Residual ISI Obtained by Blind Adaptive Equalizers and Fractional Noise

Abstract

Recently, a closed-form approximated expression was derived by the same author for the achievable residual intersymbol interference (ISI) case that depends on the step-size parameter, equalizer’s tap length, input signal statistics, signal-to-noise ratio (SNR), and channel power. But this expression was obtained by assuming that the input noise is a white Gaussian process where the Hurst exponent (H) is equal to 0.5. In this paper, we derive a closed-form approximated expression (or an upper limit) for the residual ISI obtained by blind adaptive equalizers valid for fractional Gaussian noise (fGn) input where the Hurst exponent is in the region of 0.5≤𝐻<1. Up to now, the statistical behaviour of the residual ISI was not investigated. Furthermore, the convolutional noise for the latter stages of the deconvolutional process was assumed to be a white Gaussian process (𝐻=0.5). In this paper, we show that the Hurst exponent of the residual ISI is close to one, almost independent of the SNR or equalizer’s tap length but depends on the step-size parameter. In addition, the convolutional noise obtained in the steady state is a noise process having a Hurst exponent depending on the step-size parameter.

1. Introduction

We consider a blind deconvolution problem in which we observe the output of an unknown, possibly nonminimum phase, linear system (SISO-FIR system) from which we want to recover its input (source) using an adjustable linear filter (equalizer). The problem of blind deconvolution arises comprehensively in various applications such as digital communications, seismic signal processing, speech modeling and synthesis, ultrasonic nondestructive evaluation, and image restoration [1]. Blind deconvolution algorithms are essentially adaptive filtering algorithms designed such that they do not require the external supply (training sequence) of a desired response to generate the error signal in the output of the adaptive equalization filter [2, 3]. The algorithm itself generates an estimate of the desired response by applying a nonlinear transformation to sequences involved in the adaptation process [2, 3]. Let us consider for a moment the digital communication case. During transmission, a source signal undergoes a convolutive distortion between its symbols and the channel impulse response. This distortion is referred to as ISI. Thus, a blind adaptive equalizer is used to remove the convolutive effect of the system to produce the source signal [2, 4–6].

Up to recently [7, 8], the performance of a chosen equalizer (the achievable residual ISI) could be obtained only via simulation. The equalization performance depends on the nature of the chosen equalizer (on the memoryless nonlinearity situated at the output of the equalizer’s filter), on the channel characteristics, on the added noise, on the step-size parameter used in the adaptation process, on the equalizer’s tap length, and on the input signal statistics. Fast convergence speed and reaching a residual ISI where the eye diagram is considered to be open (for the communication case) are the main requirements from a blind equalizer. Fast convergence speed may be obtained by increasing the step-size parameter. But increasing the step-size parameter may lead to a higher residual ISI which might not meet any more the system’s requirements. Up to recently, the system designer had to carry out, for a given channel and type of equalizer, many simulations in order to obtain the optimal step-size parameter for a required residual ISI. Recently [7, 8], a closed-form approximated expression was derived by the same author for the achievable residual ISI case that depends on the step-size parameter, equalizer’s tap length, input signal statistics, signal-to-noise ratio (SNR), and channel power. But this expression was obtained by assuming that the input noise is a white Gaussian process where the Hurst exponent (𝐻) is equal to 0.5. A white Gaussian process is a special case (𝐻=0.5) of the fractional Gaussian noise (fGn) model [9]. FGn with 𝐻∈(0.5,1) corresponds to the case of long-range dependency (LRD) [9]. LRD implies heavy-tailed probability density functions, which in general imply more random, see [10–13]. This point of view was recently detailed by [14, 15]. An fGn is a generalization of ordinary white Gaussian noise, and it is a versatile model for broadband noise dominated by no particular frequency band [16]. For 0.5<𝐻<1, fGn is bandlimited to (−𝜔_𝑐,𝜔_𝑐) at level 𝜀₁>0 [17].

In this paper, we derive a closed-form approximated expression (or an upper limit) for the residual ISI obtained by blind adaptive equalizers valid for fGn input where the Hurst exponent is in the region of 0.5≤𝐻<1. Please note, 𝐻=1 is the limit case, which does not have much practical sense [18, 19]. Up to now, the statistical behaviour of the residual ISI was not investigated. Furthermore, the convolutional noise for the latter stages of the deconvolutional process was assumed to be a white Gaussian process (𝐻=0.5). In this paper, we show that the Hurst exponent of the residual ISI is close to one, almost independent of the SNR or equalizer’s tap length but depends on the step-size parameter. In addition, the convolutional noise obtained in the steady state is a noise process with a Hurst exponent depending on the step-size parameter.

The paper is organized as follows. After having described the system under consideration in Section 2, the closed-form approximated expression (or upper limit) for the residual ISI is introduced in Section 3. In Section 4 the statistical behavior of the residual ISI and convolutional noise are presented and the conclusion is given in Section 5.

2. System Description

The system under consideration is illustrated in Figure 1, where we make the following assumptions.(1) The input sequence 𝑥(𝑛) belongs to a two independent quadrature carrier case constellation input with variance 𝜎²_𝑥, where 𝑥_𝑟(𝑛) and 𝑥_𝑖(𝑛) are the real and imaginary parts of 𝑥(𝑛), respectively, and 𝜎²_𝑥𝑟 is the variance of 𝑥_𝑟(𝑛). In the following we denote 𝑥_𝑟(𝑛) as 𝑥_𝑟.(2) The unknown channel ℎ(𝑛) is a possibly nonminimum phase linear time-invariant filter in which the transfer function has no “deep zeros”; namely, the zeros lie sufficiently far from the unit circle.(3) The equalizer 𝑐(𝑛) is a tap-delay line.(4) The noise 𝑤(𝑛) consists of 𝑤(𝑛)=𝑤_𝑟(𝑛)+𝑗𝑤_𝑖(𝑛), where 𝑤_𝑟(𝑛) and 𝑤_𝑖(𝑛) are the real and imaginary parts of 𝑤(𝑛), respectively, and 𝑤_𝑟(𝑛) and 𝑤_𝑖(𝑛) are independent. Both 𝑤_𝑟(𝑛) and 𝑤_𝑖(𝑛) are fractional Gaussian noises (fGn) with zero mean. Consider 𝜎²_𝑤𝑟=𝐸[𝑤²_𝑟(𝑛)], 𝜎²_𝑤𝑖=𝐸[𝑤²_𝑖(𝑛)], 𝐸[𝑤_𝑟(𝑛−𝑘)𝑤_𝑟(𝑛−𝑚)]=(𝜎²_𝑤𝑟/2)[(|𝑚−𝑘|−1)^2𝐻−2(|𝑚−𝑘|)^2𝐻+(|𝑚−𝑘|+1)^2𝐻], and 𝐸[𝑤_𝑖(𝑛−𝑘)𝑤_𝑖(𝑛−𝑚)]=(𝜎²_𝑤𝑖/2)[(|𝑚−𝑘|−1)^2𝐻−2(|𝑚−𝑘|)^2𝐻+(|𝑚−𝑘|+1)^2𝐻], where 𝐸[⋅] denotes the expectation operator on (⋅) and 𝐻 is the Hurst exponent.

Figure 1 

Block diagram of a baseband communication system.

The transmitted sequence 𝑥(𝑛) is sent through the channel ℎ(𝑛) and is corrupted with noise 𝑤(𝑛). Therefore, the equalizer’s input sequence 𝑦(𝑛) may be written as 𝑦(𝑛)=𝑥(𝑛)∗ℎ(𝑛)+𝑤(𝑛), (1) where “∗” denotes the convolution operation. The equalized output signal can be written as: 𝑧(𝑛)=𝑥(𝑛)+𝑝(𝑛)+𝑤(𝑛), (2) where 𝑝(𝑛) is the convolutional noise, namely, the residual intersymbol interference (ISI) arising from the difference between the ideal equalizer’s coefficients and those chosen in the system and 𝑤(𝑛)=𝑤(𝑛)∗𝑐(𝑛). The ISI is often used as a measure of performance in equalizers’ applications, defined by ∑𝑚𝑠𝑚̃ISI=²−|𝑠|̃²_max|𝑠|̃²_max, (3) where |𝑠|̃_max is the component of 𝑠̃, given in (4), having the maximal absolute value 𝑠̃(𝑛)=𝑐(𝑛)∗ℎ(𝑛)=𝛿(𝑛)+𝜁(𝑛), (4) where 𝛿 is the Kronecker delta function and 𝜁(𝑛) stands for the difference (error) between the ideal and the actual value used for 𝑐(𝑛). Although the ISI is often used as a measure of performance in equalizers’ applications, its statistical properties were never investigated.

Next we turn to the adaptation mechanism of the equalizer which is based on a predefined cost function 𝐹(𝑛) that characterizes the intersymbol interference, see [20–24]. Minimizing this 𝐹(𝑛) with respect to the equalizer parameters will reduce the convolutional error. Minimization is performed with the gradient descent algorithm that searches for an optimal filter tap setting by moving in the direction of the negative gradient −∇_𝑐𝐹(𝑛) over the surface of the cost function in the equalizer filter tap space [25]. Thus the updated equation is given by [25] 𝑐_eq(𝑛+1)=𝑐_eq(𝑛)+𝜇⋅−∇_𝑐eq𝐹(𝑛)=𝑐_eq(𝑛)(𝑛)−𝜇𝜕𝐹(𝑛)𝑦𝜕𝑧^∗(𝑛), (5) where 𝜇 is the step-size parameter and 𝑐_eq(𝑛) is the equalizer vector where the input vector is 𝑦(𝑛)=[𝑦(𝑛)⋯𝑦(𝑛−𝑁+1)]^𝑇 and 𝑁 is the equalizer’s tap length. The operator (⋅)^𝑇 denotes for transpose of the function (⋅).

Recently [8], a closed-form approximated expression was derived for the achievable residual ISI case (based on [7]) that depends on the step-size parameter, equalizer’s tap length, input signal statistics, SNR, and channel power and is given by ISI=10log₁₀𝑚_𝑝−10log₁₀𝜎²_𝑥𝑟, (6) where 𝜎²_𝑥𝑟 is the variance of the real part of the input sequence 𝑥(𝑛) and 𝑚_𝑝 is defined by 𝑚_𝑝=minSol^𝑚𝑝1₁,Sol^𝑚𝑝1₂forSol^𝑚𝑝1₁>0,Sol^𝑚𝑝1₂>0 (7) or 𝑚_𝑝=maxSol^𝑚𝑝1₁,Sol^𝑚𝑝1₂forSol^𝑚𝑝1₁⋅Sol^𝑚𝑝1₂<0, (8) where Sol^𝑚𝑝1₁=−𝐵₁+𝐵²₁−4𝐴₁𝐶₁𝐵2𝐴₁,Sol^𝑚𝑝1₂=−𝐵₁−𝐵²₁−4𝐴₁𝐶₁𝐵2𝐴₁,𝐴₁=𝐵45𝜎²_𝑥𝑟𝑎²₃+18𝜎²_𝑥𝑟𝑎₃𝑎₁₂+6𝑎₁𝑎₃+9𝜎²_𝑥𝑟𝑎²₁₂+2𝑎₁𝑎₁₂−23𝑎₃+𝑎₁₂+𝐵45𝑎²₃+18𝑎₃𝑎₁₂+9𝑎²₁₂𝜎𝑤²_𝑟,𝐵₁=𝐵12𝜎²_𝑥𝑟²𝑎₃𝑎₁₂+6𝜎²_𝑥𝑟²𝑎²₁₂+12𝜎²_𝑥𝑟𝑎₁𝑎₃+4𝜎²_𝑥𝑟𝑎₁𝑎₁₂+𝑎²₁+15𝐸𝑥⁴_𝑟𝑎²₃+2𝐸𝑥⁴_𝑟𝑎₃𝑎₁₂+𝐸𝑥⁴_𝑟𝑎²₁₂−2𝑎₁+3𝜎²_𝑥𝑟𝑎₃+𝜎²_𝑥𝑟𝑎₁₂+𝐵45𝑎²₃+16𝑎₃𝑎₁₂+9𝑎²₁₂𝜎𝑤⁴_𝑟+𝐵90𝑎²₃𝜎²_𝑥𝑟+36𝑎₃𝑎₁₂𝜎²_𝑥𝑟+12𝑎₁𝑎₃+18𝑎²₁₂𝜎²_𝑥𝑟+4𝑎₁𝑎₁₂−2𝑎₁₂−6𝑎₃𝜎𝑤²_𝑟,𝐶₁=2𝜎²_𝑥𝑟²𝑎₁𝑎₁₂+𝜎²_𝑥𝑟𝑎²₁+2𝐸𝑥⁴_𝑟𝜎²_𝑥𝑟𝑎₃𝑎₁₂+𝐸𝑥⁴_𝑟𝜎²_𝑥𝑟𝑎²₁₂+2𝐸𝑥⁴_𝑟𝑎₁𝑎₃+𝐸𝑥⁶_𝑟𝑎²₃+15𝑎²₃+6𝑎₃𝑎₁₂+3𝑎²₁₂𝜎𝑤⁶_𝑟+45𝑎²₃𝜎²_𝑥𝑟+18𝑎₃𝑎₁₂𝜎²_𝑥𝑟+6𝑎₁𝑎₃+9𝑎²₁₂𝜎²_𝑥𝑟+2𝑎₁𝑎₁₂𝜎𝑤⁴_𝑟+𝑎²₁+12𝑎₁𝑎₃𝜎²_𝑥𝑟+4𝑎₁𝑎₁₂𝜎²_𝑥𝑟+15𝑎²₃𝐸𝑥⁴_𝑟+12𝑎₃𝑎₁₂𝜎²_𝑥𝑟²+2𝑎₃𝑎₁₂𝐸𝑥⁴_𝑟+𝑎²₁₂𝐸𝑥⁴_𝑟+6𝑎²₁₂𝜎²_𝑥𝑟²𝜎𝑤²_𝑟,𝐵=𝜇𝑁𝜎²_𝑥^{𝑘𝑘=𝑅−1}_𝑘𝑘=0ℎ_𝑘𝑘(𝑛)²+𝜇𝑁𝜎²_𝑥,𝜎SNR₂𝑤_𝑟=𝜎²_𝑥𝑟SNR∑^{𝑘𝑘=𝑅−1}_𝑘𝑘=0ℎ_𝑘𝑘(𝑛)². (9)(10) The channel length is 𝑅, 𝜎𝑤²_𝑟 is the variance of the real part of 𝑤(𝑛), SNR=𝜎²_𝑥/𝜎²_𝑤, and 𝑎₁, 𝑎₁₂, 𝑎₃ are properties of the chosen equalizer and are found by (𝑛)Re𝜕𝐹(𝑛)=𝑎𝜕𝑧¹𝑧_𝑟+𝑎₃𝑧_𝑟³+𝑎₁₂𝑧_𝑟𝑧_𝑖², (11) where Re(⋅) is the real part of (⋅) and 𝑧_𝑟, 𝑧_𝑖 are the real and imaginary parts of the equalized output 𝑧(𝑛), respectively.

It should be pointed out that the closed-form approximated expression for the residual ISI [8] was obtained by assuming that the noise 𝑤(𝑛) is an additive Gaussian white noise with zero mean and variance 𝜎²_𝑤=𝐸[𝑤(𝑛)𝑤^∗(𝑛)]. Therefore, it is not applicable for the fGn case (for 𝐻>0.5). Thus, a new expression for the achievable residual ISI is needed.

3. Residual ISI for Fractional Gaussian Noise Input

In this section, a closed-form approximated expression (or an upper limit) is derived for the residual ISI valid for the fGn input case supported by simulation results.

3.1. Derivation of the Residual ISI

Theorem 1. Consider 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 rectangular QAM (Quadrature Amplitude Modulation) signal (where the real part of 𝑥(𝑛) is independent of 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)𝜕𝐹(𝑛)/𝜕𝑧(𝑛) can be expressed as a polynomial function of the equalized output, namely, as 𝑃(𝑧) of order three.(5)The gain between the source and equalized output signal is equal to one.(6)The convolutional noise 𝑝(𝑛) is independent of 𝑤(𝑛).(7)The added noise is fGn with zero mean.(8)The channel ℎ(𝑛) has real coefficients.(9)The Hurst exponent is in the range of 0.5≤𝐻<1.

The residual ISI expressed in dB units may be defined as (6), (8), and (9) where 𝑅 is the channel length, SNR=𝜎²_𝑥/𝜎²_𝑤, and 𝑎₁, 𝑎₁₂, 𝑎₃ are properties of the chosen equalizer and one found by (11) and 𝜎𝑤²_𝑟≅𝜎²_𝑥𝑟SNR∑^{𝑘𝑘=𝑅−1}_𝑘𝑘=0ℎ²_𝑘𝑘[𝑛]1+(𝑁−1)𝐻(2𝐻−1). (12)

Comments. Please note that for 𝐻=0.5 (Gaussian white noise case), the expression for 𝜎𝑤²_𝑟 given in (12) and (10) is equivalent. By repeating the steps in [8] for the calculation of the expression of the residual ISI, the only place where the difference between the assumption of 𝑤_𝑟(𝑛) and 𝑤_𝑖(𝑛) being Gaussian white noises or fractional Gaussian noises has a major role on the total result of the approximated derived expression for the residual ISI, is in the calculation of 𝜎𝑤²_𝑟. Thus, we bring here only the various steps that led to (12).

It should be pointed out that assumptions (1)--(6) from above are precisely the same assumptions made in [8].

Proof. The real part of 𝑤(𝑛), namely, 𝑤_𝑟(𝑛), may be expressed as 𝑤_𝑟(𝑛)=^{𝑘=𝑁−1}_𝑘=0𝑐_𝑘(𝑛)𝑤_𝑟(𝑛−𝑘). (13) Thus, the variance of 𝑤_𝑟(𝑛) is given by 𝜎𝑤²_𝑟=𝐸^{𝑘=𝑁−1}_𝑘=0𝑐_𝑘(𝑛)𝑤_𝑟(𝑛−𝑘)^{𝑚=𝑁−1}_𝑚=0𝑐_𝑚(𝑛)𝑤_𝑟(=𝑛−𝑚)_{𝑘=𝑁−1𝑘=0𝑚=𝑁−1𝑚=0}𝑐_𝑘(𝑛)𝑐_𝑚(𝑛)𝐸𝑤_𝑟(𝑛−𝑘)𝑤_𝑟(𝑛−𝑚) (14) which can be also written as 𝜎𝑤²_𝑟=𝜎²_𝑤𝑟^{𝑘=𝑁−1}_𝑘=0𝑐²_𝑘(𝑛)+^{𝑘=𝑁−1}_{𝑘=0,𝑘≠𝑚}^{𝑚=𝑁−1}_{𝑚=0,𝑘≠𝑚}𝑐_𝑘(𝑛)𝑐_𝑚(𝑛)𝐸×𝑤_𝑟(𝑛−𝑘)𝑤_𝑟(𝑛−𝑚). (15) Next, by using the assumption that the gain between the source and equalized output signal is equal to one, we may write the following expression: ^{𝑘=𝑁−1}_𝑘=0𝑐²_𝑘(𝑛)≃1∑^{𝑘𝑘=𝑅−1}_𝑘𝑘=0ℎ²_𝑘𝑘(𝑛). (16) By substituting (16) into (15) we obtain 𝜎𝑤²_𝑟=𝜎²_𝑤𝑟1∑^{𝑘𝑘=𝑅−1}_𝑘𝑘=0ℎ²_𝑘𝑘(𝑛)+^{𝑘=𝑁−1}_{𝑘=0,𝑘≠𝑚}^{𝑚=𝑁−1}_{𝑚=0,𝑘≠𝑚}𝑐_𝑘(𝑛)𝑐_𝑚(𝑛)𝐸×𝑤_𝑟(𝑛−𝑘)𝑤_𝑟(𝑛−𝑚). (17) The expression (17) can be upper limited by 𝜎𝑤²_𝑟≤𝜎²_𝑤𝑟1∑^{𝑘𝑘=𝑅−1}_𝑘𝑘=0ℎ²_𝑘𝑘(𝑛)+max𝐸𝑤_𝑟(𝑛−𝑘)𝑤_𝑟(×𝑛−𝑚)_{𝑘=𝑁−1𝑘=0,𝑘≠𝑚𝑚=𝑁−1𝑚=0,𝑘≠𝑚}𝑐_𝑘(𝑛)𝑐_𝑚(𝑛). (18) With the help of the Holder Inequality [26], we may write ^{𝑘=𝑁−1}_{𝑘=0,𝑘≠𝑚}^{𝑚=𝑁−1}_{𝑚=0,𝑘≠𝑚}𝑐_𝑘(𝑛)𝑐_𝑚(𝑛)≤(𝑁−1)^{𝑘=𝑁−1}_𝑘=0𝑐²_𝑘(𝑛). (19) Now, by substituting (16) into (19) we obtain ^{𝑘=𝑁−1}_{𝑘=0,𝑘≠𝑚}^{𝑚=𝑁−1}_{𝑚=0,𝑘≠𝑚}𝑐_𝑘(𝑛)𝑐_𝑚(𝑛)≤𝑁−1∑^{𝑘𝑘=𝑅−1}_𝑘𝑘=0ℎ²_𝑘𝑘(𝑛). (20) Thus having 𝜎𝑤²_𝑟≤𝜎²_𝑤𝑟1∑^{𝑘𝑘=𝑅−1}_𝑘𝑘=0ℎ²_𝑘𝑘(𝑛)+max𝐸𝑤_𝑟(𝑛−𝑘)𝑤_𝑟(×𝑛−𝑚)𝑁−1∑^{𝑘𝑘=𝑅−1}_𝑘𝑘=0ℎ²_𝑘𝑘(𝑛)𝐸𝑤,where_𝑟(𝑛−𝑘)𝑤_𝑟(=𝜎𝑛−𝑚)_2𝑤𝑟2(|𝑚−𝑘|−1)^2𝐻−2(|𝑚−𝑘|)^2𝐻+(|𝑚−𝑘|+1)^2𝐻. (21) This expression (21) can be therefore written as 𝜎𝑤²_𝑟≤𝜎²_𝑤𝑟1∑^{𝑘𝑘=𝑅−1}_𝑘𝑘=0ℎ²_𝑘𝑘(𝑛)×1+𝑁−12max_{𝑘≠𝑚,𝑘,𝑚=0∶𝑁−1}(|𝑚−𝑘|−1)^2𝐻−2(|𝑚−𝑘|)^2𝐻+(|𝑚−𝑘|+1)^2𝐻. (22) According to [17], 0.5(|𝑚−𝑘|−1)^2𝐻−2(|𝑚−𝑘|)^2𝐻+(|𝑚−𝑘|+1)^2𝐻≃𝐻(2𝐻−1)|𝑚−𝑘|^2𝐻−2. (23) By substituting (23) into (22) and taking into account that max_{𝑘≠𝑚,𝑘,𝑚=0∶𝑁−1}|𝑚−𝑘|^2𝐻−2=1 for 0.5≤𝐻<1, we obtain 𝜎𝑤²_𝑟≤𝜎²_𝑤𝑟1∑^{𝑘𝑘=𝑅−1}_𝑘𝑘=0ℎ²_𝑘𝑘(𝑛)[1+(𝑁−1)𝐻(2𝐻−1)]. (24) Now, since we deal with the rectangular QAM case and where 𝑤_𝑟 is independent of 𝑤_𝑖, we obtain that 𝜎²_𝑥𝑟=𝜎²_𝑥𝑖 and 𝜎²_𝑤𝑟=𝜎²_𝑤𝑖 where 𝜎²_𝑥𝑖 is the variance of the imaginary part of 𝑥(𝑛). Therefore, we may write 𝜎SNR=^2𝑥𝜎²_𝑤=2𝜎²_𝑥𝑟2𝜎²_𝑤𝑟=𝜎²_𝑥𝑟𝜎²_𝑤𝑟. (25) By substituting (25) into (24) we obtain: 𝜎𝑤²_𝑟≤𝜎²_𝑥𝑟SNR∑^{𝑘𝑘=𝑅−1}_𝑘𝑘=0ℎ²_𝑘𝑘(𝑛)[1+(𝑁−1)𝐻(2𝐻−1)]. (26) It turned out, according to simulation results, that (26) is too far away from the averaged residual ISI. Thus, it can not serve in practice as an upper limit for the expected residual ISI. In reality, we may find approximately only around √𝑁−1 coefficients in the equalizer’s tap length that may have significant values. Therefore, we may write 𝜎𝑤²_𝑟≃𝜎²_𝑥𝑟SNR∑^{𝑘𝑘=𝑅−1}_𝑘𝑘=0ℎ²_𝑘𝑘(𝑛)1+√𝑁−1𝐻(2𝐻−1). (27) This completes our proof.

3.2. Simulation

In this section we test our new proposed expression for the residual ISI for the 16QAM case (a modulation using ±{1,3} levels for in-phase and quadrature components) with Godard’s algorithm [20] for two different SNR and equalizer’s tap length values and for two different channel types. The equalizer taps for Godard’s algorithm [20] were updated according to 𝑐_𝑚(𝑛+1)=𝑐_𝑚(𝑛)−𝜇_𝐺|𝑧(𝑛)|²−𝐸|𝑥(𝑛)|⁴𝐸|𝑥(𝑛)|²×𝑧(𝑛)𝑦^∗(𝑛−𝑚), (28) where 𝜇_𝐺 is the step-size. The values for 𝑎₁, 𝑎₁₂, and 𝑎₃ corresponding to Godard’s [20] algorithm are given by 𝑎₁𝐸𝑥(𝑛=−|)|⁴𝐸|𝑥(𝑛)|²,𝑎₁₂=1,𝑎₃=1. (29) The following two channels were considered.

Channel 1 (initial 𝐼𝑆𝐼=0.44). The channel parameters were determined according to [24]: ℎ_𝑛=0for𝑛<0;−0.4for𝑛=00.84⋅0.4^𝑛−1for𝑛>0. (30)Channel 2 (initial 𝐼𝑆𝐼=0.88). The channel parameters were determined according to ℎ_𝑛=(0.4851,−0.72765,−0.4851). (31) The equalizer was initialized by setting the center tap equal to one and all others to zero.

In the following we denote the residual ISI performance according to (6) with (8), (9), and (12) as “Calculated ISI.” Figures 2, 5, 8, 11, and 13 show the ISI performance as a function of the iteration number of our proposed expression (6) (with (8), (9), and (12)) compared with the simulated results for two different channels and SNR values, various values for 𝐻, and three different step-size values and equalizer’s length. According to Figures 2, 5, 8, and 13 (for 𝐻=0.6), a high correlation is observed between the simulated and calculated results. According to Figures 11 and 13 (for 𝐻=0.8), the Calculated ISI may be considered as an upper limit for the simulated results. Figures 3, 4, 6, 7, 9, 10, and 12 show the equalizer’s coefficients in the steady state for various values for 𝐻, SNR, equalizer’s tap-length, step-size parameters, and two different channels. According to simulation results (Figures 3, 4, 6, 7, 9, 10, and 12), it was reasonable to take approximately only √𝑁−1 instead of 𝑁−1 coefficients in (12) since indeed not all the coefficients in the equalizer’s tap length have significant values.

Figure 2 

A comparison between the simulated (with Godard’s algorithm) and calculated residual ISI for the 16QAM source input going through Channel 2 for SNR=12  [dB]. The averaged results were obtained in 100 Monte Carlo trials. The equalizer’s tap length and step-size parameter were set to 13 and 0.00003, respectively.

Figure 3 

Equalizer’s coefficients in the steady state obtained with Godard’s algorithm for the 16QAM source input going through Channel 2, SNR=12 [dB] and step-size parameter equal to 0.00003. The equalizer’s tap length was set to 13.

Figure 4 

Equalizer’s coefficients in the steady state obtained with Godard’s algorithm for the 16QAM source input going through Channel 2, SNR=12 [dB] and step-size parameter equal to 0.00003. The equalizer’s tap length was set to 13.

Figure 5 

A comparison between the simulated (with Godard’s algorithm) and calculated residual ISI for the 16QAM source input going through Channel 2 for SNR=15 [dB]. The averaged results were obtained in 100 Monte Carlo trials. The equalizer’s tap length and step-size parameter were set to 13 and 0.00003, respectively.

Figure 6 

Equalizer’s coefficients in the steady state obtained with Godard’s algorithm for the 16QAM source input going through Channel 2, SNR=15 [dB] and step-size parameter equal to 0.00003. The equalizer’s tap length was set to 13.

Figure 7 

Equalizer’s coefficients in the steady state obtained with Godard’s algorithm for the 16QAM source input going through Channel 2, SNR=15 [dB] and step-size parameter equal to 0.00003. The equalizer’s tap length was set to 13.

Figure 8 

A comparison between the simulated (with Godard’s algorithm) and calculated residual ISI for the 16QAM source input going through Channel 2 for SNR=12 [dB]. The averaged results were obtained in 100 Monte Carlo trials. The equalizer’s tap length and step-size parameter were set to 27 and 0.00001, respectively.

Figure 9 

Equalizer’s coefficients in the steady state obtained with Godard’s algorithm for the 16QAM source input going through Channel 2, SNR=12 [dB] and step-size parameter equal to 0.00001. The equalizer’s tap length was set to 27.

Figure 10 

Equalizer’s coefficients in the steady state obtained with Godard’s algorithm for the 16QAM source input going through Channel 2, SNR=12 [dB] and step-size parameter equal to 0.00001. The equalizer’s tap length was set to 27.

Figure 11 

A comparison between the simulated (with Godard’s algorithm) and calculated residual ISI for the 16QAM source input going through Channel 1 for SNR=12 [dB]. The averaged results were obtained in 100 Monte Carlo trials. The equalizer’s tap length and step-size parameter were set to 13 and 0.00003, respectively.

Figure 12 

Equalizer’s coefficients in the steady state obtained with Godard’s algorithm for the 16QAM source input going through Channel 1, SNR=12 [dB] and step-size parameter equal to 0.00003. The equalizer’s tap length was set to 13.

Figure 13 

A comparison between the simulated (with Godard’s algorithm) and calculated residual ISI for the 16QAM source input going through Channel 1 for SNR=12 [dB]. The averaged results were obtained in 100 Monte Carlo trials. The equalizer’s tap length and step-size parameter were set to 23 and 0.000015, respectively.

4. The Statistical Behaviour of the Residual ISI and Convolutional Noise

The statistical behaviour of the residual ISI was never investigated in the literature. Furthermore, the convolutional noise 𝑝(𝑛) is usually assumed (in the latter stages of the deconvolution process [27]) to be a white Gaussian process. In this section we investigate the statistical behaviour of the residual ISI and convolutional noise in the steady state.

4.1. The Statistical Behaviour of the Residual ISI

In this subsection we estimate the Hurst exponent of the residual ISI from the Rescaled Range (R/S) [28] with overlapping regions. For that purpose, let us denote 𝜉 as a vector of samples of the residual ISI (obtained from a single Monte Carlo trial) with length 𝑁. The mean of 𝑛_𝑝 consecutive samples in 𝜉 may be defined as 𝜉_{𝑛𝑝,𝑖𝑖}=1𝑛_𝑝^{𝑛𝑝+𝑖𝑖}_{𝑖=𝑖𝑖+1}𝜉[𝑖],𝑖𝑖=0,1,2,3,…, (32) where 𝑛_𝑝+𝑖𝑖≤𝑁,  𝑖𝑖 denotes the 𝑖𝑖th segment in vector 𝜉 and ⟨⋅⟩ means the estimate of (⋅). Next we define 𝛾_{𝑖,𝑛𝑝,𝑖𝑖}=^{𝑖+𝑖𝑖}𝑢_=𝑖𝑖+1𝜉[𝑢]−𝜉_{𝑛𝑝,𝑖𝑖},1≤𝑖≤𝑛_𝑝,𝑅_{𝑛𝑝,𝑖𝑖}=max𝛾_{𝑖,𝑛𝑝,𝑖𝑖}−min𝛾_{𝑖,𝑛𝑝,𝑖𝑖},1≤𝑖≤𝑛_𝑝. (33) The Rescaled Range (𝑅/𝑆) [28] for the 𝑖𝑖th segment with 𝑛_𝑝 samples in the vector 𝜉 is defined as [𝑅/𝑆]_{𝑛𝑝,𝑖𝑖}=𝑅_{𝑛𝑝,𝑖𝑖}𝑆_{𝑛𝑝,𝑖𝑖}, (34) where 𝑆_{𝑛𝑝,𝑖𝑖}=1𝑛_𝑝^{𝑛𝑝+𝑖𝑖}_{𝑖=𝑖𝑖+1}𝜉[𝑖]−𝜉_{𝑛𝑝,𝑖𝑖}². (35) The averaged Rescaled Range over multiple regions of the data with length 𝑛_𝑝 is defined as [28] 𝐸𝑅_𝑛𝑝𝑆_𝑛𝑝=11+max(𝑖𝑖)^{max(𝑖𝑖)}_𝑖𝑖=0[𝑅/𝑆]_{𝑛𝑝,𝑖𝑖}≃𝐶𝑛^𝐻_𝑝, (36) where 𝐶 is a constant. Now, by applying the log₂ operator on both sides of (36) we obtain [28] log₂11+max(𝑖𝑖)^{max(𝑖𝑖)}_𝑖𝑖=0[𝑅/𝑆]_{𝑛𝑝,𝑖𝑖}≃log₂𝐶+𝐻log₂𝑛_𝑝. (37) This expression (37) may be seen as a line 𝑦=𝑏+𝑎𝑥, where 𝑦=log₂((1/(1+max(𝑖𝑖)))∑^{max(𝑖𝑖)}_𝑖𝑖=0[[𝑅/𝑆]_{𝑛𝑝,𝑖𝑖}]), 𝑥=log₂𝑛_𝑝, 𝑏=log₂𝐶 and the slope of the line is 𝐻 [28]. In order to estimate the Hurst exponent, several values for 𝑛_𝑝 are needed. Figure 14 presents the Hurst exponent value estimated for different step-size values. For each step-size value we used 100 Monte Carlo trials where for each trial the Hurst exponent was estimated. Thus, we have for each step-size value 100 estimated values for the Hurst exponent. According to Figure 14, we observe two things. (1) The averaged Hurst exponent (obtained from 100 results) is higher (closer to one) for a lower step-size value. (2) The averaged Hurst exponent obtained for the different step-size values is very high (above 0.9). This means that the residual ISI is trending. If there is an increase from time index 𝑛−1 to time index 𝑛, there will probably be an increase from time index 𝑛 to time index 𝑛+1. The same is true of decreases, where a decrease will tend to follow a decrease. The large Hurst exponent value indicates that the trend is strong. In our case, we see according to Figure 14 that the trend is stronger for lower step-size values.

Figure 14 

Hurst exponent results (belonging to the ISI) for different step-sizes by using Godard’s algorithm, a 16QAM source input going through Channel 2, a 13-tap-length equalizer and SNR=12 [dB] (the added noise was a white Gaussian process (𝐻=0.5)). 𝑁≃9000 and 𝑛𝑝=[100200300400].

Figures 15 and 16 present the Hurst exponent value estimated for different equalizer’s tap lengths and SNR values, respectively. For each tap-length and SNR value we used 100 Monte Carlo trials where for each trial the Hurst exponent was estimated. Thus, we have for each tap-length and SNR value 100 estimated values for the Hurst exponent. According to Figures 15 and 16, the equalizer’s tap-length and SNR have nearly no impact on the Hurst exponent estimated from the residual ISI series. Figure 17 describes the simulated histogram of the estimated Hurst exponent for three different step-size parameters. According to Figure 17, the histogram of the estimated Hurst exponent resembles the Gaussian shape.

Figure 15 

Hurst exponent results (belonging to the ISI) for different equalizer’s tap-length by using Godard’s algorithm, a 16QAM source input going through Channel 2, step-size parameter 𝜇=0.00001 and SNR=12 [dB] (the added noise was a white Gaussian process (𝐻=0.5)). 𝑁≃9000 and 𝑛𝑝=[100200300400].

Figure 16 

Hurst exponent results (belonging to the ISI) for different SNR values (the added noise was a white Gaussian process (𝐻=0.5)) by using Godard’s algorithm, a 16QAM source input going through Channel 2, step-size parameter 𝜇=0.00001, and with a 13-tap-length equalizer. 𝑁≃9000 and 𝑛𝑝=[100200300400].

(a)

(b)

(c)

Figure 17 

Histogram of the Hurst exponent (belonging to the ISI) for different step-size values by using Godard’s algorithm, a 16QAM source input going through Channel 2, and a 13-tap-length equalizer. 𝑁≃9000 and 𝑛𝑝=[100200300400]. The upper, bottom left, and bottom right figures were obtained for 𝜇=0.00001, 𝜇=0.00003, and 𝜇=0.00009, respectively.

4.2. The Statistical Behaviour of the Convolutional Noise

In this subsection we estimate the Hurst exponent of the real part of the convolutional noise from the Rescaled Range (R/S) [28] with overlapping regions. For that purpose, let us denote now 𝜉 as a vector of samples of the real part of the convolutional noise (obtained from a single Monte Carlo trial in the convergence region for the noiseless case) with length 𝑁. Please note that the real and imaginary parts of the convolutional noise are independent. Thus, the statistical behavior of the real and imaginary parts of the convolutional noise is approximately the same. In the following we use (37) for estimating the Hurst exponent. Figure 18 presents the Hurst exponent value estimated for different step-size values. For each step-size value we used 100 Monte Carlo trials where for each trial the Hurst exponent was estimated. Thus, we have for each step-size value 100 estimated values for the Hurst exponent. According to Figure 18, we observe that as we enlarge the step-size parameter, the estimated Hurst exponent is closer to 0.5, while for lower values for the step-size parameter, the Hurst exponent is smaller than 0.5. According to [29], a Hurst Exponent value 𝐻 between 0 and 0.5 exists for time series with “antipersistent behaviour.” This means that an increase will tend to be followed by a decrease (or a decrease will be followed by an increase). This behaviour is sometimes called “mean reversion” which means future values will have a tendency to return to a longer term mean value. The strength of this mean reversion increases as 𝐻 approaches 0. As it was already mentioned earlier in this paper, the convolutional noise is often considered as a white Gaussian process. But a white Gaussian process has a Hurst exponent value of 0.5 which in our case is achieved only for high values for the step-size parameter.

Figure 18 

Hurst exponent results (belonging to the convolutional noise) for different step-sizes by using Godard’s algorithm, a 16QAM source input going through Channel 2, the noiseless case, and a 13-tap-length equalizer. 𝑁≃9000 and 𝑛𝑝=[100200300400].

5. Conclusion

In this paper, we proposed a closed-form approximated expression (or an upper limit) for the residual ISI obtained by blind adaptive equalizers valid for the fGn input case where the Hurst exponent is in the region of 0.5≤𝐻<1. According to simulation results, a high correlation is obtained between the calculated and simulated results for the residual ISI for some cases while for others, the new obtained expression is a relative tight upper limit for the averaged residual ISI results. In this paper we investigated the statistical behavior of the residual ISI as well as the convolutional noise. We have found that the Hurst exponent of the residual ISI is close to one, almost independent of the SNR or equalizer’s tap length but depends on the step-size parameter. Since the Hurst exponent of the residual ISI is above 0.5, the residual ISI is trending. This means that if there is an increase from time index 𝑛−1 to time index 𝑛, there will probably be an increase from time index 𝑛 to time index 𝑛+1. The same is true of decreases, where a decrease will tend to follow a decrease. Concerning the convolutional noise, we have found that the convolutional noise obtained in the steady state is a Gaussian noise process having a Hurst exponent depending on the step-size parameter. Only for large values for the step-size parameter we obtain approximately a white Gaussian process (𝐻=0.5) which is the statistical model assumed in the literature for the convolutional noise for the entire range of step-size values.