© 2019 IJSRET 146
Analysis of Secant Algorithm for Optimal Path Length of Point-To-Point Microwave Link
Ezenugu, A. Isaac
Department of Electrical /Electronic Engineering Imo State University (IMSU) Owerri,
Imo State, Nigeria [email protected]
Onwuzuruike, V. Kelechi
Department of Electrical /Electronic Engineering Imo State University (IMSU) Owerri,
Imo State, Nigeria
Abstract- In this study, Secant algorithm was developed for computing the optimal path length of a point-to-point microwave.
Also, the impact of various parameters on the convergence of the algorithm is also presented. Mathlab program was developed and used to carry out sample numerical computations of optimal path length and other performance parameters for a given microwave link. The results showed that convergence cycle of 5 is achieved for a microwave link operating at the frequency of 12 GHz and in rain zone N, with percentage availability of 99.99% . Also, the results showed that the secant algorithm , as presented in this study is suitable for the occasion when rain fading dominates. In that case, the convergence cycle of the secant algorithm is quite stable, maintaining values between 5 and 6 even in the face of variations in frequency, percentage availability and rain rate. In any case, the secant method can still be improved to yield low convergence cycle in the face of multipath fading. Furthermore, the iterative secant method may not be needed when the multipath fading dominate since it is possible to develop close form mathematical solution to determine the optimal path length when the multipath fading dominates.
Keywords –Secant Method, Path Length, Microwave Link, Fade Depth, Convergence Cycle
I.
INTRODUCTION
Determination of transmission range is essential in wireless network planning and design. The maximum transmission range specifies the maximum distance between the transmitter and the receiver [1,2,3,4]. When the maximum transmission range is exceeded the specified received signal strength will be lower than expected and may drop too low and below the receiver sensitivity [5,6,7]. In that case, the link will be unavailable.; in this case, the link outage will increase beyond what is specified in the design specification.
Also, if the maximum transmission range is exceeded, the specified bit error rate may be exceeded in the case of a digital wireless link [8,9].
Most design calculations always focus on the maximum transmission range. However, recent research has introduced the concept of optimal path length [10]. In the case of maximum transmission range, the free space path loss is used to determine the transmission range.
However, in the case of optimal path length. The transmission rage obtained with the path loss model is compared with the path length that gives the fade depth equal to the available fade margin. When the two path lengths are equal, then optimal path length is achieved [10]. This ensures that the available path length at design time can accommodate the maximum fade depth
that the signal can experience at the given path length [10].The solution to optimal path length requires iterative algorithms and existing solution employed Newton Raphson method . In this paper, the Secant method is used as it does not require rigorous mathematical process which is common with the Newton Raphson method [11,12].
II. DETERMINATION OF PATH LENGTH FOR TERRESTRIAL
MICROWAVE LINK
Link budget equation is used in wireless link design to determine the received signal strength , PR as follows [10, 13,14,15,16];
PR= PT + (GT+ GR ) – (LFSP + LT + LM + LR)(1) Where GT (dBi)is transmitter antenna gain ; PT (dBm) is transmitter power ; GR (dBi) is receiver antenna gain ; PR (dBm) is received signal strength and LFSP is free space path loss (dB) given as;
LFSP = 32.4 + 20 log(f*1000) + 20 log(d) (2) where f is frequency of the emitted signal in GHz and d is the length of the link in km.
Moreover, for any given fade margin (fms), and receiver sensitivity (Ps), PR can be determined as follows:
𝑃𝑅 = 𝑓𝑚𝑠+ 𝑃𝑆 (3)
© 2019 IJSRET 147 Based on the preceding three equationsthe path length (d) is determined as follows:
d = 10 P T + GT+ GR −𝑓𝑚 𝑠−𝑃𝑆 − 32.4− 20 log (f∗1000 )
20 (4)
The ITU multipath fading model for quick planning applications specified for percentage of exceedance, 𝑃𝑜% of fade depth Amultipath (dB) in the average worst month [10, 13,14,15,16]
𝑝𝑜 = Kd3.1(1+|εP|)-1.29 𝑓0.8 10 −0.00089 ℎ𝐿− 𝑨𝑚𝑢𝑙𝑡𝑖𝑝𝑎𝑡 ℎ
𝟏𝟎
(5) Where path length , d in km, minimum antenna height,hLin meters ;frequency, f in GHz; point refractivity gradient, 𝑑𝑁1in N-units/Km; multipath fade depth, Amultipath in dB and geoclimatic factor, K which is given as:
K = 10 −4.6−0.0027 𝑑𝑁1 (6) εP is the path inclination, (in mrad). εP is calculated using the following expression
𝜺𝒑 = 𝒉𝒕−𝒉𝒓
𝒅 (7) Where: d is distance in km between the transmitter and the receiver; ℎ𝑡 is the transmitter antenna height in meters; ℎ𝑟 is the receiver antenna height in meters:
ℎ𝐿 = 𝑚𝑖𝑛𝑖𝑚𝑢𝑚 ℎ𝑡, ℎ𝑟 (8) Hence,
𝑨𝑚𝑢𝑙𝑡𝑖𝑝𝑎𝑡 ℎ= 10 −0.00089ℎ𝐿 − 10 log 𝑝𝑜
𝐾 𝑑3.1 1+ 𝜀𝑝 −1.29 𝑓0.8
(9)
The ITU-R PN.838 recommendations specified the specific attenuation originating from rainfall as γRpoin dB/km and the rain fade depth, 𝐴𝑅𝑎𝑖𝑛 which is given as a function of the rainfall rate, 𝑅𝑝𝑜 in mm/h ; path length, d (in Km ) and frequency dependent constants k and α as follows;
𝛾𝑅𝑝𝑜 = 𝑘 𝑅𝑝𝑜
𝛼 (10) 𝐴𝑅𝑎𝑖𝑛 = 𝛾𝑅𝑝𝑜 𝑑 = 𝑘 𝑅𝑝𝑜 𝛼 𝑑 (11) In practice,𝐴𝑅𝑎𝑖𝑛 is computed for the vertical and the horizontal polarization based on the values of k and α and hence the larger of the two is the effective rain fade depth. Finally, thelink maximum fade depth (𝑓𝑑𝑚) is obtained as follows;
𝑓𝑑𝑚 = 𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑨𝑚𝑢𝑙𝑡𝑖𝑝𝑎𝑡 ℎ, 𝐴𝑅𝑎𝑖𝑛 (12) The optimal path length (dmop) is obtained when the following condition is fulfilled;
𝑓𝑑𝑚 = 𝑃𝑅− 𝑃𝑆 (13)
III. THE SECANT ALGORITHM
1. The Theoretical Framework
The secant method is derived from the Newton-Raphson method [17,18]. According to Newton-Raphson method solving a nonlinear equation f(x)0 is given by the iterative formula
i i ii f x
x = x f x1
(14) One of the drawbacks of the Newton-Raphson method is the need to evaluate the derivative of the function. To overcome the drawbacks, the derivative of the function, f (x) is approximated as
𝑓′ 𝑥𝑖 = 𝑓 𝑥𝑖 −𝑓 𝑥𝑖−1
𝑥𝑖− 𝑥𝑖−𝑖
(15) Substituting Eq2 intoEq 1 gives
𝑥𝑖+1= 𝑥𝑖− 𝑓 𝑥𝑖 𝑥−𝑥𝑖−1
𝑓 𝑥𝑖 −𝑓 𝑥𝑖−1
(16) 2. Application Of The Secant Algorithm In Optimal Path Length Computation
Optimal path length of a microwave link is obtained from the given microwave link parameters, namely :frequency (f) , transmit power (PT), transmitter antenna gain (GT), receiver antenna gain (GR), fade margin (𝑓𝑚𝑠), receiver sensitivity (PS), point refractivity gradient (dN1) , link percentage outage (𝑝𝑜), rain fade constants ; 𝑘ℎ, αℎ, 𝑘𝑣, α𝑣; rain zone rain rate ( 𝑅𝑝𝑜 ), transmitter antenna height (ℎ𝑡) and receiver antenna height (ℎ𝑟 ).
From the given link parameters, the other parameters that will be used by the optimal path length algorithm will be obtained and recomputed iteratively until the optimal path length is obtained. The additional parameters include; the value of link path length, 𝑑 which is adjusted by an adjustment value (dadj) , along with other parameters such as free space path loss, LFSP, the maximum fade depth in the link, 𝑓𝑑𝑚and the received signal strength, PR. Equation 4 can be used to obtain dup , the upper value of d which is given as;
dup = d = 10 P T + GT+ GR −𝑓𝑚 𝑠−𝑃𝑆 − 32.4− 20 log (f∗1000 ) 20
(17) Rain fade depth (in Eq 11) is determined from the vertical polarization and horizontal polarization as follows;
𝐴𝑅𝑎𝑖𝑛 = maximum Kv Rpo αv ∗ dup, Kh Rpo αh ∗ dup
(18) Similarly, multipath fading (in Eq 9) is given as
𝑨𝑚𝑢𝑙𝑡𝑖𝑝𝑎𝑡 ℎ= 10 −0.00089ℎ𝐿 − 10 log 𝑝𝑜
𝐾 dup 3.1 1+ 𝜀𝑝 −1.29 𝑓0.8
(19) Then , with dup the link maximum fade depth (𝑓𝑑𝑚) is denoted as 𝑓𝑑𝑚 (𝑢𝑝 ) and it is given as
𝑓𝑑𝑚 (𝑢𝑝 )= 𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑨𝑚𝑢𝑙𝑡𝑖𝑝𝑎𝑡 ℎ, 𝐴𝑅𝑎𝑖𝑛 where 𝑑 = dup (20) In addition, LFSP(Eq 2) is computed as follows
LFSP = 32.4 + 20 log(f*1000) + 20 log(dup) (21)
© 2019 IJSRET 148 Also, PR (Eq 1) is computed as PR = PT + (GT+ GR ) – (LFSP + LT + LM + LR) Using Eq 3 to compute𝑓𝑚𝑒as follows
𝑓𝑚𝑒 = 𝑃𝑅− 𝑃𝑆 (22) Now, the lower value of d is denoted as dLw where;
dLw = 𝑓𝑚𝑒
𝑓𝑑𝑚 (𝑢𝑝 ) (23) The Secant method, shown in Figure 1 requires two initial guesses, guesses, 𝑥𝑖−𝑖and 𝑥𝑖 . With respect to the path length, 𝑥𝑖−1 and 𝑥𝑖are given as follows;
de(i)= 𝑥𝑖 = dup = 10 P T + GT+ GR −𝑓𝑚 𝑠−𝑃𝑆 − 32.4− 20 log (f∗1000 )
20 (24)
de(i−1)= 𝑥𝑖−1 = dLw = 𝑓𝑚𝑒
𝑓𝑑𝑚 (𝑢𝑝 ) (25) From Eq 16,
dadj i = 𝑓 𝑥𝑖 𝑥 −𝑥𝑖−1
𝑓 𝑥𝑖 −𝑓 𝑥𝑖−1 = 𝑓𝑑𝑚 (𝑢𝑝 ) de (i)−de (i−1)
𝑓𝑑𝑚 (𝑢𝑝 )−𝑓𝑚𝑒 (26) de(i+1)= 𝑥𝑖+1= 𝑥𝑖− dadj i (27) The iteration stops when de(i+1) − de(i) < 0.001 . Then, Optimal Free Space Path Loss (FSPLop) and the optimal fade margin (𝑓𝑚𝑜𝑝) are computed as follows;
𝑓𝑚𝑜𝑝 = 𝑓𝑑𝑚 (𝑢𝑝 ) = PRe− 𝑃𝑆 (28) FSPLop= 32.4 + 20 log(f*1000) + 20 log(dmop) (29)
IV. RESULTS FOR METHOD 3 THE SECANT METHOD
Math lab program was developed and used to perform the computation of the optimal path length and associated performance parameters for the secant algorithm. Particularly, the computation was carried out for a given LOS point to point microwave link. The convergence cycle was determined for each simulation run and the simulation was conducted for different microwave frequencies, different ITU rain zones (and their associated rain rates) and also for different link percentage availabilities.
For microwave link operating at the frequency of 12 GHz and in rain zone N, with percentage availability of 99.99% the simulation results in Table 1, Figure 1 and Figure 2 show that the convergence cycle is 5.
Furthermore, the value of the optimal path length is 5.8905 km, the value of optimal free space path loss is 129.40 dB, the value of optimal fade margin the system can accommodate is 30.57 dB while the value of optimal fade depth is 30.65 dB. Essentially, a maximum fade depth of 30.60 dB can be accommodated at the optimal path length.
Table 1 the value of give free space between optimal fade depths.
Fig.1 Secant Method: The variation of the link parameters values with iteration, n.
1. The effect of frequency on the performance of the Secant algorithm for optimal path length- The variation shown in the values of various link parameters when frequency is varied from 12 GHz to 45 GHz are in Table 2 and Figure 2.. The convergence cycle for the Secant algorithm varies from 6 at 12 GHz to 5 at 45 GHZ. In essence, the convergence cycle of secant algorithm increased slightly with increase in signal frequency.
© 2019 IJSRET 149 Table 2 Secant Method Impact of frequency on the
initial and the optimal path length and the Convergence cycle.
2. The effect of percentage availability on the performance of the Secant algorithm for optimal path length-With respect to the Secant method, Table 3 and Figure 3 show the variation of the various link parameters with different values of percentage availability of the link, from 99.0% to 99.999%.
Particularly, the convergence cycle for the Secant algorithm varied from 39 at 99.0% link availability to 5 at 99.7% and 99.9% link availability and then to 6 at 99.97%, 99.99%,99.997% and 99.999% link availability. Again, the result means that the convergence cycle of secant algorithm increased slightly with increase in link percentage availability.
Fig.2 Secant Method: Impact of frequency on the initial and the optimal path length and the convergence cycle.
Table 3 Secant Method: Effect Of Percentage Availability On The Initial And Optimal Path Length
As Well As The Convergence Cycle.
Fig.3 Secant Method: Effect of Percentage Availability On The Initial And Optimal Path Length As Well As
The Convergence Cycle.
3.The effect of rain zone rain rate on the performance of the Secant algorithm for optimal path length-With respect to the Secant method, Figure 4 show how rain rate of the various rain zones, from rain zone A,C,E,…,Q affect the values of the link parameters and the convergence cycle. The convergence cycle for the Secant algorithm remained at 34 for rain zone A to rain zone E ; at 5 for rain zone G to rain zone L and at 6 for rain zone N to rain zone Q.
It can be seen that the convergence cycle is extremely high when the effective fading is multipath fading.
However, the convergence cycle is relatively small when rain fading dominates. In essence, the secant method here is suitable for such link where the dominant fading if the rain fading. However, the good news is that a close form solution that does not require iteration can be used to determine the optimal path length when the dominant fading if the multipath fading.
As a whole, the Secant method is suitable for computing the optimal path length of microwave links especially when the rain fading dominates. As long as the rain fading dominates, the secant algorithm has quite stable
© 2019 IJSRET 150 convergence cycle which it maintains between 5 and 6 even in the face variations in frequency, percentage availability and rain rate.
Fig. 4 Secant Method: Variation of convergence cycle, initial and optimal path length and with rain rate of the
various rain zones.
V. CONCLUSION
Secant algorithm for computing optimal path length of a microwave link is presented and the impact of various parameters on the convergence of the algorithm is also presented. The results showed that the secant algorithm as presented in this study is suitable for the occasion when rain fading dominates. In any case, the secant method can still be improved to yield low convergence cycle in the face of multipath fading. Furthermore, the iterative secant method may not be needed when the multipath fading dominate since it is possible to develop close form mathematical solution to determine the optimal path length when the multipath fading dominates.
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