Many propagation models exist in scientific literature, we present only the Okumura-Hata, free space and K factors models on which we relied for this work. The distance d is expressed in km and the frequency f in MHz.
1) Description
The general shape of the K factor model [
L = K 1 + K 2 × log ( d ) + K 3 × h m + K 4 × log ( h m ) + K 5 × log ( h b ) + K 6 × log ( h b ) log ( d ) + K 7diff + K clutter (1)
The values of the K parameters vary according to the type of terrain and the characteristics of the propagation environment of the hm, hb; Valid heights of the BS antenna and the MS antenna in meters.
d is the distance between the base station and the mobile station (km);
The previous equation can be rewritten as follows:
L = ( K 1 + K 7diff + K clutter ) + K 2 × log ( d ) + K 3 × h m + K 4 × log ( h m ) + K 5 × log ( h b ) + K 6 × log ( h b ) log ( d ) (2)
By taking, K ′ 1 = ( K 1 + K 7diff + K clutter ) , the equation of the model K factor becomes:
L = K 1 + K 2 × log ( d ) + K 3 × h m + K 4 × log ( h m ) + K 5 × log ( h b ) + K 6 × log ( h b ) log ( d ) (3)
The previous Equation (3) can be written into two forms, a factorized form as a function of a column vector that will be specified in the sequence and a linear form but in the context of our study, we consider only the factorized form.
2) Factorized Form of the Propagation Model K Factors
The factorized form of the model K factors is written
L = [ K 1 K 2 K 3 K 4 K 5 K 6 ] × [ 1 log ( d ) h m log ( h m ) log ( h b ) log ( h b ) log ( d ) ] (4)
In the previous equation letting K = [ K 1 K 2 K 3 K 4 K 5 K 6 ] and
M = [ 1 log ( d ) h m log ( h m ) log ( h b ) log ( h b ) log ( d ) ] (5)
| K parameters | K1 | K2 | K3 | K4 | K5 | K6 | K7diff | Kclutter |
|---|---|---|---|---|---|---|---|---|
| Parameter value | 149 | 44.9 | −2.49 | 0.00 | −13.82 | −6.55 | −0.8 | 0 |
It follows that, the propagation model K factors can be written as
L = K × M . (6)
This expression will be considered as the factorized form of the propagation model.
The Okumura-Hata and free space models are special cases of the K factor models.
The expression of the Okumura-Hata model is written as follows:
L = 69.55 + 26.16 log ( f c ) − 13.82 log ( h b ) + ( 44.9 − 13.82 log ( h b ) ) log ( r ) − E (7)
with
= 3.2 ( log ( 11.75 h m ) ) 2 − 4.497 . (8)
hb is the height of the base station. r the coverage radius of a site. h m = 1.5 m represents User’s quipment antenna the height. For this value of hm, E = −9.2 × 10−4 which is negligible.
Subsequently, E will therefore be assimilated to the value E = 0.
The free space model is expressed by the relation:
L = 32.45 + 20 log ( f c ) + 20 log ( r ) (9)
The K values for these two models are as follows in
The word “ion” is a Greek term. The English physician Michael Faraday introduced this term in 1834. In general, charged particles are called ions and can be divided into two types:
Anions: negatively charged ions (−);
Cations: positively charged ions (+).
The main inspiration for the IMO algorithm is the fact that ions with similar charges tend to repel each other, while ions with opposite charges tend to attract to each other (
The attraction and repulsion forces between anions and cations are also depicted in
| Model | K1 | K2 | K3 | K4 | K5 | K6 |
|---|---|---|---|---|---|---|
| Okumura hata | 69.55 + 26.16log(fc) − E | 44.9 | 0 | 0 | 13.82 | −6.55 |
| Free Space | 32.45 + 20log(fc) | 20 | 0 | 0 | 0 | 0 |
charges. The ions are evaluated based on their fitness, so the fitness of ions is proportional to the value of the objective function. Needless to say, anions move toward the best cation, whereas cations move toward the best anion. Their amount of movement depends on the attraction/repulsion forces between them. The magnitude of this force specifies the momentum of each ion. So far, the movement of ions in the IMO algorithm can guarantee the improvement of all ions throughout iterations. However, there are no mechanisms for diversification and intensification. To require the ions to diversify and intensify, we assume that the ions can be in two completely different phases: liquid versus solid.
In the IMO algorithm, the movement of ions is carried out in two main phases, in particular the liquid phase representing the diversification and the crystalline phase representing the intensification.
1) Liquid phase (Diversification)
In the liquid phase the ions have greater freedom of movement. In addition, the forces of attraction between ions of opposite charges are greater than the forces of repulsion between ions of similar charges. This is why in the IMO algorithm the forces of repulsion are ignored in this phase to explore the search space. The mathematical model is presented as follows:
- Calculation of the forces of attraction between the ions
The calculation of the forces of attraction is a function of the distance that separates the ions and the best ions from opposite charges.
A F i , j = 1 1 + e − 0.1 / A D i , j (10)
C F i , j = 1 1 + e − 0.1 / C D i , j (11)
with
A D i , j = | A i , j − C b e s t j | (12)
and
C D i , j = | D i , j − A b e s t j | (13)
i indicates the index of the ion and j the dimension.
A D i , j represents the distance between the ith anion and the best cation in dimension j; C D i , j represents the distance between the ith cation and the best anion in dimension j. A F i , j is the resultant force of attraction of the anions and C F i , j the resulting force of attraction of the cations.
According to Equations (10) and (11) above, the forces between ions are inversely proportional to the distances between them. The greater the distance, the lower the force of attraction. In other words, when the distance of the ions from the best ion of opposite charge becomes greater, a weaker force of attraction is applied to them.
According to these same equations, the force of attraction varies between 0.5 and 1 [
- Updating the positions of Anions and Cations
After the attraction force calculation, the position of the anions and cations is updated as follows:
A i , j = A i , j + A F i , j × ( C b e s t j − A i , j ) (14)
C i , j = C i , j + C F i , j × ( A b e s t j − C i , j ) (15)
According to these two previous equations, only the force element determines the movement of each ion to the best ion of opposite charge. Cbestj and Abestj indicate the best cation and the best anion respectively.
We can deduce from previous mathematical models that there is no random component involved in the liquid phase. However, since the movements are proportional to the distances, the ions diversify around the search space. As ions tend to be attracted to other ions with opposite charges, exploration/diversification can be guaranteed. With the increase in the number of iterations, more ions interact and converge to the best ions with opposite charges and diversification is gradually reduced. The displacement of ions to the best opposite charge ions in the liquid phase is as follows:
The search agents of the IMO algorithm also enter the crystalline phase which means that they eventually converge on a solution in the search space. The crystalline phase is presented below.
2) Crystalline phase (Intensification)
In the Crystalline Phase, the ions have converged at an optimal point and convergence occurred. However, due to the unknown shape of the research space, it could be that the convergence in question took place at a local optimum. Therefore, a mechanism is implemented for ions in the crystalline phase to excite the trapped ions out of the local optimum. The mechanism implemented in the crystalline phase is illustrated by the following diagram (
The algorithm representing the mathematical modeling of this phenomenon to solve the local trapping of the optimum is as follows:
Q1 and Q2 are random numbers within the range [−1, 1] and rand ( ) is a function that returns a random number within the interval [0, 1].
According to the pseudo-code here is a condition for moving from the liquid state to the solid state the average ability of the worst ions must be less than or equal to the ability of the best ions. If this condition is met, the anions and cations are randomly distributed around the best cation and anion respectively.
CworstFit and CbestFit are respectively the ability of the worst and the best cation. In addition, AworstFit and AbestFit are the ability of the worst and the best anion. To increase diversity, several anions and cations are eventually reset with low probability.
In order to better understand the behavior of the proposed IMO algorithm, some comments on the novel concepts and ideas as well as their effects on the performance of the proposed algorithms are as follows:
● The proposed method of interaction between the search agents (ions) of the IMO algorithm using attraction and repulsion forces are novel [
● The proposed equations for simulating liquid phase cause convergence of search agents toward the recent best solutions found so far. This approach guarantees the convergence of the IMO algorithm toward the best solutions during optimization [
● The proposed equations and conditions for simulating the crystal phase are useful for resolving local optima stagnation. The liquid phase may cause the search agents to be trapped in local solutions quickly. However, the crystal phase resolves such cases by relocating the search agents around the search space suddenly and stochastically.
The different phases of the IMO algorithm have been clearly explained above, we present here the organization chart of the sequence of these different phases (
In this section, we will present the IMO algorithm can be applied in the optimization of the Okumura-Hata propagation model.
To optimize the Okumura-Hata model, the IMO algorithm will be considered as a black box that will take a vector K as input as defined by the factorized form of the K factor model and the data collected from the radio measurements. The methodological approach is presented by the following organizational chart.
In this chart of
Radio measurements and the factorized form of the K-factor model are used to define the evaluation function.
| Criterion | Distance (m) | Power (dBm) |
|---|---|---|
| Minimum | 100 | −110 |
| Maximum | 1000 | −40 |
The idea here is to show that using drive test data, the k-factors model equation and IMO algorithm, we can obtain a suitable propagation model for a given frequency and a specific environment. This is illustrated below in
The evaluation function accurately represents the input parameter of the IMO algorithm because it integrates the K factor model and radio measurement data. Using this evaluation function as an input parameter of the IMO algorithm aims to minimize the Euclidean distance between field measurements (real measurements) and those of predictions from the new model ( L = K × M ).
Let L M = { L i } i = 1 : N a vector representing the set of values of the losses measured for N points at N given distances. Note also that the vector M of equation (10) depends on the distance d for d seen as a variable, that is a function of d; M= f(d). Thus, for N measuring points at different distances d i , M will become a matrix of 6 rows and N columns.
The objective function will therefore be:
f = min { 1 N ∑ i = 1 N ( L M ( i ) − ( K × M i ) ) 2 } (17)
The formula of the link balance between a measuring point and the BTS provides the received power presented as follows:
P received = P BTS + G BTS + G MS − ∑ L f − L Measured (18)
thus,
L Measured = P BTS + G BTS + G MS − ∑ L f − P received (19)
Preceivedrepresents the power received at each measuring point; PBTS the transmission power of the BTS; GBTS the gain of the BTS antenna; GMS the gain of the mobile station; Lf losses induced by cables and connectors and LMeasured the value of measured losses. In addition, the values of the different distances are also implicit and are obtained from (the longitude and latitude of the BTS considered) and (the longitude and latitude of each measuring point). The method of determining the distance is as follows:
- BTS-MS Distance Assessment
Consider two points A and B of GPS coordinates presented as below (
The calculation of the Euclidean distance between A and B is given by the relation below:
D = R × arcos ( sin ( a ) × sin ( b ) − cos ( a ) × cos ( b ) × cos ( c − d ) (20)
Considerations relating to the processing of data from radio measurements; the establishment of the evaluation function; to the generation of the basic family made within the framework of the city of Yaoundé are the same used in the city of Bertoua. However, given the fact that we are now working in 4G, the link budget will be different.
➢ Link Budget for eLTE network
The total power of an enodeB is evenly distributed across all available block resources.
The number of resources (RB) block NumberRB = 5 × B (only for B= 3, 5, 10, 15 and 20) With B the available spectrum width, If the total power of the eNodeB is PeNodeB in W, the power per subcarrier will be:
| Points | A | B |
|---|---|---|
| Latitude | a | b |
| Longitude | c | d |
P sub-carrier ( W ) = P enodeB N sub-carrier , (21)
where Nsub-carrier is the number of subcarriers
Then
P sub-carrier ( dB ⋅ m ) = P enodeB ( dB ⋅ m ) − 10 log 10 ( N sub-carrier ) (22)
with
N sub-carrier = Number RB × Number sub-carrierperRB (23)
Each RB consists of 12 LTE subcarriers. It is the power per subcarrier that will then be used in the link balance.
P received = P sub-carrier + G eNodeB + G MS − ∑ L f − L Measured − P penetration − M Interference − M Fading (24)
Thus,
L Measured = P sub-carrier + G eNodeB + G MS − ∑ L f − P received − P generation − M Interference − M Fading (25)
Ppenetration represents the penetration losses, MInterference represents the interference margin and MFading the fading margin. The power of the eNodeBs that was considered is 46 dBm and the frequency 400 Mhz.
At the end of the execution of the algorithm for each of the three districts of Bertoua we will have the vectors KCentral; KCRTV; KLycée each consisting of six parameters in the format [ K 1 K 2 K 3 K 4 K 5 K 6 ] we will have a vector
K Bertoua = [ K 1Mean K 2Mean K 3Mean K 4Mean K 5Mean K 6Mean ] (26)
whose value of each of its parameters comes from the average of the parameters of the three cities considered above.
The research space is between the standard Okumura-Hata model and the free space propagation model that characterizes barrier-free propagation.
The base family will be generated according to the algorithm below:
Algorithm
This pseudo-code is very important because it defines integrity constraints for each of the parameters K1, K2, K3, K4, K5 and K6. Fixing the order of magnitude of their respective values.
In this code, K1ok represents the parameter K1 in the Okumura-Hata model; K1el represents the parameter K1 for the free space model. P is the matrix representing the generated population and the population size corresponds to the number of distances measured.
The execution of the IMO algorithm is stopped only when it is certain that it has reached the overall optimum. That is to say, a global convergence in the research space.
This optimum is marked by obtaining the best ion in other words the ion for which its values of the six K parameters give the most minimal value to the objective function. At the optimum, the value of the anion is identical to that of the cation since they converged towards the same point.
Acceptance criterion for an optimized propagation model
An optimized propagation model is accurate if the square root of the mean quadratic error between actual and prediction measurements is less than 8 dB [
RMSE = 1 N ∑ i = 1 N ( L M ( i ) ) − ( K × M ( i ) ) 2 (28)
The solution is saved only if RMSE < 8 dB which materializes at the end of the optimization process [