Symbol Error Rate for Nonblind Adaptive Equalizers Applicable for the SIMO and FGn Case

Abstract

A nonzero residual intersymbol interference (ISI) causes the symbol error rate (SER) to increase where the achievable SER may not answer any more on the system’s requirements. Recently, a closed-form approximated expression was derived by the same author for the residual ISI obtained by nonblind adaptive equalizers for the single-input single-output (SISO) case. Up to now, there does not exist a closed-form expression for the residual ISI obtained by nonblind adaptive equalizers for the single-input multiple-output (SIMO) case. Furthermore, there does not exist a closed-form expression for the SER valid for the SISO or SIMO case that takes into account the residual ISI obtained by nonblind adaptive equalizers and is valid for fractional Gaussian noise (fGn) input where the Hurst exponent is in the region of 0.5≤𝐻<1. In this paper, we derive a closed-form approximated expression for the residual ISI obtained by nonblind adaptive equalizers for the SIMO case (where SISO is a special case of SIMO), valid for fGn input where the Hurst exponent is in the region of 0.5≤𝐻<1. Based on this new expression for the residual ISI, a closed-form approximated expression is obtained for the SER valid for the SIMO and fGn case.

1. Introduction

We consider a nonblind deconvolution problem in which we observe the multiple output of a finite impulse-response (FIR) single-input multiple-output (SIMO) channel (unknown channel) from which we want to recover its input using adjustable linear filters (equalizers) and training symbols. In the field of communication, SIMO channels appear either when the signal is oversampled at the receiver or from the use of an array of antennas in the receiver [1–6]. It should be pointed out that, for the SIMO case, the same information is transmitted through different subchannels and all received sequences will be distinctly distorted versions of the same message, which accounts for a certain signal diversity [6, 7]. Therefore, it is reasonable to assume that more information about the transmitted signal will be available at the receiver end [6, 7]. SIMO transmission is widely replacing single-input single-output (SISO) approach to enhance the performance via diversity combining [6, 8]. It is well known that ISI is a limiting factor in many communication environments where it causes an irreducible degradation of the bit error rate (BER) and SER, thus imposing an upper limit on the data symbol rate [1, 6]. In order to overcome the ISI problem, an equalizer is implemented in those systems [6]. For the blind SIMO channel equalization case, we may find the following methods [9–13] which exploit various properties of the input signal.

In this paper we consider the nonblind adaptive equalizer where training sequences are needed to generate the error that is fed into the adaptive mechanism which updates the equalizer’s taps [14–17]. The nonblind adaptive approach yields in most cases better equalization performance considering convergence speed and equalization quality compared with the blind adaptive version [18]. In addition, the blind adaptive version has a higher computational cost compared to its nonblind approach [18].

Recently [19], a closed-form approximated expression (or an upper limit) was derived for the residual ISI obtained by nonblind adaptive equalizers for the SISO case that depends on the stepsize parameter, equalizer’s tap length, input signal statistics, signal-to-noise ratio (SNR), and channel power and is valid for the fGn case where the Hurst exponent is in the region of 0.5≤𝐻<1. But this expression is not valid for the SIMO case. Thus, there is still no answer on the question of how the number of receiving antennas contributes to the residual ISI obtained by nonblind adaptive equalizers in the convergence region for the fGn input case with 0.5≤𝐻<1. Since up to now there does not exist a closed-form expression for the residual ISI obtained by nonblind adaptive equalizers for the SIMO case, no closed-form expression exists either for the SER valid for the SIMO case that takes into account the residual ISI.

In this paper, we propose a closed-form approximated expression for the residual ISI obtained by nonblind adaptive equalizers for the SIMO case (where SISO is a special case of SIMO). This new expression depends on the stepsize parameter, equalizer’s tap length, input signal statistics, SNR, channel power, and the number of receiving antennas and is valid for the fGn case where the Hurst exponent is in the region of 0.5≤𝐻<1. Please note that 𝐻=1 is the limit case, which does not have much practical sense [20–22]. It should be pointed out that a white Gaussian process is a special case (𝐻=0.5) of the fGn model [23]. FGn with 𝐻∈(0.5,1) corresponds to the case of long-range dependency (LRD) [23]. Thus, the new obtained expression for the achievable residual ISI is not only valid for the special case of white Gaussian process but covers also those cases that correspond to the case of LRD. The Hurst exponent 𝐻 may be estimated with the rescaled range (R/S) method [24, 25]. Based on the new proposed expression for the residual ISI, a closed-form approximated expression is obtained for the SER valid for the SIMO and fGn case where nonblind adaptive equalizers are used in the system.

The paper is organized as follows. After having described the system under consideration in Section 2, the closed-form approximated expressions for the residual ISI and SER are introduced in Section 3. In Section 4 simulation results are presented and the conclusion is given in Section 5.

2. System Description

The system under consideration, illustrated in Figure 1, is the same system described in [6]. In this paper, we make the following assumptions as done in [6].(1)The source sequence 𝑥[𝑛] belongs to a real or two independent quadrature carrier case constellation inputs with variance 𝜎²_𝑥 where 𝑥_𝑟[𝑛] and 𝑥_𝑖[𝑛] are the real and imaginary parts of [𝑛], respectively.(2)The unknown subchannel ℎ^(𝑚)[𝑛] (𝑚=1,2,3,…,𝑀 where 𝑀 is the number of subchannels) is a possibly nonminimum phase linear time-invariant filter. There is no common zero among all the subchannels. There are 𝑀 subchannels since there are 𝑀 receiving antennas.(3)Each equalizer 𝑐^(𝑚)[𝑛] (𝑚=1,2,3,…,𝑀) is a tap-delay line.(4)The noise 𝑤^(𝑚)[𝑛] (𝑚=1,2,3,…,𝑀) consists of 𝑤^(𝑚)[𝑛]=𝑤_𝑟^(𝑚)[𝑛]+𝑗𝑤_𝑖^(𝑚)[𝑛] where 𝑤_𝑟^(𝑚)[𝑛] and 𝑤_𝑖^(𝑚)[𝑛] are the real and imaginary parts of 𝑤^(𝑚)[𝑛], respectively, and 𝑤_𝑟^(𝑚)[𝑛] and 𝑤_𝑖^(𝑚)[𝑛] are independent. Both 𝑤_𝑟^(𝑚)[𝑛] and 𝑤_𝑖^(𝑚)[𝑛] are fractional Gaussian noises (fGn) with zero mean.

Figure 1 

Block diagram of a baseband SIMO communication system.

 For 𝑚=𝑘 we have 𝐸𝑤_𝑟^(𝑚)[𝑛]𝑤_𝑟^(𝑘)[𝑛]=𝜎²_𝑤𝑟𝛿[𝑛],𝐸𝑤𝑛−_𝑖(𝑚)[𝑛]𝑤_𝑖^(𝑘)[𝑛]=𝜎²_𝑤𝑖𝛿[𝑛−𝑛]. (1) For 𝑚≠𝑘 we have 𝐸𝑤_𝑟^(𝑚)[𝑛]𝑤_𝑟^(𝑘)[𝑛=𝜎_2𝑤𝑟]2(|𝑚−𝑘|−1)^2𝐻−2(|𝑚−𝑘|)^2𝐻+(|𝑚−𝑘|+1)^2𝐻[𝑛],𝐸𝑤×𝛿𝑛−_𝑖(𝑚)[𝑛]𝑤_𝑖^(𝑘)[𝑛=𝜎_2𝑤𝑖]2(|𝑚−𝑘|−1)^2𝐻−2(|𝑚−𝑘|)^2𝐻+(|𝑚−𝑘|+1)^2𝐻×𝛿[𝑛−𝑛], (2)where 𝐸[⋅] denotes the expectation operator on (⋅), 𝛿 is the Kronecker delta function, and 𝐻 is the Hurst exponent. In the following, we assume that 𝜎²_𝑤𝑖=𝜎²_𝑤𝑟 and 𝜎²_𝑤=𝜎²_𝑤𝑖+𝜎²_𝑤𝑟.

According to Figure 1, the 𝑚th observation 𝑦^(𝑚)[𝑛] (𝑦^(𝑚)[𝑛]=𝑥[𝑛]∗ℎ^(𝑚)[𝑛]+𝑤^(𝑚)[𝑛]) is the result of a linear convolution between the source signal 𝑥[𝑛] and the corresponding channel response ℎ^(𝑚)[𝑛], corrupted by noise 𝑤^(𝑚)[𝑛], where “∗” denotes the convolution operation. The equalizer’s output 𝑧[𝑛] is derived as follows: 𝑧[𝑛]=^𝑚=𝑀_𝑚=1𝑧^(𝑚)[𝑛]=^𝑚=𝑀_𝑚=1𝑦^(𝑚)[𝑛]∗𝑐^(𝑚)[𝑛]=^𝑚=𝑀_𝑚=1𝑥[𝑛]∗ℎ^(𝑚)[𝑛]∗𝑐^(𝑚)[𝑛]+𝑤^(𝑚)[𝑛]∗𝑐^(𝑚)[𝑛]=𝑥[𝑛]+𝑝[𝑛]+𝑤[𝑛], (3) where 𝑝[𝑛] is the convolutional noise (𝑝[𝑛]=𝑥[𝑛]∗𝜉[𝑛]), 𝜉[𝑛]=∑^𝑚=𝑀_𝑚=1ℎ^(𝑚)[𝑛]∗𝑐^(𝑚)[𝑛]−𝛿[𝑛], and 𝑤[𝑛]=∑^𝑚=𝑀_𝑚=1𝑤^(𝑚)[𝑛]∗𝑐^(𝑚)[𝑛]. Next we turn to the adaptation mechanism of the equalizer in each subchannel which is based on training symbols [14–17, 26] 𝑐^(𝑚)[𝑛+1]=𝑐^(𝑚)[𝑛]−𝜇^(𝑚)(𝑧[𝑛]−𝑥[𝑛])𝑦^(𝑚)∗[𝑛]=𝑐^(𝑚)[𝑛]−𝜇^(𝑚)𝑃[𝑧[𝑛]]𝑦^(𝑚)∗[𝑛], (4) where 𝜇^(𝑚) is the stepsize parameter in the subchannel, 𝑃[𝑧[𝑛]]=𝑧[𝑛]−𝑥[𝑛],𝑐^(𝑚)[𝑛] is the equalizer vector where the input vector is 𝑦^(𝑚)[𝑛]=[𝑦^(𝑚)[𝑛]⋯𝑦^(𝑚)[𝑛−𝑁+1]]^𝑇, and 𝑁 is the equalizer’s tap length. The operators (⋅)^𝑇 and (⋅)^∗ denote for transpose and conjugate of the function (⋅), respectively.

3. Residual ISI and SER for the SIMO and Fractional Gaussian Noise Case

In this section, a closed-form approximated expression is derived for the residual ISI valid for the SIMO and fGn case. Based on this new expression, the SER is obtained.

3.1. Derivation of the Residual ISI

In this subsection we derive the residual ISI valid for the SIMO and fGn case.

Theorem 1. For the following assumptions:(1)the convolutional noise, 𝑝[𝑛], is a zero mean, white Gaussian process with variance 𝜎²_𝑝=𝐸[𝑝[𝑛]𝑝^∗[𝑛]]. The real part of 𝑝[𝑛] is denoted as 𝑝_𝑟[𝑛] and 𝐸[𝑝²_𝑟[𝑛]]=𝑚_𝑝;(2)the source signal 𝑥[𝑛] is a square QAM (quadrature amplitude modulation) signal (where the real part of 𝑥[𝑛] is independent with the imaginary part of 𝑥[𝑛]) with known variance and higher moments;(3)the convolutional noise 𝑝[𝑛] and the source signal are independent;(4)the gain between the source and equalized output signal is equal to one;(5)the convolutional noise 𝑝[𝑛] is independent with 𝑤[𝑛];(6)the added noise is fGn as defined in the previous section in assumption (4);(7)the Hurst exponent is in the range of 0.5≤𝐻<1,

the residual ISI expressed in [dB] units may be defined as ISI=10log₁₀𝑚_𝑝−10log₁₀𝜎²_𝑥𝑟, (5) where 𝑚_𝑝=𝜎𝑤²_𝑟𝑁𝜎²_𝑥∑^𝑚=𝑀_𝑚=1𝜇^(𝑚)∑^{𝑘=𝑅−1}_𝑘=0ℎ_𝑘^(𝑚)[𝑛]²+(1/SNR)2−𝑁𝜎²_𝑥∑^𝑚=𝑀_𝑚=1𝜇^(𝑚)∑^{𝑘=𝑅−1}_𝑘=0ℎ_𝑘^(𝑚)[𝑛]²+(1/SNR) (6) with 𝜎𝑤²_𝑟≅^𝑀_𝑚=1𝜎²_𝑥𝑟𝑀²SNR∑^{𝑘=𝑅−1}_𝑘=0ℎ_𝑘^(𝑚)[𝑛]²×(1+𝐻(2𝐻−1)(𝑀−1)), (7) where SNR=𝜎²_𝑥/𝜎²_𝑤 and 𝜎²_𝑥𝑟 is the variance of 𝑥_𝑟[𝑛].

Comments. It should be pointed out that assumptions (1)--(7) from above are precisely the same assumptions made in [6]. In addition, assumptions (1), (3), and (5) were also used in [18, 27–33] where [31] explained why assumptions (1) and (3) can hold for the latter stages of the deconvolution process for which our residual ISI is derived. Since the noise and the source signal are independent in practical applications, assumption (5) holds. For a square QAM constellation, the real and imaginary parts of the constellation input are always independent and, in practical applications, the update mechanism of the equalizer is given by (4) where the gain between the equalized output signal 𝑧[𝑛] and the source signal 𝑥[𝑛] is equal to one. Thus, assumptions (2) and (4) are reasonable and are met in practical applications. FGn gains wide applications in various fields, ranging from geosciences to telecommunications [34]. FGn is a commonly used model of network traffic with LRD [35]. This traffic may for instance play as an interferer on other communication links. Thus, assumptions (6) and (7) are also reasonable and can be met in practical applications.

Proof. The real part of 𝑤[𝑛], namely, 𝑤_𝑟[𝑛], may be expressed as 𝑤_𝑟[𝑛]=^𝑀_𝑚=1^{𝑘=𝑁−1}_𝑘=0𝑐_𝑟^(𝑚)[𝑘]𝑤_𝑟^(𝑚)[𝑛−𝑘]−𝑐_𝑖^(𝑚)[𝑘]𝑤_𝑖^(𝑚)[𝑛−𝑘], (8) where 𝑐_𝑟^(𝑚)[𝑘] and 𝑐_𝑖^(𝑚)[𝑘] are the real and imaginary parts of 𝑐^(𝑚)[𝑘], respectively. The variance of 𝑤_𝑟[𝑛] may be expressed by 𝜎𝑤²_𝑟=𝐸^𝑀_𝑚=1^{𝑘=𝑁−1}_𝑘=0𝑐_𝑟^(𝑚)[𝑘]𝑤_𝑟^(𝑚)[𝑛−𝑘]−𝑐_𝑖^(𝑚)[𝑘]𝑤_𝑖^(𝑚)[]⋅𝑛−𝑘_{𝑀𝑝=1𝑘𝑘=𝑁−1𝑘𝑘=0}𝑐_𝑟^(𝑝)[𝑘𝑘]𝑤_𝑟^(𝑝)[𝑛−𝑘𝑘]−𝑐_𝑖^(𝑝)[𝑘𝑘]𝑤_𝑖^(𝑝)[𝑛−𝑘𝑘] (9) which can be written according to [6] as 𝜎𝑤²_𝑟=^𝑀_𝑚=1𝜎²_𝑤𝑟^{𝑘=𝑁−1}_𝑘=0𝑐^(𝑚)[𝑘]²+^𝑀𝑀_{𝑚=1,𝑚≠𝑝𝑝=1,𝑝≠𝑚}𝜎²_𝑤𝑟2𝑚−𝑝−1^2𝐻−2𝑚−𝑝^2𝐻+𝑚−𝑝+1^2𝐻⋅^{𝑘=𝑁−1}_𝑘=0𝑐_𝑟^(𝑚)[𝑘]𝑐_𝑟^(𝑝)[𝑘]+𝑐_𝑖^(𝑚)[𝑘]𝑐_𝑖^(𝑝)[𝑘]. (10) This expression (10) was shown in [6] to be equal to (7) with the help of the triangle inequality [36], the Holder inequality [36], and the approximation given by [34] 0.5𝑚−𝑝−1^2𝐻−2𝑚−𝑝^2𝐻+𝑚−𝑝+1^2𝐻≃𝐻(2𝐻−1)𝑚−𝑝^2𝐻−2. (11) Next we turn to the expression for 𝑚_𝑝. According to [1], 𝐸Δ𝑝²_𝑟=−2𝐸𝑝_𝑟𝑃_𝑟[𝑧[𝑛]]^𝑚=𝑀_𝑚=1𝜇^(𝑚)𝑦^(𝑚)[𝑛]²+ℜ^𝑚=𝑀_𝑚=1Δ₍₎^(𝑚)_𝑤∗𝑐+𝐸−𝑃_𝑟[𝑧[𝑛]]^𝑚=𝑀_𝑚=1𝜇^(𝑚)𝑦^(𝑚)[𝑛]²−ℜ^𝑚=𝑀_𝑚=1Δ₍₎^(𝑚)_𝑤∗𝑐², (12) where Δ(𝑝²_𝑟)=𝑝²_𝑟[𝑛+1]−𝑝²_𝑟[𝑛], ℜ(⋅) is the real part of (⋅), 𝑃_𝑟[𝑧[𝑛]] is the real part of 𝑃[𝑧[𝑛]], and Δ^(𝑚)_{(𝑤∗𝑐)}=𝑤^(𝑚)[𝑛]∗𝑐^(𝑚)[𝑛+1]−𝑤^(𝑚)[𝑛]∗𝑐^(𝑚)[𝑛]=𝑤^(𝑚)∗(𝑐^(𝑚)[𝑛+1]−𝑐^(𝑚)[𝑛]). By using (3) and (4), we may have for the nonblind case 𝑃_𝑟[𝑧[𝑛]]=𝑝_𝑟[𝑛]+𝑤_𝑟[𝑛]=𝑝_𝑟+𝑤_𝑟[𝑛] (where 𝑝_𝑟=𝑝_𝑟[𝑛]). According to [1], when the equalizer has converged we have ∑^𝑚=𝑀_𝑚=1Δ^(𝑚)_{(𝑤∗𝑐)}→0 and 𝐸[Δ(𝑝²_𝑟)]≅0. Thus, after substituting 𝑃_𝑟[𝑧[𝑛]]=𝑝_𝑟+𝑤_𝑟[𝑛] and 𝐸[Δ(𝑝²_𝑟)]=0 into (12) we obtain −2𝐸𝑝_𝑟𝑝_𝑟+𝑤_𝑟[𝑛]^𝑚=𝑀_𝑚=1𝜇^(𝑚)𝐸𝑦^(𝑚)[𝑛]²+𝐸𝑝_𝑟+𝑤_𝑟[𝑛]²𝐸^𝑚=𝑀_𝑚=1𝜇^(𝑚)𝑦^(𝑚)[𝑛]²²=0. (13) According to [1], 𝐸[(‖𝑦^(𝑚)[𝑛]‖²)²] may be approximated as (𝐸[‖𝑦^(𝑚)[𝑛]‖²])². Thus we obtain −2𝑚_𝑝𝐸^𝑚=𝑀_𝑚=1𝜇^(𝑚)𝑦^(𝑚)[𝑛]²+𝑚_𝑝+𝜎𝑤²_𝑟𝐸^𝑚=𝑀_𝑚=1𝜇^(𝑚)𝑦^(𝑚)[𝑛]²²=0. (14) From (14) we have 𝑚_𝑝=𝜎𝑤²_𝑟𝐸∑^𝑚=𝑀_𝑚=1𝜇^(𝑚)𝑦^(𝑚)[𝑛]²²2𝐸∑^𝑚=𝑀_𝑚=1𝜇^(𝑚)𝑦^(𝑚)[𝑛]²−𝐸∑^𝑚=𝑀_𝑚=1𝜇^(𝑚)𝑦^(𝑚)[𝑛]²²=𝜎𝑤²_𝑟𝐸∑^𝑚=𝑀_𝑚=1𝜇^(𝑚)𝑦^(𝑚)[𝑛]²2−𝐸∑^𝑚=𝑀_𝑚=1𝜇^(𝑚)𝑦^(𝑚)[𝑛]². (15) Next we substitute [1] 𝐸[‖𝑦^(𝑚)[𝑛]‖²]=𝑁𝜎²_𝑥(∑^{𝑘=𝑅−1}_𝑘=0|ℎ_𝑘^(𝑚)[𝑛]|²+(1/SNR)) into (15) and obtain the expression for 𝑚_𝑝 given in (6). This completes our proof.

3.2. Derivation of the SER for the SIMO Case

In this subsection, we derive the SER for a source signal 𝑥[𝑛] belonging to a rectangular QAM constellation, applicable for the SIMO and fGn case.

Theorem 2. Based on the assumptions given in the previous subsection, the SER for the nonblind adaptive version valid for the SIMO and fGn case may be defined as SER_QAM=4𝑀−1𝑀𝑄𝑑𝜎_𝑇1−𝑀−1𝑀𝑄𝑑𝜎_𝑇, (16) where 𝑀=𝑀_QAM and 𝑀_QAM is the number of signal points for a 𝑀_QAM-ary QAM constellation, 𝑑 is half the distance between adjacent 𝑀_QAM-ary PAM (pulse amplitude modulation) signals: 𝜎_𝑇=𝑚_𝑝+𝜎²𝑤_𝑟;𝑄𝑑𝜎_𝑇=1√2𝜋^∞𝑑/𝜎_𝑇𝑒^−𝑢2/2𝑑𝑢 (17) and 𝑚_𝑝, 𝜎𝑤²_𝑟 are according to (6) and (7), respectively.

Proof. According to [6], the equalized output for the blind adaptive version for the SIMO case is given by 𝑧[𝑛]=𝑥[𝑛]+𝑝[𝑛]+𝑤[𝑛]. This expression is quite similar to the nonblind adaptive version given in (3). The difference between the two cases (blind and nonblind case) lies on the fact that the variance of 𝑝[𝑛] is very different. For each case, the expression for the variance of the real part of 𝑝[𝑛] (named in this paper as 𝑚_𝑝) is different. However, the expression for the variance of the real part of 𝑤[𝑛] (named in this paper as 𝜎𝑤²_𝑟) is the same for both cases. Recently [6], a closed-form approximated expression was obtained for the SER (blind adaptive version), based on 𝑧[𝑛]=𝑥[𝑛]+𝑝[𝑛]+𝑤[𝑛], valid for the SIMO and fGn case where 𝐻 is in the region of 0.5≤𝐻<1. The recently obtained expression for the SER [6] is recalled in (16) and (17). In order to obtain a closed-form approximated expression for the SER valid for the nonblind adaptive, SIMO and fGn case, the recently obtained expression for the SER [6] was used with 𝑚_𝑝 and 𝜎𝑤²_𝑟 ((6) and (7), resp.), valid for the nonblind adaptive, SIMO and fGn case. This completes our proof.

4. Simulation

In this section we test our new proposed expression for the SER for the 16QAM case (a modulation using ±{1,3} levels for in-phase and quadrature components) for different values of SNR and equalizer’s tap length and for four different channel cases. The equalizer taps were updated according to (4) where we set for simplicity 𝜇^(𝑚)=𝜇. In the following we consider four channel cases: channel case A which is a four-channel model with channel coefficients determined according toℎ₁=,ℎ−0.0318+𝑗0.23300.3128−𝑗0.3693−0.3609+𝑗0.38100.6490−0.1110+𝑗0.0396₂=,ℎ0.3005−𝑗0.02470.67840.6248−𝑗0.13160.1282−𝑗0.1411−0.0590−𝑗0.0366₃=,ℎ−0.1351−𝑗0.2061−0.1274+𝑗0.58760.6401−0.1818−𝑗0.33540.0871−𝑗0.1216₄=0.2835+𝑗0.02040.67990.5936+𝑗0.09860.1938+𝑗0.2101−0.0333+𝑗0.1095 (18) channel case B which is a two-channel model with channel coefficients determined according toℎ₁=_{−0.40000.80000.32000.12800.05120.02050.00820.00330.00130.00050.00020.00010.0000}ℎ₂=1.30000.6000−𝑗0.636400000000000; (19) channel case C which is a four-channel model with complex channel coefficients. Each subchannel has a five-channel tap length with randomly generated complex coefficients at each time where no common zero among all the subchannels is generated; channel case D which is a two-channel model with complex channel coefficients. Each subchannel has a three-channel tap-length with randomly generated complex coefficients at each time where no common zero among all the subchannels is generated.

Please note that channel cases A and B were also used in [6]. The equalizers were initialized by setting the center tap equal to one and all others to zero.

In the following we denote the SER performance according to (16) as “Calculated with Equalizer.” In addition, we wish to show the SER performance for the case where the residual ISI is not taken into account. Therefore, we denote in the following the SER performance that does not take into account the residual ISI as “Calculated without Equalizer.” Figure 2 to Figure 16 show the SER performance for different values of 𝐻 and four channel cases as a function of SNR of our proposed expression (16) compared with the simulated results and with those calculated results that do not take into account the residual ISI. According to Figures 2, 3, 4, 6, 7, 10, 11, 12, 14, and 15 the calculated results (16) describe closer the simulated results in comparison to the results that do not take into account the residual ISI. According to Figures 5, 8, 9, 13, and 16 the proposed expression for the SER (16) may be considered only as the upper bound for the simulated results. According to Figure 10 to Figure 16, the proposed expression for the SER (16) for 𝐻=0.7 is a very accurate approximation for the actual SER. Thus, the proposed expression for the SER (16) for 𝐻=0.7 can be very useful for the system designer when dealing with time varying channels where the averaged SER is needed with high accuracy. But, if the averaged SER is not needed with such high accuracy for the time-varying channel case, then the proposed expression for the SER (16) is also very useful for the range of 0.6≤𝐻<0.7 where it provides the system designer a more accurate approximation for the averaged SER compared to the case where the expression for the SER is used that does not take into account the residual ISI. According to Figure 1, the equalized output 𝑧[𝑛] is the sum of the equalizer’s output from each subchannel. For 𝐻>0.5, the noise components from the different subchannels are dependent (please refer to assumption (4) in the system description section). In addition, according to (11), the noise dependency is stronger for higher values of 𝐻. Thus, it is reasonable to think that the SER might increase as the value for 𝐻 increases due to the noise dependency from the different receiving paths. But, according to simulation results (Figure 2 to Figure 9), improved SER performance is seen for higher values of 𝐻. This outcome was also seen in [6]. The reason for having improved SER performance (Figure 2 to Figure 9) for higher values of 𝐻 may be explained via (10). The second part of the right side of (10) defined by ^𝑀𝑀_{𝑚=1,𝑚≠𝑝𝑝=1,𝑝≠𝑚}𝜎²_𝑤𝑟2𝑚−𝑝−1^2𝐻−2𝑚−𝑝^2𝐻+𝑚−𝑝+1^2𝐻⋅^{𝑘=𝑁−1}_𝑘=0𝑐_𝑟^(𝑚)[𝑘]𝑐_𝑟^(𝑝)[𝑘]+𝑐_𝑖^(𝑚)[𝑘]𝑐_𝑖^(𝑝)[𝑘] (20) can obtain negative values via ∑^{𝑘=𝑁−1}_𝑘=0[𝑐_𝑟^(𝑚)[𝑘]𝑐_𝑟^(𝑝)[𝑘]+𝑐_𝑖^(𝑚)[𝑘]𝑐_𝑖^(𝑝)[𝑘]]. By using the approximation of (11) in (20), we can see that, for certain cases where ∑^{𝑘=𝑁−1}_𝑘=0[𝑐_𝑟^(𝑚)[𝑘]𝑐_𝑟^(𝑝)[𝑘]+𝑐_𝑖^(𝑚)[𝑘]𝑐_𝑖^(𝑝)[𝑘]] achieves a negative value, this negative value is multiplied by a positive value depending on 𝐻 (𝐻(2𝐻−1)|𝑚−𝑝|^2𝐻−2). Thus, for higher values of 𝐻 and for the case where ∑^{𝑘=𝑁−1}_𝑘=0[𝑐_𝑟^(𝑚)[𝑘]𝑐_𝑟^(𝑝)[𝑘]+𝑐_𝑖^(𝑚)[𝑘]𝑐_𝑖^(𝑝)[𝑘]]<0, 𝜎𝑤²_𝑟 (10) decreases which causes 𝜎_𝑇 (17) to decrease which leads to improved SER. Since the equalizer’s coefficients 𝑐_𝑟^(𝑚)[𝑘], 𝑐_𝑟^(𝑝)[𝑘], 𝑐_𝑖^(𝑚)[𝑘], and 𝑐_𝑖^(𝑝)[𝑘] are not available to us, ∑^{𝑘=𝑁−1}_𝑘=0[𝑐_𝑟^(𝑚)[𝑘]𝑐_𝑟^(𝑝)[𝑘]+𝑐_𝑖^(𝑚)[𝑘]𝑐_𝑖^(𝑝)[𝑘]] was upper limited (please refer for more details to [6]) in order to supply the system designer an expression for 𝜎𝑤²_𝑟 depending only on available parameters to him. Thus, the outcome was that 𝜎𝑤²_𝑟 as appears in (7) increases for higher values for 𝐻 for all cases of ∑^{𝑘=𝑁−1}_𝑘=0[𝑐_𝑟^(𝑚)[𝑘]𝑐_𝑟^(𝑝)[𝑘]+𝑐_𝑖^(𝑚)[𝑘]𝑐_𝑖^(𝑝)[𝑘]], namely, also for the case where ∑^{𝑘=𝑁−1}_𝑘=0[𝑐_𝑟^(𝑚)[𝑘]𝑐_𝑟^(𝑝)[𝑘]+𝑐_𝑖^(𝑚)[𝑘]𝑐_𝑖^(𝑝)[𝑘]]<0. Thus, the accuracy of our proposed expression for the SER (16) is dependent on 𝐻.

Figure 2 

SER comparison with the following parameters: 𝑑=1, the stepsize parameter 𝜇=0.0005, 𝑁=17, 𝐻=0.6, and channel case A; the results were obtained for 1600000 symbols.

Figure 3 

SER comparison with the following parameters: 𝑑=1, the stepsize parameter 𝜇=0.0005, 𝑁=17, 𝐻=0.7, and channel case A; the results were obtained for 1600000 symbols.

Figure 4 

SER comparison with the following parameters: 𝑑=1, the stepsize parameter 𝜇=0.0005, 𝑁=17, 𝐻=0.8, and channel case A; the results were obtained for 1600000 symbols.

Figure 5 

SER comparison with the following parameters: 𝑑=1, the stepsize parameter 𝜇=0.0005, 𝑁=17, 𝐻=0.9, and channel case A; the results were obtained for 1600000 symbols.

Figure 6 

SER comparison with the following parameters: 𝑑=1, the stepsize parameter 𝜇=0.002, 𝑁=13, 𝐻=0.6, and channel case B; the results were obtained for 1600000 symbols.

Figure 7 

SER comparison with the following parameters: 𝑑=1, the stepsize parameter 𝜇=0.002, 𝑁=13, 𝐻=0.7, and channel case B; the results were obtained for 1600000 symbols.

Figure 8 

SER comparison with the following parameters: 𝑑=1, the stepsize parameter 𝜇=0.002, 𝑁=13, 𝐻=0.8, and channel case B; the results were obtained for 1600000 symbols.

Figure 9 

SER comparison with the following parameters: 𝑑=1, the stepsize parameter 𝜇=0.002, 𝑁=13, 𝐻=0.9, and channel case B; the results were obtained for 1600000 symbols.

Figure 10 

SER comparison with the following parameters: 𝑑=1, the stepsize parameter 𝜇=0.0005, 𝑁=9, 𝐻=0.6, and channel case C; the averaged results were obtained in 50 Monte Carlo trials for 128000 symbols.

Figure 11 

SER comparison with the following parameters: 𝑑=1, the stepsize parameter 𝜇=0.0005, 𝑁=9, 𝐻=0.65, and channel case C; the averaged results were obtained in 50 Monte Carlo trials for 128000 symbols.

Figure 12 

SER comparison with the following parameters: 𝑑=1, the stepsize parameter 𝜇=0.0005, 𝑁=9, 𝐻=0.7, and channel case C; the averaged results were obtained in 50 Monte Carlo trials for 128000 symbols.

Figure 13 

SER comparison with the following parameters: 𝑑=1, the stepsize parameter 𝜇=0.0005, 𝑁=9, 𝐻=0.8, and channel case C; the averaged results were obtained in 50 Monte Carlo trials for 128000 symbols.

Figure 14 

SER comparison with the following parameters: 𝑑=1, the stepsize parameter 𝜇=0.0005, 𝑁=19, 𝐻=0.6, and channel case D; the averaged results were obtained in 50 Monte Carlo trials for 128000 symbols.

Figure 15 

SER comparison with the following parameters: 𝑑=1, the stepsize parameter 𝜇=0.0005, 𝑁=19, 𝐻=0.7, and channel case D; the averaged results were obtained in 50 Monte Carlo trials for 128000 symbols.

Figure 16 

SER comparison with the following parameters: 𝑑=1, the stepsize parameter 𝜇=0.0005, 𝑁=19, 𝐻=0.8, and channel case D; the averaged results were obtained in 50 Monte Carlo trials for 128000 symbols.

5. Conclusion

In this paper, we proposed for the real and two independent quadrature carrier case, closed-form approximated expressions for the residual ISI and SER obtained by nonblind adaptive equalizers that are applicable for the SIMO and fGn (0.5≤𝐻<1) input case. The expression for the SER depends on the system’s parameters (stepsize parameter, equalizer’s tap length, input constellation statistics, channel power, number of receiving antennas used in the SIMO system and on 𝐻). Thus, there is no need anymore to carry out any simulation in order to find those system’s parameters that will lead to the required SER performance. According to simulation results, the proposed expression for the SER can be very useful for the system designer when dealing with time-varying channels where the averaged SER is needed and 𝐻≤0.7. For channel case A and 0.6≤𝐻≤0.8 or channel case B and 0.6≤𝐻≤0.7 the proposed expression for the SER provides better results for the simulated results in comparison to the results that do not take into account the residual ISI. For channel case A and 1>𝐻>0.8 or channel cases B, C, and D and 1>𝐻>0.7 the new proposed expression for the SER may be considered only as the upper bound for the simulated results. FGn with 𝐻∈(0.5,1) corresponds to the case of LRD. Thus, it could be thought that the SER might increase as the value for 𝐻 increases due to the noise dependency from the different receiving paths. But, according to simulation results, improved SER performance is seen for higher values of 𝐻.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The author would like to thank the reviewers for their helpful comments.