
+ e;{D
a24Ed1-cos(:)
a24E& - cos(;)
+D i
D2(a) = I @ e,
s:
j 1
i ep
+ -j-{D
9
a24Edl-cos(:))
d4EJl- cos(;)
(2)
+3D
1
9
(3)
29.2.2. 0537
Here @ is modulo 2 addition, D = exp(-
s), and
-
2
I
i
p p is the fade amplitude of the i-th channel
-
Here, = E b R is the average symbol energy-toNo noise ratio, and R is the code rate.
corresponding to the coded bit 3 .
The generalized Ro expressed as
We will now examine the performance of the proposed system operating over a Gaussian and a Rayleigh channel.
Ro = 3-10g2{l+D~}-210g~(l+D2~
For the Gaussian Channel, p
PP
for these channels can be
and
(a)= 6(a-I), for all
p. Thus, Eq. (3) can then be expressed in the form (4)
for the Gaussian and the Rayleigh channels, respectively.
where
Figures 4 and 5 show the upper bounds on the values D1=
D
(I- cos& +D
(1- cos(;)
Eb ofG that are required to transmit at R= Ro, for Gray
mapping and suboptimal metric assignment. We see from these theoretical results that the cost of using this suboptimal decoding metric is approximately 1.3
I::[
dB and 1.7 dB for operating at Ro = 2 -, over a
Es D = exp(- ZN,),
(5)
and W i is the Hamming weight of the error pattern of
(c&.).
At high the i th encoder output bit sequence, signal-to-noise ratio, the dominant term in the pairwise error performance is the one with the minimum Hamming distance. Thus,
Gaussian channel and a Rayleigh channel, respectively. The error performance of the proposed system which uses practical codes can be evaluated with the aid of a new enumerating function. Let us denote by T(W1, W 2 , W 3 , L, I ) a new generating function of a convolutional code, given by m 1 t w2, w3, L , 0
Similarly, for the Rayleigh channel one can derive bound on the painvise error probability
where a
nt.ij k=1,2,3
Es
I+-sin NO
-
2 w (F)
0.5
Es
I+-sin No
2 w
No
8
(-)
-
Es
1 + -sin
No
I
2 w (-) 4
1.5
+
Es
I + -sin
2 a (-) 4
where W 2 = W3, and Wi = Di or Wi =Di for the Gaussian and Rayleigh channels, respectively. In Figure 7, the upper bound on the bit error rate vs. &,/No is shown for the proposed system, employing
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a,
k=l, 2, Hamming weight nk on the subsequence 3, which have length i, and j information bits associated with the codeword C. This function can be enumerated in analogy to a regular convolutional code. By combining the pairwise error bounds and the union bound, an upper bound on the bit error rate is obtained. The bounds may then be written in the form
where
D2 =
is the number of codewords with
the best convolutional codes, for different numbers of decoder states. It is seen that 8-PSK modulation requires EdNo=145 dB for Pb = 10-5, for a code With 64 states. On the other hand, we note from Figure 6 that for the corresponding baseline coded systems, the requirements is 15.7 dB, respectively, which makes a difference of 1.2 dB relative to the "optimal" code.
S-F'SKMobla
IV. Acknowledgments
I
Oil
I
Discussion with A. J. Viterbi and Ayal Bar-David were valuable in the course of the work reported here. We want to acknowledge the suggestions and recommendations received from Y. Kofman, which have greatly contributed to the quality of this paper. REFERENCES
I
G. Ungerboeck, "Channel Coding with
MultilevelPhase Signals," IEEE Trans. Inform. Theory, Vol. IT-28, No. 1, pp. 55-67, Jan. 82.
S. G. Wilson, and Y. S. Leung, " Trellis-Coded Phase Modulation On Rayleigh Channels," in Proc. I987 Znt. Conf. Commun., pp. 21.3.121.35, Seattle, Wa, June 7-10, 1987. D. Divsalar and M. K. Simon, "Trellis Coded Modulation for 4800-9600 bitds Transmission Over a Fading Mobile Satellite Channel," IEEE Jr. on Sel. Areas in Commun., Vol. SAC-5, No. 2, pp. 162-175, Feb. 87. M. K. Simon, J. K. Omura, R. A. Scholtz and B. K. Levitt, Spread Spectrum Communication, Vol I, Ch. 4; Computer Science Press, 1985. E. Zehavi and J. K. Wolf, "On the Error Performance of Trellis Codes," IEEE trans. Inform. The~ry,Vol. IT-33, NO. 2, pp. 196-202, Mar. 87.
T
A. J. Viterbi, E. Zehavi, R,. Padovani and J. K. Wolf, "A Pragmatic Approach to Trellis Coded Modulation," will be published in the to IEEE Communication Magazine. S. Lin and D. J. Costello, Jr. Error Control Codinn: Fundamentals and Applications, .. FkntickHall, Inc.,1983.
Bit 2
Bit 1
0
O1O0
ow,
0110* 111 101
Figure 3. Partitionsof thc signal set r~ subsus.
29.2.4.
Bit3
-
Figure 5.
$ vs. 1&
Figure 4.
For 8-PSK and 16-PSK modulation over a Rayleigh channel.
2
vs. I&
for 8-PSK and 16-PSK modulation over a Gaussian channel.
Eb
16 0
14 0
12 0
10 0
80
60
40
1.00
0.10
10.00
liRo [synbolslbiiJ 1 M = 2
OM-4
*M=B
0.I
Figure 6. Upper bounds on the Bit Error Rate Vs. E W o of optimal trellis codes, rate
U3.8-PSK. over a Rayleigh channel, for various number of states. Pb
Figure 7. Upper b u n d s on the Bit Error Rate Vs. EbMo of the propsed system, rate 213, 8-PSK. over a Rayleigh channel, for various number of states. Pb
I.E%l
1.E-02
LE43
1.E-03
LE44
l.E-04
LE45
1.E-05
LE46
1.E-
LE47
l.E-07
1.E10
I1
12
13
IS
14
EbiNo =S=4
0 S=8
- S=16
16
17
L
I8
L
A
19
20
[dBl OS32
t S=b4
& S=128 I
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