Advances in Statistical Channel Modeling for Wireless CommunicationsView this Special Issue
Research Article | Open Access
Characterization of Path-Loss Disparity in Virtual Multiple-Input-Single-Output Links
Cooperative transmission (CT) is an effective technique to achieve spatial diversity in fading environments, where spatially separated wireless nodes collaborate to form a virtual antenna array or virtual multiple-input-multiple-output (VMISO) link. Many authors model the path loss of the VMISO link as though the elements in the virtual antenna array are colocated, even though they are spread out. In this paper, we show that the spreading causes a signal-to-noise-ratio (SNR) penalty of up to 3 dB. Moreover, in the high SNR regime, we show that the performance degradation caused by the path-loss disparity can be characterized equivalently by log-normal distribution. We use these two observations to propose a new CT link model, expressed in closed form as an outage probability based on the log-normal shadowing model. The quality of the model is evaluated by a statistical test based on Kolmogorov-Smirnov method.
For small wireless nodes with limited power, where collocated antennas (a real antenna array) cannot be deployed, CT is an alternative way to achieve spatial diversity in fading channels [1, 2]. CT provides an SNR advantage through array and diversity gains by creating a VMISO link that connects a transmitting cluster (multiple nodes) with a single receiver node. Based on the SNR advantage of CT, various higher layer protocols have been proposed, in which the VMISO links provide gains at higher layers such as throughput improvement, energy saving, energy balancing, and range extension [3–16].
In multihop ad hoc networks, the CT-based protocols are divided into two (decentralized and centralized architectures) depending on how the cooperative clusters are activated and supported . In the decentralized architecture such as [3–8], the cluster is activated randomly in a distributed manner, where all the nodes in a cluster have an equal role. On the other hand, in the centralized cluster architecture, which has a cluster head in each cluster, the cluster head recruits its cooperators and triggers the group transmission to the cluster head in the next cluster [12–16]. Because the member nodes follow their cluster heads’ decision and the cluster size can be readily controlled, a cluster can work as a single node. Therefore, the centralized cluster architecture is highly compatible with the conventional non-CT or single-input-single-output- (SISO-) based systems.
In network layer research using CT, the nodes in the centralized cooperative cluster are often assumed to be colocated [12–15]; in this model, the disparate path losses caused by the different distances between the transmitting nodes in a cluster to the receiver node are ignored. The authors in  realized through simulation of some specific topologies that there can be a significant error (i.e., SNR penalty) incurred for making the colocated assumption.
Motivated by this fact, in this paper, we focus on the error in the colocated assumption, for the centralized cluster architecture with finite node degree. To be specific, we analyze the impact of path-loss disparity, which is ignored in the colocated approximation. When relays are randomly located around a cluster head, their disparate path losses cause variation in the statistics of received SNR or power at the receiver of the VMISO link compared to the colocated (or real) MISO link. Even though the centralized CT is highly compatible with the classical SISO-based system, this SNR variation ignored in the colocated assumption limits accurate design and evaluation of high-layer protocols based on CT. For this reason, we quantify the error of the colocated assumption in the presence of path-loss disparity by comparing the required SNRs to achieve a certain outage probability.
Beaulieu and Hu  derived a closed form expression for the outage probability at the destination for the case of the decode-and-forward (DF) relays, where the location of the relays is assumed to be known. The paper  has an intermediate result, where the number of relays that successfully decode is assumed to be known, while the final result in  allows for the opportunistic case, in which the number of relays that successfully decode is not known a priori. However, this final expression in  is long, complicated, and numerically sensitive . Moreover, the SNR penalty for the colocated assumption is not considered in , nor are random locations of nodes taken into account.
In this paper, we allow the node locations to be random and analyze two-hop DF cooperative networks with multiple relays. Cho et al.  compared outage performances of random and best relay selection schemes assuming random relay locations, but they only assumed single-relay amplify-and-forward (AF) system. Wang et al.  also analyzed outage probabilities with the single-relay DF two-hop network, where the relay is selected by different criteria. Zhai et al.  proposed a cooperative automatic repeat request (ARQ) scheme, in which randomly located relays try to retransmit in response to a negative acknowledgement (NACK) from the destination. However, if more than one relay accesses the channel simultaneously, collision occurs and the retransmission is unsuccessful. Aalo et al. in [24, 25] analyzed outage and average bit error rate (ABER) of multihop AF relay networks with randomly located nodes. However, it only assumed a single-relay node for each hop in the interference-limited scenario focusing on signal-to-interference ratio (SIR) instead of SNR.
On the other hand, Wang et al.  derived the outage rates for both single- and multirelay scenarios with a two-hop DF network. The authors assumed transmit power control for different nodes and maximum ratio transmission (MRT) , which is not practical because channel state information (CSI) should be known at both the transmitters (i.e., source and relays) and receiver (i.e., destination). Moreover, for some types of networks such as sensor networks, constraints on node (e.g., transmit power, processor, and memory) may preclude optimal power control. Also, the presence of multiple time and frequency offsets in CT may make channel estimation very challenging , especially when a large number of nodes do CT as in . For the same reason, in this type of network, distributed beamforming schemes in [29, 30], which require CSI at the transmitters, are also difficult to implement in practice.
Therefore, we assume equal transmit power for any transmitting nodes with cooperative DF relaying, which does not require CSI at the transmitters. Moreover, while  provides an approximate outage rate, which is complex to interpret, we capture the impact of the random relay locations in an intuitive form. To be specific, for the known number of successfully decoding relays, we show that the random locations of nodes produce a random received power that, when averaged over multipath fading, has a log-normal distribution as in shadowing, when averaged over multipath fading. We also treat the opportunistic case and derive the outage performance for the case when the number of relays that successfully decode is not known a priori by assuming that the first-hop errors are independent and identically distributed (i.i.d.). The conference version of this paper  presents that a SNR penalty of up to 3 dB should be included, when making colocated assumption to achieve a certain target outage probability. This paper extends the analysis in  with a single approximate outage probability based on the log-normal approximation, which is more accurate than the upper and lower bounds in . Moreover, the log-normal distribution model of the path-loss disparity, which is the key contribution of this paper, is analyzed in more detail by the statistical normality test based on the Kolmogorov-Smirnov method. In other words, we evaluate the statistical validity of the log-normal model with different system parameters.
The organization of the paper is as follows. In Section 2, the system models including network topology, channel, and outage are presented. The following three sections correspond to three stages for the outage capacity approximation. In Section 3, first-hop error is approximated to decouple the first and second-hop errors. Then, the log-normal distribution to model the path-loss disparity in the second hop is presented in Section 4. Also, the equivalent log-normal model is evaluated by the cumulative distribution function (CDF) comparison and the statistical normality test. Section 5 is the final stage to derive the outage approximation in a closed form, in which first and second moments of channel gain parameter are derived under the random relay locations. Section 6 provides the simulation results that show the error of the colocated assumption and the accuracy of the proposed approximate outage capacity .
2. System Model
We consider a VMISO communication in each hop consisting of two phases as shown in Figure 1, where the source, which is indicated by the left black dot at the center of the dotted lined circle, first transmits a packet to the destination (the right black dot) in Phase 1. After that, the multiple relays (the white-filled circles) around the source decode and then forward using orthogonal channels to the destination in Phase 2 . We note that, instead of orthogonal transmission, space-time block codes (STBCs) can be used, in which case the analysis in this paper is still applicable with a slight modification. This cluster architecture is referred to as the centralized cluster , where each cluster has a cluster head that recruits its cooperating relays and triggers the group transmission.
2.1. Network Topology
We consider a static VMISO network as shown in Figure 2, where the source node is located at the origin with a distance to the destination. Also, there are number of cooperating relays around the source, which are located in the dotted lined circle with a radius centered at the source node. can be regarded as the SISO range, over which the source collects its cooperators. As in [12–14], we assume the “CT range extension case,” where , which was demonstrated in . We note that maximum range extension happens when radios do not decrease their transmit powers when cooperating and they transmit using either orthogonal channels or space-time block coding, so the link benefits from diversity gain . The colocated VMISO model in [12–14] assumes that the links between the source and the relays are error-free, which is another factor that causes the error of the colocated assumption.
We assume that relays are independently and identically distributed in the circle with the radius following the uniform distribution. As shown in Figure 2, relay represented by the white-filled circle exists at a distance of from the source with an angle with respect to the line connecting the source and destination. Also, relay is away from the destination, which determines the path loss between the relay and the destination. It follows that the polar coordinates of relay have the probability distribution functions (PDFs): where and , respectively.
2.2. Channel Model
We assume mutually independent Rayleigh fading for any links between two nodes (the source, relays, and destination). The node indices of the source and destination are zero and , and the relay index satisfies . The complex channel gains are denoted by (from the source to relay ), (from the source to the destination), and (from relay to the destination). Hence, follows the exponential distribution with a parameter that is determined by the locations of nodes and . Therefore, when the node locations are given, the cumulative distribution function (CDF) of given is expressed as where the conditional mean and variance are and , respectively. If the path-loss exponent is , then for the source-relay link and for the relay-destination link.
2.3. Outage Model
In this section, we first consider the outage probability for a deterministic network topology (i.e., the relay locations are given), the closed form of which is provided in . Next, we consider the random spatial distribution of the relays to capture how random path-loss disparities impact outage probability.
Assuming the same transmission powers for the source and relays, the conditional mutual information between the source and relay given that is where is the transmit SNR of each node . Also, the normalization factor is for orthogonal DF transmissions of the source and relays, where the destination achieves diversity gain. Therefore, for a certain transmission rate (bit/Hz/sec), the probability that relay fails to decode the received signal from the source is given by where is the conditional CDF of , given , and .
Suppose that, for a given network topology, is a particular set of relays that successfully decode the source transmission, where . Therefore, the conditional mutual information of the VMISO link, conditioned on , is given by Thus, the outage probability of the VMISO communication for a given network topology indicated by two parameter vectors and that satisfy and , with is expressed as where . This equation is the same as in  except that we suppose is the probability conditioned on the certain network topology (i.e., and ). Considering the randomness of the relay locations, the outage rate averaged over the random network topology has -fold integral as where is the joint PDF of and , which can be obtained by the variable transformation from and .
3. First-Hop Error Approximation
It is difficult to obtain the closed form expression of the outage probability under the random relay locations in (8), because the first hop (between the source and the relays) and second hop (between the relays and the destination) depend on each other by the random relay locations. Therefore, to obtain a single estimate of the outage probability with random relay locations, we first approximate the first-hop error rates of the relays by independent Bernoulli trials. Then, assuming that out of relays successfully decode the first-hop transmission from the source, we derive the second-hop outage probability for a given (deterministic) relay topology.
3.1. Bernoulli Trial Approximation of First-Hop Error
The first-hop outage rate of relay in (5) satisfies that , because . To decouple the outage rates between the first (source-relays) and second (relays-destination) hops, we approximate the first-hop error rates of the relays by independent Bernoulli trials with the same probability . If all relays have the same constant first-hop error rate of , where , in Section 2.3, which does not depend on anymore, is simplified as Also, if , then that is the conditional CDF of for given and a truncated vector with elements out of , . Therefore, the final outage in (8) is approximated by -fold integral by
The upper and lower bounds of outage capacity in  are obtained using and , respectively. Alternatively (i.e., new in this paper), to obtain a single estimate of the outage rate with higher accuracy than the two bounds, we will use the average first-hop outage rate , which is derived by (1) and (5) as where is th order exponential integral , which is given by We note that this exponential function can be easily calculated by popular numerical tools such as MATLAB, MATHEMATICA, and MAPLE .
3.2. Second-Hop Error and Path-Loss Disparity
In this section, we look at the second-hop outage rate for a given relay topology, assuming the number of active relays participating in the second-hop transmission is . As a general case with path-loss disparities, when for , in (10) follows the hypoexponential distribution as where and .
On the other hand, when , which corresponds to the colocated antenna array with elements, is the CDF of the gamma distribution as Therefore, as shown in the equation, can be simplified into with the single condition variable instead of the condition vector . Also, the outage probability in (10) can be expressed with a single integral as We note that, for the colocated assumption, without any integral, because all the links have the same distances of to the destination. Compared to (10), the outage equation in (15) is much simpler, because there is only one integral. Also, the conditional CDF in (13) is numerically sensitive to compute . Therefore, ultimately to capture the impact of path-loss disparity, we use an approximation to project the vector into a single variable assuming high SNR, while keeping the influence of the random relay locations on the outage rate, in the following section.
4. Path-Loss Disparity Modeling by Log-Normal Distribution
In this section, assuming high SNR, we will first approximate the hypoexponential distribution in (13) using the gamma distribution in (14), because the hypoexponential distribution is long, complicated, and numerically sensitive . Also, the approximation into the gamma distribution enables us to characterize the path-loss disparity as the log-normal distribution, which means the impact of the path-loss disparity on the outage rate can be captured by the equivalent log-normal shadowing. Therefore, the final outage rate formula derived in this section is the same as the outage capacity in the conventional composite channel with Rayleigh multipath fading and log-normal shadowing. This log-normal approximation is validated by the illustrative CDF comparison and the statistical normality test using the Kolmogorov-Smirnov (KS) method.
4.1. Gamma Approximation of into
Many studies on CT focus on the asymptotic performance by the limit of . In , the author proposes a simple but accurate way to calculate outage performance of CT with dissimilar (deterministic) path losses, where the outage probability based on hypoexponential distribution in (13) can be approximated by the outage probability computation using gamma distribution in (14) with a negligible error. This enables us to use the traditional notion of the performance analysis based on the real multiple-antenna array system to the VMISO link. The key result in this approximation in  is
In other words, in this gamma approximation, the single parameter is equal to the geometric mean of the distinct for . Please see the appendix for the proof, which is relatively simpler than .
4.2. Approximation of by Log-Normal Distribution
To calculate the outage rate with the random relay locations in (15) with the gamma approximation, we need to obtain the conditional PDF of for given . In dB, (16) is expressed as where is deterministic, while are i.i.d. random variables, when , , and are given.
Because (17) is the sum of i.i.d variables, we model as a Gaussian distribution. In other words, is approximated by log-normal distribution, which becomes more accurate, as increases by the Central Limit Theorem (CLT) . The log-normal approximation is also convenient, because it enables us to use the conventional composite channel model (Rayleigh multipath fading and log-normal shadowing). By the log-normal approximation, we have the PDF as below: where , . Also, because , where and are derived in Section 5, which presents the distribution of .
Using this log-normal approximation, the influence of the random relay locations can be characterized by the log-normal approximation of the PDF of , which means that the random separations have the same effect as log-normal shadowing on the outage. Based on this approximation, the outage probability in (15) can be simplified in a summation form using the Gauss-Hermite method : where , are weight factors, are the roots of the Hermite polynomial, and is the order of the Hermite polynomial. We test this log-normal approximation in the following section.
4.3. Log-Normal Approximation Test
In this section, we test the quality of the log-normal approximation in (18) with different system parameters. First, we compare empirical CDF of in (17) and its normal CDF fit by matching their mean and variance.
(1) CDF Comparison Example. Figure 3 shows three CDFs of in dB, , when and . The results in the three subfigures are achieved with different distances from the source to destination and the number of active relays . To be specific, Figures 3(a), 3(b), and 3(c) correspond to , , and , respectively. In each figure, the red solid curve indicates the real empirical CDF with samples, while the black dotted line with the “o”-markers represents the normal fit of (i.e., the log-normal fit of ).
In Figure 3(a) obtained with , the two CDF curves are almost identical, which means the log-normal approximation in (18) is highly accurate. On the other hand, when the number of active relays decreases to one as in Figure 3(b), the normal fit shows a significantly greater error from the empirical CDF compared to Figure 3(a). Also, with and , Figure 3(c) shows a smaller error of the normal fit compared to Figure 3(b). On the other hand, compared to Figure 3(a) with and , the error of the normal fit in Figure 3(c) is greater. Thus, comparing the three figures, we can conclude that the quality of the log-normal approximation in (18) becomes more accurate, as and increase. We note that the quality of the log-normal approximation does not depend on , because , where is a product term.
However, the outage probability is determined by the tail property of the channel distribution, and the log-normal approximation has a longer tail than the original conditional PDF , especially when is small. Therefore, this outage rate based on the log-normal approximation is higher than (15), which is used to derive the upper bound of the outage rate in the following section.
(2) Statistical Normality Test: Kolmogorov-Smirnov. In this section, we use a well-known statistical normality test: the KS tests  at significance level 5%. The KS method is widely used to assess a null hypothesis that distribution of the data is normal in various applications [38–40]. We note that the KS test tends to be more sensitive near the center of the distribution than at the tails. However,  shows that the KS method is as powerful as other tests (e.g., Shapiro-Wilk, Anderson-Darling, and Lilliefors), when the number of samples is large enough (more than or equal to 1,000 samples).
We perform the KS test using 2,000 samples with different and , when and . Table 1 shows the values averaged over iterative tests. The six rows correspond to from 1.5 to 4.0 with the same (0.5) interval, while the four columns represent , 2, 4, and 8, respectively. In the table, the value increases, as and increase.
Figure 4 shows the corresponding probability of passing test, which is averaged over KS tests, with the significance level of 5%. As in Table 1, we use and . In the figure, the horizontal axis indicates the distance between the source and destination . On the other hand, the vertical axis indicates test-passing probability, which means the probability that value is greater than the 5% significance level over iterative KS tests. Also, the blue solid line, the red dashed line, and the black dotted line represent , 4, and 8, respectively. This figure shows the same trends as in Table 1 with respect to and ; it becomes more likely to pass the test, as and increase. In other words, the log-normal approximation in (18) becomes a better fit, as and increase. Moreover, when and , the probability of passing is greater than 90%, which means the log-normal fit is good.
5. Distance to Destination under Random Relay Locations
To derive the approximated outage probability in the previous section, the disparate path losses, which depend on the distances to destination , need to be calculated. In particular, to obtain the complete outage rate in (20), and should be obtained for and . In this section, we first derive the PDF of , , which characterizes the distance from relay to the destination. However, because it is difficult to obtain closed form solutions of and , therefore, as the final stage to derive an approximate outage rate in the presence of the path-loss disparity caused by the random relay locations, we approximate and derive the corresponding and in closed forms.
5.1. Original Probability Distributions of
Figure 5 illustrates the variable assignments, which are used to derive the PDF of , where a relay node is randomly located inside the left circle centered at the origin , where the source exists. Also, the destination node is at the distance from the origin on the -axis, and the relay node is away from the destination. The locations , , , and indicate some important intersection points that will be used in the derivation of the probability distribution. Using the illustration, the CDF of can be the ratio of the area of the shaded region () to the total area of the left circle, which is given by where , and the area can be expressed as where , . Hence, the PDF of is given by where is the length of the arc in Figure 5. However, this original PDF of does not give the closed form expressions of its mean and variance.
5.2. Approximation of
For the lack of the closed form expressions of the two statistics, we propose an approximated PDF of by using the chord , which is indicated by the dotted straight line in Figure 5, instead of the arc . When is large enough compared to in Figure 2 (i.e., the arc and chord are almost the same), the original PDF of , can be simplified into where .
As an example, in Figure 6, when and , the empirical PDF, obtained with random samples, is indicated by the blue “o”-markers. Also, the original PDF is represented by the black solid line, where is indicated by the black dotted line. As shown in the figure, the empirical PDF is identical to in (23), which verifies its derivation. Moreover, in the domain , which confirms the validity of .
Moreover, this approximated PDF has the mean and variance in closed forms as and . Therefore, by the Taylor expansion, and . Therefore, we can obtain and as By plugging these two parameters with the average first-hop probability in (11) into (20), the outage rate in (20), which is approximated form of the real in (8), can be obtained.
6. Simulation Results
In this section, we first compare the outage probability curves of three cases: the real outage obtained by Monte Carlo simulation, the colocated assumption outage, and the proposed estimate based on the three-stage approximations in the previous sections. Then, we measure the error of the colocated assumption by calculating an SNR gap to achieve a certain outage rate . Also, the SNR gap caused by the path-loss disparity is compared with the SNR gap calculated by the proposed outage rate approximation in (20) by the log-normal distribution.
6.1. Outage Probability
Figure 7 shows the outage simulation results with , , , and . In the figures, the horizontal axis is in dB, while the vertical axis indicates the outage probability. There are two groups of the curves: the group having the higher outage rates with the steeper slopes corresponds to , while the lower group corresponds to . We note that the heights of the group are greater than the other group with because we assume orthogonal channel transmissions of the relays. If space-time block codes are used, the outage curves of the two groups are parallel, with a translation distance corresponding to the code rate, which can be readily calculated.
The blue dashed lines represent the outage rates based on the colocated assumption that ignores the path-loss disparity and first-hop errors. Also, the red solid curves are the true outage probabilities obtained by the Monte Carlo simulation (MC), while the solid line with the “o”-markers indicates the proposed outage approximation by log-normal (LN) approximation.
Therefore, the gaps between the blue dashed lines and the red solid lines are the errors caused by the colocated assumption. For example, when the target outage rate is , the errors are about 1.75 and 1.8 dB for and , respectively. In other words, to achieve the outage probability of considering the relay separations and the source-relay link errors, more transmission power (e.g., 1.75 and 1.8 dB for and , resp.) is required than is indicated by the colocated assumption.
Moreover, in the both groups, the real outage curves obtained by Monte Carlo simulation are very close to in the high SNR regime (when the outage is less than ), which implies the proposed approximation has good accuracy. Therefore, the SNR penalty of the colocated assumption (i.e., the error in terms of SNR to achieve the target outage rate) can be estimated by that characterizes the impact of the path-loss disparity by the equivalent log-normal shadowing model, following the conventional notion of the composite channel.
6.2. SNR Penalty Depending on System Parameters
In this section, we look at the error of the colocated assumption in terms of the SNR penalty or gap to achieve the outage probability of depending on three system parameters: the distance ratio , the path-loss exponent , and the number of relays . In the simulation results shown in Figures 8, 9, and 10, there are two SNR gaps to the colocated assumption: the real error based on MC (the red solid lines) and the estimated errors by with the log-normal (LN) approximation (the black curve with the “o”-markers, resp.). In the three figures, the real error is much closer to the error based on , compared to the upper and lower bounds in the conference paper version of this paper .
Figure 8 shows the SNR penalty in dB indicated by the -axis for the different distance ratio represented by the -axis, when , , and . The two SNR gaps decrease as increases, because the impact of the path-loss disparity becomes negligible for high , which is also noticed by that the two variances and of the equivalent log-normal shadowing model are decreasing functions of . Also, the first-hop error ignored in the colocated assumption is relatively much smaller than the second-hop error, when is large. However, considering that and are the CT range extension ratios widely assumed in the VMISO-based protocol studies, the corresponding SNR gaps are large enough to degrade the protocols designed and operated with the colocated assumption. Moreover, if comparing the two SNR gap curves, the proposed approximation curve grows closer to the MC curve as increases, because the quality of the log-normal approximation becomes more accurate as shown in Section 4.3.
Figure 9 displays the impact of the path-loss exponent on the SNR penalty, when , , and . The two SNR gaps based on the MC and the LN approximation () have the convex curves that have the minimum heights at around . The height decreases in the two curves for are because the first-hop error relative to the second-hop error increases as decreases for a fixed , which means that the first-hop error cannot be ignored in this range. On the other hand, when , both curves increase, as increases, because the path-loss disparity within the VMISO cluster becomes significant ( and are increasing functions of ). Also, the height difference between the two curves increases as increases. By testing first-hop error-free case , we conclude that it is because the first-hop error, which is approximated by i.i.d. Bernoulli trials in Section 3.1, is overestimated. In other words, when is large, the first-hop error relative to the second-hop error in the real outage (MC) is much smaller than the Bernoulli approximation, since .
The last figure, Figure 10, shows the impact of the number of relays on the SNR gap, when , , and . As shown in the figure, the SNR gap increases as increases, because the path-loss disparity ignored in colocated approximation increases. Moreover, the height difference between the MC and the LN approximation is caused by the log-normal approximation error with the small as discussed in Section 4.3. As shown in Figure 8, for , the gap between the two curves is smaller.
Motivated by the limited accuracy of the colocated assumption, which ignores the path-loss disparity in VMISO link, we analyze the impact of the path-loss disparity on the outage probability. Simulation results show that the SNR penalty (i.e., error) up to 3 dB should be considered, when the physically separated relays are assumed to be colocated. We investigate how the error changes depending on the system parameters such as , , and . Moreover, by the asymptotic analysis () of the outage capacity, we show that the performance loss by the path-loss disparity has the same model as log-normal shadowing. The proposed outage approximation is accurate enough to estimate the SNR gap (or the error of the colocated assumption) caused by the first-hop error and the path-loss disparity. Therefore, higher-layer protocol design and analysis with large-scale CT networks can be facilitated by the proposed approximation.
Based on the asymptotic outage behavior in the high SNR regime, we propose the limit matching method that extracts the single parameter by matching the limiting conditional outage rates of two distributions (13) and (14) as follows. First, when , the conditional CDF in (13) satisfies that On the other hand, the limiting for the gamma distribution in (14) is given by Thus, if we match two limits (A.1) and (A.2), then we have which gives the projection equation in (16).
The material in this paper is presented in part at the International Conference on Communications (ICC), June 2013, Budapest, Hungary.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors gratefully acknowledge partial support for this research from the National Science Foundation, Grant no. CNS-1017984.
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Copyright © 2014 Haejoon Jung and Mary Ann Weitnauer. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.