CODED MODULATIONS AND DIVERSITY FOR SATELLITE

CELLULAR COMMUNICATIONS Ezio Biglieri, Giuseppe Caire, Giorgio Taricco*

Javier Ventura-Traveset**

ABSTRACT

* Politecnico di Torino Corso Duca degli Abrtlz::i 24, 1-10129 Torino (Italy)

** Etlmpean Space Agency / ESTEC P.O. Box 299. 2200 AG Noordwijk (The Netherland.s)

Two detection schemes are considered for the fading channel with co-channel interfer­ence. Both schemes exploit the synergy of diversity and coding with maximum-rat.io combining and perfect channel state information. The first one - conventional receiver - reduces the error floor by the use of diversity and coding. The second - multi-user receiver - eliminates the error floor by exploiting the structure of the interfering sig­nal which is no more treated as an unknown source of disturbance [6]. Performance results include bit error probability bounds and simulation results for both schemes envisaged.

Keywords: PCS Systems, Satellite Communications, Coding. Diversity.

1 INTRODUCTION

Most Personal Communication Systems (PCS) include coding and diversity to protect the transmitted signal from the severe performance degradation associ­ated with fading in the mobile radio channel. Their simultaneous use allows the designer to cope with stringent power requirements and co-channel interference (CCI) typical of cellular environments. However, only a few works address the interaction of coding and diversity, which is the main focus of this paper.

This work was supported in part by the Human Capital and Mobility Program of t.he Commission of the European Union.

57

58

Different kinds of diversity have been considered for a long time: space, fre­quency, direction-of-arrival, polarization, time, and multi path diversity [8]. Coded modulations can be regarded themselves as kind of time diversity [9]. It is then natural to try to exploit coding and diversity together in order to reduce receiver complexity.

The design of coded modulations for the fading channel [3. 4] without diversity is generally more complex than designing coded modulation ~or ,t.he AWGN channel. Codes are selected according to different criteria than the st.andard minimum Euclidean distance used for the AWGN channel, so t.hat. good codes for the AWGN channel may perform poorly on a fading channel and vicf' versa. However, as the order of diversity increases, a fading channel approaches all AWGN channel [12, 13, 11]. This allows us to use powerful codes designed for the AWGN channel ~ such as t.he well known Ungerboeck codes [10] - after introducing a suitable amount of diversity.

Focusing on space diversity, we consider a NORmalized Diversity (NORD) re­ceiver based on an antenna array of 1'.1 elements, each having equivalent area Ae. Assuming that the transmitted power is Pt , we identify two equivalent cases of NORD: i) Ae = A/M, Pt = P (fixed total antenna gain and t.rans­mitted power) and ii) Ae = A, Pt = P/M (antenna gain increased by 1'.1 and transmitted power reduced by 1'.1). In both cases, the product of each ant.enna element's gain by the transmitted power is constant. The average signal-t.o­noise ratio (SNR) per diversity branch is given by r /1'.1 where r = E /No is the total average SNR at the receiver input, E is the energy per symbol and No is the unilateral power spectral density of the AWGN.

Cellular PCSs are mostly limited by co-channel interference (CCI), which de­termines the system capacity [5]. Therefore, we will concentrate on system performance vs. CCI. We consider two different receiver schemes based on i) coherent detection with maximum-ratio combining (MR.c). hereafter referred to as the conventional receiver, suboptimal in the presence of eCI and ii) a novel detection scheme based on the multi-user approach introduced in [6], hereaft.er referred to as the multi-user receiver. In both cases, we present result.s based on error probability bounds and simulat.ion.

The paper is organized as follows. Section 2 describes the channel model. Sec­tion 3 considers the coded system with diversity and the conventional recpiver. Section 4 considers the same system with the multi-user receiver. In both cases, error probability performance is studied based on union bounds. Section 5 gives numerical results on the performance of a specific code. Finally, our conclusions are collected in Section 6.

Coded Modulations and Diversity ... 59

2 SYSTEM MODEL

The basic transmission system under analysis is described in Fig. 1. We have a source generator of binary symbols mk,' mapped by a coded modulator to channel symbols Xk from the q-ary PSI{ modulation set .1:'q = {e:i2n,,/q}~-::~. The encoded symbols are then interleaved and shaped to produce the signal x(t) = VEL-k x~p(t - kT), where E is the energy per symbol, T is the symbol time, and p(t) is a unit-energy pulse chosen in order that, when passed through a matched filter p* (-t), the synchronously-sampled output is inter-symbol in­terference (lSI) -free. The transmitted signal x(t) is sent through M different faded channels (diversity branches). Interleaving allows us to assume the case of stationary slow Rician fading (see, e.g., [4]). The received signal over each branch i is given by

where, gi(t) and hi(t) are independent complex Gaussian random processes with E[gi(t)] = E[hi(t)] = JK/(K + 1), E[lgi(t)12] = E[lhi(tW] = 1, and var(gi(t)) = var(hi(t)) = 1/(2(K + 1)). K is the Rician parameter defined as the line-of-sight to diffuse multipath component power ratio; ni(t) is an inde­pendent white Gaussian complex noise process with two-sided power spectral density No/2; J(t) is the CCI signal of the same type as x(t) but statistically independent from it and with energy per symbol E' = E / (3 where (3 is the signal-to-interference ratio (SIR). In our model, we assume a single dominant CCI signal and perfect channel state information (CSI) on both the gi (t) and the hi (t). This strong assumption can be well approximated, for instance, by including non-overlapping pilot tones, multiplexed pilot symbols or multibeam receivers in the return link of a satellite mobile communication system.

After demodulation, matched filtering, and synchronous sampling with ideal timing (which provides lSI-free samples), we obtain the following discrete-time channel model:

(1.1)

where

• gk and hk are independent random M -vectors whose components g£ and h{ are complex Gaussian random variables (RVs) with unit second mo­

ment E[lg£12] = E[lhLI2] = 1, mean E[g£] = E[hL] = J/(/(1{ + 1) and covariance E[g£(h~)*]- E[g£]E[(h{)*] = Cg(k - £, i - j).

• llk is a random M-vector whose components n£ are i.i.d. complex Gaussian RVs with zero mean and variance 1/2.

60

• 10 = r /!v! and 11 = r II!vI are tllP avprage SNR per branch of the mwflll and interfering signaL respectively, and r = £ jIV'1I and r J = £' /NII are the corresponding total average SNRs.

• h is the CCI given by h = L£ WrC/h'-f, where 11'( are the samples of the total channel impulse response ll'(t) = p(t) * p*( _t)e.iH sampled with an arbitrary delay with respect to tllf' optimal sampling timp and rotated by the phase difference e between the signal and the interference carriers, and the C/k E Xq are symbols from the eel source.

3 CONVENTIONAL RECEIVER

Here, we assume that CCI is an additive disturbance treated as independent Gaussian noise, thus obtaining a suboptimal detection scheme. From thp per­fect CSI assumption, Yk is conditionally Gaussian with mean .JYO .r" anel co­variance matrix HIIE[lhI2] + l)IM . Since the modulated symbols .r" have constant magnitu~de, maximum-likelihood (ML) coherent detection consists of

maximizing the additive metric Lk 2 Re [gkYk x;:] over all the possible coded sequences x (here the dagger t denotes Hermitian transposp). '<\-'e call first

form the combined diversity-channel output rk = gl.Yk, and then use it to generate the metric associated to the branch of the cocle trellis labeled by 7/,-: m(rk,xk) = 2 Re[rkxZ]. This is equivalent to perform maXimllll1-ratiocolllhill­ing (MRC) detection before Viterbi decoding. The bit error probability call 1w obtained by the union bound [1]

where m is the number of information bits entering the encoder at each trellis step; S is the number of states; the pair (xes, Xf,s) denotes an error event with span f starting from state 5, i.e., a pair of distinct sequences stemming from the same state 5 and merging after C 2: 1 discrete-time instants; B(XI'.8' X(.,) is the Hamming distance between the information bit sequences corresponding to xe,s and Xf,s; P(xe,s --t xe,s) is the pairwise error probability (PEP) of t,]lP

sequences xe,s and Xf,s'

Several methods have been proposed to calculate (l.:l) in closed form for thp AWGN channel [1] but none applies for the fading channel. In this case, we have to resort to numerical approximation and consider a limited number of terms in (l.2) [3].

Coded Modulations and Diversity ... 61

The PEP P(x -+ x) is calculated as follows. The difference bptween the metrics of two sequences x and x is given by

~ = L 2 Re [g1.n dZ] (1.:3 ) kEKL

where we defined dk = Xk - Xk and KL = {iI, ... , jL} = {k : J:k of :rd· L is the Hamming distance between x and x. The smallest Hamming distance between two encoded sequences is usually referred to as the "code diversity." We will examine the interaction between antenna and code diversity. If the code is not catastrophic and L is finite, the span of the error event is also finite, and the PEP is given by

P(x --+ x) = P(/)" :::; 0). ( 1.4)

The right-hand side (RHS) of (1.4) can be computed as follows. First, evalu­ate the Laplace transform of the probability density function of ~, <I>~(s) E[e-S~]. Then, we have from [7]

1 lc+j~ ds P(~ :::; 0) = -, -. <I>~(s)-

27rJ c-joo S (1.5 )

where c is chosen so as to let (1.5) converge. Numerical evaluation of (1.5) can be carried out by means of Gauss-Chebyshev quadrature, as explained in [2].

3.1 Performance of the conventional receiver

Following the approach of [12] with lSI-free cel (lhl 2 = 1), we obtain

<I>~(s) = II [1 + v00ldk l2s - hI + 1)ldk l2s2r M ( 1.()) kEKL

and then we get from (1.5):

Alternatively, applying the Chernoff bound to the RHS of (1.6), we obtain

( 1.8)

62

Letting r -+ 00, the Chernoff bound (l.8) approaches the limit value

(U))

This follows from the existence of an irreducible bit error probability, with CCI and fading, as the SNR increases without limits - an enor floor - which is exponentially reduced with the product of time and space diversities. The effect is mostly remarkable for weak codes (up to uncoded modulation). In the following, we will see a more powerful approach that is able to delete the error floor, which we call multi-user detection.

4 MULTI-USER RECEIVER

With multi-user detection, we recognize that CCI cannot be assimilated to Gaussian noise and deal with it as another modulated signal. We assume that the receiver has perfect knowledge of both gk and hk . The discrete-channel output sample can be written as Yk = Gkbk +Uk where G k = [v'FQ gA" -/hhkJ and bk = (Xk, h f. Then, the ML decoder branch metric associated to ;rk is given by

m(Yk,xk) ...:.. InP(Yklxk,gk,hk) == In{Eh[P(Yklxk,gk, h,hk)]}

...:.. In {Eh [exp[-IYk - Gk b kl 2]J}

(1.10)

where == stands for metric equivalence (ml == m2 if and only if ml = al 1112 + a2

for some constant al > 0 and (2) and Hk = G! Gk. We further assume that the CCI signal is not affected by lSI, thus wanted and interfering signals are symbol­synchronous, but not frame-synchronous. Therefore the (coded) sequence (h) is received as a sequence of independent, uniformly distributed, symbols from Xq and we have

m(Yk,xk) == In [L eXP[Sl(Xk,h)l] ~ Slmax(Xk) = max Sl(Xk,!k) (1.11) hEX.

hEXq

where we defined Sl(Xk,h) == Sl(bk) = 2Re[y!Gkb A,]- b!Hkbk and used the approximation In[Li exp(zd] ~ maxi Zj. The metric (l.11) is formally identical to the multi-user metric derived in [6] for two-user CDMA in AWGN and Gaussian spreading sequences gk and hk. The detection scheme based on (l.11) defines what we call m ulti- user receiver.

Coded Modulations and Diversdy ... 63

4.1 Performance of the multi-user receiver

Our approach to the evaluation of the PEP is based on the union bound. Let us consider the error event (x, x) with Hamming distance L and let KL = {it, .. ·,jd = {k : Xk "# :rd· Let Ie denote the Cth interfering sequence from Xf, for £ = 0, 1, ... , qL - 1. From the branch metric (1.11), we can write the path metric increment as follows l :

!1max (x) = L !1max (Xk) = mtx!1(x,I) kEICL

( 1.12)

where I = (Ijl' ... , Ih) and the last equality derives from the independence of the CCI samples in I. Then, the PEP is given as

(1.13)

In order to evaluate this probability, we first condition on the (random) trans­mitted CCI sequence I and we call Imax the CCI sequence attaining the maxi­mum of !1(x, I):

P(x -7 x I I) P { !1(x, Imax) ::; mfx!1(x, I)} (1.14)

< P { !1(x, I) ::; myx!1(x,l) }

p rQ' {O(x, I) - O(S<, I,) ~ OJ} (1.15 )

qL_l

< L P {!1(x, I) - !1(x, Ie) :S O} (1.16) (=0

where (1.15) is greater than (1.14) because !1(x, I) :2 !1(x, Imax) and (1.16) is obtained from (1.15) by the union bound. The branch metric difference can be written as

~ t t ilk = !1(bk) - !1(bk) = 2Re[zkdkl + dkHkdk ( 1.17)

where dk = bk - bk and

Zk = GkYk - Hkbk = Gk(Gkbk + llk) - Hkbk = Gk llk

since Hk = GkGk. Conditionally on Gk, Zk is a vector of complex Gaussian RVs with zero mean and covariance matrix ~Hk. Then, still conditionally on

1 We do not consider the positions k such that Xk = ?k in the path metric since, in this case, rl.max(Xk) = rl.max(?k).

64

G k , ZLdk is a complex Gaussian RV with zero mean and variance ~d!Hkdk. It is then clear that the conditional PEP (1.15) does not depend on I but only on the difference Ie - I, so that, when If runs over all possible sequences from Xf, we can remove conditioning and choose. arbitrarily, I = 1, the seqlwllce of all ones. Then, we can write

qL_ 1

P(x -+ x) ::; L P {0(x, 1) - 0(x, Ie) ::; O} ( 1.18) (=0

Now, to calculate the probabilities P{0(x,1) - 0(x, Ie) ::; O}, we advocate the Laplace transform technique already used in Section ::l. First., by iterating expectations, we obtain

EGk [EZk [exp (-05 (2 Re [d!Zk] + dl,Hk dk)) I G k ]]

t .) EGk [exp( -dkHI,.ddo5 - o5~)))

[1 + (rold!.J' + "'flld~I::?)(o5 - o52)rM

with dk = Xk - Xk and d~, = 1 - h. Then, from the independpl1cP of the c:'k'S,

setting ~ = LkEKL ~k, we have

<I>A (s) = E[exp( -s~)) = IT [1 + (ro Idk I::? + "Ylld~12)(o5 - 82)]-M (1.19) kEKL

Applying (1.5) to (1.19) and inserting the result in bound (1.18), we get.

P(x --+ x)

where Jl = 1 - exp(j27r£/ q). This probability can be evaluated numerically by resorting to Gauss-Chebyshev quadrature, as described in Appendix A. Alter­natively, applying the Chernoff bound to each term in (1.18), we get

(1.21 )

Bound (1.21) shows that the multi-user receiver suppresses t.he errol' floor, irrespectively of the value of SIR and for any finite diversity order AI. Wit.h

Coded Modulations an.d Diurrsily ... 65

strong CCI, i.e .. for (3 -+ 0, (l.21) approache~ the limit

with Je = 1 - exp(j27rJi/ q). This bound is independent of the coele as we assumed that the CCI interferer is not frame-synchronous and tlH~reforp ap­pears uncoded. For weak interfering signal, i.e., for !3 -+ 00, the performancE' approaches the CCI-free case, since the effect of eCI becomes negligible.

Bound (l.16) is tight when the interference-to-noise ratio per branch /1 IS so high that there is a CCI sequence that attains a sharp maximum in nmax(x). This is not true when the SNR is lower than or around the SIR and. in this case, the bound is loose. However, performance cannot be worse than with the conventional receiver, so that we obtain a tighter general bound as follows

P(x-+X) ::::;min{bound (l.7). bound (1.l9)} ( 1.22)

5 RESULTS

In this section we collect results obtained by our analysis. We consider an uncorrelated flat Rayleigh fading channel affected by CCI and AWGN.

Performance with conventional coherent detection. In the following, results are relevant to Ungerboeck's rate 2/3 coded 8-PSK scheme with 4 states (U4). Fig. 2 shows the performance ofU4 with conventional detection. Results, obtained by analytic bounds and simulation, show excellent agreement below the level BER = 10-2 • Fixing a target BER = 10-3 , Fig. 3 shows the minimum Eb / No required to obtain the target performance vs. SIR for different values of the diversity order M. These results are obtained by analytic bounds, which proved to be extremely accurate in this case, as mentioned above. Finally, Fig. 4 shows how diversity progressively eliminates the error floor, following the asymptotic behavior of bound (1.9).

Impact of multi-user detection. An alternative to diversity is multi-user detection. We analyzed it in conjunction with diversity for the same code U4. Figs. 5 to 8 show the BER vs. Eb/ No for different values of SIR (-10,0,10 and 20 dB) and diversity order. First, we notice that the error floor has been deleted

66

for any diversity order. Second, the curves show the same good performance at. all values of SIR, as if the negative impact of SIR were totally removed. This can be explained by noting that multi-user detection exploit.s t.he struct.ure of the CCI disturbance, provided a reliable overall estimation of the CSI. This also explains why intermediate levels of SIR (around 5 dB) achieve the worst. performance, as neither the wanted nor the interfering signal are dominant. wit.h this level of CCI. The figures also show t.he looseness of bOil \ld (l.1 g) at low values of SNR. Simulation results are in good agreement. with analytic bounds below t.he level BER = 10-3 . The goodness of approximat.ion is better for low diversity order. Fig. 9 shows the minimum Eb/No required t.o obt.ain a target BER = 10-3 vs. SIR for different values of t.he diversit.y order M. As a consequence of the elimination of the error floor we can all ways achieve the target performance, irrespectively of the SIR.

As a final observation, the angular points of the bounds for the mult.i-user re­ceiver derive from the fact that the multi-user bound (l.22) is a minimum oft.wo bounds, the second of which is looser for low SNR. However, from a practical point of view, this is acceptable since it produces a conservative performance analysis.

6 CONCLUSIONS

We have studied diversity r.eceivers for a coded communicat.ion system affected by fading and co-channel interference, namely a typical cellular PCS. Two de­tection schemes have been considered here: coherent detection wit.h maximum­ratio combining (MRC) (conventional receiver) and a novel detection scheme based on the multi-user approach introduced in (multiuser receiver) [6]. The performance of the conventional receiver is limited by the implicit assumption that the co-channel interference is an unknown disturbance, which inevitably introdeces an error-floor in the BER performance. The multi-user receiver overcomes this limitation and provides rather interesting results.

Further study is required to assess how close we can approach t.he perfect. CSI assumption, fundamental to the operation of both receiver, in the multipath environment. This is mostly important for t.he multi-user receiver which uses the CSI of both useful and interfering signals.

Coded Modulations and Diversity ... 67

REFERENCES

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[2] E. Biglieri, G. Caire, G. Taricco, and J. Ventura-Traveset, "Simple method for evaluating error probabilities," Electronics Letters, Vol. 32, No.3, pp. 191-192, Feb. 1996.

[3] P. Ho and D. K. P. Fung, "Error Performance of Interleaved Trellis Coded PSK Modulations in Correlated Rayleigh Fading Channels," IEEE Trans. Communications, Vol. 40, No. 12, Dec. 1992.

[4] S. H. Jamali and T. Le-Ngoc, Coded-Modulation Techniques for Fading Channels. New York: Kluwer Academic Publishers, 1994.

[5] W. C. Y. Lee, "Spectrum Efficiency in Cellular," IEEE Trans. Veh. Tfch­nol., vol. 38, pp. 69-75, May 1989.

[6] H. V. Poor and S. Verdu, "High-speed Digital Signal Processing for Satel­lite Communications," Final Report European Space Agency P. O. 184422, Feb. 1995.

[7] J.Proakis, Digital Communications. New York: McGraw-Hill, 1983.

[8] M. Schwartz, W. R. Bennett, and S. Stein, Communications Systems and Techniques. New York: McGraw-Hili, 1966.

[9] C.E. W. Sundberg and N. Seshadri, "Coded modulation for fading chan­nels: An overview," European Transactions on Telecommunications, Vol. 4, No.3, pp. 325-334, May-June 1993.

[10] G. Ungerboeck, "Channel Coding with Multilevel/Phase Signals," IEEE Trans. on Inf. Theory, Vol. IT-28, Jan. 1982.

[11] J. Ventura-Traveset, G. Caire, E. Biglieri, and G. Taricco, "Impact of diver­sity reception on fading channels with coded modulation. Part I: Coherent detection," submitted to IEEE Trans. Communications., June 1995.

[12] J. Ventura-Traveset, G. Caire, E. Biglieri, and G. Taricco, "A multi-user approach to combating co-channel interference in narrowband mobile com­munications," 7th Tyrrhenian International Workshop on Digital Commu­nications, Viareggio (LU), Italy, 10-14 Sept., 1995.

68

Source Generator

T R A

C N

H S

A M I

N

S N

S E

I L

0 N

Viterbi Decoder

rk

Coded Modulator

:q.

hI (t)

hM(t)

Deinterleaver

Int.erleaver

+

+

Matched

D Filtering

E T

I E rk

C 1M T Yk

I 0 N

Figure 1 Model of a coded transmission system with fading. diversity. and coherent detection combining. affected by CCl and AWGN.

[13] J. Ventura-Traveset, G. Caire, E. Biglieri, and G. Taricco, "Synergy of diversity reception and coded modulation in mobile communication chan­nels," European Personal Mobile Communications Conference, Bologna, Italy, 30 Nov., 1995.

Coded lvlodulations and Diversity ...

10"

10'\

10.2

10.1

Hr"

ffi 10.5 CQ

10'"

10.7

Hr"

Hr"

10 20 30

Eh/N" (dB)

Figure 2 Bit error rate of U4 vs. Eb/NO with coherent detection. SIR=1O dB. Solid curves: TUB, dots: simulation results.

25

20 f+···················· .j .........•...........•............ .;+

o

~ 15 I-I .............. \. UJ

10

10 20 30

SIR (dB)

Figure 3 Eb/NO vs. SIR at Pb = 10-3 of U4 with coherent detection for different values of M (diversity order).

69

70

10" 10,1

10,2

Hr' 10-4 10,5

10-6 10,7

Hr'

~ Hr"

Ltl 10,111 CO

10,11

10,12

10,13 10,14

10,15

10,16

10,17

IO'IH

10,1'

10,2"

0 10 20 3D

SIR (dB)

Figure 4 "Error floor" of U4 vs, SIR with coherent detection for diversity order M = 1,2,4,8,16,

10,2

10-6 ~, ................ : ....... , ......... : ................. .

10" ~"'C ", .. ,: •.. ", ,', •. , ...•. : .....

10.111 L~~-'_-'-_~_-'--_.i.---'~-'_-'-_-'--_-'-_~"--'~-'_..J

o 10 20

EtfN" (dB)

Figure 5 BER vs, Eb/NO of U4 for SIR=-l0 dB and M curves: minimum upper bounds, dots: simulation results.

3D

1,2,4. Solid

Coded Modulations and Diversity ...

10"

10.2

104

t>: ~ <C

Hrn

Hr"

10-10

0 10 20 30

Figure 6 BER vs. Eb/NO of U4 for SIR=O dB and M = 1,2,4. Solid curves: minimum upper bounds, dots: simulation results.

10-2

104

10-10 '--~_~_~_~----1._~_~~~~_....L.._~~~~--"-~-----'

o 10 20 30

Figure 7 BER vs. Eb/NO ofU4 for SIR=10 dB and M = 1,2,4. Solid curves: minimum upper bounds, dots: simulation results.

71

72

lO"

10-2

10-4

e< Ul a:l

)(r"

IO-X

)(r 11l

0 III 211 311

Figure 8 BER vs. E b / No of \)4 for SIR=20 dB and !vI = 1,2,4. Solid curves: minimum upper bounds, dot.s: simulat.ion results.

SIR (dB)

Figure 9 Eb/No vs. SIR at Pb = 10-3 of \)4 with multiuser detection for different values of M (diversity order).

364

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[6] M. Aicardi, G. Casalino, and F. Davoli. Independent Stations Algo­rithm for the maximization of one-step throughput in a nmltiarcess chan­nel. IEEE Transactions 011 CommulIicatZ01l8, COM-:35(8):795-800. Angust. 1987.

[7] K.W. Ross. Multiservice Loss Models for Broadballd Tr[(communicatioll Networks. Springer, 1995.

[8] S. Jordan and P.P. Varaiya. Control of multiple service, multiple resollrcp communication networks. IEEE Trans. Communication, 42:2979--2988. November 1994.

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