Abstract
Keywords: Bisection method, Rain rate, Rain attenuation, Path length, Multipath fading.
Received: 16 October 2018 / Revised: 21 November 2018 / Accepted: 24 December 2018/ Published: 11 February 2019
Contribution/ Originality
This study applies Bisection algorithm to determine the optimal path length of line of site wireless communication link. The Bisection method presented in this paper is simpler and easier to be applied in more situations than the Newton Raphson method. Used in Emenyi, et al. [1].
1. INTRODUCTION
As the adoption of wireless communication across the globe increases, wireless communication link designers seek to adopt measures to ensure optimal performance of the system. One factor that significantly impacts on system performance is the communication link path length [2-5] . Generally, the maximum link path is determined from the knowledge of the link parameters applied in the link budget equation [6-12] . With link budget, the expected received signal power can be determined and the percentage availability of the link under different weather conditions can also be ascertained.
In most cases, the focus has been on the maximum path length. However, research has found that such approach should be reconsidered given that it gives rise to more signal outages than what is expected by the design specifications [1]. In view of this , the concept of optimal path length is has been proposed in Emenyi, et al. [1]. Optimal path length is the transmitter to receiver distance at which the available fade margin is equal to the maximum fade depth expected from the fade mechanisms to which the signal will be subjected to as it propagates from the transmitter to the receiver. Previous studies have tried to determine the optimal path length using Newton Raphson iterative algorithm [1]. However, such approach requires differentiation of the operating mathematical expression before it can be applied. In this paper, the Bisection method is used. It is simple to apply to diverse situations without the complex mathematical exercises required in Newton Raphson’s method.
2. METHODOLOGY
2.1. Analytical Model for Determination of the Optimal Path Length of Terrestrial Microwave Link
In wireless link design, the Free Space Path Loss (LFSP) is given as Aba [13]; Shamanna [14]; Tsai [15]; Mämmelä, et al. [16]; De Bruyne, et al. [17];
LFSP = 32.4 + 20 log(f*1000) + 20 log(d) (1)
where f is frequency of the emitted signal in GHz and d is the length of the link in km. Also, in wireless link design, for any given fade margin (fms) and receiver sensitivity (Ps), the received signal (PR) is given as:
Also, based on link budget equation for wireless link, PR can also be determined as follows [13-19] :
PR = PT + (GT + GR ) – (LFSP + LT + LM + LR) (3)
where;
PR = Received Signal Power (dBm)
PT = Transmitter Power Output (dBm)
GT = Transmitter Antenna Gain (dBi)
GR = Receiver Antenna Gain (dBi)
LFSP = Free Space Path Loss (dB).
LT = Losses from Transmitter (cable, connectors etc.) (dB)
LR = Losses from Receiver (cable, connectors etc.) (dB)
LM = Misc. Losses (fade margin, polarization misalignment etc.) (dB)
Based on Eq 1 to Eq 3the path length (d) can be obtained as follows:
2.2. Bisection Method of Solving a Nonlinear Equation
2.2.1. The Theorem behind the Bisection Method:
Figure-1. The root of a nonlinear equation by the bisection method.
Sorce: Kaw [25]
2.2.2. Application of Bisection Algorithm for Calculating the Optimal Path Length
3. NUMERICAL COMPUTATION OF THE OPTIMAL PATH LENGTH USING THE BISECTION METHOD ALGORITHM
4. RESULTS AND DISCUSSION
The numerical computation results for the optimal path length, the convergence cycle of the algorithm and the effect of various parameters on the convergence cycle are given in this section. In Table 1 to Table 3, as well as Figure 2 to Figure 3, frequency is 12 GHz and the rain zone is N, with percentage availability of 99.99%. The convergence cycle for the bisection algorithm is 17. That means, as shown in Table 1, Table 2, and Table 3, (as well as, Figure 2, Figure 3, and Figure 4), the bisection algorithm is iterated for 17 times before the optimal path length is obtained. Also, the optimal path length is 5.8905 km, the optimal free space path loss is 129.43 dB, the optimal fade margin the system can accommodate is 30.57 dB while the optimal fade depth is 30.65 dB. In essence, at the optimal path length, a maximum fade depth of 30.57 dB can be accommodated by the link. However, the maximum fade depth the rain and multipath fading can present at the optimal path length of 5.8905 km is 30.65dB which is 0.08 dB above the optimal fade margin.
Table-1. Bisection Method: Rain Fading, Multipath Fading , Free Space Path Loss , Effective Fade Margin , Effective Maximum Depth and Effective Path Length vs Number of Iterations
Number of Iterations (n) |
Effective Rain Fading |
Multipath Fading |
Free Space Path Loss |
Effective Fade Margin |
Effective Fade Depth |
Effective Path Length |
0 |
104.03 |
26.59 |
140.04 |
19.96 |
104.03 |
19.99 |
1 |
19.98 |
0 |
125.71 |
34.29 |
19.98 |
3.84 |
2 |
19.98 |
0 |
125.71 |
34.29 |
19.98 |
3.84 |
3 |
30.49 |
5.3 |
129.38 |
30.62 |
30.49 |
5.86 |
4 |
30.49 |
5.3 |
129.38 |
30.62 |
30.49 |
5.86 |
5 |
30.49 |
5.3 |
129.38 |
30.62 |
30.49 |
5.86 |
6 |
30.49 |
5.3 |
129.38 |
30.62 |
30.49 |
5.86 |
7 |
30.49 |
5.3 |
129.38 |
30.62 |
30.49 |
5.86 |
8 |
30.49 |
5.3 |
129.38 |
30.62 |
30.49 |
5.86 |
9 |
30.65 |
5.4 |
129.43 |
30.57 |
30.65 |
5.89 |
10 |
30.65 |
5.4 |
129.43 |
30.57 |
30.65 |
5.89 |
11 |
30.65 |
5.4 |
129.43 |
30.57 |
30.65 |
5.89 |
12 |
30.65 |
5.4 |
129.43 |
30.57 |
30.65 |
5.89 |
13 |
30.65 |
5.4 |
129.43 |
30.57 |
30.65 |
5.89 |
14 |
30.65 |
5.4 |
129.43 |
30.57 |
30.65 |
5.89 |
15 |
30.65 |
5.4 |
129.43 |
30.57 |
30.65 |
5.89 |
16 |
30.65 |
5.4 |
129.43 |
30.57 |
30.65 |
5.89 |
17 |
30.65 |
5.4 |
129.43 |
30.57 |
30.65 |
5.89 |
18 |
30.65 |
5.4 |
129.43 |
30.57 |
30.65 |
5.89 |
19 |
30.65 |
5.4 |
129.43 |
30.57 |
30.65 |
5.89 |
20 |
30.65 |
5.4 |
129.43 |
30.57 |
30.65 |
5.89 |
Figure-2. Bisection Method: Rain Fading, Multipath Fading , Free Space Path Loss , Effective Fade Margin , Effective Maximum Depth and Effective Path Length vs Number of Iterations (n)
Table-2. Bisection Method:Differential Fade Depth and Effective Path Length vs Number of Iterations
Number of Iterations (n) |
Differential Fade Depth |
Effective Path Length (de) |
0 |
84.067 |
19.99 |
1 |
-14.307 |
3.84 |
2 |
-14.307 |
3.84 |
3 |
-0.132 |
5.86 |
4 |
-0.132 |
5.86 |
5 |
-0.132 |
5.86 |
6 |
-0.132 |
5.86 |
7 |
-0.132 |
5.86 |
8 |
-0.132 |
5.86 |
9 |
0.079 |
5.89 |
10 |
0.079 |
5.89 |
11 |
0.079 |
5.89 |
12 |
0.079 |
5.89 |
13 |
0.079 |
5.89 |
14 |
0.079 |
5.89 |
15 |
0.079 |
5.89 |
16 |
0.079 |
5.89 |
17 |
0.08 |
5.89 |
18 |
0.08 |
5.89 |
19 |
0.08 |
5.89 |
20 |
0.08 |
5.89 |
Figure-3. Bisection Method: Differential Fade Depth and Effective Path Length vs Number of Iterations (n)
Table-3. Bisection Method: Initial and Optimal Values For Free Space Path Loss, Fade Depth , Fade Margin, Received Power , Differential Fade Depth , Differential Path Length , Path Length and Convergence Cycle
n |
Free Space Path Loss (in dB) |
Fade Depth (in dB) |
Fade Margin (in dB) |
Received Power (in dBm) |
Path Length (in km) |
|
Initial Value |
0 |
140.04 |
104.03 |
19.96 |
-60.04 |
19.9903 |
Optimal Value |
17 |
129.43 |
30.65 |
30.57 |
-49.43 |
5.8905 |
Figure-4. Bisection Method: Initial and Optimal Values For Free Space Path Loss, Fade Depth , Fade Margin, Received Power , Path Length and Convergence Cycle
Effect of Frequency on the convergence cycle the Bisection algorithm: With respect to the Bisection method, Table 4 and Figure 5 show how the various link parameters vary with frequency, from 3 GHz to 45 GHz. The convergence cycle for the Bisection algorithm varies from 17 at 12 GHz to 15 at 45 GHZ. Essentially, the bisection method converges faster as frequency increases. It can also be inferred from table 4 that the initial path length decreases as the frequency increases. As such, it can be said that the bisection algorithm converses faster as the initial path length decreases.
Table-4. Bisection Method: Initial Path Length, Optimal Path Length and Convergence Cycle vs frequency
f (GHz) |
Convergence Cycle |
Initial Path Length (km) |
Optimal Path Length (km) |
12 |
17 |
19.99 |
5.88 |
15 |
17 |
15.99 |
4.22 |
18 |
16 |
13.33 |
3.29 |
21 |
16 |
11.42 |
2.7 |
24 |
16 |
10 |
2.31 |
27 |
16 |
8.88 |
2.03 |
30 |
16 |
8 |
1.82 |
33 |
15 |
7.27 |
1.66 |
36 |
15 |
6.66 |
1.54 |
39 |
15 |
6.15 |
1.44 |
42 |
15 |
5.71 |
1.36 |
45 |
15 |
5.33 |
1.3 |
Figure-5. Bisection Method: Initial Path Length, Optimal Path Length and Convergence Cycle vs frequency
Effect of Percentage Availability on the convergence cycle the Bisection algorithm: With respect to the Bisection method, Table 5 and Figure 6 show how the various link parameters vary with seven different values of percentage availability of the link, from 99.0% to 99.999%. In Table 5 and Figure 6 , the convergence cycle for the Bisection algorithm varies from 18 at 99.0% link availability to 15 at 99.9% , and eventually to 17 to 15 at 99.999% link availability.
It is observed that as the percentage availability increases the rain fade depth increases and the optimal path length decreases. The convergence cycle is affected by both the rate at which the rain rate varies with link percentage availability and the value of the optimal path length. As such, the variation of the convergence cycle with link percentage availability is not linear.
Table-5. Bisection Method: Initial Path Length, Optimal Path Length and Convergence Cycle vs Percentage Availability
Percentage Availability, |
Convergence |
Initial Path |
Optimal Path |
Pa (%) |
Cycle |
Length (km) |
Length (km) |
99 |
18 |
15.99 |
30.63 |
99.7 |
15 |
15.99 |
19.45 |
99.9 |
16 |
15.99 |
9.92 |
99.97 |
16 |
15.99 |
5.88 |
99.99 |
17 |
15.99 |
4.22 |
99.997 |
17 |
15.99 |
2.99 |
99.999 |
17 |
15.99 |
2.38 |
Figure-6. Bisection Method: Initial Path Length, Optimal Path Length and Convergence Cycle vs Percentage Availability ,Pa (%)
Effect of Rain Zone on the convergence cycle the Bisection algorithm: With respect to the Bisection method, Table 6 and Figure 7 show how the various link parameters vary with fifteen different values of rain zone , from rain zone A, C,E,…,Q. The convergence cycle for the Bisection algorithm is constant at 15 for rain zone A to G; 16 for rain zone J to L and then 17 for rain zone N to Q. Further examination shows that for a given link percentage availability the rain rate varies with the various rain zones; it has the lowest value in rain zone A and the highest value with rain zone Q. Again, smaller rain rate amounts to smaller rain fade depth and smaller optimal path length. It can be inferred from Table 6 that for the given link percentage availability , the initial path length is constant but the convergence cycle increases with increasing rain rate; from rain zone A with the lowest rain rate to rain zone Q with the highest rain rate.
Table-6. Bisection Method: Initial Path Length, Optimal Path Length and Convergence Cycle vs Rain Zone
Rain Zone |
Rain Zone(#) |
Convergence Cycle |
Initial Path Length (km) |
Optimal Path Length (km) |
A |
1 |
15 |
15.99 |
13.86 |
C |
3 |
15 |
15.99 |
13.86 |
E |
5 |
15 |
15.99 |
13.86 |
G |
7 |
15 |
15.99 |
11.25 |
J |
9 |
16 |
15.99 |
9.92 |
L |
11 |
16 |
15.99 |
6.3 |
N |
13 |
17 |
15.99 |
4.22 |
Q |
15 |
17 |
15.99 |
3.57 |
Figure-7. Bisection Method: Initial Path Length, Optimal Path Length and Convergence Cycle vs Rain Zone
In all, the bisection method can be used to determine the optimal path length of terrestrial microwave link. However, the convergence cycle of the algorithm is affected by various link parameters.
5. CONCLUSION
Development of bisection method for determination of optimal path length of terrestrial microwave link is presented along with performance analysis of the algorithm in terms of the convergence cycle of the algorithm. It was found from the analysis that the convergence cycle of the algorithm varies linearly with frequency and it varies non linearly with percentage availability of the link. Also, for a given frequency and link percentage availability, the convergence cycle increases with increase in rain rate.
| Funding: This study received no specific financial support. |
| Competing Interests: The author declares that there are no conflicts of interests regarding the publication of this paper. |
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