Page 122 - ITU Journal Future and evolving technologies Volume 2 (2021), Issue 1

ITU Journal on Future and Evolving Technologies, Volume 2 (2021), Issue 1

where Ω is given by (21), and Proposition 10 Algorithm 1 converges to an equilibrium.
0
supp( ) ≜ { ∈ ∶ > 0}, (43) The proof can be found in Appendix 9.12.

2
2
ℐ ≜ { ∈ ∶ ℎ / = min ℎ / }. (44) The complexity involved in designing the equilibrium

strategies by Algorithm 1 is log ( / ) log ((Ω −Ω )/ ).
∈
2 2 0 ∞
The proof can be found in Appendix 9.10.
In particular, Proposition 7 and Proposition 8 imply that 7. DISCUSSION OF THE RESULTS
with large or small total jamming power resources, the
jammer’s strategy and the transmitter’s strategy are in‑ In this section we illustrate Algorithm 1 using a system
sensitive to the TCC. consisting of = 5 receivers, with fading gains from
the transmitter to the receivers ℎ = (1, 2, 3, 4, 5), fading
5.4 Large or small threshold of communica‑ gains from the jammer to the receivers = (3, 2, 1, 4, 1),
tion connectivity the background noises at the receivers = (3, 2, 1, 4, 1),
the total transmitter power budget = 2 and the to‑
In this section we consider the cases where the TCC is ei‑ tal jamming power budget = 3. Fig. 1(a) illustrates
ther small or large. that an increase in the total transmission power leads to
an increase in the PCC (i.e., in ), while an increase in
Proposition 9 (a) Let the TCC be large. Then the Nash
equilibrium ( , ) can be approximated by (37) and the jamming power reduces the PCC. Fig. 1(b) illustrates
(38) of Proposition 6. the transmitter’s normalized strategies, i.e., / , while
Fig. 1(c) illustrates the jammer’s strategies for the total
(b) Let the TCC be small. Then, the Nash equilibrium power transmitter budget ∈ {0.1, 1, 10, 100}. It shows
( , ) can be approximated by (34) and (33) of Propo‑ that the jammer’s strategy for a small total transmitter
sition 5. power budget tends to a uniform strategy (Proposi‑
tion 6), while for a large total transmitter power budget
The proof can be found in Appendix 9.11.
the jammer’s strategy tends to water illing‑form strate‑
gies given by (33). Due to the water illing form of Equa‑
6. ALGORITHM TO ARRIVE AT THE EQUI‑ tion (33), smaller ℎ / calls for applying larger jam‑
2

LIBRIUM ming efforts. Here we have that
In this section, an algorithm based on superposition of ℎ/ = (0.3, 1, 3, 1, 5). (45)
2
two bisection methods to arrive at equilibrium strategies
is given. For this reason, the largest jamming effort is focused on
Algorithm 1 The algorithm for deriving the equilibrium receiver 1 while receiver 2 and receiver 4 face approxi‑
strategies ( , ) and ( , ), where > 0 is a tolerance for mately equal‑level of interfering signals. Fig. 2(a) illus‑
the algorithm. trates that an increase in the total jamming power leads
Procedure Strategies() to a decrease in the PCC, while an increase in the to‑
let = 0 and = tal transmission power reduces such negative effect from
repeat
jammer’s interference. Fig. 2(c) illustrates normalized
let = ̃ (Ω( ), ) −
jammer’s strategies, i.e., / , while Fig. 2(b) illustrates
let = ̃ (Ω( ), ) −
let = ( + )/2
transmitter’s strategies for total jamming power budget
let = ̃ (Ω( ), ) −
if < 0 then ∈ {0.1, 1, 10, 100}. By (44) and (45), we have that
let = ℐ = {1}. That is why, jamming efforts for small total
else
let = power jamming budget is focused on receiver 1 (Propo‑
end if
until − > sition 8), while jamming efforts for large total jamming
return strategies (Ω( ), ) and (Ω( ), ) power budget tends to uniform distribution over all the
End Procedure
Procedure Ω( ) receivers (Proposition 7). Fig. 3(a) illustrates that an in‑
crease in the TCC leads to a decrease in the PCC, while
let = Ω ∞
let = Ω 0
repeat Fig. 3(b) and Fig. 3(c) illustrate that the jammer’s strat‑
let = ( , ) egy focuses jamming efforts on receiver 1 due to a wa‑
let = ( , )
let = ( + )/2 ter illing form of Equation (33) and (45) for a small TCC.
let = ( , ) For a large TCC, jammer’s strategy tends to a uniform one
if ( − )( − ) < 0 then
(Proposition 9). In all of the cases, the transmitter tries
let =
else to communicate with each receiver (i.e., > 0 for all

let =
end if ). This observationmakes the MLCC problem remarkably
until − > different from standard OFDM communication scenarios
return
End Procedure where the transmitter, lacking suf icient transmission re‑
sources, must avoid transmission in some of the channels.
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