System Spectral Efficiency Analysisof a 2-link Ad Hoc Network

Nikola Serafimovski∗

∗School of Engineering and ScienceJacobs University Bremen28759 Bremen, Germany

n.serafimovski@jacobs-university.de

Abstract

This paper analyzes system spectral efficiency of a2-link ad hoc network. It is proven that the optimal operatingpoints are obtained at the maximum power for either simultaneous or single transmission. Exact conditions under whichthe system spectral efficiency is maximized are derived as a function of pathlosses, transmit and noise power levels so thata clear choice between simultaneous and single transmission can be made. Furthermore, an exhaustive numerical searchover a wide range of pathloss parameters shows that the single transmission is better in about two thirds of the cases. Also,it is demonstrated in simulation that even when the simultaneous transmission is better than the single one, the gains aretypically not substantial. We also show that the power control in many cases causes only marginal degradation for systemspectral efficiency. Finally, the system spectral efficiency normalized with respect to the expended energy is considered intheory and in simulation.

I. I NTRODUCTION

In recent years, there has been substantial research activity in the area ofad hoc networks.Ad hoc networks attractconsiderable interest since they aim to provide wireless communication without any preplanned infrastructure. However,this lack of centralized planning also opens many challenging research questions. Present research focuses on thefundamental problems ofad hoc systems such as routing, radio resource allocation, dynamic self-configuration, detectionalgorithms, user admission etc. System capacity, which provides an upper bound on the sum of pairwise data rates,certainly belongs among important questions.

The landmark paper by Gupta and Kumar [1] introduced notion of transport capacity (in bit-meters per second) andinvestigated how it asymptotically scales with the number of nodes. Essentially, it is shown that per user throughputdiminishes to zero as the number of users is increased. One obvious implication is that the nodes should try to mostlycommunicate with their neighbors to reduce detrimental effects of the interference. In [2], a further result is shown:for the assumption of equal transmit powers per node, the network capacity is maximized when nodes transmit withmaximum power. In [3], the authors investigate capacity regions for wirelessad hoc networks with respect to specifictransmission protocols, which themselves may be suboptimal. Here, via simulation, the results are shown to improvefor multihop routing, power control and SIC (successive interference cancellation). However, even for more than eightnodes, computational complexities cause the developed computer programs to become impractical in finding the capacityregions.

Ad hoc networks are power-constrained networks with even smallerpower limitations than the current cellular networks.The fundamental limits of capacity in energy-constrained networks, the ability of power control to significantly improveand ensure desired SINR (signal-to-interference-plus-noise ratio), as well as its use in defining the neighboring nodes, areall areas of concern in [4, 5]. Therefore, power regulation represents a substantial part in improvement and implementationof ad hoc networks. In [6], the uniform throughput capacity per node of power constrainedad hoc network is shownto increase as the number of nodes increases, under the assumption of UWB (ultra-wide band) communication model.This asymptotic result is in sharp contrast with the result in [1] which illustrates that the change in the system modelassumptions causes drastically different results.

In the light of the above, we propose to consider a simple caseof two simultaneously communicating links. Wedemonstrate that, from the system spectral efficiency pointof view, it is optimal to always transmit with maximumpower, either both links simultaneously or only one active link. To this end, there exist only three operation modesthat maximize capacity, independent of the pathloss conditions, and these are the combinations where both links areconnected with maximum power or one of both links communicates with maximum power while the other link is silent.This extends the findings of [2] in the way thatexact conditions, based on the channel characteristics, are derived whichdetermine the optimum power allocation and selection of active links. Results are presented that illustrate the spectralefficiency analysis for a wide range of system parameters. Finally, we investigate, in theory and via simulation, systemspectral efficiency per Watt which has important implications for the transmission costs of power-constrained networks.

In Section II the problem and the notation are introduced, inSection III the special case of symmetric pathlosses isanalyzed, in Section IV maximizers of the system spectral efficiency are found for the general case and normalizationper Watt is also analyzed. In Section V simulation results over a wide range of pathlosses are shown and Section VIfinishes the paper with the conclusions and the future work.

II. PROBLEM DESCRIPTION

We consider a2-link communication scenario where nodes Tx1 and Rx1 as well as nodes Tx2 and Rx2 form a link, asshown on Fig. 1. If two links are active at the same time, theircommunication is not only limited by noise, but also bythe mutual interference. From the system spectral efficiency point of view, the objective is to identify the transmissionscheme that maximizes system capacity: Is it simultaneous or sequential transmission at the maximum powers or, perhaps,at some other power level?

Fig. 1. 2-link system. Vertical arrows show interference, while horizontal ones show intended communication links.

We start with the system spectral efficiency expression:

C(x, y) = log(1 + γ1) + log(1 + γ2) (1)where

γ1 =L21

L11

x

y + NL21and γ2 =

L12

L22

y

x + NL12

denote SINRs at receivers Rx1 and Rx2. Moreover,Lab denotes the pathloss between transmitter Txa and receiverRxb, a, b∈{1, 2}, andN accounts for additive white Gaussian noise (AWGN). Now, we would like to find values oftransmit powers,x, y ∈ [0, P ], whereP is the maximum power, for whichC(x, y) is maximum. After finding first partialderivatives ofC(x, y) with respect tox andy and setting them to zero, we obtain:

1

L11L212

x2 +2N

L11L12x +

N2

L11−

1

L21L22L12y2 +

N(L12 − L11)

L11L22L12y = 0 (2)

1

L22L221

y2 +2N

L21L22y +

N2

L22−

1

L11L21L12x2 +

N(L21 − L22)

L11L21L22x = 0 (3)

This system of two quadratic equations is equivalent to finding the intersection points between two hyperbolas. Aftersolving one of the equations and inserting the solution intothe other equation, a quartic equation is obtained whichafter lengthy transformations reduces to a quadratic equation. Before the derivation of the general case for arbitrarypathlosses in Section IV, we first consider symmetric pathloss cases for finding the pair(x, y) which maximizesC(x, y)in Section III.

III. SPECIAL CASE OFSYMMETRIC PATHLOSSES

We now consider special case whereL11 = L22 andL21 = L12. In other words, pathlosses on two desired and twointerfering links are the same pairwise. The system of equations (2)-(3) then becomes

x2

L11L221

+2Nx

L11L21+

N2

L11−

y2

L11L221

+N(L21 − L11)y

L211L21

= 0 (4)

y2

L11L221

+2Ny

L11L21+

N2

L11−

x2

L11L221

+N(L21 − L11)x

L211L21

= 0 (5)

After adding equations (4) and (5), we get

x + y = −2NL11L21

L11 + L21

which implies thatx or y (or both) are negative. Therefore, there is no stationary point in the first quadrant. Now, welook what valuesC takes whenx or y or both reach maximum powerP . Whenx = P andy = 0 or x = 0 andy = P ,we have

C(P, 0) = C(0, P ) = log

(

1 +P

NL11

)

(6)

However, when bothx andy are equal toP , we have

C(P, P ) = 2 log

(

1 +PL21

L11(P + NL21)

)

(7)

Becauselog(·) is monotonously increasing function, arguments of twolog functions can be compared directly so todetermine whenC(P, 0) is superior toC(P, P ):

1 +P

NL11≥

(

1 +PL21

L11(P + NL21)

)2

(8)

After simplification, we get

P 2 −L2

21

L11NP − L2

21N2 ≥ 0 (9)

Let P1 andP2 (P1 ≤ P2) denote solutions of the left-hand side quadratic. Then, for P /∈ (P1, P2) inequality (9) is true.However, since

P1 =N

2

(

L221

L11−

√

L421

L211

+ 4L221

)

< 0

P2 =N

2

(

L221

L11+

√

L421

L211

+ 4L221

)

> 0

it is obvious thatP1 is always negative andP2 is always positive. Finally, sinceP must be positive, we conclude thatfor P ≥ P2 we have thatC(P, 0) ≥ C(P, P ). Clearly, available power budget impacts the optimal resource allocation.For example, whenL23 ≫ L12 andP is small,C(P, P ) is the maximum.

IV. GENERAL CASE OFPATHLOSSES

The objective is to find values ofx andy (0 ≤ x, y ≤ P ) which maximizeC(x, y). In order to maximizeC(x, y) in(1), after adding the twologs into the singlelog of the product, it is sufficient to maximize the argument of that singlelog since it is monotonically increasing function. We first showthat for fixed power of one of the transmitters, the otherone should use either no or full power to maximizeC(x, y). Specifically, we first show that fory = y0, C(x, y) achievesmaximum when eitherx = 0 or x = P . The argument inside the log ofC(x, y) in (1) can be written as:

f(x) =Ax2 + Bx + D

Ex + F(10)

whereA =

L21

L11,

B =L21

L11NL12 +

L12

L22y0 + y0 + NL21,

D = (y0 + NL21)(

NL12 +L12

L22y0

)

,

E = y0 + NL21,

andF = (y0 + NL21)NL12.

After variable transformationz = Ex + F , expression (10) becomes

g(z) =A1z

2 + B1z + D1

z(11)

where

A1 =L21

L11(y0 + NL21)2

B1 =y0 + NL21 +

L21

L11

L12

L22y0 −

L21

L11NL12

y0 + NL21

D1 =L12

L22y0

(

y0 + NL21 −L21

L11NL12

)

z ∈ [NL12(y0 + NL21), (NL12 + P )(y0 + NL21)]

In order to find maxima ofg(z), we solveg′(z) = 0. SinceA1 > 0 and if D1 > 0, stationary points arez+,− =±√

D1/A1. The negative solution is discarded as it is not physically valid. Finally, the second derivative isg′′(z) =2D1/z3. SinceD1 > 0 and z+ > 0, it follows that g′′(z+) > 0, which meansz+ is minimizer and the maximizer isat the boundaries. IfD1 ≤ 0, stationary points do not exist, so we checkg(z) at the boundaries only. Therefore, in allcasesg(z) is maximized at the boundary values ofz. Hence, the maximum ofC(x, y) in (1) for y = y0 is attained foreitherx=0 or x=P .

Similar reasoning can also be applied if we fixx = x0 to show that the argument of thelog of C(x, y) achievesmaximum when eithery=0 or y=P . Therefore, the system spectral efficiency can possibly reach maximum only at threepoints(x, y): (0, P ), (P, 0) and (P, P ) since the point(0, 0) is obviously a minimizer.

A. Finding the maximum between simultaneous and single transmission

In order to maximize the system spectral efficiency, nodes must transmit with maximum power. The remaining problemis to derive the conditions to choose between simultaneous transmission where two links simultaneously transmit, orsingle transmission where only one link is active. The threeremaining candidates that possibly reach maximum systemcapacity are given by

C(P, 0) = log(1 +P

NL11) (12)

C(0, P ) = log(1 +P

NL22) (13)

C(P, P ) = log

(

1 +L21

L11

P

P+NL21

)

+ log

(

1 +L12

L22

P

P+NL12

)

(14)

It is easily shown thatC(P, 0) ≥ C(P, P ) when

P

N≥

1

2L22(L12(L11 + L21 − L22)) +

1

2L22

√

L212(L22 − L11 − L21)2 + 4L11L21L22L12 (15)

Likewise,C(0, P ) ≥ C(P, P ) when

P

N≥

1

2L11(L21(L22 + L12 − L11)) +

1

2L11

√

L221(L11 − L22 − L12)2 + 4L11L21L22L12 (16)

Finally, C(P, 0) ≥ C(0, P ) whenL11 ≤ L22.

B. Comparison Between Sequential and Simultaneous Transmissions in per Watt Sense

In the case of a simultaneous transmission, when two nodes are broadcasting at maximum powerP , the system poweris double that of when the nodes are transmitting sequentially, each at maximum powerP . In the following we showthat, in per Watt sense, sequential transmission always exhibits superior system spectral efficiency than simultaneoustransmission. Mathematically speaking, we wish to show that the following always holds:

C(P, 0) + C(0, P )

2P>

C(P, P )

2P(17)

We note that on the left hand side one link transmits with power P , while on the right two links are active so that thetotal power amounts to2P . After some algebraic manipulation the inequality (17) canbe transformed to:

P 2 + PN(L11 + L21 + L22 + L12) + N2(L11L21 + L22L12) > 0 (18)

It is obvious that (18) always holds, as all quantities are positive. We stress here, however, that sequential transmissionis not always better than simultaneous transmission. In other words,[C(P, 0) + C(0, P )]/2 > C(P, P ) does not alwayshold.

C. Fair-Power Comparison of Simultaneous and Sequential Transmission

The aim of this part is to show and analyse the conditions under which sequential is preferable to simultaneoustransmission for a constant system power thus enabling a fair spectral efficiency and power comparison of the twoscenarios. While previous work looked into these conditions, the derived results for power efficiency still slightly favoredsequential transmission since the logarithm of the power was the actual system power and was then divided by just thenormal power value.

1) Condition derivation for sequential versus simultaneous transmission: We try to show

C(P, 0) ≥ C

(

P

2,P

2

)

(19)

so we establish:

log

(

1 +P

NL11

)

≥ log

(

1 +L21

P2

L11

(

P2 + NL21

)

)

+ log

(

1 +L12

P2

L22

(

P2 + NL12

)

)

. (20)

After some math we obtain

L22P2

2N+ P

(

L21L22

2+ L22L12 −

L11L12

2−

L21L12

2

)

+ NL21L12 (L22 − L11) ≥ 0. (21)

Thus once solved we can establish that forC(P, 0) ≥ C(

P2 , P

2

)

, we must havePN

≥P1,2

N, where P1,2

Nare solutions

to the quadratic formula and are given as:

P1,2

N=

−b ±√

b2 − 2L21L22L12 (L22 − L11)

L22, (22)

whereb = L21L22

2 + L22L12 −L11L12

2 − L21L12

2 .From the above solution it can be seen that (19) always holds true if b2 < 2L21L22L12 (L22 − L11). If this is

not the case, however, then there are four distinct pairs of possible solutions which must be analyzed. Given thatb2 ≥ 2L21L22L12 (L22 − L11), then given thatP1 < P2 correspond to the ‘negative’ and ‘positive’ solutions of (22)respectively, we establish:

• For b > 0 and

– L22 > L11 – Both of the solutions are negative which implies that for positive values of P, (19) must alwayshold true.

– L22 < L11 – Only P2 is positive which implies that (19) only holds true forP > P2.

• For b < 0 and

– L22 > L11 – Both solutions of (22) are positive, which implies that forpositive values of P, (19) must holdtrue just beforeP1 and just afterP2. That is, sequential transmission will be preferable at some very smallvalues of P and very high values of P.

– L22 < L11 – Both solutions of (22) are positive, which implies that (19) again holds true forP < P1 andP > P2.

With these observations we can show the same for

C(0, P ) ≥ C

(

P

2,P

2

)

. (23)

This is nothing more than the second case of the symmetric sequential transmission pair so we have:

P1,2

N=

−b ±√

b2 − 2L11L21L12 (L11 − L22)

L11, (24)

whereb = L1L12

2 + L1L21 −L21L22

2 − L21L12

2 .Since (23) is exactly symmetrical to (19), however, it can beshown that the same conditions hold true for (23) as for

(19), with the exception thatL11 is replaced withL22, L21 with L12 and vise-versa.Having derived the conditions for sequential versus simultaneous transmission where only the corner values are

considered and their system power is kept constant, we now look at the average spectral efficiency of sequentialtransmission and compare it to simultaneous transmission again at equal system power.

2) Conditions derivations for average sequential versus simultaneous transmission: In this part we attempt to show:

C(P, 0) + C(0, P )

2≥ C

(

P

2,P

2

)

. (25)

We begin by considering:

log(

1 + PNL11

)

+ log(

1 + PNL22

)

2≥ log

(

1 +L21

P2

L11

(

P2 + NL21

)

)

+ log

(

1 +L12

P2

L22

(

P2 + NL12

)

)

. (26)

After significant algebraic manipulation the solution to (26) was obtained as a full fifth order polynomial, that is:

A5P5 + A4P

4 + A3P3 + A2P

2 + A1P + A0 ≥ 0 (27)

whereA5, A4, A3, A2, A1, A0 ∈ R. This fact alone shows that any analytical solution to the system similar to thesolutions presented above would not be feasible due to the system complexity. From math theory we know that full-fifthorder systems are not analytically solvable in general. What adds to the complexity of the analysis that eachAi is afunction of the four pathlosses,Lij wherei, j ∈ {1, 2}, and additive white Gaussian noise. While characterization of thesystem is possible in certain situations, the probability of those states is too small to be of any practical consideration.Remaining with the conclusion that an analytical solution to (25) is of little consequence, we can now attempt tocharacterize the system through simulations. Since (25), however, is only for theoretical analysis, system simulationsresembled those of (19) and (23).

V. SIMULATION RESULTS

Although the theory above determines conditions under which either single or simultaneous transmission maximizessystem spectral efficiency, it requires perfect system knowledge which for some applications, such asad hoc networks,may not be feasible. Also, we would like to know how often, fortypical pathlosses,C(P, P ) is smaller than(C(0, P )+C(P, 0))/2. For these reasons, we analyze the system behavior over the wide range of possible pathlosses by means ofsimulations. The simulation is performed in steps of1 dB for all combinations ofL11, L21, L22 andL12 where each isin the range from30 to 120 dB, so to compare the system spectral efficiencies values at the three corners(0, P ), (P, 0)and (P, P ).

In the first scenario, the maximum system spectral efficiencies in single transmission cases,C(P, 0) andC(0, P ), arecompared with the simultaneous transmission case,C(P, P ):

Power [W] 0.1 0.5 1

Pr(

CP > CPP

)

[%] 88.15 90.59 91.76

where CP = max{

C(P, 0), C(0, P )}

and CPP = C(P, P ). However, this scheme implies that the transmitter withlower quality link would never be able to transmit. Thus, in the next table, results of the simulation of the sequentialtransmission are shown, where the two links transmit independently on orthogonal channels:

Power [W] 0.1 0.5 1

Pr(

CP > CPP

)

[%] 65.32 71.90 74.86

whereCP = 0.5 (C(P, 0) + C(0, P )) and CPP = C(P, P ). Yet, in both of the above cases, to obtainCPP the systemconsumes twice as much power. Since this provides a significant bonus for the simultaneous transmission, a thirdcomparison was made such that the system spectral efficiencyis compared, subject to an overall fixed power:

Power [W] 0.1 0.5 1

Pr(

CP > CPP

)

[%] 66.72 72.61 75.34

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

90

100Normalized relative increase from [C(P,0) + C(0,P)]/2 to C(P/2,P/2)

Num

ber

of p

oint

s (

% )

Relative increase from [C(P,0) + C(0,P)]/2 to C(P/2,P/2)

P = 0.5WP = 0.8WP = 1W

Fig. 2. Normalized increase in capacity fromCP = C(P,0)+C(0,P )2

to CPP = C“

P2

, P2

”

. The transmission power was set to 0.5 W

where CP = 0.5 (C(P, 0) + C(0, P )) and CPP = C(

P2 , P

2

)

. As the above results demonstrate, in more than65%sequential transmission proves to be superior, implying that in a system where pathloss conditions are not known,sequential transmission is the preferable choice.

Fig. 2 shows the normalized increase in capacity when switching from sequential transmission,CP = C(P,0)+C(0,P )2 ,

to simultaneous transmission,CPP = C(

P2 , P

2

)

. Over 50% of all points whereCPP > CP exhibit a relative increase ofless than 30% when compared toCP and whereP = 0.5W. Similar results are obtained forP = 0.8W andP = 1W.However, as the power increases the curve bends closer to theorigin, while as the power decreases, the curve becomesmore straight. This behavior implies that for lower maximumpower valuesCPP becomes more efficient when comparedto CP .

Figs. 3, 4, 5 and 6 show several typical system spectral efficiency plots over the available power domain. It is seen thatthe maximum occurs at(P, P ) when interference pathlosses are relatively large when compared with communicatinglink pathlosses, as shown in Fig. 6. We also note that the power control does not degrade system spectral efficiencysignificantly. This observation is important from a practical point of view, especially for power constrainedad hocnetworks. Although maximum capacity is only achieved by transmitting with maximum power, the gains tend to bemarginal for simultaneous transmission. When the interference pathlosses are smaller, the maximum occurs at(0, P ),(P, 0) or both, as shown in Figs. 3, 4 and 5. Now, transmitting with maximum power, either by(0, P ) or (P, 0),provides significant capacity gains. Hence, when single transmission provides maximum system capacity, transmittingwith maximum power is beneficial, even for power constrainednetworks.

Fig. 3. System spectral efficiency whereC(0, P ) > C(P, 0) forP = 0.5W

Fig. 4. System spectral efficiency whereC(0, P ) < C(P, 0) forP = 0.5W

Fig. 5. System spectral efficiency whereC(0, P ) = C(P, 0) forP = 0.5W

Fig. 6. System spectral efficiency whereC(P, P ) > C(P, 0) andC(P, P ) > C(0, P ) for P = 3W

In retrospect, while the derivation of (19) and (23) exactlydemonstrate which conditions must hold true for sequentialtransmission to be preferred to simultaneous, they may be impractical for a real system since they requires perfectknowledge of the system. In an attempt to supplement the theoretical results, simulations were performed for differentpower levels.

Three different power levels were considered and as expected at lower powers, simultaneous transmission was preferredsince the lower the transmit power the smaller the interference to the receiving nodes. The obtained results in table (I)demonstrate this conclusion whereCP = C(P,0)+C(0,P )

2 and CPP = C(

P2 , P

2

)

.

TABLE I

SEQUENTIAL TRANSMISSION PREFERABLE TO SIMULTANEOUS.

Power [dBm] -30 0 30

Pr{

CP ≥ CPP

}

[%] 45.12 54.14 75.34

While table (I) demonstrates that for lower power value it isin fact better to transmit simultaneously, it does not sayanything about how much better this transmission is. In an attempt to demonstrate the percentage increase in systemspectral efficiency when simultaneous transmission is preferable, the difference between simultaneous and sequentialtransmission spectral efficiency has been plotted. This is presented in Fig. 7 and as can be seen, for smaller values of P,simultaneous transmission is preferable but significantly. Fig. 7 has been obtained with the same method as Fig. 2 andthus again shows the previously made prediction that as the power decreases, the slope of the curve becomes more andmore linear.

Since the simulation results indicate that for lower powerssimultaneous transmission is preferable and we know thatat most it is only double the efficiency of any given sequential transmission at the same conditions, the next step was toshow the reverse. That is, how much greater is the spectral efficiency of sequential transmission when it is preferable.Utilizing the same algorithm that generated Fig. 2, and Fig.7, we were able to obtain a similar graph for the reversesituation. As can be seen from Fig. 8, however, in the cases when sequential outperform simultaneous transmission, itdoes so significantly. We see that even for P= −30dBm, over 45% of the points show more than double increase inspectral efficiency over simultaneous transmission. Additionally we see that for greater powers, that is for P= 0dBmand P= 30dBm, a doubling of spectral efficiency occurs for over 65% of the points.

From these results it can be seen that sequential transmission is significantly more preferable to simultaneous, andeven when simultaneous transmission is more efficient, the information needed to make such a decision is not worth thegain in performance. Although the results indicate that forsmaller power values simultaneous transmission is preferable,the gain from transmitting simultaneously rather than sequentially is not very significant, which implies that even forlow powered networks with limited system knowledge, sequential transmission is preferable.

VI. CONCLUSIONS ANDFURTHER RESEARCH

In this paper we studied the system spectral efficiency of a2-link ad hoc network. We have proved that the transmitpower values at three corners,(0, P ), (P, P ) and(P, 0), are the only possible maximizers of the system spectral efficiency.We then established conditions, which are functions of pathlosses, power and noise level, that determine the maximum

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

90

100Normalized relative increase from [C(P,0) + C(0,P)]/2 to C(P/2,P/2)

Num

ber

of p

oint

s (

% )

Relative increase from [C(P,0) + C(0,P)]/2 to C(P/2,P/2)

P = 0dBmP = −30dBmP = 30dBm

Fig. 7. Normalized increase in system spectral efficiency from CP to CPP .

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 520

30

40

50

60

70

80

90

100

Normalized relative increase from C(P/2,P/2) to [C(P,0) + C(0,P)]/2

Num

ber

of p

oint

s (

% )

Relative increase from C(P/2,P/2) to [C(P,0) + C(0,P)]/2

P = −30dBmP = 30dBmP = 0dBm

Fig. 8. Normalized increase in system spectral efficiency from CPP to CP .

among the three corners. Furthermore, the system spectral efficiency evaluated at(P, P ) and normalized by the usedpower is shown to always be smaller than the corresponding values at the other two corners. Equivalent power analysiswas also performed and conditions for the best transmissionmethod were derived. It was also shown that analyticalsolutions to the timesharing sequential versus simultaneous transmission is of no practical value. Simulations show thatthe single or sequential outperforms the simultaneous transmission in more than65% of the tested scenarios and in theremaining cases, it is not significantly worse. Further research is now aimed at finding a similar result for the significantlymore complex multi-node system. Since knowledge of the system as a whole is not practical, a probabilistic approachwill be used.

VII. A CKNOWLEDGMENTS

I would like to express my gratitude to Dr. Sinan Sinanovic,Prof. Dr. Harald Haas and Dr. Gunther Auer for the workand contributions on which this paper is based.

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