International Symposium on Information Theory and its pplications, ISIT2004 Parma, Italy, October 10 13, 2004 Spectral Efficiency-Connectivity Tradeoff in d Hoc Wireless Networks Gianluigi FERRRI,, Bernardo BRUFFINI, and Ozan K. TONGUZ Dipartimento Ingegneria Informazione Università di Parma, 43100 Parma, ITLY E-mail: gianluigi.ferrari@unipr.it, URL: http://www.tlc.unipr.it/ferrari Dept. of Electrical and Computer Engineering Carnegie Mellon University Pittsburgh, P 15213-3890, US E-mail: tonguz@ece.cmu.edu URL: http://www.ece.cmu.edu/ tonguz bstract In this paper, reservation-based ad hoc wireless networks are investigated. The performance of the considered ad hoc wireless networking schemes is evaluated in terms of effective transport capacity, which represents the actual bandwidth-distance product carried by the network. The maximum of this quantity, for any network traffic load and transmission data-rate, is the transport capacity. The focus of this paper is on the analysis of the impact of the modulation format on the network performance, especially in terms of the relationship between spectral efficiency and connectivity. We evaluate the performance offered by three different reservation-based medium access control (MC) protocols in a realistic scenario affected by inter-node interference (INI). comparison with the performance obtained in an ideal (no INI) scenario provides valuable insights into the pros and cons of the proposed MC protocols. 1. Introduction d hoc wireless networks represent a new communication paradigm which will have ever-increasing applications in future wireless communication systems [1]. Several parameters affect the performance of this type of networks, such as transmission delay and average life-time of a node. The concept of transport capacity has been introduced in [2], from an information-theoretic perspective, to quantify the maximum bandwidth-distance product which can be supported by the network. In this sense, the transport capacity represents a very good, albeit not comprehensive, indicator of the system s potential to support information transfer across the network. In [3,4], a new communication-theoretic approach to the analysis of ad hoc wireless networks has been proposed. In particular, in [3] the concept of effective transport capacity has been introduced to quantify the actual bandwidthdistance product supported by the network. In [3], the evaluation of the effective transport capacity is carried out both for (i) an ideal network communication scenario, i.e., not affected by inter-node interference (INI) and for (ii) a realistic (with INI) network communication scenario. In the INI case, the following two MC protocols are considered. The first MC protocol, defined as reserve-and-go (RESGO), is such that a node, after reserving a route to its destination, transmits without sensing the channel. 1 The second MC protocol, defined as reserve-listenand-go (RESLIGO) is characterized by the fact that a node, after reserving a multi-hop route to its destination, senses the channel before transmitting: if no transmission is going on, then the node starts transmitting. 2 2. Motivation In the last years, wireless communication systems have experienced a continuously growing need to transport large quantities of data. Hence, it is reasonable to expect that future ad hoc wireless networks will follow the same trend. Use of spectrally efficient modulations will therefore play an important role. While the impact of the modulation formats is well-understood for single hop point-to-point communications, their impact in the case of multi-hop wireless network communications is not. The analysis in [3] is limited to binary modulations. In this paper, we investigate, in a systematic manner, the impact of the modulation format on the effective transport capacity. It will be shown that there exists a clear trade-off between spectral efficiency and connectivity. In particular, in an ideal case, i.e., without INI, use of high-order modulations can increase the effective transport capacity, but connectivity is preserved only for small bandwidth values. novel MC protocol, which is an extension of RESGO MC protocol and exploits the spectral 1 This MC protocol was referred to in [3] as loha MC protocol, for its resemblance, in terms of route activation independent from the activity of other nodes in the network, to the classical loha MC protocol [5]. However, there are significant differences which make the proposed protocol different from the classical loha MC protocol: (i) multi-hop route reservation; and (ii) no use of retransmission techniques. 2 This MC protocol was referred to in [3] as per-route carrier sense multiple access (PR-CSM) MC protocol, for its resemblance, in terms of route activation after sensing, to the classic CSM MC protocol [6].
efficiency of high-order modulations to reduce the INI, will be proposed. 3. Network Communication Model The considered ad hoc wireless networking scenario can be characterized as follows. 3 The route creation phase is not explicitly considered, and we assume that it is perfectly accomplished. lthough simulation results show that this phase is the most troublesome in ad hoc wireless networking [1], the focus of this paper is on performance analysis from a communication-theoretic perspective. Peer-to-peer multi-hop communication with disjoint routes is considered. This means that a node can not act as a relay node in more than one route simultaneously. We assume that N nodes are placed uniformly, at the vertices of a square grid, in a circular area. It can be easily shown that two neighboring nodes are at distance r L 1/ ρ S, where ρ S N/ (units: m 2 ) is the node spatial density [3]. Each node transmits a fixed power and multi-hop routes are constituted by a sequence of minimum length links. No buffer is considered at the nodes. In other words, if a node needs to communicate with another node, it must first reserve a multi-hop route and it can then transmit. Packet generation/transmission at each node is described by a Poisson process with parameter λ (dimension: [pck/s]). The generated packets have a fixed length of L bits. Denoting by R b the transmission data-rate of the nodes, a necessary condition that needs to be satisfied for network communications is λl R b. No retransmission mechanism, such as automatic repeat request (RQ), is used. This assumption is valid for real-time communications and networking scenarios where delay is important and/or battery power of nodes is limited. It is straightforward, however, to extend the framework in this paper to scenarios where such assumptions are relaxed and RQ techniques are used. 3 Some of the assumptions indicated in Section 3 are not realistic. However, they allow to isolate and accurately take into account communicationtheoretic aspects of ad hoc wireless networking. 4. Communication-Theoretic Preliminaries Indicating by BER L the BER at the end of a single link, and assuming that (i) there is regeneration at each intermediate node and (ii) the errors at the end of a link are not corrected in the following links (this is a reasonable assumption at large link signal-to-noise ratio, SNR), it is possible to show that the BER at the end of an n-hop route can be written as BER (n) 1 (1 BER L ) n. (1) The value of BER L is related to the link SNR through a function which depends on the modulation format and channel model. For all the modulation formats considered in the remainder of this paper, closed form expressions of the link BER can be found in the literature [7, 8]. Uncoded transmission will be considered. simple approach to obtain an average route BER consists in evaluating the route BER in (1) in correspondence to an average number of hops. It is possible to show that the average number of hops is proportional to N. In particular, for the given circular topology, one can show that the average number of hops can be written as n h = N/π, where indicates the integer value closest to the argument [3]. The average BER can therefore be written as BER BER (n h) 1 (1 BER L ) N/π. (2) ccording to the proposed reservation-based network communication model, it is intuitive to associate to each multi-hop route a per-route effective transport capacity, given by the product of the average information generation rate at the source node of the route and the route length. ccording to our assumption of disjoint multi-hop routes, the effective transport capacity is obtained by simply adding the per-route transport capacities and can be given the following expression [3]: C T,e = }{{} λl n sh r L N }{{} ar (3) b/s m where N ar Nπ is the maximum (average) number of disjoint routes (each route is formed, on average, by n h = N/π hops) and n sh is the average number of sustainable hops [3]:, defined as { ln(1 BER max } ) n sh min, n h. (4) ln(1 BER L ) t this point, one can pursue the proposed derivation for two cases: 1. ideal case: the nodes do not interfere with each other (for example, all disjoint communication routes use perfectly orthogonal spreading codes);
2. realistic case: the nodes interfere with each other. These two different scenarios will be analyzed in the following, by suitably expressing the link SNR on which the link BER depends. 4.1. Ideal Case In the ideal case of INI absence, the link SNR, defined as the ratio between the energy per information bit and the noise power spectral density, is independent of the modulation format. ssuming free space propagation loss, the link SNR can be written as follows: SNR ideal L = αρ s F kt 0 R b (5) where F is the noise figure, k = 1, 38 10 23 J/K is the Boltzmann s constant, T 0 300 K is the room temperature, and α (G t G r c 2 )/[(4π) 2 f l fc 2 ], G t and G r are the transmitter and receiver antenna gains; f c is the carrier frequency; c 3 10 8 m/s is the speed of light; and f l 1 is a loss factor. We remark that all the results presented in this paper can be extended to other propagation channel models, provided that the model is properly taken into account in the link SNR expression. 4.2. Realistic Case In a realistic (INI) scenario, assuming that the interference noise is independent from the thermal noise, the total noise power spectral density can be written as follows: N = N 0 + I (units : [W/Hz]) (6) where I is the power spectral density of the interference noise, which is assumed to be constant over the transmission band this is equivalent to treating the INI as additive white noise. In other words, we will assume that I = P INT B eq (units : [W/Hz]) (7) where B eq is the equivalent receiver bandwidth (which is equal to the transmission band for all considered modulation formats 4 ) and P INT is the interference power, which depends on the MC protocol and the spatial node distribution, but not on the modulation format. The link SNR can therefore be written as follows: αρ s SNR L = (8) F kt 0 R b + P INT η where η R b /B T (units: [b/s/hz]) is the spectral efficiency and B T is the transmission bandwidth. Obviously, the higher the spectral efficiency of the modulation format, the larger the impact of the INI (η acts as an amplification factor for the INI power). 4 The reader should observe that the equivalent bandwidth depends on the particular modulation format, and there are some formats for which the equivalent bandwidth is not equal to the transmission bandwidth. SNR L 80 60 40 20 =6x10-7 W λ=10 pck/s L=10 b/pck N=5000 0 N=15000 N=20000 0 10 3 10 4 10 5 10 6 R b [b/s] Figure 1: SNR L versus R b, for binary modulation formats with B T = R b. Several values of N are considered. 5. nalysis of the Link SNR In order to better understand the behavior of the effective transport capacity, it is useful to first evaluate the dependence of the link SNR on the fundamental network parameters. Since in the case with RESLIGO MC protocol the interference power is basically zero [3], we will limit the discussion to the case of RESGO MC protocol. The proposed analysis can be extended in a straightforward fashion to the case of RESCHOGO MC protocol, which will be described in more detail in the following section. In the case of RESGO MC protocol, it is possible to show (based on a worst-case scenario analysis [3]) that ( ) PINT RESGO α ρ S 1 e λl R b (N) (9) where (N) N/2 i=1 6 i 1 i 2 + 8 j=1 1 i 2 + j 2 1. (10) Given the expression of the interference power (9), we first analyze the dependence of SNR L on the data-rate R b. The results, in the case of binary modulations with B T = R b, are shown in Figure 1. The major network parameters are set as indicated in the figure: in particular, the area is fixed. s one can see, there is an optimal value of R b which maximizes the link SNR. In fact: for low values of R b, the interference noise dominates and the link SNR is very low; for intermediate values of R b, any SNR L curve reaches its maximum, which corresponds to larger values of R b for an increasing number of nodes; 5 for large values of R b, the thermal noise dominates, and the link SNR tends to zero. 5 Note that, since the area is fixed, this corresponds to a higher node spatial density.
60 SNR L 50 40 30 20 10 =6x10-7 W λ=10 pck/s L=10 b/pck 0 M=2 M=4 M=8 M=16 M=64 0 10 3 10 4 10 5 10 6 R b [b/s] max C T,e [bit-m/s] 10 9 10 8 10 7 =10-7 W =10 6 m 2 BPSK DBPSK QPSK Gray MSK 4-PM Gray 8-PSK 16-QM 16-PSK 64-QM Figure 2: SNR L versus R b, for modulation formats with η = R b /B T = log 2 M. Several values of M are considered. t this point, it is interesting to evaluate the behavior of the link SNR as a function of the spectral efficiency. ssuming that η = R B /B T = log 2 M, where M is the cardinality of the transmitted symbols (this assumption is valid for all considered modulation formats), Figure 2 shows the dependence of SNR L on R b for various values of the parameter M. ccording to the observations made in the previous paragraph, the interference plays a crucial role for low values of R b. This is confirmed in Figure 2, where the dependence on M is very pronounced for low/intermediate values of R b, whereas it reduces significantly for large values of R b. 6. Effective Transport Capacity fter analyzing the behavior of the link SNR as a function of the number of nodes and the spectral efficiency, we now evaluate the effective transport capacity in the ideal case and in a realistic case with RESGO MC protocol. Finally, a comparative analysis will be considered. 6.1. Ideal Case We limit our analysis to the case of an additive white Gaussian noise (WGN) channel. The extension to the case of fading channels is straightforward, provided that the correct link BER expressions are considered [8]. In Figure 3, the maximum effective transport capacity is shown as a function of the transmission bandwidth B T, considering the indicated modulation formats. For each value of B T, the corresponding value of the effective transport capacity is obtained by maximizing over the per-node offered load λl in [3], it is shown that the largest effective transport capacity is obtained by fixing λl = R b. For low values of B T, the largest data-rate R b = ηb T is supported by high-order modulation formats (e.g., 64-ary quadrature amplitude modulation, 64-QM or 16-ary phase shift keying, 16-PSK), and the corresponding maximum effective trans- 10 2 10 3 10 4 10 5 10 6 B T [Hz] Figure 3: Comparison, in terms of the maximum effective transport capacity versus B T, between several modulation formats in an ideal (no INI) network communication scenario. port capacity curve is the highest. However, for increasing values of B T connectivity is lost earlier when using spectrally efficient modulations. 6.2. Realistic Case: RESGO MC Protocol We first analyze the performance with RESGO MC protocol, highlighting the impact, on the effective transport capacity, of the communication channel. In particular, we consider the case of transmission over an WGN channel and the case of transmission over a Rayleigh fading channel. 6 6.2.1. WGN Channel In Figure 4, the behavior of the effective transport capacity (maximized with respect to λl), as a function of B T, is shown for various modulation formats. s opposed to the ideal case, an energy efficient modulation format (e.g., binary phase shift keying, BPSK) guarantees highest effective transport capacity at all values of B T. From equation (8), it is clear that for low values of R b (and, therefore, B T ) the larger noise component is the interference noise. For low values of B T, C T,e increases linearly with B T. This can be explained as follows: once B T is fixed, R b is also fixed (for a given spectral efficiency) and P INT depends on λl/r b. In other words, if B T increases linearly, R b has the same trend and it is possible to increase λl by the same quantity still preserving connectivity. For higher values of B T, the larger noise term is the thermal noise, so that it is not possible to increase λl 6 The expressions of the link SNR for the various modulation formats can be found in [7, 8]. They are not reported for lack of space.
max C T,e RESGO [b-m/s] 10 8 10 7 10 6 10 5 10 4 10 3 10 2 10 1 =6x10-7 W f c =2,4 GHz 10 0 10 0 10 2 10 4 10 6 10 8 B T [Hz] BPSK DBPSK QPSK Gray MSK 4-PM Gray 8-PSK 16-QM 16-PSK 64-QM C T,e [b-m/s] 3.0 10 7 2.5 10 7 2.0 10 7 1.5 10 7 1.0 10 7 5.0 10 6 λl=200 Ideal Case =10 6 m 2 =6x10-6 W R b =5x10 6 b/s f c =2.4 GHz λl=100 RESGO 0.0 0 5000 10000 15000 20000 N Figure 4: C T,e (maximized with respect to λl) versus B for several modulation formats and RESGO MC protocol. further and maintain the network fully connected. Note that there is a saddle point in each curve in Figure 4 after the corresponding maximum. Our analysis shows that for values of B T above that corresponding to the saddle point there is loss of connectivity. This is due to the fact that for large values of B T, very low values of λl are required in order for the network to be fully connected, and this leads to values of C T,e which are lower than the maximum. The indicated behavior shows also that around (both before and after) the maximum there is connectivity, i.e., the network is stable if one operates in that regime. In order to better understand the impact of the interference, in Figure 5 the effective transport capacity is analyzed as a function of the number of nodes N (for fixed area ), for two values of per-node traffic load: λl = 100 b/s and λl = 200 b/s, respectively. The data-rate is the same in both cases (R b = 5 10 6 b/s). The considered modulation format is 64-QM. For comparison, the behavior in the ideal case is also shown. major difference can be immediately noticed, comparing ideal and realistic cases: in the ideal case, an increase of the per-node traffic load λl does not change the value of the connectivity threshold; in a realistic case, an increase of λl (for fixed datarate R b ) leads to an increase of the interference power, and this leads to an increase of the connectivity threshold, in terms of the number of nodes N. 6.2.2. Rayleigh Fading Channel The analysis considered for transmission over an WGN channel can be extended straightforwardly to the case of a fading channel. In Figure 6, the behavior of the effective transport capacity is shown as a function of the datarate R b. s expected from the behavior of the link SNR Figure 5: Effective transport capacity versus N, in the case with 64-QM modulation format. The transmission channel is WGN. analyzed in Section 5, the effective transport capacity falls to zero for very low or very large values of R b. The modulation format which guarantees highest effective transport capacity for the largest R b interval is BPSK. 7 Observe that with the chosen network parameters, use of spectrally efficient modulation formats, such as 16-PSK or 64-QM, do not guarantee full connectivity for any value of R b. 6.3. RESCHOGO MC Protocol: the Idea In this subsection, we describe RESCHOGO MC protocol. This protocol is derived from RESGO, and its key idea consists in trading spectral efficiency with reduction of INI, rather than to increase the data-rate. Suppose that (i) all nodes transmit at fixed data-rate R b, (ii) they use the same modulation format (with spectral efficiency η) and (iii) there is a fixed bandwidth B R b available in the network. The transmission bandwidth is B T = R b /η. ssuming, for simplicity, that B = R b, it follows that for increasing spectral efficiency the transmission bandwidth B T becomes a smaller portion of the total bandwidth B. Hence, the overall network bandwidth could be divided in slots of width B/η, and a node could choose a specific slot to transmit. In the case of uniform distribution 8 of the slots among the nodes, one can assume that the interference noise power is reduced by a factor η, so that the amplification effect of the interference noise power, due to the spectral efficiency, disappears. This MC protocol will therefore be referred to as reserve-choose-and-go (RESCHOGO). 6.4. Comparative nalysis 7 lthough QPSK with Gray coding has the same energy efficiency, it is more sensitive to the interference because of its higher spectral efficiency. 8 In order for the slot assignment to be uniform over the nodes, a node could sense the channel and choose the slot where the received power is lowest.
C T,e RESGO [b-m/s] 1500 1000 500 0 =6x10-5 W N=15000 λl=0,01 b/s BPSK DBPSK FSK QPSK 8-PSK 16-PSK 64-QM 10 2 10 4 10 6 10 8 R b [b/s] max C T,e [b-m/s] 10 10 10 8 10 6 10 4 10 2 Ideal RESGO RESCHOGO RESLIGO =6x10-7 W =10 6 m 2 10 0 10 0 10 1 10 2 10 3 10 4 10 5 10 6 λl [b/s] Figure 6: CT,e RESGO versus R b for several modulation formats. The transmission channel is affected by Rayleigh fading. comparison among the considered MC protocols, in terms of effective transport capacity versus traffic load, is shown in Figure 7 in the case with 64-QM. s one can see, RESCHOGO MC protocol improves the performance of RESGO MC protocol, in the sense that the obtained effective transport capacity coincides with that of the ideal case even for larger values of the traffic load. However, as already observed in [3], RESLIGO MC protocol guarantees the overall highest effective transport capacity, even with respect to RESCHOGO MC protocol. In the case with fading, a behavior similar to that in Figure 7 is observed. 7. Conclusions In this paper, we have investigated the relationship between spectral efficiency and connectivity in reservationbased ad hoc wireless networks. systematic analysis of various modulation formats has highlighted characteristic behaviors in the cases with and without INI. In the ideal case, for low transmission bandwidths high-order modulations lead to higher effective transport capacity. However, the price paid for high spectral efficiency is an early loss of connectivity for relatively low transmission bandwidth (i.e., datarate). In a realistic (INI) case with RESGO MC protocol, energy efficient modulation formats are to be preferred for all B T values. novel MC protocol, defined as RESCHOGO, has also been proposed for realistic (INI) scenarios where spectrally efficient modulations are considered. Our results show that RESCHOGO MC protocol outperforms RESGO MC protocol: the key idea of the former MC protocol is to exploit the spectral efficiency in order to reduce the INI. Figure 7: Comparison, in terms of the maximum (with respect to R b ) effective transport capacity, versus traffic load λl for a 64-QM modulation using several MC protocols. In the figure, the maximum effective transport capacity in the ideal case is also shown. References [1] C. E. Perkins, d hoc Networking. Upper Saddle River, NJ, US: ddison-wesley, 2001. [2] P. Gupta and P. R. Kumar, The capacity of wireless networks, IEEE Trans. Inform. Theory, vol. 46, pp. 388 404, March 2000. [3] G. Ferrari and O. K. Tonguz, MC protocols and transport capacity in ad hoc wireless networks: loha versus PR-CSM, in Proc. IEEE Military Comm. Conf. (MILCOM), Boston, US, October 2003. [4] O. K. Tonguz and G. Ferrari, d Hoc Wireless Networks: Communication-Theoretic Perspective. John Wiley & Sons, 2005, to be published. [5] N. bramson, The throughput of packet broadcasting channels, IEEE Trans. Commun., vol. 25, pp. 117 128, January 1977. [6] L. Kleinrock and F.. Tobagi, Packet switching in radio channels: Part I Carrier sense multipleaccess modes and their throughput-delay characteristics, IEEE Trans. Commun., vol. 23, no. 12, pp. 1400 1416, December 1975. [7] J. G. Proakis, Digital Communications, 4th Edition. New York: McGraw-Hill, 2001. [8] M. K. Simon and M.-S. louini, Digital Communication over Generalized Fading Channels: a Unified pproach to the Performance nalysis. John Wiley & Sons, 2000.