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

Authors: Shu-Ming Tseng; Yung-Chung Wang; Cheng-Wei Hu; Tseng-Chun Lee
Journal: Mathematical Problems in Engineering
 (2018)

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## 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.


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## Body

## 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 overj represents the extra dimension of the memory impulse noise state, and additional multiplicative termhj ’,j represents the state transition of the memory impulse noise from statej’ to statej. The summation overj = 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 Section2, 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 j=1 represents the Gaussian noise only, and the channel state j=2 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 l is(1)yl=xl+nl,where xl is the transmitted signal and has the bit energy Eb and nl is the noise. Then, the probability density functions (PDF) of nl are as follows:(2)Pnl∣j=1=12πσG2exp⁡-nl22σG2(3)Pnl∣j=2=12πRσG2exp⁡-nl22RσG2where σG2 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:(4)hMN=Pj′′=N∣j=MM,N=1,2where j′′ is the next state.Therefore, the state probabilities can be replaced as follows:(5)P1=Pj=1=h21h12+h21(6)P2=Pj=2=h12h12+h21We define the channel state transition probability matrixH as follows:(7)H=h11h12h21h22The channel memory parameterω is defined as [31](8)ω=1h12+h21Whenω=1, the channel is memoryless. When ω>1, the channel has a persistent memory. The rationale is presented in the Appendix. In this study, we consider ω>1, 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 denoteG (packet/packet duration) as the offered load andS (packet/packet duration) as the Tp=L/γ throughput. (sec) is the packet duration, where γ is the data rate (bits/sec).We assume a CDMA system withk interfering users, because of memory impulse noise. The bit error probabilities Pb for the Gaussian noise (j=1) are given by [10–12](9)Pbk,j=1=23Qk3N+N02Eb-0.5+16QkN/3+3σN2+N02Eb-0.5+16QkN/3-3σN2+N02Eb-0.5Note the enlarged noise varianceRσG2 in (3), and we obtain the bit error probabilities Pb for impulse noise (j= 2) as follows:(10)Pbk,j=2=23Qk3N+RN02Eb-0.5+16QkN/3+3σN2+RN02Eb-0.5+16QkN/3-3σN2+RN02Eb-0.5where(11)σ2=kN223360+N120+k-136-120-k-136and N0/2 denotes the noise power spectral density and Q(x) is defined as (12)Qx=12π∫x∞exp⁡-u22du


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

We use theM/M/∞ queuing models to model the CDMA/ALOHA networks with fixed packet length [11, 12]. The validity of the M/M/∞ 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 themeanof the packet length only, not thedistributionof 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 probabilityPk,j which is the probability that there are k interfering packets and the state is j on the 1st bit.(13)Pk,j=Pje-λ/μλ/μkk!,k≥0,j=1,2,where Pj is the state probabilities defined in (5) and (6) and μ is the death rate defined as(14)μ=kTpFurthermore, λ is the arrival rate defined as (15)λ=GTpP s ( k , j , i ) is defined as the probability that there are k interfering packets, and on thei-th bit the Markov-Gaussian channel is in the state j 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 dimensionj=1 or 2, compared to previous CDMA/ALOHA throughput analysis papers.Figure 1
Two-dimensional state transition diagram in Markov-Gaussian channels.The initial conditionPs(k,j,1) is a (16)Psk,j,1=Pk,j,k=0,1,2…,j=1,2.We can recursively derive Ps(k,j,i) by(17)Psk,j,i=∑j′=121-λΔt-kμΔt·Psk,j′,i-1·1-Pbk,j′·hj′,j+k+1μΔt·Psk+1,j′,i-1·1-Pbk+1,j′·hj′,j+λΔt·Psk-1,j′,i-1·1-Pbk-1,j′·hj′,jwherej’ is the state in thei-1th bit. Note that summation over j represents one extra dimension and hj′,j represent the state transition of the memory impulse noise fromj’ to j. These did not exist in the previous CDMA/ALOHA throughput analysis papers [10–12] because they did not consider memory impulse noise.From (16) and recursive form of (17), we can obtain Ps(k,j,2), Ps(k,j,3), and finally Ps(k,j,L).Then, we obtain the packet success probabilityQs(18)Qs=∑j=12∑k=0∞1-Pbk,jPsk,j,LFinally, we obtain the throughput S(19)S=G×Qs


## 5. Throughput Analysis in Markov-Gaussian Channels with CLSP

The channel load sensing protocol (CLSP) allows packet transmission when the current number of users is smaller than the thresholdα and denies packet transmissions otherwise. Therefore, the maximum number of transmitting packets is α.Next, we definePsclsp(k,j,i) as the probability that there are k interfering packets and at thei-th bit; the Markov-Gaussian channel is in the state j and the first (i-1) bits are all correct. The state transition diagram of the CDMA/ALOHA networks with CLSP in Markov-Gaussian channels is shown in Figure 2. The Markov-Gaussian channels introduce an additional dimensionj=1 or 2, compared to previous CDMA/ALOHA throughput analysis papers. Figures 1 and 2 are different because the maximum interfering packets are limited to α-1 owing to CLSP in Figure 2.Figure 2
Two-dimensional state transition diagram with CLSP in Markov-Gaussian channels.The initial conditionPsclsp(k,j,1) is as follows: (20)Psclspk,j,1=Psk,j,1∑k=0α-1Psk,j,1.Further, we derive Psclsp(k,j,i) in recursive form(21)Psclspk,j,i=∑j′=121-λΔt-kμΔt·Psclspk,j′,i-1·1-Pbk,j′·hj′,j+λΔt·Psclspk-1,j′,i-1·1-Pbk-1,j′·hj′,j+k+1μΔt·Psclspk+1,j′,i-1·1-Pbk+1,j′·hj′,jwherej’ is the state in thei-1th bit. Note that summation over j represents one extra dimension and hj′,j represent the state transition of the memory impulse noise from j’ to j. These do not exist in the previous CDMA/ALOHA throughput analysis papers [10–12] because they do not consider memory impulse noise.We obtain the packet success probabilityQclsp(22)Qclsp=∑j=12∑k=0α-11-Pbk,jPsclspk,j,LWe compute the throughputSclsp as(23)Sclsp=Gclsp×Qclspwhere we define Gclsp as the mean offered load given by(24)Gclsp=∑k=0α-1k·Psk,j,1∑k=0α-1Psk,j,1


## 6. Numerical Results

The system parameters are the same as [10–12] to facilitate comparisons. The packet lengthL= 1,000 bits, SNR = 30 dB, processing gainN = 30, and the data rate γ= 9.6 kbps. The channel memory parameter ω=10, and the state probability P2=0.1 [27]. The channel state transition probability matrixH is given in (25), and the average noise ratio between impulse noise and Gaussian noise, R = 10 and R = 100; these parameters are the same as [27, 28]. (25)H=0.990.010.090.91In Figure3, we compare the throughput in Markov-Gaussian channels for R=1-150, where R=1 represents the case of no impulse noise (Gaussian noise only). We observe that the network throughput is worse when the impulse noise is larger (R is larger). For example, the peak throughputS= 4.21 when the offered loadG=6 for R=1, and the peak throughputS= 3.14 when offered loadG=6 for R=150. The analytic results in lines and the simulated results in symbols are close; thus, the accuracy of the throughput analysis is confirmed.Figure 3
Simulation and numerical results of the throughput for various R.In Figures4 and 5, we present the analytic throughput (in lines) and simulated throughput (in symbols) in Markov-Gaussian channels with CLSP for R=10.Figure 4
Throughput for R=10 with CLSP (α≤8).Figure 5
Throughput for R=10 with CLSP (α≥7).In Figures6 and 7, we present the analytic throughput (in lines) and simulated throughput (in symbols) in Markov-Gaussian Model with CLSP for R=150.Figure 6
Throughput for R=150 with CLSP (α≤7).Figure 7
Throughput for R=150 with CLSP (α≥6).According to Figures4–7, if α is too small, the packet will be refused transmissions. If α is too large, the multiple access interference (MAI) of CDMA systems is more severe, so the throughput gets worse. When the offered load is high, the CLSP can reduce the MAI, so the throughput is higher than that without CLSP. In Figures 4 and 5, we find the optimum throughput when α=7 for R=10, and in Figures 6 and 7, we find the optimum throughput when α=6 for R=150. All the optimal CLSP thresholds α for R=1-150 are given in Table 1. We note that the optimal CLSP threshold α becomes smaller when R=150. This can be explained as follows: a larger impulse noise (R=150) increases the packet error rate and thus decreases the number of users allowed in the CLSP system.Table 1
Optimal CLSP threshold for different R.
average noise ratio R thresholdα 1 7 10 7 50 7 100 7 150 6Finally, in Figure8 we compare the throughput in Markov-Gaussian channels with CLSP and the same threshold α=7 for R=1-150. We can see that, for the same CLSP threshold α, the system with CLSP throughput is smaller, when the impulse noise is larger (R is larger).Figure 8
Throughput performance with CLSP (α=7).For example, the throughputSclsp= 5.5 when offered load Gclsp=20 for R=1, and the throughput Sclsp= 3.9 when offered load Gclsp=20 for R=150.


## 7. Conclusions

In the study, we analyzed the CDMA/ALOHA network in Markov-Gaussian channels to characterize the bursty nature of the impulse noise. The previous papers about CDMA/ALOHA network did not consider memory impulse noise. The state transition diagram in Markov-Gaussian channels has one extra dimension representing whether the impulse noise is present or not. In the throughput analysis, the additional summation over j represents one extra dimension and additionalhj′,j represents the state transition of the memory impulse noise fromj’ to j. These were not presented in previous papers [10–12].We derived the exact throughput analysis based on M/M/∞ queuing model without CLSP and with CLSP. In addition, based on the analytic results, we found the optimal CLSP thresholdα for various R (the average noise ratio between the impulse noise and the Gaussian noise). The optimal CLSP threshold α for R=150 (large impulse noise) was smaller than that for R=1 (no impulse noise). We also found that the throughput is smaller when R is larger. For CLSP threshold α=7 and offered load G=20, the peak throughput decreased from 5.5 to 3.9 when R increased from 1 (no impulse noise) to 150 (large impulse noise).



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