Page 124 - ITU Journal Future and evolving technologies Volume 2 (2021), Issue 1

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

Also, in OFDM communication, the transmitter’s strategy only if the following condition holds:
becomes uniform in a high SINR mode, i.e., large trans‑

mission power, [25], while in the MLCC problem the jam‑ ℒ , ( ) = + − = 0, > 0,

mer’strategytendstoauniformoneifeitherthetotaljam‑ ( + ℎ ) ℎ 2 ≤ 0, = 0.

ming resource budget is large, or the total transmission (48)
resource budget is small. By (47), the NLP (13) is a convex problem. Thus, to solve
the NLP (13) we introduce Lagrangian ℒ ( ) with is
,
8. CONCLUSIONS a Lagrange multiplier as follows: ℒ , ( ) = − ( , ) +

− ∑ . Then, similarly, for a ixed ∈ , ∈
A problem of multi‑link communication connectivity un‑ =1
der jamming of a transmitter with a group of receivers is the best response if and only if the following condition
when the channels are affected by Rayleigh fading has holds:
been formulated as a zero‑sum power resource alloca‑ ℒ , ( ) = 0, > 0,

tion game. Existence and uniqueness of the equilibrium = − (49)
+ ℎ ≤ 0, = 0.

in power allocation strategies have been proven. Thus,
in contrast to Colonel Blotto games, if channels are af‑ By (48), we have that
fected by Rayleigh fading, then the stability of commu‑
nication connectivity in a multi‑link system can be main‑ > 0 for any . (50)

tained without introducing a random factor for a decision Then, by (48) and (49), > 0 and > 0 correspondingly.
maker. Also, the problem of designing the equilibrium By (48), only two cases arise to consider: (I) > 0, = 0

power allocation strategies has been reduced to the prob‑ and (II) > 0, > 0.

lem of inding a ixed point of a real‑valued function. An (I) Let > 0 and = 0. Then, by (48),

algorithm based on the bisection method for inding the
ixed point has been developed and its convergence has
been proven. = ℎ . (51)

ACKNOWLEDGEMENT Substituting (51) into (49) implies
This work was supported in part by the U.S. National Sci‑ 2 ≤ . (52)
2
ence Foundation under Grant ACI‑1541069, Grant CCF‑ ℎ

1908308 and Grant CNS‑1909186.
(II) Let > 0 and > 0. Then, by (48) and (49), we have

that

9. APPENDIX + = (53)
( + ℎ ) ℎ 2

9.1 Proof of Theorem 1 and

+ ℎ = . (54)
Note that ( , ) is an additively separable function of
( , ), ∈ and Thus,

1 ℎ
2

( , ) = − ( + 2ℎ ) − 2 < 0 (46) = − . (55)

2 2
2 ( + ℎ ) ℎ 3 Substituting (55) into (53) implies

and 1 ℎ
2 2
2
( , ) + ℎ 2 = + . (56)

2 = ( + ℎ ) 2 > 0. (47) Solving (56) on implies that

Thus, ( , )isconcavein andconvexin , andtheresult
follows from the Nash’s theorem [2] since sets and of 1 + 1 + 4 + ℎ
feasible strategies of the transmitter and the jammer are ℎ
compact. = ℎ . (57)

2 +

9.2 Proof of Proposition 1 Thus,
By (46), the NLP problem (12) is a concave problem. Thus,
to solve the NLP (12) we introduce Lagrangian ℒ , ( ) 1 + 1 + 4 + ℎ
with is a Lagrange multiplier: ℒ , ( ) = ( , ) + 1 ℎ ℎ
= − . (58)

− ∑ =1 . Then, for a ixed ∈ , following [21] ℎ

and the KKT Theorem, ∈ is the best response if and 2 +
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