Page 120 - ITUJournal Future and evolving technologies Volume 2 (2021), Issue 1

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

apply a non‑linear modi ication for the Gaussian elimi‑ 4.3 An auxiliary ixed point equation
nation method suggested in [23] of a search game, and, In this section we reduce Equation (16) and Equation (17)
further, in [24], applied for an Orthogonal Frequency‑ to a ixed point equation. First note that multiplying both
Division Multiplexing (OFDM) jamming game and, in [22], sides of Equation (17) by implies that Equation (17) is
for a multi‑user OFDM game. The issue is that although equivalent to
we can establish monotonicity in one direction by both
variables for one of the functions (in our case, ), the ̃ ( , ) = , (23)
other function (in our case, ) generally is not mono‑
tonic in opposite directions of its variables. Instead of that where
approach, we reduce the non‑linear equations (16) and
(17) to a ixed point equation on and prove that it has ̃ ( , ) ≜ ( , ). (24)
a unique solution and then develop an algorithm to ind
this ixed point. Moreover, by (15) and (24), we have that

4.2 Auxiliary monotonicity properties ⎢ ℎ ℎ ⎥
⎢ 1 + 1 + 4 ℎ + ⎥
In this section we establish monotonicity properties of ⎢ ⎥
⎢
( , ) with respect to its parameters and derive a bijec‑ ̃ ( , ) ≜ 1 − ℎ ⎥ .
tive relation between and based on (16). ∈ ⎢ 2 + ⎥
⎢ ⎥
Proposition 2 Function ( , ) has the following proper‑ ⎣ ⎦ +
ties: (25)

(a) ( , ) is continuous in > 0 and ≥ 0; 1 In the following proposition we establish monotonicity of
̃ ( , ) on parameters and , and reduce Equation (16)
(b) for a ixed > 0, ( , ) is decreasing in from in in‑ and Equation (17) to a ixed point equation.
ity for ↓ 0 to zero for ↑ ∞;
Proposition 3 Function ̃ ( , ) has the following proper‑
(c) for a ixed > 0, ( , ) is decreasing in from ties:

̃
1 (a) ( , ) is continuous in > 0 and ≥ 0;
( , 0) = 1 + 1 + 4 for = 0
2 ℎ (b) for a ixed > 0, ̃ ( , ) is decreasing in from for
∈
(18) = 0 to zero for ≥ where is the unique positive
to root of the equation
3/2
( , ∞) = for ↑ ∞; (19) + = (26)
ℎ

∈
with
(d) for a ixed ≥ 0 there exists the unique = Ω( ) such 1/2
that = 2Ω , = 2 max ℎ and = min ℎ ;

(Ω( ), ) = . (20) 0 ∈ ∈ 3/2
Moreover, Ω( ) ∈ [Ω , Ω ] and Ω( ) can be found via (27)
∞
0
the bisection method;
(c) for a ixed > 0, ̃ ( , ) is increasing in and tends
(e) Ω( ) is continuous and decreasing from Ω for = 0 to for ↑ ∞;
0
to Ω ∞ for ↑ ∞, where Ω is the unique root of the
0
equation: (d) function
1 ( ) ≜ ̃ (Ω( ), ) (28)
1 + 1 + 4 Ω = (21)
0
2Ω 0 ℎ
∈ is decreasing in from for = 0 to zero for ≥ ,
and (e) The following ixed point equation has the unique pos‑
2 itive root :
∗
Ω ∞ = 2 . (22)
∗
∗
∈ ℎ ( )/ = . (29)
This root can be found via the bisection method with
The proof can be found in Appendix 9.3.
[0, ] as the initial localization interval for such .
∗
1 ↓ denotes that tends to decreasingly. Similarly, ↑ denotes
that tends to increasingly. The proof can be found in Appendix 9.4.

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