Mathematical Problems in Engineering

Volume 2018, Article ID 9373468, 8 pages

https://doi.org/10.1155/2018/9373468

## Performance Analysis of CDMA/ALOHA Networks in Memory Impulse Channels

^{1}Department of Electronic Engineering, National Taipei University of Technology, No. 1, Sec. 3, Zhongxiao E. Rd., Taipei 10608, Taiwan^{2}Department of Electrical Engineering, National Taipei University of Technology, No. 1, Sec. 3, Zhongxiao E. Rd., Taipei 10608, Taiwan

Correspondence should be addressed to Yung-Chung Wang; wt.ude.tutn@gnawcy

Received 2 January 2018; Accepted 13 May 2018; Published 7 June 2018

Academic Editor: Filippo Cacace

Copyright © 2018 Shu-Ming Tseng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Previous CDMA/ALOHA network performance analysis papers do not consider memory impulse channels. The Markov-Gaussian model can characterize the bursty nature of the memory impulsive noise in some wireless channels and power line communication (PLC) channels. In this paper, we propose the use of the Markov-Gaussian model in the throughput analysis of CDMA/ALOHA networks. The state transition diagram for the throughput analysis has one extra dimension to represent if the impulse noise is present or not. We derive the exact throughput analysis in memory impulse channels with and without channel load sensing protocol (CLSP). The analytic results demonstrate that if the level of the impulsive noise is higher, performance degradation is more and vice versa. Based on the analytic results, we also find the optimal CLSP threshold alpha for various impulse noise conditions. The analytic results of the throughput performance are all matched by the simulation results.

#### 1. Introduction

Code division multiple access (CDMA) is popular in wireless networks and power line communication (PLC) networks [1–7]. ALOHA is a popular media access control (MAC) protocol in wireless and PLC networks [8, 9]. The performance of the CDMA/ALOHA networks has been analyzed in wireless communication systems [10–14], RFID systems [15], neighbor discovery in wireless networks [16], and satellite communications [17, 18]. However, these previous studies considered only memoryless thermal noise (additive white Gaussian noise) and did not consider the memory impulse noise.

Impulse noise occurs in indoor wireless networks [19], industrial wireless sensor networks [20, 21], and PLC networks [22, 23]. The source of memory impulse noise includes printers, cash registers, microwave ovens [19], motors, ignition systems, heavy machineries, and electric switch [21]. They are often described by Middleton Class A model [24] or Bernoulli-Gaussian model [25]. However, these models are memoryless, so the bursty nature of the memory impulse noise for several consecutive samples cannot be represented [22]. Markov chains are therefore used to modify the two aforementioned memory models. The Markov-Middleton model [26] is based on the Middleton Class A model and the Markov-Gaussian model [21, 23, 27, 28] is based on the Bernoulli-Gaussian model [29].

In this paper, we propose to apply the Markov-Gaussian channel to model the memory impulse noise and then analyze the throughput of the CDMA/ALOHA networks. The Markov-Gaussian channel adds one extra dimension to the state transition diagram in the previous papers about performance analysis of the CDMA/ALOHA networks. Additionally, we consider the channel load sensing protocol (CLSP) [10, 30]. In CLSP, the base station or access point measures the number of transmitting users; this is defined as the channel load. If the number of transmitting users is smaller than a certain value, we allow the packet transmission. Otherwise, we deny the transmission of the packet. The advantage of CLSP is that it can improve the throughput in high offered load.

The contributions of this paper are as follows:(1)Based on the Markov-Gaussian channel and the two-dimensional state transition diagram with one extra dimension representing the memory impulse noise, we derive the exact throughput analysis of the CDMA/ALOHA network without CLSP and with CLSP in memory impulse channels. In the throughput analysis equations, the additional summation over represents the extra dimension of the memory impulse noise state, and additional multiplicative term* h*_{j ’,j} represents the state transition of the memory impulse noise from state* j*’ to state* j*. The summation over* j* = 1,2; thus, the complexity is doubled. These do not exist in previous papers [10–12], which considered only memoryless thermal noise.(2)Based on the throughput analysis results, we determine the optimal CLSP threshold for various impulse noise conditions.

The remainder of this paper is organized as follows. In Section 2, the Markov-Gaussian channel model is introduced. We describe system model in Section 3. The throughput analysis in the Markov-Gaussian channel without CLSP is presented in Section 4. In Section 5, the throughput with CLSP is analyzed. In Section 6, we present the numerical results of the throughput analysis without and with CLSP. The conclusion is presented in Section 7.

#### 2. Markov-Gaussian Channel Model

The Markov-Gaussian channel model [27] is a hybrid Markov chain with two states to model memory (bursty) impulse noise. The channel state represents the Gaussian noise only, and the channel state represents the impulse noise. The Markov-Gaussian channel model provides a simple method to describe the bursty nature of the channel state. First, the received signal at time index iswhere is the transmitted signal and has the bit energy and is the noise. Then, the probability density functions (PDF) of are as follows:where denotes the Gaussian noise variance and R denotes the average noise ratio between the impulse noise and the Gaussian noise and R>>1. The channel state transition probabilities can be written as follows:where is the next state.

Therefore, the state probabilities can be replaced as follows:

We define the channel state transition probability matrix** H** as follows:

The channel memory parameter is defined as [31]

When =1, the channel is memoryless. When , the channel has a persistent memory. The rationale is presented in the Appendix. In this study, we consider , a persistent memory channel.

#### 3. System Model

A CDMA/ALOHA network with memory (bursty) impulse noise is considered. Let N denote the CDMA processing gain and L denote the packet length in bits. We assume one hub station and infinitely many users. We denote* G* (packet/packet duration) as the offered load and* S* (packet/packet duration) as the throughput. (sec) is the packet duration, where is the data rate (bits/sec).

We assume a CDMA system with interfering users, because of memory impulse noise. The bit error probabilities for the Gaussian noise (*j*=1) are given by [10–12]

Note the enlarged noise variance in (3), and we obtain the bit error probabilities for impulse noise (*j *= 2) as follows:whereand denotes the noise power spectral density and is defined as

#### 4. Throughput Analysis for CDMA/ALOHA Network in Markov-Gaussian Channel without CLSP

We use the queuing models to model the CDMA/ALOHA networks with fixed packet length [11, 12]. The validity of the queuing model, as stated in [11, 12], is justified as follows. Theorem 4.1 in [32] shows that the number of users in M/G/∞ queuing system is a function of the* mean *of the packet length only, not the* distribution *of the packet length. Thus, M/M/∞ (exponential packet length) is equal to M/D/∞ (fixed packet length) in CDMA/Unslotted ALOHA throughput analysis because the bit error probability (and thus packet success probability) depends on the number of users only, as shown in (9) and (10).

First, we define the steady state probability which is the probability that there are interfering packets and the state is on the 1st bit.where is the state probabilities defined in (5) and (6) and is the death rate defined asFurthermore, is the arrival rate defined as

is defined as the probability that there are interfering packets, and on the* i*-th bit the Markov-Gaussian channel is in the state and the first (*i*-1) bits are all correct. The state transition diagram in Markov-Gaussian channels is shown in Figure 1. The Markov-Gaussian channels introduce an additional dimension* j*=1 or 2, compared to previous CDMA/ALOHA throughput analysis papers.