Transmiission Range Optimiization for

FH-CDMA Networks in Time-Varying Channels

Haichang Sui and James R. Zeidler

Dept. of Electrical & Computer Engineering,Univ. of California, San Diego, La Jolla, CA 92093-0407Emails: haichangsui@ gmail.com, zeidler@ece.ucsd.edu

ABSTRACT

Abstract- Optimization of the transmission range for max-imizing information efficiency is studied in this paper formobile ad hoc networks with frequency-hopped (FH) CDMAand multiple antennas. Realistic channel models are employedto accountfor path-loss, log-normal shadowing, and Rayleighfading. The shadowing and fading are assumed time-varyingwith different time-scales. The receiver performs decision-feedback demodulation for differential unitary space-timemodulation and erasure insertion decoding for Reed-Solomoncodes. The decoding error probability is derived based ondistributions of the multiple access interference power andthe SIR for ground propagation model (path-loss exponentequal to 4). The trade-off between information efficiency andtransmission range is studied in detail and insight is obtainedinto the impact of various system parameters.An important design issue in mobile ad hoc wireless

networks is to determine the optimum transmission range[1]. For networks where nodes are randomly distributedon the plane according to a two-dimensional Poissondistribution with density A, the transmission range Rcan be specified by No AAwR2, which is related tonetwork connectivity [1]. Optimizing R, or No equiva-lently, has been studied for such networks with slottedALOHA protocol in previous literature [2]-[6] [8]. Formaximizing the expected forward progress, it is shownthat the optimum No scales with the spreading gainin DSCDMA networks where nodes are equipped withsingle-user receivers and no power control is employed[2], [3]. On the other hand, more insight into networkdesign may be revealed by considering the product of theexpected forward progress and the spectral efficiency, orthe so-called information efficiency (IE) [4]. Optimizing

(This work is supported by, or in part by, the Office of NavalResearch (Code 313) and the U S Army Research Office under theMulti-University Research Initiative (MURI) grant # W91 IlNF-04- 1-0224.

No to maximize IE for DS-CDMA networks is studiedin [4] for a channel with no shadowing. Compared toDS-CDMA, an important feature of frequency-hopped(FH) CDMA is that, due to hopping and interleaving,the coded symbols corresponding to the same codewordare subject to different SINR levels. Such "interferencediversity" can be exploited by proper coding and inter-leaving to effectively suppress MAI [8]-[11]. In contrast,symbols in the same packet are usually subject to thesame SINR level in narrow-band or DS-CDMA networksand the packet error probability is commonly determinedby a threshold test on the SINR [2], [3], [4], [7]. Thismakes FH-CDMA very robust to the near-far problem,while DS-CDMA requires either accurate power-controlor multi-user detection.A Reed-Solomon (RS) coded FH-CDMA transceiver

with MIMO is developed in [10], [11], where differ-ential unitary space-time modulation (DUSTM) is usedto achieve spatial diversity noncoherently. Decision-feedback demodulation (DFD), decision-directed adap-tive estimation, and erasure insertion (El) decoding oper-ate interactively to mitigate time-varying fading, estimatesymbol reliability, and suppress interference. The supe-rior near-far resistance observed in [10], [11] motivatesus to further investigate transmission range optimizationwith the proposed FH transceiver for maximizing IE.Previous studies of similar problems include [5], [6],which consider FSK and RS coding without El. Other re-lated works include [8], [9], where bit-interleaved codedmodulation and coherent demodulation are assumed.Our study assumes a realistic time-varying channel

model, which accounts for path-loss, log-normal shad-owing, and Rayleigh fading. Different time-scales areused to model the time-variation of different channeleffects. Results in this paper extend previous studies[2]-[9] to include the impact of the time-varying fading

1-4244-151 3-06/07/$25.00 ©2007 IEEE I _f

and shadowing on network design. The optimum Nofor maximizing IE is shown to be linearly proportionalto the spreading gain q in FH-CDMA networks, whileit scales as q in DS-CDMA networks without multi-user detection or power-control. The trade-off betweentransmission range and IE critically depends on the mod-ulation and coding scheme (MCS). Optimum MCSs arefound for different system settings by numerical analysis,which offers guidance for adaptive transmissions froma network perspective. Our results further shed lighton how the network performance depends on variousdesign parameters and channel properties, including thenumber of antennas, receiver configuration, dwell length,Doppler frequency, and shadowing spread/speed.The rest of the paper is organized as follows. The

system model is presented in Section I. In Section II,packet error probability and IE are derived. The trade-off between IE and the transmission range, together withthe effects of various system parameters, are studied indetail in Section III and Section IV concludes the paper.

I. SYSTEM MODEL

A. Network Model

We consider a network where nodes are randomlydistributed over the plane according to a two-dimensionalPoisson process with density A and access the channelaccording to the slotted-ALOHA protocol with transmis-sion probability p. The transmitting and receiving nodesof interest are denoted by S and D respectively, whichare separated by the transmission range R. AWGN isignored and the transmit power of all nodes is set tobe unity [5]. Random hopping patterns are assumed inthis paper. The total number of carrier frequencies thata hopped signal can be transmitted on, i.e. the spreadinggain, is denoted by q.

B. Transceiver and Channel Model (PHY)

The PHY transceiver structure is specified in detail in[10], [11]. At the transmitter shown in Fig. 1, informa-tion bits are coded by (L, K) RS code and the codedsymbols are then modulated by DUSTM. For simplicity,we assume the codeword length L is also the size ofthe DUSTM constellation V = {Vo, ..., VL1}, where{VI }fL- are NT X NT unitary matrices and NT is thenumber of transmit antennas. By denoting the Tth codedsymbol as ZT (zT=. . .,L-1), the corresponding space-time signal to be transmitted is an NT X NT unitarymatrix generated by differential modulation SO,T =

Vz,So,T1. The length of a DUSTM block is denotedby T8 and SO,T is transmitted in the Tth block by one of

the q carrier frequencies. The hopping rate is assumedto be 1 for some positive integer NC > 1 (slow FH)and the interval between hops is referred as a dwell.The interleaver and packet format are the same as in [5],[10], [11]. Specifically, a packet contains multiple frameswhile each frame consists of NC RS codewords, whichare interleaved such that the L coded symbols from thesame codeword are transmitted in different dwells toachieve frequency diversity.

lnf6, BA9 RS Ekc |. ,.i=b 13,t} |s [LterDSI B .4 i

Fig. 1. Transmitter model

The wireless channel between two arbitrary nodesis subject to deterministic path-loss and time-varyingshadowing and fading. Formally, the channel gain can bewritten as d-b/2 S (t) h (t) where d is the distancebetween the two nodes, b is the path-loss exponent, S (t)is a log-normal r.v. representing shadowing, and h (t)is circular symmetric complex (c.s.c.) Gaussian due toRayleigh fading. The pdf of S(t) is given by

1 _n2yfs(s) ef2FOs

(1)

where l0 078 is the shadowing spread expressed indecibel. The received space-time signal RT in the 7thDUSTM block after dehopping can be written as anNT X NR matrix. If the shadowing So(t) and the fadingho (t) between S and D are assumed time-invariantwithin each DUSTM block of length T8, we have

RT R b2SO(7T)SO,THT + RMAI,T ' (2)

where RMAI,T represents MAI, and the NT x NR chan-nel matrix Ho, contains the Rayleigh fading coefficientsbetween all NTNR antenna pairs, which are i.i.d. c.s.c.Gaussian r.v.s with zero mean and unit variance.The receiver proposed in [10], [11] performs DFD,

decision-directed adaptive estimation, and El decod-ing interactively, as shown in Fig.2. The DFD de-modulate the transmitted symbol ZT by zTarg maxk=0,...,L- 1AT,k [10], [11], where

2 vH'TZaj,TAT,k=-^2 Vk RT-Ea, ( l VS{) RT_j!('P,T =t= -+

(3)The coefficients alT,, .. ., ap,T and 7p4T in (3) are adap-tively updated by the decision-directed RLS algorithm in

2 of 7

Fig.2. Details of the algorithm can be found in [10], [11]and will not be presented here. In this paper, El basedon the so-called effective SIR (ESIR) [10] is considered.The ESIR accounts for the time-correlation of Rayleighfading and the feedback length P, which is given by [17]

Pef(P) =(PR + c7' ) /472 -1 (4)

where PR = R bS0 is the received signal powerand cr2 is the P-th order minimum linear predic-tion error variance of the Gaussian random process{Ho, + RMAI,}T. The value of cr2and the theoreticalvalues of a1,T,..., ap, can be solved from the Yule-Walker equations (c.f. [17] (8)). In practice, it is esti-mated by PefT L tot 1 where Ptot.T =UPtot,- +

NT-NF{. IIRT 112 is an estimate of total received signal plusinterference power [10] and ,u is the forgetting factor.The demodulated symbol ZT is erased if Peff, < Peth iSsatisfied. Since the reliability of DFD can be measuredby the ESIR [17], the decoding error probability maybe significantly improved by choosing Peth appropriately[10]. The IE with and without the ESIR threshold test(ESTT) El will be analyzed in Section II.

D 1 RLS D Demtodulatedt Symbols &

ReliabilityDe- { Ey'asure

MeasuresDcoder nterleaver Inserion

F[g. 2. Receiver model

An important consideration in performance analysis isthe time-variation of the channel. The relative distancebetween nodes is assumed invariant at least in a packetduration. The Rayleigh fading is time-varying acrossDUSTM blocks with time-correlation parameterized bythe normalized Doppler frequency fdT8 according toJakes' model. We consider two different time-scales forshadowing, which generally has less time-variation thanRayleigh fading. In fast shadowing, So (r) is assumedconstant in a dwell and independent across dwells. Inslow shadowing, So (T) is assumed to be constant for thewhole packet as in [5], [6], [8]. It is clear that whetherthe shadowing is fast or slow time-varying depends notonly on the mobility but also on the length of a dwell.

C. Interference Model

For clarity, we assume dwell synchronization so thatthe MAI process {RMAI,T}T is stationary in a dwell.

The MAI RMAI,T can be formally defined as RMAI,T =

Iima0 RMAI,T (a), where RMAI,T (a) denotes theMAI resulted from nodes in a circle centered at nodeD with radius a, which is given by

RMAI,T (a)[A'-Fa2]

Z dkbd2 Sk (T)Sk,THk,Tk

(5)

[NO]In (5), the notation E refers to a sum whose number ofterms is a Poisson r.v. with mean NO and A' -Ap/q isthe density of interfering nodes. The r.v. dk and Sk (r)in (5) are the distance and log-normal shadowing (withpdf (1)) of the kth interferer-D link respectively. Theinterference signal Sk,T is NT x NT,k and the Rayleighfading matrix Hk,T is NT,kXNR. Gaussian approximationis employed for elements in Sk,THk,T, whose varianceis normalized to unity to account for equal transmitpower of nodes. It is further assumed that Sk,THk,Tare independent for different ks and uncorrelated inspace and time. Thus, by conditioning on the distanceand shadowing for all interferer-D links, elements inRMAI,T (a) are i.i.d. c.s.c. Gaussian r.v.s with zero mean

and variance given by[A'wFa2]

7

(a) k dkbSk. (6)

In (6), time dependence of the shadowing is not explicitlyshown because it is constant within a dwell in boththe fast and slow shadowing model. The MAI process{RMAI,T}T in a dwell is thus modelled to be stationaryc.s.c. Gaussian and uncorrelated in space/time whenconditioned on its variance

2 A lim 2(a),a-*oo (7)

which is itself an i.i.d. r.v. for each dwell. Distributionof cr2 will be derived in Section Il-B.

II. INFORMATION EFFICIENCY AND OPTIMUMTRANSMISSION RANGE

In this section, we use IE as the objective functionto optimize the transmission range R, or No = AwrR2equivalently. The IE is defined in [5] as

(8)

where T (p) = (1 -p) (1 -e-P) is the "tendency topair-up" [2]. In (8), c( = §qL 1g2 (L) is the spectralefficiency (bits/sec/Hz). The packet error probability isdenoted by PE in (8) and will be approximated by thedecoding error probability PC.

3 of 7

IEA T (p) - (1 -PE)-.-R ,

In the rest of this section, we will first review theresults on ESIR and symbol error probability derived in[17] by conditioning on the received SIR p R bS0,where So and u72 are the shadowing and MAI power ina given dwell. Then the pdf of ca2 is derived. Finally, thedistribution of p is derived and the decoding error prob-ability Pc is obtained based on the resulting distributionand the conditional symbol error/erasure probabilities.

A. Symbol error/erasure probability

When feedback errors are ignored, the pairwise errorprobability is given in [17] by

Pk,l (p) - Pr{z klzT = }-F21T2 - N

1 lIIP1[ Peff(P)<O'm,k,1 Et'

7Tfl ~ 4 cos2 0Jwhere Jm,k,l (m = 1, . .. , NT) is the mth singularvalue of the matrix Vk- V. The pre-EI symbol errorprobability can be approximated by the union bound as

Pe (P) PLk, (P)1=0 k=o,kAl

For the purpose of analysis, the ESTT El is equivalentto a threshold test on p with a certain threshold Pth >0.The post-El symbol error and erasure probabilities canbe expressed as

Pa (P, Pth)

Pe (P, Pth) {

1 p < Pth0, P > Pth

0, P < PthPe(p) , P>Pth

The analytic decoding error probability of ESTT Elbased on (10), (11) is shown to be in good accuracy withlink simulation results, where the estimated ESIR is usedfor El [10]. The optimum threshold test is also derivedin [10] for El and shown to be robust to the choice ofthreshold. Results in [10] further showed that analysis ofESTT El with optimized Pth is close to the performanceof the optimum El in interference dominated environ-ment. This justifies the network analysis based on ESTTEl with optimized threshold in this paper.

B. Distribution of the MAI power

Due to the nature of Poisson processes on a plane,interfering nodes are uniformly distributed in a circlecentered at D with an arbitrary radius a [13]. That is,the pdf of dk in (6) is given by

fd (r) = (r < a).

In the following, we derive OMAI (w), the characteristicfunction (CF) of 2 , based on the influence functionmethod [13]. Results in [13], [14], [15] cannot be directlyapplied to obtain the CF of cr2 considered in this paperbecause the log-normal r.v. Sk in (6) is neither zero-meannor spherically symmetric.By definition, we can write SMAI (w) E [eiwU ] as

[A' wFa2]

OMAI () = lim E{ exp (jw E d Sk)I)}.k

(13)

Since dk, Sk are i.i.d. and the number of terms in (13)is a Poisson r.v., we have

YMAI (w)lim E0 (A7ra2)m Em [exp (jwd-bS)]aoo eA'-a2m!a*om=O

lim exp{A/7a2 [E (ejwd-bS)a-oo

1]}

(14)

By denoting the CF of S to be Os (.) and integrating bypart, the expectation in (14) becomes

E (eiwdObs) = S(a-b) -limr 2s(r bw)a'2o r-bLoaj r dclS() ) dr (15)a2 dr

The second term in (15) is zero if Os (w) is boundedfor arbitrary L. Furthermore, by changing variables andapplying L'Hopital's rule, we have

(I11) lim A'w7a2 K5s (a-b ) 1]A'Fj

E [S lim xb- 2

2 X*

(16)If we assume the path-loss exponent satisfies b > 2, wehave the following from (14)-(16):

OMAI (w) = exp (A'w j0dA s(t)t- 2/bdt) (17)

By denoting a = 2/b and noticing that E [SL]exp (o2 ,Tj) for log-normal r.v. S, the integral in (17)can be further simplified as

00dos (ti)~ 1ta

ii ldt )t-dt j= E [S1 eJStWt-dtj1-(1 v) j) E [S¶ 8)

where F(x) is the Gamma function. Consequently, theCF of cr2 for b > 2 reduces to

In OMAI (W)

(12)

A'w17(l a) el 2,2 aIF

x [1 -jsgn()tan 2 ] (19)

4 of 1

from which it is clear that o2 is an alpha-stable r.v. [12].Unfortunately, the pdf of cr2 can be expressed in

closed-form only when b 4, in which case thedistribution coincides with the Levy distribution, whosepdf and the cumulative distribution function (cdf) are

23

fMAI (X) = a 2a FMAI() fc(20)

for x > 0 with? A-w3 2eC2 8/ 2. We will focus on theground wave propagation model (b= 4) in the sequel.

C. Decoding error probabilityThe decoding error probability under El is

LLE LIPeP (1 -Pe Pa) (21)

i=O j=jo(i)

where jo(i) - 1max(L-K+1-2i,0) and Pa:Pe arethe probabilities of demodulated symbols being erasedor erroneous, respectively. Pa, Pe can be obtained byaveraging (10), (11) over the distribution of p, which isderived in the next for slow and fast shadowing.

1) Slow shadowing: In slow shadowing, transmittedsymbols corresponding to the same codeword experiencethe same realization of log-normal shadowing. Therefore,the symbol erasure/error probabilities have to be derivedby conditioning on the shadowing first. The averagedecoding error probability is then obtained by averagingover the log-normal distribution (1).

Based on (20) and conditioning on So, it is easy toderive the pdf and cdf of p to be

fpIso (t)

FpIso (t)

N/e a2 8 (

2S exp

erf(N

where No A A'wR2 =0 ~~~~~qover the distribution (22),

PaSo (Pth) =erf(

PeISo (Pth) =IAP>Pth

Noe/

The decoding error probabi

PC s. (Pth)L L-i

i=O j=jo(i)

x Oj -JPC

42FS24So

8/Fi2/8 \

¾4t , (22)

By optimizing over Pth, we get Pcso = minPth Pqso (4th)-The decoding error probability is obtained by

Pc = Es, [Pc so]1 00 2

00

eZ PCIe -2-, dz .(271)2) Fast shadowing: In fast shadowing, the SIR in a

given dwell has a pdf fp (t) = Es, [fplsO (t)] wherefpls0 (t) is given in (22). By applying Gaussian quadra-ture rule with Hermite polynomials again, we have

fp (t) _ er8 s fpex (t)dxoc-\22-F 2

No' e¢ 4 -x No12 e3,2 /4-EHXne 4N02 nt

Based on (28), the symbol erasure probability isrPth

Pa (Pth) = h fp (t) dt .

and the post-El symbol error probability is

Pe (Pth) =IAP>Pth

Pe (p) fp (p) dp

s n t(.28)

(29)

(30)

The decoding error probability is again given by Pc =

min Pth PC (Pth) where Pc (Pth) is given by (21).D. Optimizing IE

From the analysis in Section Il-C, we notice that thedecoding error probability depends on

No' = -Noq

After some manipulation, we can write

AIE

(31)

T(p) 10g2(L) KwFp Vq L [1 -PC(NoW No(2

/_ ), (23) The optimal ALOHA transmission probability is given2 soT(p)V\ SoJ by Popt arg maxp 0.27, which is consistent

No. By averaging (10), (11) with [2], [3], [5].we have The role of spreading gain q can be readily observed

2/8 from (31) and (32). Specifically, when other parameters(Noe F (24) are fixed in (32), the optimal No is proportional to

2 0SoPJ q due to (31), while the maximum achievable IE is

172 N127r,s2/4 inversely proportional to q. This is in contrast with8s- 4SO the results obtained in [2] [3] for DS-CDMA networks2 /Sop e (p) dp . (25) with single-user receiver and no power-control, where

lity conditioned on So is thus the authors showed that the optimum No is proportionalto q. The corresponding IE is therefore on the order

1 3L!Pe50 (pth)Pals0(Pth) of 1 q 4 2q 2* This may suggest theqi!j! (L -i-j)! advantage of FH-CDMA over DS-CDMA in terms of

both the transmission range and the IE, when the totalelso (Pth) -PalSo (Pth)] L26) bandwidth constraint is the same for both. Such an

5 of 1

Inx10 =4.3dB, fdT =0.05, q=512, NR=2, P=1, 2x 2 DUSTM (TJ), Fast shadowing model

Lw

L=4, K/L=1/4- L=4, K/L=1/2

L=4, K/L=3/4- L=64, K/L=1/4-c- L=64, K/L=1/2-- L=64, K/L=3/4

55ko15 2

N0/q

Fig. 3. AIE versus Nq for different MCSs (N = 2, P = 1,07 = 4.3dB, fast shadowing, fdT, = 0 05)

advantage comes from the inherent near-far resistanceof FH-CDMA waveform compared to DS-CDMA withsingle-user receiver and no power control.

III. NUMERICAL RESULTS

In this section, the trade-off between AIE andNo together with the impact of various parameters arestudied in more detail. We set q = 512 as a baseline forcomparison, but as discussed in Section II-D, the resultscan be easily interpreted for other values of q. Twotransmit antennas are considered in simulation and theDUSTM constellations proposed in [16] are employed.Also Popt = 0.27 is always assumed.

In Fig.3, curves of AIE versus NO are plotted forq

DUSTM constellation size 4 and 64 with code rate

4' 12 4' respectively. It is observed that the MCS impactsthe trade-off significantly. Observed from the peaks ofthe curves in Fig.3, lower maximum IE is achieved asthe code rate or constellation size decreases.The next two figures shows the effects of different

channels. Curves in Fig.4 correspond to different shad-owing models. The maximum achievable IE is higherfor fast shadowing than for slow shadowing becausehigher time diversity is available under the fast shad-owing model. Interestingly, the comparison reverses asthe transmission range extends beyond a certain pointand the MAI becomes more dominant. This is mainlybecause, in fast shadowing, multiple independent shad-owing realizations have to meet certain SIR threshold fora packet to be successfully decoded, while the packetis subject to the same shadowing in slow shadowingand the probability of satisfying certain SIR requirement

x 10-3 fdTs=0.05, q=512, NR=2, P=1, 2x 2 DUSTM (TJ), MCS5(16,1/4)41 ~

2

1.5

1 ;w

0.5, 1-

0 0.5 1 1.5 2 2.5

N0/q

Fig. 4. AIE versus No for different shadowing models (Lq

KIL = ,N =2,P =lfdl' 0%0)16,

can be much greater for a single shadowing realizationthan for multiple realizations when the interference isdominant. In terms of dwell length, this indicates thatwhether long or short dwell is advantageous depends onthe transmission range among other factors. As expected,the gap between the fast and slow shadowing is reducedas o7. decreases and the shadowing becomes more deter-ministic.The effect of different mobility is shown in Fig.5. It is

observed that higher mobility results in lower maximumachievable IE and reduced optimum transmission range.However, the performance may be significantly improvedby increasing the DFD feedback length P to alleviatethe time-varying Rayleigh fading. It is further shown inFig.6 that the spatial diversity from receive antenna arrayand interference suppression capability from El are veryvaluable for improving the network performance, whichcorroborates their importance in the link performancereported in [10], [1 1].

IV. CONCLUSION

In this paper, information efficiency and transmissionrange optimization is studied for FH-CDMA ALOHAnetworks with randomly distributed nodes. The MIMOtransceiver proposed in [10], [11] is considered, whereRS codes and DUSTM are employed with DFD and Eldecoding. The time-varying channel between any twonodes is subject to path-loss, log-normal shadowing,and Rayleigh fading. We have shown that the expectednumber of nodes in the optimum transmission rangeand the corresponding maximum information efficiency

6 of 1

I`6.4

'*.-i

are proportional to q and q- 1/2 respectively, where q isthe spreading gain. This implies higher information effi-ciency and increased transmission range for FH-CDMArelative to DS-CDMA networks at large spreading gainwhen a single-user receiver is used without power con-trol. It is also shown that the performance critically de-pend on the MCS. Insight is obtained into optimizing theMCS. Significant improvement in network performanceis observed with higher spatial diversity order and/orerasure insertion decoding. Effects of different Dopplerfrequency shifts due to mobility and different time-scaleof shadowing variation are also evaluated.

x 10-, 5 =4.3dB, fdT =0.05, q=512, NR=2, 2x 2 DUSTM (TJ), MCS5(16,1/4),Fast shadowingRI

35 fdTs=001 ,P=1fdTs=0.05, P=1fdTs=0.10,P=1

3- 1--V fdTs=0.10,P=5

25

0 0.5 1 1 5 2 2.5 3N0/q

Fig. 5. VAIE versus Ntfor different mobility and DFD feedbackq

length P (L = 16, KIL = 4 N = 2, a, = 4.3dB, fast shadowing)

x 10-3 (4 3dB, fdTs=0 05, q=512, 2x 2 DUSTM (TJ), MCS5(16,1/4), Fast shadowing45r

NR=2, ESTT ElNR=2, no ElNR=4' ESTT ElNR=4' no El35

WL] 2.5

22

1.5

051 d

0.5 1.5

N0/q

21 25

_

v wf_ ~~~~~~~~~~~~

Fig. 6. AIE versus I for ditterent receive antenna numbers andwith/without El (L 16 KL = , 4 3dB, fast shadowing,45

fdli 0%0)

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