First, to modify the classical Egli model, we introduce some constant free adaption parameters into (9), resulting in eq (10):
$$ {Pl}_{Egli}\left(\mathrm{dB}\right)={A}_1+{A}_2{\log}_{10}\left({f}_{t(MHz)}\right)-{A}_3{\log}_{10}\left({H}_{tr}\right)-{A}_4{\log}_{10}\left({H}_r\right)+{A}_5{\log}_{10}\left({d}_c\right) $$
(10)
Now, let Ymdefine the measured signal propagation loss and \( {\hat{Y}}_m \) be the Egli model with A1,A2, A3, A4, andA5 being the adaptation parameters to be obtained based on the field signal propagation measurements in a non-linear square sense. It is given by eq (11):
$$ H=\sum \limits_{m=1}^n{\left({Y}_m-{\hat{Y}}_m\right)}^2=\sum \limits_{m=1}^n{\left({Y}_m-{A}_1-{A}_2{\log}_{10}\left({f}_{t(MHz)}\right)+{A}_3{\log}_{10}\left({H}_{tr}\right)-{A}_4{\log}_{10}\left({H}_r\right)+{A}_5{\log}_{10}\left({d}_c\right)\right)}^2 $$
(11)
where n indicates measured signal propagation data number. From (11), the challenge of determining the values of A1,A2,A3, A4, and A5 based on field measurement can be transformed into an optimisation problem as given by (12). The evolving partial derivatives are given in eq.(13) to (17). Details of the mathematical procedures are available [35,36,37]:
$$ \underset{A_1,{A}_2,..,{A}_n}{\min }H\left({A}_1,{A}_2,..,{A}_n\right)=\underset{A_1,{A}_2,..,{A}_n}{\min}\sum \limits_{m=1}^n{\left({Y}_m-{\hat{Y}}_m\right)}^2 $$
(12)
$$ \frac{\partial H}{\partial {A}_1}=2\sum \limits_{i=1}^n\;\left({Y}_m-{\hat{Y}}_m\;\right)\;\frac{\partial {\hat{Y}}_m}{\partial {A}_1}=0 $$
(13)
$$ \frac{\partial H}{\partial {A}_2}=2\sum \limits_{i=1}^n\;\left({Y}_m-{\hat{Y}}_m\;\right)\;\frac{\partial {\hat{Y}}_m}{\partial {A}_2}=0 $$
(14)
$$ \frac{\partial H}{\partial {A}_3}=2\sum \limits_{i=1}^n\;\left({Y}_m-{\hat{Y}}_m\;\right)\;\frac{\partial {\hat{Y}}_m}{\partial {A}_3}=0 $$
(15)
$$ \frac{\partial H}{\partial {A}_4}=2\sum \limits_{i=1}^n\;\left({Y}_m-{\hat{Y}}_m\;\right)\;\frac{\partial {\hat{Y}}_m}{\partial {A}_4}=0 $$
(16)
$$ \frac{\partial H}{\partial {A}_5}=2\sum \limits_{i=1}^n\;\left({Y}_m-{\hat{Y}}_m\;\right)\;\frac{\partial {\hat{Y}}_m}{\partial {A}_5}=0 $$
(17)
By setting \( \frac{\partial H}{\partial {A}_1}={f}_1\left({A}_1,{A}_2,{A}_3\right)\frac{\partial H}{\partial {A}_2}={f}_2\left({A}_1,{A}_2,{A}_3\right) \), and \( \frac{\partial H}{\partial {A}_3}={f}_3\left({A}_1,{A}_2,{A}_3\right) \)
The expressions in (13) to (17) are converted as given in (18):
$$ \left\{\begin{array}{l}{f}_1\left({A}_1,{A}_2,{A}_3,{A}_4,{A}_5\right)=0\\ {}{f}_2\left({A}_1,{A}_2,{A}_3,{A}_4,{A}_5\right)=0\\ {}{f}_3\left({A}_1,{A}_2,{A}_3,{A}_4,{A}_5\right)=0\\ {}{f}_3\left({A}_1,{A}_2,{A}_3,{A}_4,{A}_5\right)=0\\ {}{f}_3\left({A}_1,{A}_2,{A}_3,{A}_4,{A}_5\right)=0\end{array}\right. $$
(18)
Equation (18) expresses the non-linear equation; solving for the values of A1, A2, A3, A4, and A5 by employing analytical techniques is generally a complex task. A commonly used method of solving the above complex equation is the Gauss-Newton method [38]. Still, its application during the iterative implementation process requires a full rank matrix, thus becoming a significant limitation of the algorithm.
In this work, we engaged the Levenberg-Marquart method [39,40,41] to upturn the limitation of the Gaussian Newton algorithm in resolving the non-linear equation in (18). Thus, by employing the Levenberg-Marquart algorithm, eq (18) is transmuted as given in (19):
$$ \underset{A\in {\Re}^3}{\min }f(A)=\frac{1}{2}{\left\Vert f(A)\right\Vert}^2=\frac{1}{2}\sum \limits_{i=1}^5{f}_i^2(A),A=\left({A}_1,{A}_2,{A}_3,{A}_4,{A}_5\right) $$
(19)
The Levenberg-Marquart method can be resolved with the following eq (20)–(21):
$$ \left(J{\left({x}_q\right)}^TJ\left({x}_q\right)+{\mu}_qI.\right)\Delta {x}_q=-J{\left({x}_q\right)}^Tf\left({x}_q\right),\mu \ge 0 $$
(20)
$$ \Rightarrow \Delta {x}_q={\left(J{\left({x}_q\right)}^TJ\left({x}_q\right)+{\mu}_qI.\right)}^{-1}\left(-J{\left({x}_q\right)}^Tf\left({x}_q\right)\right)\mu \ge 0 $$
(21)
J(xq) = fI(A)= Jacobian Matrix
where
I ∈ ℜm × m and μ express the damping term and the identity matrix introduced by Leverberg-Marquart into the classical Gauss-Newton algorithm [38] to improve its performance.
Accordingly, utilizing the Levenberg-Marquart method, the parameters A = (A1, A2, A3, A4, A5) can be obtained iteratively using eq (22):
$$ {x}_{q+1}={x}_q+{\left(J{\left({x}_q\right)}^TJ\left({x}_q\right)+{\mu}_qI.\right)}^{-1}\left(-J{\left({x}_q\right)}^Tf\left({x}_q\right)\right),\mu \ge 0 $$
(22)
Thus, for μ = 0, the expression in eq (20) becomes the classical Gauss-Newton algorithm given by eq (23):
$$ \left(J{\left({x}_q\right)}^TJ\left({x}_q\right)\right)\Delta {x}_q=-J{\left({x}_q\right)}^Tf\left({x}_q\right) $$
(23)
Similarly, eq (22) is simplified as given in (24):
$$ {x}_{q+1}={x}_q+{\left(J{\left({x}_q\right)}^TJ\left({x}_q\right)\right)}^{-1}\left(-J{\left({x}_q\right)}^Tf\left({x}_q\right)\right) $$
(24)
To implement the LM algorithm, the following steps are engaged intuitively:
LM algorithm implementation steps
-
I.
Initialise guess parameters,xo for x at iteration q = 0, 1, 2, …
-
II.
Select the Lagrange multiplierλqfor each step q
-
III.
Calculate theΔxqwith its expression in eq. (21)
-
IV.
Calculate xq + 1 = xq + Δxq
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V.
Evaluate Δxq = xq + 1 − xqat the initial parameter,xo
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VI.
For smaller Δxqvalues, check the rate of convergence
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VII.
If convergence rate is acceptable, stop the calculation or else go a step (IV).