Receiver Diversity Combining Using Evolutionary Algorithms in Rayleigh Fading Channel

Abstract

In diversity combining at the receiver, the output signal-to-noise ratio (SNR) is often maximized by using the maximal ratio combining (MRC) provided that the channel is perfectly estimated at the receiver. However, channel estimation is rarely perfect in practice, which results in deteriorating the system performance. In this paper, an imperialistic competitive algorithm (ICA) is proposed and compared with two other evolutionary based algorithms, namely, particle swarm optimization (PSO) and genetic algorithm (GA), for diversity combining of signals travelling across the imperfect channels. The proposed algorithm adjusts the combiner weights of the received signal components in such a way that maximizes the SNR and minimizes the bit error rate (BER). The results indicate that the proposed method eliminates the need of channel estimation and can outperform the conventional diversity combining methods.

1. Introduction

Diversity techniques are among the prominent ways to improve the reliability of wireless communication systems [1, 2]. These techniques, which fundamentally amount to transmitting signals over independent fading channels, are used in reality to fight against fading. The main idea of diversity is to extract information from the received signal components transmitted over multiple fading channels to improve the received signal-to-noise ratio (SNR) [3, 4]. The large-enough spacing is essential in order to make sure that the received signals are independent, which is a vital requisite to acquire the full benefit of the diversity receiver [5]. It is obvious that there would be a small probability that all the received versions of signal are in a deep fade. Therefore, these techniques assume independent fading effects over the different signal paths. Out of the three mechanisms, namely, path loss, large scale, and small scale fading, the first two are somehow similar and can be mitigated by the power control over a long period of time. Diversity techniques are particularly intended to overcome the small scale fading.

In the past decades, different kinds of diversity receivers functioning over a variety of fading channels have been comprehensively reviewed in the literature [5]. The widely used diversity techniques include maximal ratio combining (MRC), equal gain combining (EGC), and selection combining (SC) [6, 7]. The aim of these techniques is to find a set of weights 𝝎=[𝜔₁,𝜔₂,…,𝜔_𝑀], as shown in Figure 1, which optimizes a specific objective function. Here, the weights are selected to minimize the effect of fading on the received multiple signal components for each individual user. In MRC, the received signals are weighted accordingly so that the SNR at the output of the combiner is the sum of the average SNR of each branch. In EGC, on the other hand, the received signals are weighted equally and then added. In SC, the branch with the highest SNR is selected. In all cases, we consider that the receiver has the necessary information of channel fading.

Figure 1 

Diversity combining block diagram.

The performance of these methods has been extensively examined in the literature for Rayleigh fading. If the channel is perfectly estimated at the receiver, MRC can be applied to maximize the output SNR and minimize the bit error rate (BER) [8]. However, since the channel estimation is often imperfect in practice, the estimation error will decay the system performance. While this problem has long been investigated [9, 10], the recent evolutions in mobile communication systems have renewed the attention in comprehending and mitigating the effect of imperfect channel estimation on diversity techniques [11]. The error performance of MRC in Rayleigh fading environment with independent and identically distributed (i.i.d.) diversity branches is investigated in [12]. In [13], the SNR distribution is given for similar scenarios. In [14], the error performance of MRC with independent but not identically distributed (i.n.d.) branches is studied. In [15, 16], a comparison of hybrid SC/MRC scheme with SC and MRC schemes over Rayleigh fading channels in two scenarios of flat and exponentially decaying multipath intensity profile (MIP) has been done. In [17], the hybrid diversity scheme is studied as such selection combining and MRC are at the first and second stages, respectively. In [18], 𝐿 out of 𝑁 diversity branches was selected and combined using MRC over Rayleigh fading channel. The performance study of conventional MRC receiver in the presence of cochannel interference has also been a substantial interest of researchers [19–24]. Particularly in [24], the effect of the number of interferers on the diversity gain has been investigated in the context of frequency-selective Rayleigh fading. The study, however, has been done with the assumption of the prefect channel estimation of a desired user, which may not be the case in practice. The impact of imperfect channel estimation on the performance of diversity receivers in noise-limited circumstances has been presented in [25–30]. However, considering the frequency-nonselective fading, the investigation has been widened to circumstances with multiple cochannel interferers [31–33].

In this paper, to overcome the effect of imperfect estimation of channel state information, a diversity combining technique based on the imperialistic competitive algorithm (ICA) is proposed in which the signals received by the antennas are iteratively weighted based on ICA operation. The channel model used is slow flat Rayleigh fading. It should be noted that Rayleigh model is the simplest and the most controllable model, but it is not effective in all circumstances. However, since this paper basically aims at studying the use of evolutionary algorithms on receiver diversity, the authors believe that Rayleigh model is enough. Hence, the results given in this paper are only a ballpark figure of pros and cons of diversity methods and different algorithms to improve them. It is shown that the proposed combining method does not require the channel estimation, and it outperforms the MRC when channel estimation is imperfect. On the other hand, it has almost the same performance as MRC when channel estimation is assumed to be perfect. The ICA method shows faster convergence speed when compared with particle swarm optimization- (PSO-) and genetic algorithm- (GA-) based methods. This makes ICA a promising solution for the real-time applications.

2. System Model

In this paper, it is assumed that the information bits are modulated by binary phase-shift keying (BPSK) modulation. The channel is assumed to be frequency nonselective and slowly fading over the length of the transmitted symbol. We also assume that 𝑀 diversity branches are employed at the receiver for reception. In addition, this research work assumes that the diversity branches are sufficiently far apart from each other, so that the received signals are statistically independent with negligible correlation. This is a vital requisite to acquire the full advantage of the diversity receiver [5]. The received signal at the 𝑖th branch is given by 𝑟_𝑖(𝑡)=𝑔_𝑖𝑆(𝑡)+𝑛_𝑖,𝑖=1,2,…,𝑀, (1) where 𝑆(𝑡) is the unit-power transmitted signal and 𝑔_𝑖 denotes the complex channel gain with uncorrelated and Gaussian distributed real and imaginary parts, each with zero mean and variance 𝜎²_𝑔𝑖. The noise random variable 𝑛_𝑖 is complex additive white Gaussian noise (AWGN) with zero mean and variance 𝜎²_𝑛=𝑁₀/2. The channel gain 𝑔_𝑖 at two different diversity branches is assumed to be identically distributed. It is also assumed that 𝑔_𝑖 and 𝑛_𝑖 are uncorrelated. The signal power over one symbol period 𝑇_𝑠, at 𝑖th path, is 1𝑝=𝑇_𝑠^𝑇𝑠₀𝑔_𝑖²|𝑆(𝑡)|²𝑑𝑡=𝑔_𝑖²1𝑇_𝑠^𝑇𝑠₀|𝑆(𝑡)|²𝑑𝑡=𝑔_𝑖². (2) Since we are assuming slow fading, the term |𝑔_𝑖|² remains constant over a symbol period and can be taken out of the integral. 𝑆(𝑡) is assumed to have unit power. As a result, the instantaneous SNR at the 𝑖th path is 𝛾_𝑖=𝑔_𝑖²𝜎²_𝑛. (3) Since we are considering Rayleigh fading, 𝑔_𝑖=|𝑔_𝑖|𝑒^{𝑗∠𝑔𝑖} where ∠𝑔_𝑖 is uniformly distributed over [2𝜋,0] and 𝑔_𝑖 has a Rayleigh pdf. Therefore, |𝑔_𝑖|² and hence 𝛾_𝑖 have exponential pdf. Consider 𝑔_𝑖∼2𝑔_𝑖𝑃₀𝑒^{−|𝑔𝑖|2/𝑃0},𝛾_𝑖∼1Γ𝑒^{−𝛾𝑖/Γ},Γ=𝐸𝛾_𝑖=𝐸𝑔_𝑖²𝜎²_𝑛=𝑃₀𝜎²_𝑛. (4)𝑃₀ is the statistical average of |𝑔_𝑖|² and Γ represents the average SNR at each individual branch, which serves as a basic parameter to improve the SNR at the receiver.

The bit error rate (BER) in a BPSK system, given an SNR of 𝛾_𝑖, is identified by erfc2𝛾_𝑖, where erfc(𝑥)=(2/√𝜋)∫^∞_𝑥𝑒^−𝑡2𝑑𝑡 [12]. Therefore, the BER averaged over the Rayleigh fading in (4) is given by [13] BER=^∞₀2𝑔_𝑖𝑃₀𝑒^{−|𝑔𝑖|2/𝑃0}erfc2𝑔_𝑖𝜎_𝑛𝑑𝑔_𝑖=121−Γ.1+Γ (5) The physical model assumes the fading to be independent from one branch to the next. Each branch, therefore, acts as an independent sample of the random fading process (here, Rayleigh). It means each branch receives an independent copy of the transmitted signal. Our goal here is to combine these independent samples in a way to achieve the desired goal of increasing the SNR and reducing the BER.

3. Conventional Weighting Schemes

In this section, different combining schemes, such as selection combining (SC), equal gain combining (EGC) and maximal ratio combining (MRC) are investigated.

3.1. Selection Combining

In selection combining (SC), the branch with the greatest SNR is chosen as output SNR to be used in the next step: 𝝎_𝑖=1𝛾_𝑖=Max0otherwise. (6) The average output SNR for SC is defined as [14] 𝛾_𝑇=Γ^𝑀_𝑖=11𝑖1≅Γ𝐶−ln𝑀+,2𝑀 (7) in which 𝐶 is Euler’s constant. The final approximation is valid for 𝑀≥3. The overall BER is obtained by bringing together the conditional BER at a certain SNR. In BPSK modulation, the conditional BER is erfc2𝛾_𝑇 and the total BER is BER_𝑇=^∞₀𝑒𝑟𝑓𝑐2𝛾_𝑇𝑀Γ𝑒^𝛾𝑇/Γ1−𝑒^{𝛾𝑇/Γ𝑀−1}𝑑𝛾_𝑇. (8)

3.2. Equal Gain Combining

Equal gain combiner (EGC) sets unit gain at each branch to increase the average SNR in the system. In the equal gain combiner, 𝜔_𝑖=𝑒^{𝑗∠𝑔𝑖}⟹𝜔_𝑖∗𝑔_𝑖=𝑔_𝑖⟹𝝎𝐆^𝑇=^𝑀−1_𝑖=0𝑔_𝑖,𝐆=𝑔₁,𝑔₂,…,𝑔_𝑀,𝛾_𝑖=∑^𝑀−1_𝑖=0𝑔_𝑖²𝑀𝜎²_𝑛,𝛾_𝑇=𝐸∑^𝑀−1_𝑖=0𝑔_𝑖²𝑀𝜎²_𝑛=()𝜋1+𝑀−14Γ. (9) There is no closed form solution for the BER for general 𝑀, but several researchers have investigated the BER performance in several kinds of fading channels [15, 16].

3.3. Maximal Ratio Combining

In MRC, receiver linearly combines the received signal 𝑟_𝑖(𝑡) with 𝜔_𝑖, which is the weighting coefficient of the 𝑖th branch. The output signal 𝑟(𝑡) of the linear diversity combiner is then given by 𝑟(𝑡)=^𝑀_𝑖=1𝜔_𝑖𝑟_𝑖(𝑡)=𝑆(𝑡)^𝑀_𝑖=1𝜔_𝑖𝑔_𝑖+^𝑀_𝑖=1𝜔_𝑖𝑛_𝑖. (10) Since 𝑆(𝑡) is assumed to have unit power, SNR at the output of combiners is 𝛾_𝑇(𝝎)=1𝜎²_𝑛∑^𝑀_𝑖=1𝜔_𝑖𝑔_𝑖²∑^𝑀_𝑖=1𝜔_𝑖²=𝝎𝐆^𝑇2𝐸𝝎𝐍^𝑇2,𝐸𝝎𝐍^𝑇2=𝐸𝝎𝐍^𝑇𝝎^𝑇𝐍=𝜎²_𝑛𝝎². (11)(12) According to the Cauchy-Schwarz inequality, MRC with perfect channel estimation has maximum output SNR among all methods if 𝜔 is linearly proportional to 𝐺. If 𝝎=𝐆⇒𝛾_𝑇=|𝐆𝐆^𝑇|²/𝜎²_𝑛𝐆^𝑇𝐆=𝐆𝐆^𝑇/𝜎²_𝑛⇒𝛾_𝑇=∑^𝑀_𝑖=1|𝛾_𝑖|, the output SNR is, therefore, the sum of the SNR at each element. By using the above assumption, the expected value of the output SNR is therefore 𝑀 times the SNR at each branch.

For the case of imperfect channel estimation, which is the main issue in practice, it is observable that the SNR is highly dependent on 𝜔_𝑖. Therefore, the optimal solution is the weighting vector, which maximizes the objective function 𝛾_𝑇 in (11). We assume 𝑝_𝑖 is the estimate of the complex gain 𝑔_𝑖 on the 𝑖th diversity branch and 𝑒_𝑖 is the estimation error with zero mean and variance 𝜎²_𝑒=𝜎²_𝑔(1−𝜌²) where 𝜌∈[0,1] is the normalized estimation error correlation coefficient. Under Gaussian error model, 𝑔_𝑖 and 𝑝_𝑖 are related as 𝑔_𝑖=𝑝_𝑖+𝑒_𝑖 [16]. According to the diversity combining rule, the combiner’s weights take on the 𝜔_𝑖=𝑝^∗_𝑖 for MRC diversity, which is based on the Cauchy-Schwartz inequality, maximizes (11) if the channel is perfectly estimated (i.e., 𝜌=1). However, since channel estimation is often imperfect in practice, the MRC is a suboptimal solution [17–37].

4. Evolutionary Algorithm-Based Weighting Schemes

In this paper, the optimization problem is to maximize the output SNR of the combiner 𝛾_𝑇(𝝎) in (11) where 𝝎=[𝜔₁,𝜔₂,…,𝜔_𝑀] and 𝑀 is the number of variables (number of branches) of 𝛾_𝑇(𝝎) with 𝜔^𝑙≤𝜔≤𝜔^𝑢 where 𝜔^𝑙=0 and 𝜔^𝑢=1 are lower and upper limits on 𝜔, respectively. Thus, we propose to use evolutionary algorithms at the combiner so that all possible weighting vectors 𝝎 are investigated and the optimal one, which maximizes the output SNR in (11), is obtained. Hence, the need for estimating the channel state information is eliminated. As mentioned earlier, a simple Rayleigh channel model is satisfactory to illustrate the efficiency of the method. We mainly introduce an imperialist competitive algorithm (ICA) to find the optimal 𝝎 and compare its performance with two other iterative algorithms, namely, genetic algorithm (GA) and particle swarm optimization (PSO) to prove its effectiveness. The three algorithms of GA, PSO, and ICA are presented in the next sections.

4.1. Genetic Algorithm-Based Weighting Scheme

In the genetic algorithm (GA), a group of chromosomes will be arbitrarily generated. Equation (11) is used as the fitness function to evaluate the SNR of randomly generated chromosomes of the initial population. Then, a new population from the former population will be reproduced based on the fitness scores (output SNR values) of its chromosomes and the process is repeated until a predefined termination criterion is met [36]. Better populations can be continually formed due to the concept of surviving the fit/best chromosomes. In GA terminology, the evolutionary process of forming an offspring population from a parent population is called generation [37]. The number of produced generations is predetermined by the designer or self-set based on the quality of obtainable solutions. The algorithm is configured to maximize the SNR and it is outlined as follows.

Step 1. Randomly generate a population of pops chromosomes.

Step 2. Decode each chromosome into its corresponding weighting vector 𝝎_𝑗=[𝜔_𝑗1,𝜔_𝑗2,…,𝜔_𝑗𝑀]^𝑇, where 𝜔_𝑗𝑖∈[0,1], 𝑖=1,2,…,𝑀, and 𝑗=1,2,…,𝑝𝑜𝑝𝑠.

Step 3. Compute the SNR value of every decoded weighting vector 𝝎_𝑗 using (11) and rank and identify the best ⌊𝑝𝑜𝑝𝑠∗𝑒𝑙𝑖𝑡𝑒⌋ chromosomes that have maximized SNR. elite is a parameter that determines a fraction of pops, that is, 𝑒𝑙𝑖𝑡𝑒∈[0,1), and ⌊⋅⌋ denotes the floor operation.

Step 4. After large-enough generations (runs of the algorithm), if the output SNR of the system converges to a stable value at each iteration, the procedure is terminated. Otherwise, increase the generation number by one.

Step 5. Reproduce ⌈𝑝𝑜𝑝𝑠∗(1−𝑒𝑙𝑖𝑡𝑒)⌉ new chromosomes where ⌈⋅⌉ denotes ceiling operation, and construct new population by concatenating the newly ⌈𝑝𝑜𝑝𝑠∗(1−𝑒𝑙𝑖𝑡𝑒)⌉ reproduced chromosomes with the best ⌊𝑝𝑜𝑝𝑠∗𝑒𝑙𝑖𝑡𝑒⌋ found in Step 3. Jump to Step 2.

Finally, the optimal weighting vector (decoded chromosomes) that leads to the highest stable value of the output SNR can be indicated and used.

4.2. Particle Swarm Optimization-Based Weighting Scheme

PSO algorithm is abstracted from the social behavior of swarm of fish and birds. The behavior of these social organizations is emulated by the PSO algorithm. Each particle in PSO algorithm functions based on its own knowledge as well as the group knowledge and has two main features: position and velocity. In each iteration, information about the best position is cooperatively exchanged among the particles. The steps involved in the PSO algorithm are as follows.

Step 1. Randomly generate 𝑁 number of particle positions (weighting vectors) as 𝝎_𝑠=[𝜔₁,𝜔₂,…,𝜔_𝑀]^𝑇, (𝑠=1,…,𝑁) and 𝑁 number of length-𝑀 velocity vectors 𝐯_𝑠^(𝑗), which are initially set to zero. Here, particle position and velocity at iteration 𝑗 are demonstrated by 𝝎_𝑠^(𝑗) and 𝐯_𝑠^(𝑗), respectively.

Step 2. Calculate the objective function (SNR in (11)) for particle positions as 𝛾_𝑇(𝝎₁^(𝑗)), 𝛾_𝑇(𝝎₂^(𝑗))⋯𝛾_𝑇(𝝎_𝑁^(𝑗)). Find the maximum SNR and name its corresponding position as 𝑃_best,𝑗. The best experienced particle position among all iterations is called global best position and is expressed by 𝐺𝑙_best.

Step 3. Update the velocity of the particles by 𝐯_𝑠^(𝑗)=𝐯_𝑠^(𝑗−1)+𝑐₁𝑟₁𝑃_best,𝑗−𝝎_𝑠^(𝑗−1)+𝑐₂𝑟₂𝐺𝑙_best−𝝎_𝑠^(𝑗−1), (13) where individual and social learning acceleration coefficients are, respectively, denoted by 𝑐₁ and 𝑐₂ and 𝑟₁ and 𝑟₂ which are the random numbers between 0 and 1.

Step 4. Update the position of particles as follows: 𝝎_𝑠^(𝑗)=𝝎_𝑠^(𝑗−1)+𝐯_𝑠^(𝑗). (14)

Step 5. Check the convergence. The output SNR in (11) is regularly checked at each iteration. After a large-enough number of iterations, if the algorithm results in the same output SNR in each iteration, the procedure is terminated. Otherwise, set 𝑗=𝑗+1 and the process is repeated from Step 2.

Therefore, the value of the 𝐺𝑙_best is the optimal weighting vector that maximizes the SNR at the output of the combiner.

4.3. Modified Imperialist Competitive Algorithm-Based Weighting Scheme

It is considerably obvious that genetic and physical evolution does not happen as fast as the communal and the academic evolution of human being. Due to this fact, some developing algorithms have applied the cultural side of social life in order to reach well outcomes. Imperialistic competition and human’s sociopolitical evolution inspire ICA [38–40]. ICA algorithm has not been deep rooted in refining diversity combining issue to the best knowledge of the author. Hence, checking the effectiveness of the algorithm in comparison to other techniques is the main disquiet of this research. The main steps of ICA are explained as follows.

Step 1. Generate 𝑁_pop numbers of countries (combiner’s weighting vector shown in Figure 1) as 𝝎_𝑘=[𝜔₁,𝜔₂,…,𝜔_𝑀]^𝑇 where 𝑘=1,…,𝑁_pop. The SNR value of each country, based on (11), is calculated and sorted.

Step 2. 𝑁_imp of most powerful (in terms of SNR) countries are chosen as imperialists to form empires and the rest of 𝑁_col countries are called colonies. Figure 2(a) depicts the initial colonies for each empire. The initial number of colonies for an empire is randomly selected from 𝑁_col with respect to the empire’s imperialist power (𝑝_imp), which is its corresponding normalized SNR: Initialnumberofcoloniesinanempire=𝑁_𝑐=round𝑁_col⋅𝑝_imp. (15)

(a) Imperialists and colonies in each empire

(b) Movement of colony towards imperialist

Figure 2 

Current and future position of imperialists and colonies in imperialistic competitive algorithm.

Step 3. Colonies in an empire start to move in the search space towards an imperialist state in different directions (assimilation). 𝐱_𝑗=[𝑥₁,𝑥₂,…,𝑥_𝑀]^𝑇(𝑗=1,…,𝑁_𝑐) is the transferred distance of the 𝑗th colony, which is randomly chosen from the interval of [𝟎,^′Υ⋅𝐝_𝑗] where 𝟎 is a 1-by-𝑀 zero vector, ^′Υ is the assimilation coefficient (0<^′Υ≤2), and 𝐝_𝑗=[𝑑₁,𝑑₂,…,𝑑_𝑀]^𝑇 is the distance between the imperialist and 𝑗th colony in an empire, which is calculated by 𝐝_𝑗=𝝎_imp−𝝎_col𝑗=𝜔_1,imp−𝜔_1,col𝑗,𝜔_2,imp−𝜔_2,col𝑗,…,𝜔_𝑀,imp−𝜔_𝑀,col𝑗^𝑇. (16) Therefore, the new position of the 𝑗th colony is calculated as follows [41]: 𝝎_new,col𝑗=𝝎_old,col𝑗+𝐱_𝑗+𝐫⋅tan(𝜃), (17) where 𝐫 is a 1-by-𝑀 random vector, whose values are uniformly distributed on (−1,+1) and 𝜃 is assimilation deviation which can be chosen from −𝜋/2<𝜃<𝜋/2. Figure 2(b) depicts how colonies transfer to their related imperialist.

Step 4. The cost of each colony in the new position is again computed based on (11). Position exchange between a colony and imperialist can happen in this step. In other words, if a colony in its new position has a higher SNR than that of the imperialists, it has the chance to take the control of empire by replacing the existing imperialist. Consider 𝛾_𝑇𝝎_new,col𝑗>𝛾_𝑇𝝎_imp⟹𝑗thcolonywillbecometheimpersialist. (18)

Step 5. Imperialistic competition is being performed in this step. The colony with the lowest SNR value from the empire with the weakest power is chosen and provided to one of the best empires. The total power (in terms of SNR) of an empire is calculated as follows: Totalempire’spower=𝛾_𝑇𝝎_imp∑+𝜉^{𝑁𝑐𝑗=1}𝛾_𝑇𝝎_col𝑗𝑁_𝑐, (19) where positive number, 𝜉, is equal to or less than one (0<𝜉≤1).

Step 6. When all colonies of an empire move to other powerful empires and just imperialist remains, this imperialist automatically joins best empire as a simple colony. This empire will then be removed.

Step 7. Stop condition will satisfy, if only one empire remains. In other words, after a while, only one empire with the highest total power (as in (19)) remains, which controls all the colonies. In this condition, all of the colonies and the imperialists have the same position (weighting vector) and cost (SNR at (11)). Otherwise, algorithm jumps to Step 3.
The equivalent weighting vector of the final imperialist is the best vector that maximizes the output SNR of our diversity problem here.
Figure 3 abstractly shows the flowchart of ICA, which explains how ICA is applied to improve the reliability of the wireless communication systems.

Figure 3 

ICA-based flowchart that maximizes the quality of the received signal in diversity combining technique.

5. Numerical Results and Discussion

In this section, Monte-Carlo simulation is employed to present the performance of the proposed ICA-based diversity combining technique and compare it with PSO, GA, MRC, EGC, and SC methods in two different scenarios of the perfect and imperfect channel estimation. It is assumed that the average symbol energy 𝐸_𝑠=1 and channel gain and AWGN variances are 𝜎²_𝑔=𝜎²_𝑛=0.5 per dimension. The parameters for the PSO are 𝑁=25 and 𝑐₁=𝑐₂=2. Figure 4 compares the normalized output SNR of ICA-, PSO-, and GA-based combining with MRC, EGC, and SC in terms of different numbers of diversity branches when the channel is perfectly estimated (𝜌=1) [42]. As expected, it has been observed that the MRC provides the best performance when channel estimation is perfect. However, the ICA- and PSO-based solutions demonstrate almost the same SNR gain as MRC without the need for channel estimation. Since the parameters in each algorithm are generally problem-dependent, the set-and-test approach is used in this work to obtain the optimal values for them. In other words, 𝑐₁𝑟₁ and 𝑐₂𝑟₂ in PSO or 𝜃,^′Υ in ICA guarantee that the particles or colonies would fly over the target about half the time. In this respect, the environment has been tested separately for parameters as mentioned in Table 1 and the optimal value is found in Table 2.

Table 1 

Different parameters values used for testing.

Table 2 

Optimal parameter values for ICA, PSO, and GA algorithms which maximize the output SNR.

Figure 4 

Normalized output SNR of MRC-, ICA-, PSO-, GA-, EGC-, and SC-based methods when the channel is perfectly estimated.

The comparison between the iterative based algorithms and MRC methods in the case of imperfect channel estimation (𝜌=0,0.5,0.75) [43, 44] is illustrated in Figure 5. It can be seen that ICA- and PSO-based methods outperform MRC when channel estimation is imperfect. The achieved improvement can be justified by the ability of the algorithms to investigate the search space thoroughly and evaluate the objective function in (11) to maximize the output SNR. As it is shown in Figures 4 and 5, PSO and ICA results are quite close to each other. However, on the other hand, Figures 6 and 7 present the superiority of ICA over PSO in terms of achievable BER and SNR, respectively. These two metrics declare that the quality of the diversity performance achieved by ICA is quite better than that of PSO. However, 𝑡-test has been carried out to provide an evidence of statistical significance in the difference of means of these two algorithms. With a significance level of 0.10, it has been found that the two-tailed 𝑃 value is 0.0805, which means that the results are considered statistically significant.

Figure 5 

Comparison of normalized output SNR of ICA-, PSO-, GA-, MRC-, EGC-, and SC-based methods for imperfect channel estimation.

Figure 6 

Error performance of ICA-, PSO-, and MRC-based methods for different numbers of diversity branches.

Figure 7 

Convergence performance of the iterative algorithms.

Considering the BPSK modulation and imperfect channel estimation, the error performance of the MRC-, ICA- and PSO-based methods for 1, 2, and 3 diversity branches is illustrated in Figure 6. It is observable that the bit error rate of the ICA-based technique is considerably lower than that of the MRC. For instance, for a two-branch diversity, the MRC approximately requires almost 3 dB higher SNR than that of ICA-based to achieve a 𝐵𝐸𝑅=10⁻⁴. In addition, as it is shown, increasing the number of branches results in improved error performance.

Next, Figure 7 compares the convergence of ICA, PSO, and GA algorithms used in the diversity method. The number of diversity branches is assumed to be 8. The mean and max of each algorithm are achieved when the algorithms run for 100 times. The average of all results is called mean and the best one among these 100 simulations, which results in the maximum output SNR, is named as max. As it is shown in the figure, max curve in ICA method converges after 18 iterations whereas about 31 iterations of PSO algorithm are needed for convergence. This indicates the higher convergence speed of the ICA compared to PSO.

Table 3 shows the details of convergence speed for each method. The term NA in Table 3 indicates that the iteration number for that specific condition is not available. For instance, ICA-based method with 5 countries cannot converge in 100 iterations. Moreover, the number of fitness evaluations as a parameter to compare the complexity of iterative algorithms has been provided in Table 3. The number of fitness evaluations is simply the product of the number of generations by which the maximum SNR fitness is achieved multiplied by the number of fitness evaluations performed in every iteration. The latter equals the population size of any of these algorithms. For instance, with ICA, the number of iterations required to achieve the maximum SNR is 18 and the number of countries is 25. This means that the number of fitness evaluations to find the optimal setting is 450, which is considerably low with the advancement of signal processing and computing cores.

Table 3 

Performance comparison of ICA and PSO assisted for different population numbers.

The SNR variances of ICA, PSO, and GA are shown in Table 4 and are recorded every five iterations until the 55th iteration after which the variances are zeroed when all colonies, particles, and chromosomes of ICA, PSO, and GA, respectively, converge to the same optima. Considering the values in the table and calculating standard deviations at each iteration, one can conclude that ICA, with all of its fluctuations around its mean, can still outperform the other two algorithms. This validates the superiority of this algorithm in comparison with the other methods.

Table 4 

Variance of SNR of all algorithms when the population size is 25.

6. Conclusion

One of the most important issues in reception antenna diversity occurs when the channel is imperfectly estimated. This defective estimation results in obtaining a vector of the weighting coefficient of the combiner that deteriorates the SNR and BER performance of the system at the receiver. To address the issue, an ICA-based diversity combining method is proposed to optimize the weighting vector, which is used to combine the received signals at the receiver. Simulation results validate that the proposed method provides better SNR and error performance than that of other evolutionary algorithms, such as GA and PSO and conventional MRC when channel estimation is imperfect. On the other hand, in the perfect channel estimation environment, the proposed method performs as effectively as the MRC.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This research work is supported by the University of Malaya High Impact Research (HIR) Grant (no. 51) sponsored by the Ministry of Higher Education (MOHE), Malaysia.