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An Operation Reduction Using Fast Computation of an Iteration-Based Simulation Method with Microsimulation-Semi-Symbolic Analysis^{ †}

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## Abstract

**:**

## 1. Introduction

## 2. Problem Statement

_{k}/a

_{k}tends to 0 as k tends to infinity [19]. The complete procedure is shown in Figure 2.

`FullForm`command, and the counting is performed using the

`StringPosition`command. Since we had sums where the numbers are repeated q times, the WL code for completely counting the operations is:

InnerOperations=q*FullForm[aK[z,q];]StringPosition[InnerOperations,{"Times","Power","Plus","Rational", "BesselI","Log","Exp"}];

`Times`,

`Plus`,

`BesselI`,

`Log`, and

`Exp`are functions used in close form expressions. Similarly, substituting

`s`for

`ak[z,q]`, we obtained the number of operations in the accelerated algorithm.

## 3. Applications of the Accelerating Procedure and Operation Reduction with Microsimulation Semi-Symbolic Analysis

#### 3.1. Non-Coherent Amplitude Shift Keying (ASK) with Shadowing, Interference, and Correlated Noise

_{i}(t) = x

_{i}(t)·cos(ωt) − y

_{i}(t)·sin(ωt). The receiver is sheltered, and no optical visibility exists toward the transmitter, but interference i

_{1}(t) = A

_{1}·cos(ωt) is present. If the system sends logical zero, then the signal s

_{0}(t) = a

_{0}·cos(ωt) has been sent, but if the system sends a logical unit, then the signal s

_{1}(t) = a

_{1}·cos(ωt) has been sent. The parameters a

_{0}and a

_{1}are the signal elements from which the code words are formed. The receiver detects information signal b

_{0}·cos(ωt) and b

_{1}·cos(ωt) with envelops z

_{0}and z

_{1}after passing through a transmitting channel. The b

_{m}(m = 0, 1) are the elements of the detected signals. The receiver system includes a filter and detector envelope. In the receiver input, the signal is:

_{m}(t) = b

_{m}·cos(ωt) + A

_{1}·cos(ωt) + x

_{m}·cos(ωt) − y

_{m}·sin(ωt) = z

_{m}·cos(ωt + φ

_{m}), m = 0, 1

_{0}and z

_{1}, and phases φ

_{0}and φ

_{1}, respectively.

_{0}, r

_{1}/b

_{0}, b

_{1}, φ

_{0}, φ

_{1}, A

_{1}) = p(x

_{0}, y

_{0}, x

_{1}, y

_{1})·|J|

_{n}(x) = I

_{-n}(x), it follows that:

_{1}.

_{0}and φ

_{1}for all values between −π and π.

_{0}and b

_{1}. So, when the code word |ij| (i = 0, 1; j = 0, 1) has been sent, marked with H

_{i}H

_{j}in Equation (18), and when the same is detected in the input of the receiver marked with D

_{i}D

_{j}, the detection of the signals is described as:

_{1}values is necessary, according to Equation (15). The general form of the condition joint probability density function is defined in Equation (14), and is described in Figure 7.

`s`is marked variance σ,

`R`is the correlation coefficient, and

`v`is the order of the iterations. The finalization of IBSM obtains closed form expressions of the probability density function, and outage probability in term of iterations (Figure 8).

_{outage}in Figure 8 provides the next parameters: iteration q, h

_{0}, and h

_{1}are the resolution of the iteration, z

_{0}and z

_{1}are envelopes, R is the coefficient of correlation, and σ is variance. This expression cannot be manually obtained by using numerical tools. The resultant closed form solution of P

_{outage}is an expression that is ready for further processing. Accordingly, the viewpoint is an insight into the parameters and variables that participate in obtaining all the features of this case study. Drawing the characteristics is now possible, but this calculation would take too long, regardless of the chosen accuracy. On the other hand, for greater accuracy, a number of iterations is required, which is not beneficial for this form of expression.

_{outage}is shown in Figure 9.

_{k}represents a general member of the series in P

_{outage}, from the closed form solution in Figure 9.

_{k}verified that:

_{k}, with respect to the convergence theorems that have been mentioned above. The new member becomes b

_{k}→ a

_{k}+ c

_{k}, so:

_{k}is general term in Equation (21). We obtain the general member of P

_{outage}marked as a

_{k}in Figure 10, separating it from Figure 9. Following the next step in MSSA, we derived the term ρ (Figure 11).

_{outage}obtained by the IBSM. Otherwise, a large number of iterations are required to calculate the closest exact values of P

_{outage}, but the computation is time consuming. Then, the resulting P

_{outage}equalizes with a new series obtained by the Kummer's transformation, and performs point matching for the various values of the envelopes, followed by a new reduced number of iterations. After that, the verification of the obtained results was performed by checking the relative error, which determined the degree of adjustability of the algorithm [29]. Finally, we checked the number of operations of calculations in the expression in Figure 9, and then obtained a reduced number of operations with a new decreased number of iterations.

_{outage}in Figure 9, and the resolution of the iteration was h

_{0}= 0 and h

_{1}= 1. We also used z

_{0}= z

_{1}= z to simplify the analysis. The next step was calculating the new numbers of iterations that are reduced for various values of the envelope z. This was performed using the command

`FindRoot[s==Poutage,{q,1}]`.

`s`is a new expression obtained by Kummer’s transformation in Equation (22), and P

_{outage}is a closed form solution in Figure 8. We took the range of values z = {1, 15} for a concrete case [29]. Experiments were performed for various values of the coefficient of correlation R (R = 7/10,8/10) and the variance σ (σ = 2, 3). All calculations were performed with a precision of 10

^{−6}. All tests were performed on a PC with: Intel

^{®}Core™ i5-6500 CPU@ 3.2 GHz, 8 GB RAM, 64-bit Operating System, Windows 10, and Mathematica Wolfram 11.1. The reduced number of iterations are shown in Table 1.

_{outage}and s are shown. The accelerated algorithm s is marked as P

_{e,approx}.

_{outage}required 1193.97 s, or 19 min and 54 s, so the average time per iteration was 70.2335 s. The sped up algorithm’s total calculation time for the accelerated formula was 1.25 s, so the average time per iteration was 0.0735294 s. Wolfram language code for time consumed is:

`Table[Timing[N[Poutage]],{z,15}] // Total`. Command

`Table`provides a calculation for any value of envelope

`z`, and command

`Timing`provides the exact time of calculation. Command

`Total`summarizes total time per envelope. Similarly, changing the parameter

`Poutage`with

`s`for the accelerated algorithm in the previous WL command line provides the time consumed for fast computation. Our algorithm is accelerated as:

_{outage}because we initially assumed that this number of iterations was satisfied for the closest exact value of P

_{outage}. The number of operations for fast computation of IBSM is less than P

_{outage}. For 500 iterations, we counted 120,000 math operations for P

_{outage}. The number of math operations changes in the range of 9000 to 34,000, which is the result of variety in the number of iterations for fast computation.

#### 3.2. Second-Order Statistics in Wireless Channels

_{Z}(z), is defined as the signal speed crossing through level z with a positive derivative at the intersection point z. The ADF, marked as T

_{Z}(z), represents the mean time for which the signal overlay is below the specified z level.

_{Z}(z ≤ Z) represents the probability that the signal level Z(t) is less than the level z. Evaluation and calculation of LCR and ADF are trivial in an environment where no large reflections exists with a large number of transmission channels and shadowing, which simplifies the mathematical description of the distribution of the signal. However, in complex environments, obtaining LCR and ADF characteristics is time-consuming. An example of a complex environment is described in Stefanovic et al. [20]. In this example, the LCR and ADF expressions were obtained. Their analytical shapes are closed forms, but the complexity shows a long computation time. Thus, the LCR value is normalized by the Doppler shift frequency f

_{d}[20] through Equation (15):

_{i}is $({m}_{i}{N}_{i}^{2})/{r}_{i}$, m

_{i}is the Nakagami-m fading severity parameter, N

_{i}denotes the number of identically assumed channels at each microlevel, r

_{i}is related to the exponential correlation ρ

_{i}, c

_{i}denotes the order of Gamma distribution, Ώ

_{0i}is related to the average powers of the Gamma long-term fading distributions, and K

_{v}(x) is the modified Bessel function of the second order. Similarly, the AFD is obtained as [20] per Equation (16):

_{k}from Equation (27), shown in Figure 16.

_{3}(0, e

^{−1})–1), where ϑ

_{a}(u, x), (a = 1,…,4) is the theta function, defined as [30]:

_{Z}) in terms of the number of iterations q for fast computation. The number of iterations was fixed at q = 100 for LCR

_{orig}because we initially assumed that this number of iterations satisfied the closest exact value of LCR

_{orig}. For 100 iterations, we counted 20,200 math operations for LCR

_{orig}. The number of math operations was 1184 for LCR

_{accelerated}calculated in the first iteration using fast computation. Using the same method, the AFD was obtained by applying Equation (22). Figure 20 shows the comparative characteristics of AFD and accelerated AFD. A small deviation in the range of −35 ≤ z ≤ −28 was observed, perceived through the relative error in Figure 21.

_{orig}required 19,553.1 s, or 5 h and 25 min, so, the average time per iteration was 279.33 s, or 4 min and 19.33 s. The sped up algorithm total calculation time with the accelerated formula was 1.29688 s, so, the average time per iteration was 0.0185268 s. An obvious difference in the time calculation exists because the number of sums for AFD increased in Equation (27), where we have sums for k, l, and q. In this case, our algorithm is accelerated as:

^{6}math operations for AFD

_{orig}. The number of math operations was 5619 for LCR

_{accelerated}calculated in the first iteration for fast computation.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 4.**Condition joint probability density function using Wolfram language for shadowing and interference.

**Figure 7.**Log-normal distribution for non-coherent ASK in the presence of shadowing and interference.

**Figure 8.**Closed form solution of probability density function (PDF

_{outage}) of a non-coherent ASK system.

**Figure 9.**Closed form solution of outage probability P

_{outage}of a non-coherent ASK system with shadowing and interference.

**Figure 15.**Number of operations in terms of number of iterations q for fast computation. The number of iterations is fixed with q = 500 for P

_{outage}.

z | q_{1} R = 7/10; σ = 2 | q_{2} R = 7/10; σ = 3 | q_{3} R = 8/10; σ = 2 | q_{4} R= 8/10; σ = 3 |
---|---|---|---|---|

1 | 25 | 32 | 21 | 30 |

2 | 20 | 27 | 16 | 25 |

3 | 19 | 25 | 14 | 22 |

4 | 18 | 24 | 12 | 21 |

5 | 17 | 23 | 11 | 20 |

6 | 17 | 23 | 10 | 20 |

7 | 18 | 23 | 9 | 19 |

8 | 18 | 23 | 9 | 19 |

9 | 20 | 24 | 9 | 20 |

10 | 21 | 25 | 9 | 19 |

11 | 23 | 26 | 10 | 20 |

12 | 27 | 27 | 11 | 21 |

13 | 29 | 28 | 12 | 21 |

14 | 32 | 30 | 14 | 22 |

15 | 36 | 31 | 16 | 23 |

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**MDPI and ACS Style**

Mladenovic, V.; Milosevic, D.; Lutovac, M.; Cen, Y.; Debevc, M.
An Operation Reduction Using Fast Computation of an Iteration-Based Simulation Method with Microsimulation-Semi-Symbolic Analysis. *Entropy* **2018**, *20*, 62.
https://doi.org/10.3390/e20010062

**AMA Style**

Mladenovic V, Milosevic D, Lutovac M, Cen Y, Debevc M.
An Operation Reduction Using Fast Computation of an Iteration-Based Simulation Method with Microsimulation-Semi-Symbolic Analysis. *Entropy*. 2018; 20(1):62.
https://doi.org/10.3390/e20010062

**Chicago/Turabian Style**

Mladenovic, Vladimir, Danijela Milosevic, Miroslav Lutovac, Yigang Cen, and Matjaz Debevc.
2018. "An Operation Reduction Using Fast Computation of an Iteration-Based Simulation Method with Microsimulation-Semi-Symbolic Analysis" *Entropy* 20, no. 1: 62.
https://doi.org/10.3390/e20010062