A Low-Overhead Cooperative Retransmission Scheme for IR-UWB Networks

Abstract

The UWB unique properties such as fine ranging and immunity to small scale fading are utilized in order to exploit the multiuser diversity in UWB networks. The optimal cooperation strategies in the absence of control packet overhead are analyzed in the proactive and reactive settings. It is shown that the proposed method achieves a considerable diversity gain while minimizing the overhead of control packet exchange that is required for coordination among the relays.

1. Introduction

Due to the large bandwidth occupied by pulses, UWB signals are considered robust to small scale fading effects. In addition, UWB enables high accuracy ranging which can be used for the design of location-aware MAC and routing mechanisms. We exploit these properties of UWB, that is, availability of ranging information and immunity to small scale fading, in the design of an UWB-based Cooperative Retransmission Scheme (UCoRS) [1].

Most of the existing distributed relay selection schemes, such as [2–4] rely on the Priority-Based Backoff Timer (PBT) mechanism to discover which relay is the best one at a time instance by sending a flag message. We note that like PBT, the other existing mechanisms such as CMAC [5] also require the exchange of the RTS/CTS and other control messages for every transmission. These cooperative methods may be inefficient for UWB networks. This is because the standard IR-UWB MAC protocol is ALOHA [6] and the exchange of RTS/CTS packets is not required prior to the data transmission in UWB. Furthermore, it is preferred to exchange fewer control packets due to the complex and costly UWB receiving procedure.

2. System Model

Figure 1 shows the system model. As can be seen, there are a source 𝑆 and a destination 𝐷, and 𝑁 relays 𝑅_𝑖, 𝑖=1,2,…,𝑁, in a slotted time domain, and each time slot consists of 2 subslots. At the transmission subslot (Tx), the source node sends data to its destination. At the Cx subslot, the relays retransmit the source data. In particular, the ith relay, 𝑅_𝑖, decides to cooperate (i.e., retransmit the data) with probability 𝑎_𝑖.

(a) Network model

(b) Time slot model

Figure 1 

System model.

Since accurate ranging information is available through UWB physical layer, we also presume that when 𝑅_𝑖 finds its distance to 𝑆 and 𝐷, it broadcasts a packet to inform other nodes about these ranging information. Note that as long as the nodes do not move, the process of ranging and informing other nodes should be performed only once, which incurs much less overhead compared to sending control packets for every transmission.

The link success probabilities are denoted by 𝑃_𝑖 and 𝑄_𝑖, as can be seen in Figure 1. The success probability of the 𝑆-𝐷 link is denoted by 𝑃₀. To calculate these values, we note that in time-hopping pulse position modulation, TH-PPM, the transmitted signal by node 𝑖 is given by 𝑠^𝑖(𝑡)=∑^{∞𝑗=−∞}√𝐸_𝑝𝜔(𝑡−𝑗𝑇_𝑓−𝑐^𝑖𝑗𝑇_𝑐−𝛿𝑏^{𝑖⌊𝑗/𝑁}_𝑆⌋), where 𝐸_𝑝 is the transmission energy per pulse, 𝑇_𝑓 and 𝑇_𝑐 are the frame and chip durations, 𝑏^{𝑖⌊𝑗/𝑁}_𝑆⌋∈{0,1} is the information bit to be sent, 𝜔(𝑡) is the monocycle pulse, and 𝛿 determines the time shift in the chip when the data bit is 1. Each frame consists of 𝑁_ℎ chips, that is, 𝑇_𝑓=𝑁_ℎ𝑇_𝑐. Moreover, each bit is repeated in 𝑁_𝑆 frames with different time hopping codes, 𝑐^𝑖𝑗∈{0,1,…,𝑁_ℎ−1}, which results in additional (random) time shifts and hence increases the pulse immunity to interference.

The received signal from user 𝑖 at node 𝑗 is given by [7] 𝑟^𝑖𝑗(𝑡)=𝛼^𝑖𝑗∑^𝐶𝑐=1∑^𝐿𝑙=1𝛽_𝑐𝑙𝑠^𝑖(𝑡−𝜏_𝑐𝑙)+𝑛(𝑡), where 𝑛(𝑡) is AWGN with the power spectral density 𝑁₀/2, and 𝛼^𝑖𝑗 denotes the 𝑖-𝑗 link gain. Since UWB pulses are robust to small scale fading effects, we consider only the channel pathloss, as defined in [1, 6]. Then, the bit error probability (BEP) in the absence of interference can be approximated by 𝑃_be(𝑑_𝑖𝑗)=(1/2)erfc(√(𝛼^𝑖𝑗𝐸_𝑝𝑁_𝑆/2𝑁₀)(1−𝜌(𝛿))) [7], where 𝜌(𝛿) is the autocorrelation function of the monocycle pulse, 𝜔(𝑡). From the above-mentioned model, the probability that a packet with length 𝐿 bits is successfully transmitted can be represented as follows:𝑃_𝑠(𝑑)=1−1−𝑃_be(𝑑)^𝐿.(1)This equation can be used to determine the values of 𝑃_𝑖 and 𝑄_𝑖 as a function of the relays' distances to 𝑆 and 𝐷. Having obtained 𝑃_𝑖 and 𝑄_𝑖s from (1), the next problem is to find the cooperation probabilities 𝑎_𝑖 in order to maximize the 𝑆-𝐷 throughput.

3. Analysis

We assume that the packet level collision occurs if the signal strength of more than one packet is above the threshold at the receiver. Therefore, 𝐷 successfully receives a useful data packet if either the 𝑆-𝐷 transmission in the Tx subslot is successful, or the transmission from one and only one of the relays in the Cx subslot is successful.

We consider two different settings, namely, the proactive and reactive modes. In the proactive mode, the decision is made prior to the source transmission. In the reactive mode, all relays listen for the data first and then decide to cooperate. Note that since message exchange between relays is not performed in UCoRS, a relay 𝑅_𝑖 is unable to find out the set of relays which have successfully decoded the packet from 𝑆 at time slot 𝑡, denoted by 𝐹(𝑡). In fact, the global optimum of the relay selection problem would be obtained if 𝐹(𝑡) were available to the nodes.

In the proactive case, the expected success probability in a time slot is given by𝑈(𝐴)=𝑃₀+1−𝑃₀^𝑁_𝑖=1[𝑎_𝑖𝑃_𝑖𝑄_{𝑖𝑁𝑗=}_{1,𝑗≠𝑖}1−𝑎^𝑗𝑃_𝑗𝑄_𝑗].(2)In order to find the optimal solution of (2), we use Lemma 1 in the Appendix. The following theorem gives the optimal solution.

Theorem 1. Consider a cooperative network with one 𝑆-𝐷 pair and 𝑁 relays. The optimal cooperation strategy to maximize the 𝑆-𝐷 throughput (𝑈(𝐴) in (2)) is 𝐴^(𝐾)={𝑎_𝑖=1,𝑖≤𝐾;𝑎_𝑖=0,𝑖>𝐾}, where 𝐾 satisfies: ∑^{𝐾−1𝑖=1}(𝑃_𝑖𝑄_𝑖/(1−𝑃_𝑖𝑄_𝑖))<1, and ∑^𝐾𝑖=1(𝑃_𝑖𝑄_𝑖/(1−𝑃_𝑖𝑄_𝑖))≥1, where relays are sorted in descending order according to the values of 𝑃_𝑖𝑄_𝑖 (i.e., 𝑖≤𝑗⇔𝑃_𝑖𝑄_𝑖≥𝑃_𝑗𝑄_𝑗); and 𝐴^(𝐾) denotes a binary vector whose first Kth elements are 1. (Proof is straightforward from Lemma 1.)

Here, we mention that if 𝑃₁𝑄₁≥0.5, then 𝐾=1, and only 𝑅₁ will be active. In this special case, the result is in agreement with [3]. The reactive and global optimum cooperation strategies can be derived using the same reasoning as Theorem 1, as discussed in detail in [1].

4. Performance Evaluation

Figure 2 compares the packet delivery ratio (PDR) for different scenarios. As can be seen, the proactive performance is near to the maximum achievable throughput. Furthermore, as expected, both reactive and proactive methods outperform the noncooperative case.

Figure 2 

Comparison of packet delivery ratio (PDR) for different schemes.

Figure 3(a) shows the effect of increasing the number of relays on the achieved PDR in UCoRS for different 𝑆-𝐷 link qualities. As can be seen, adding one relay can significantly increase the PDR of the direct link. However, the achieved PDR in UCoRS is upper bounded by a function of 𝑑_𝑆𝐷, regardless of number of available relays.

(a) Effect of increasing number of relays on PDR

(b) Asymptotic throughput achievable by UCoRS and PBT as 𝑁→∞

Figure 3 

Simulation results.

Figure 3(b) shows the asymptotic achievable throughput of UCoRS, PBT, and noncooperative schemes as a function of 𝑃₀ when 𝑁→∞. As stated previously, the throughput advantage of PBT over UCoRS is at the expense of control packet exchange for every data transmission, which may not be efficient in UWB. More details can be found in [1].

5. Conclusion

We introduced UCoRS, a simple UWB-based Cooperative Retransmission Scheme, that utilizes the unique properties of IR-UWB technology for achieving multiuser diversity in UWB in the proactive and reactive settings. The amount of control packet overhead is minimized in UCoRS in order to eliminate the corresponding energy cost at the UWB receivers.

Appendix

Lemma 1. Assume a set of variables 𝑍={𝑧_𝑖},𝑖=1,2,…,𝑛, that can take on real values between 0 and 1>𝑚₁≥𝑚₂≥⋯≥𝑚_𝑛, respectively. Then, the maximum value of 𝑋(𝑍)=∑^𝑛𝑖=1(𝑧_𝑖)∏_𝑗≠𝑖(1−𝑧_𝑗) is obtained when 𝑧_𝑖=𝑚_𝑖, 𝑖≤𝐾, and 𝑧_𝑖=0, 𝑖>𝐾, where 𝐾 satisfies ∑^𝐾𝑖=1(𝑚_𝑖/(1−𝑚_𝑖))≥1, and ∑^{𝐾−1𝑖=1}(𝑚_𝑖/(1−𝑚_𝑖))<1.

Proof. Taking the partial derivative of 𝑋(𝑍), we have𝜕𝑋(𝑍)𝜕𝑧_𝑖=_𝑗≠𝑖1−𝑧^𝑗−_𝑗≠𝑖(𝑧_𝑗=_𝑗≠𝑖_{𝑘≠𝑖,𝑗}1−𝑧^𝑘)1−𝑧^𝑗(1−_𝑗≠𝑖𝑧_𝑗1−𝑧_𝑗).(A.1)
Therefore, 𝜕𝑋(𝑍)/𝜕𝑧_𝑖>0⇔∑^{𝑛𝑗=1,𝑗≠𝑖}(𝑧_𝑗/(1−𝑧_𝑗))<1, and 𝜕𝑋(𝑍)/𝜕𝑧_𝑖>𝜕𝑋(𝑍)/𝜕𝑧_𝑗⇔𝑧_𝑖>𝑧_𝑗. According to these two results, in order to maximize 𝑋(𝑍), the 𝐾 “best” variables (with looser bounds) should be set to their maximum values and other variables should be set to 0. The required conditions on 𝐾 are also clearly observed from the above-mentioned equations. Note that if 𝑚₁≥0.5, then 𝐾=1.

Acknowledgments

An earlier version of paper [1] has won the Best Student Paper award in the IEEE International Conference on Ultra-Wideband (ICUWB), 10-12 September 2008, Germany. This work is done under the USCAM-CQ project which is a part of the Ultra Wide Band-enabled Sentient Computing (UWB-SC) Research Program funded by Science and Engineering Research Council (SERC), A^*STAR, Singapore.