Closed Form Capacity Expression Of Multiuser Diversity Computer Science Essay

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Abstract-Diversity is a good way to mitigate fading penalty on modulation performance. cooperative diversity "COD" is one method where the user receives many versions of the same signal from the direct link and relayed links. Another form of diversity is multiuser diversity "MUD" where it operate in the MAC layer instead of the physical layer. The gain obtained by this diversity is a result of using a certain scheduling mechanism. COD and MUD can be combined to gain more system capacity. however, this made mathematical derivation more complicated. We proposed to use Weibull distribution as an approximation to the COD link signal to noise ratio PDF in case Nakagami-m fading environment is assumed. Then we plugged in this approximation into MUD mathematical derivation to obtain system average ergodic capacity. We studied a cooperative system where the user receives two-version of the signal, one from the direct link and another from a dual-hop amplify-and-forward semi-blind relay. We found that the deduced capacity equation did fit simulation result with a maximum error not more than 5%.

Keywords-component; Multiuser diversity; Cooperative diversity; Nakagami; capacity


Fading induces a very large power penalty on the performance of modulation over wireless channels. Diversity techniques are well-known concepts to mitigate signal fading due to multi-path propagation. Diversity works by improving the reliability of packets delivery by employing two or more communication channels with different characteristics in that delivery, utilizing the fact that independent signal paths have a low probability of experiencing deep fades simultaneously.

In point-to-point communication links, reliability can be enhanced directly using time, frequency and spatial diversity. There is a different trade-offs between realization cost and spectral efficiency among these schemes. spatial diversity (multiple Antenna or cooperative) has drawn considerable attention recently as it can be easily combined with other types of diversity to obtain higher diversity orders. Multiple Antennas diversity requires a number of antennas either on the receiver side or the transmitter side to create an antenna array. The physical limitation on users' handset and the requirements of spatial distances between antennas increase the difficulty of employing such type of diversity [1],[2]. In the cooperative diversity general form, Instead of multiple antennas deployed at the same user handset, users can share their individual antenna in one effort to create "virtual arrays" [3].

Users are required to relay whatever message received to other users in these "virtual arrays". This requires users to chip-in from their own handset resources to participate in it. So, instead of using users handset to relay the message, a central relay station is used to relay the signal to other users, thus an individual user would receive two version of the same packet. Relay adds more robustness and coverage to individual links used in cooperative diversity. Performance and cost changes according to the what relay scheme the system is using [3]. This paper mostly focus on downlink dual-hop cooperative network. Where a user would receive the packet directly from the base station and another version after it hop twice, one hop from the source to the relay station and another one from the relay station to the target user. All the above was considering diversity on the physical layer of the network where we exploit the different characteristic between different paths to our advantage.

We have another form of diversity that is not on the physical layer, but in the media-access control (MAC) layer. In point-to-multipoint communication almost all users would fade independently. Thus if we try to send the packet to the user with the best signal condition at the moment of transmission, we would maximize the overall system throughput eventually as we will maximize the chance of selecting higher modulation schemes every time a packet is sent. The gain obtained over all would be described as "multiuser diversity (MUD)" [4]. And as this mechanism would always try to serve the best user, some users may suffer delays while other may never have the chance to be served at all. Thus, Different scheduling mechanism would cause different trade-offs between system throughput, delay and fairness. So, in comparison to the relay cooperative diversity working on the physical layer, scheduling is a MAC layer technique [5]. This type of diversity uses the same principle of water-filling, but across users instead of time. Water-filling gives more resources when channel condition is more favorable [1]. Similarly multiuser diversity gives more resources to the user who has the most favorable channel condition. MUD became a building block on some recent multiuser access technology such as WiMAX to the extent that "adaptive subscriber allocation" in OFDMA system was motivated by MUD [4]. In general there are 4 major types of scheduling [6]‎[7].

a. Round-Robin scheme RR:

In this scheme, the time of transmission is divided equally among the users. This division is regardless of user SNR. This provides the best fairness situation, but in the expense of a questionable overall cell capacity cost, as not all users do have the same SNR. This scheme would not provide any MUD gain.

b. Max-SNR scheme "MS":

This scheme chooses the user with the highest SNR to serve. This can give the highest overall cell capacity, in the expense of fairness among users as some user may never have the chance to be served at all.

c. Weighted-SNR scheme "WS" or Proportional Fairness "PF":

This scheme chose the user with the best signal behavior. So, it will choose the user with the highest SNR-to-mean-SNR ratio. This scheme gives a reasonable capacity with a reasonable proportional fairness; this is why it is called proportional fairness scheme PF and it is widely deployed in commercial networks.

d. Hybrid scheme "HS":

This method exploits the advantages of above schemes in a controlled balance between throughput and fairness. It is based on grouping users in a tournament fashion and then using a two-step selection process to chose either to apply one scheme or another.

In systems where cooperative diversity is combined with MUD, mathematical derivation of system capacity can be a an irresolvable problem. This due to that fact that one would apply certain mathematical derivation on the physical layer that would be injected as an input to another mathematical derivation at the MAC layer. This can get more and more complicated if Nakagami fading model is assumed. Hee-jin et al work resorted to approximate the SNR PDF of point-to-point communication under downlink dual-hop cooperative single relay scheme to Gamma PDF. Then the resultant approximated SNR PDF is used as an input to the MUD MS scheme study. The result would be an approximated SNR PDF of the whole system. Then it can be easily used in Shannon theory to get system capacity. But Hee-jin et al assumed Rayleigh fading environment and they did not explore other approximation options. In this paper, we would assume Nakagami-m fading environment using Weibull PDF as an approximation option in a way to get a closed-from expression of system capacity employing MUD and COD in the same system. The system would consist of single-link dual-hop semi-blind relays and it will use PF MUD as a scheduling method.

H. Shin et al derived the "moment-generating function (MGF)" of a COD dual-hop link "signal-to-noise (SNR)". They assumed that relay is using semi-blind "amplify-and-forward (AF)" gain. This scheme of relay is more practical as it does not require a continues CSI information at the receiver side. It rather take the average of the CSI reported info as a way to obtain the gain required to relay the message.

System model

We consider a downlink of a single-cell wireless system with K users in a frequency flat Nakagami block fading channel. We assume a single semi-blind AF relay that assists the transmission of one data packet from the source node "S" to the destination node "D". One packet would occupies two time slots.

Link model

We consider a dual-hop cooperative diversity system with a source, an AF relay using single antenna for all elements as in Fig. 1. At first time slot the source transmits to the destination and to the relay. And in the second time slot the relay will amplify and retransmit to the destination the signal received at the previous time slot. The signals received by the nodes are:

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Where and denote the unit energy transmitted signal and the signal received at the destination and the relay in the first time slot, and the signal received by the destination in the second time slot respectively. and , are channels coefficient of source-relay, source-destination and relay-destination respectively presented in (2). These coefficients are assumed to be Nakagami-m distributed random variable with the fading severity parameter ≥ .5 and average fading power , i.e. . In additionand are the average energy transmitted at the source and relay. and , are zero-mean additive white Gaussian noises in the corresponding channels having the same variance, i.e. . represents the gain used by the relay where . Hence the SNR received by the destination on the relay link is where and [8],[9].

Combining Output

If we assume a "maximal ratio combiner (MRC)" , then the combined output SNR is the simple addition of the SNR on the direct branch and the SNR on the relay branches in as in (2).

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System Model

Multiuser Diversity

As there areusers, then there should be a mechanism to choose to which user the data should be sent. This mechanism must be a tradeoff among the total capacity achieved, fairness among users and delay. The increase in capacity is a result of exploiting the instantaneous SNR differences among users at a giving time. The combined instantaneous SNR of the links selected using WS scheduler will be denoted as. The distribution of this SNR varies with the type of scheduling scheme used, if we assume a fixed link model.

Approximation of the SNR distribution

We will assume that the approximation of PDF presented in (2) follow Weibull distribution and then use that assumption to get the system SNR with multiuser scheduling that will result in a closed form capacity formula for the system. So, the resultant has a PDF and CDF of (3).

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Or more simply as , where c > 0 is the shape parameter and b >0 is the scale parameter of the distribution and,

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From the MGF obtained from [9] we can get the mean and variance as in (5)

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We can solve (4) and (5) simultaneously using numerical methods to get the Weibull distribution coefficient b and c. In WS multiuser diversity the system decide to send the packet to the th link with the highest SNR-to-mean-SNR ratio. So, we can model the chosen user as follow:


To simplify the study we would introduce in (7)


whererepresents the SNR-to-mean-SNR ration of the th user. And using identity (8) we get (9)

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We would also defineto be the combined of the scheduled links over time. can be computed using order statistics as follows [10]:

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And using (9) and (10) we would get:

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As our interest is the combined SNR distribution of the chosen links we can see that from (12), we would multiply the combined ratio by the combined means of the scheduled links to get :

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As we are studying a proportional fair system and for the purpose of simplicity we would assume that all links have the same characteristics. This means that all candidates have equal access time and they all share the same average SNR . This result that , and this can simplify (12) to:

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And would have a distribution of the following:

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The PDF of is then obtained using (13) and (14) with the identity (8), as follow:

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The average channel capacity is represented in (16) [10]. As two time slots are used to send one packet, we will divide the capacity by 2, hence:

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We would substitute (11) in (15) and use (16) to get:

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Simulated average system capacity with the theoretical average capacity using (17) where.

Relative error on capacity between the simulation and the calculated value in percentage against number of links K when .

Relative error on capacity between the simulation and the calculated value in percentage against variations where .

Simulation results

We assumed that the average fading Power on the direct link to be 10 and that at each link and. We average the transmission rate over 10,000 channel realization. The number of links range from 1 to 1694, andrange from 0dB to 20dB. We also assumed that the average channel gain including shadowing effect is the same among the users with mean of 10. We assumed also that the Nakagami coefficient of all the links is same and equaland . The simulated average capacity of this system along with the calculated average capacity using (17) is presented in Fig. 2. The Relative error between the simulation and the calculated value in percentage is shown in Fig 3. We can see that overall approximation using Weibull do fit the simulation to a maximum relative error of 5% at = 0 dB and the number of links is 1694. It is also observed that as increase above 5 db, error start to decrease to virtually zero over all the 1694 links. This is due to the diminishing role the direct link do play in case of large . Also, for a particular value, the error is directly proportional to the number of links. for that reason, Fig 4. concentrate on the relative error against delta when the number of links is limited to 400 links. We can see that the error reaches its peak when is about 4 dB, and flat out beyond 20 dB. This is also seen on Fig 4. as the error on =3 dB is higher than the error on =0 dB.

Summary and conclusions

Cross layer diversity has great advantages in reducing some physical requirement by method of scheduling. This cause more complexity in theoretical analysis. The Nakagami-m environment can add more complexity as it deals with more general cases rather than the conventional Rayleigh studies. As we are interested in system capacity, we are no longer working with a specific modulation or coding scheme, thus we need to work directly with the SNR of the signal regardless. We showed that the resultant SNR can be approximated as a Weibull distribution. The coefficients of this distribution can be deduced from the MGF derived by [9]. This would allow an approximate closed form solution to the capacity calculation. Overall, Weibull distribution gave good approximation to the problem in hand. It is also observed that as increase above 5 db, error start to decrease to virtually zero.