A Nonlinear Diversity Combiner of Binary Signals ... - Semantic Scholar
A Nonlinear Diversity Combiner of Binary Signals in the Presence of Impulsive Interference Khodr A. Saaifan and Werner Henkel Transmission Systems Group (TrSyS) Jacobs University Bremen Bremen 28759, Germany {k.saaifan, w.henkel}@jacobs-university.de
Abstract—A Middleton Class-A (MCA) model is one of the most accurate statistical-physical models for narrowband impulse noise. The previous studies show that time diversity can efficiently be used to reduce the impact of MCA noise. The optimum combiner in such noise consists of a nonlinear preprocessor followed by a conventional combiner. Since an MCA noise process consists of an infinite number of noise states, there is no closedform solution of the optimum nonlinearity. In this paper, we adopt a two-term model for the MCA process, which is further approximated to a simpler noise model. Therefore, we introduce a closed-form approximation of the optimum nonlinearity in the presence of real-valued MCA noise. In fading channels, we use a complex extension of an MCA model. We show how the nonlinearity operation maintains the diversity advantage in such a noise model. Index Terms—impulse noise, Middleton Class-A model, diversity combiner, fading channels.
I. I NTRODUCTION Diversity is a well-known technique for combating the detrimental effects of fading and interference. The idea behind this technique is to supply the receiver with several replicas of the same information signal transmitted over channels of different levels of fading and interference. These signals can be combined at the receiver by using a diversity combiner, which improves the average signal-to-noise ratio (SNR). In additive white Gaussian noise (AWGN) channels, maximumratio combining (MRC) is the optimum diversity combiner [1]. However, communication systems are often subject to natural and man-made interference such as atmospheric noise, interference from car ignition circuits, ingress from RF and power lines sources. Such interference has an impulsive appearance, which can be modeled by a Middleton Class-A (MCA) density [2]. In the absence of fading, diversity combining in statistically independent MCA channels has been a topic of interest of several papers [3]–[5]. Since the MCA model contains an infinite number of noise states, a closed-form expression of the optimum combining is not available in the literature. Under a small signal assumption, it has been shown in [3], [6] that a linear combiner preceded by a logarithmic nonlinearity approximates the optimum combiner. However, it has a very poor performance at high SNRs. In [5], we introduced a decision boundary evaluation to analyze the behavior of a
locally optimum detector (LOD) and a clipping nonlinearity for two statistically independent observations. In the presence of fading, the performance analysis of MRC, equalgain combining (EGC), and post-detection combining has been considered in [7]. So far, there has been no investigation how the optimum nonlinearity should look like over fading channels. The basic objectives of this paper can be summarized by two contributions. The primarily contribution is to derive an approximation for the optimum nonlinearity in the presence of MCA noise. The second contribution is to generalize this nonlinearity for Rayleigh fading channels. This paper is organized as follows. Section II briefly describes the mathematical model of binary phase-shift keying (BPSK) with diversity. In Section III, we derive the optimum nonlinearity of binary signals corrupted by MCA noise. In Section IV, we introduce a closed-form approximation for the optimum nonlinearity, and we show how it looks like in the presence of fading. Finally, simulation results and concluding remarks are presented in sections V and VI, respectively. II. S YSTEM M ODEL We start by describing the general mathematical model of a binary signal transmission with diversity. The interference processes among the L channels are assumed to be statistically independent, which can, e.g., be realized by transmitting the signal in L different time slots. The interference process in the lth channel consists of AWGN superimposed to impulsive interference, which represents the interference of various manmade or natural sources. Hence, the received noise is given by zl (t) = nl (t) + wl (t) ,
(1)
where nl (t) and wl (t) are assumed to be statistically independent. For simplicity, we assume that the transmitter uses a rectangular transmit pulse x(t) within a bit duration 0 ≤ t ≤ Tb . The equivalent low-pass received signal in one signaling interval is √ 2Eb xk (t)+zl (t) , l = 1, 2, · · · , L , k = 1, 2 rl (t) = LN0 (2) where x1 (t) = +x(t) and x2 (t) = −x(t) correspond to a binary 1 and 0, respectively. N0 is the one-sided noise power
spectral density. For real signal transmission, zl (t) is a realvalued noise process with zero mean and unit variance. At the receiver, the received signal rl (t) is passed through a matched filter followed by a sampler. Thus, we have rl = Bk + zl , l = 1, 2, · · · , L , k = 1, 2 (3) √ √ 2Eb 2Eb where B1 = LN0 and B2 = − LN0 . zl represents the real-valued noise samples at the output of the matched filter in the lth diversity channel. When the mean duration of the interference waveforms comprising wl (t) is comparable to a bit duration Tb , the noise samples zl can be modeled by an MCA density as [2] ∞ z2 ∑ βm − 2σl2 √ p(zl ) = (4) e m , 2 2πσ m m=0 where βm =
e−A Am , m!
(5)
and
m/A + Γ . (6) 1+Γ This model is defined using two parameters A and Γ. The parameter A reflects the impulsiveness of the noise process, e.g., at small values of A, the statistics of the noise samples are characterized as a summation of a few interfering waveforms and the interference has an impulsive appearance. For a large number of interferers, i.e., A ≫ 1, the noise statistic is almost Gaussian. The parameter Γ represents the power ratio of a Gaussian part to an impulsive part of noise. Regarding (4), the MCA model can be seen as a noise process of an infinite number of noise states [8]. The first state, m = 0, represents AWGN samples, whereas the remaining states, m ≥ 1, model the noise samples contaminated by impulses. The state m has a 2 Gaussian probability density function (pdf) with variance σm , 2 2 2 such that σ0 ≤ σ0 ≤ · · · ≤ σ∞ . m is a Poisson-distributed random variable such that the probability of being in a given state m is equal to βm . [Using the variance ] this ∑∞description, 2 of the noise process is E zl2 = m=0 βm σm , which can be easily shown to equal 1. Similar to the conventional model [2], we assume that the interfering waveforms creating the impulse are independent over time. Then the noise observations zl , ∀l = 1, · · · , L, are statistically independent. Therefore, the pdf of the received noise vector z = [z1 · · · zL ] can be expressed as
Fig. 1.
Model of binary digital communications with diversity
where hl = αl e−jθl is a complex-valued channel gain with Rayleigh distributed envelope and uniformly distributed phase. In this model, zl is a complex-valued noise process. Since the in-phase and quadrature (IQ) components of zl are generated by the same physical process creating the impulse, the noise samples zl = zI,l + jzQ,l can be modeled by a bivariate MCA density as [8], [10] ∗ ∞ ∑ βm − z2σl 2zl p(zl ) = e m . 2 2πσ m m=0
2 σm =
p(z) =
L ∑ ∞ ∏
β √ m e 2 2πσm l=1 m=0
z2 − 2σl2 m
l = 1, 2, · · · , L ,
III. O PTIMUM C OMBINER The optimum detector in real-valued MCA channels is quantitatively studied in [3]. Given the received vector r = [r1 · · · rL ], assuming equally likely transmitted signals, the optimum detector computes the following statistics {∏ } L H1 p(r |B ) l 1 ≥ ΛM L = log ∏l=1 (10)
Abstract—A Middleton Class-A (MCA) model is one of the most accurate statistical-physical models for narrowband impulse noise. The previous studies show that time diversity can efficiently be used to reduce the impact of MCA noise. The optimum combiner in such noise consists of a nonlinear preprocessor followed by a conventional combiner. Since an MCA noise process consists of an infinite number of noise states, there is no closedform solution of the optimum nonlinearity. In this paper, we adopt a two-term model for the MCA process, which is further approximated to a simpler noise model. Therefore, we introduce a closed-form approximation of the optimum nonlinearity in the presence of real-valued MCA noise. In fading channels, we use a complex extension of an MCA model. We show how the nonlinearity operation maintains the diversity advantage in such a noise model. Index Terms—impulse noise, Middleton Class-A model, diversity combiner, fading channels.
I. I NTRODUCTION Diversity is a well-known technique for combating the detrimental effects of fading and interference. The idea behind this technique is to supply the receiver with several replicas of the same information signal transmitted over channels of different levels of fading and interference. These signals can be combined at the receiver by using a diversity combiner, which improves the average signal-to-noise ratio (SNR). In additive white Gaussian noise (AWGN) channels, maximumratio combining (MRC) is the optimum diversity combiner [1]. However, communication systems are often subject to natural and man-made interference such as atmospheric noise, interference from car ignition circuits, ingress from RF and power lines sources. Such interference has an impulsive appearance, which can be modeled by a Middleton Class-A (MCA) density [2]. In the absence of fading, diversity combining in statistically independent MCA channels has been a topic of interest of several papers [3]–[5]. Since the MCA model contains an infinite number of noise states, a closed-form expression of the optimum combining is not available in the literature. Under a small signal assumption, it has been shown in [3], [6] that a linear combiner preceded by a logarithmic nonlinearity approximates the optimum combiner. However, it has a very poor performance at high SNRs. In [5], we introduced a decision boundary evaluation to analyze the behavior of a
locally optimum detector (LOD) and a clipping nonlinearity for two statistically independent observations. In the presence of fading, the performance analysis of MRC, equalgain combining (EGC), and post-detection combining has been considered in [7]. So far, there has been no investigation how the optimum nonlinearity should look like over fading channels. The basic objectives of this paper can be summarized by two contributions. The primarily contribution is to derive an approximation for the optimum nonlinearity in the presence of MCA noise. The second contribution is to generalize this nonlinearity for Rayleigh fading channels. This paper is organized as follows. Section II briefly describes the mathematical model of binary phase-shift keying (BPSK) with diversity. In Section III, we derive the optimum nonlinearity of binary signals corrupted by MCA noise. In Section IV, we introduce a closed-form approximation for the optimum nonlinearity, and we show how it looks like in the presence of fading. Finally, simulation results and concluding remarks are presented in sections V and VI, respectively. II. S YSTEM M ODEL We start by describing the general mathematical model of a binary signal transmission with diversity. The interference processes among the L channels are assumed to be statistically independent, which can, e.g., be realized by transmitting the signal in L different time slots. The interference process in the lth channel consists of AWGN superimposed to impulsive interference, which represents the interference of various manmade or natural sources. Hence, the received noise is given by zl (t) = nl (t) + wl (t) ,
(1)
where nl (t) and wl (t) are assumed to be statistically independent. For simplicity, we assume that the transmitter uses a rectangular transmit pulse x(t) within a bit duration 0 ≤ t ≤ Tb . The equivalent low-pass received signal in one signaling interval is √ 2Eb xk (t)+zl (t) , l = 1, 2, · · · , L , k = 1, 2 rl (t) = LN0 (2) where x1 (t) = +x(t) and x2 (t) = −x(t) correspond to a binary 1 and 0, respectively. N0 is the one-sided noise power
spectral density. For real signal transmission, zl (t) is a realvalued noise process with zero mean and unit variance. At the receiver, the received signal rl (t) is passed through a matched filter followed by a sampler. Thus, we have rl = Bk + zl , l = 1, 2, · · · , L , k = 1, 2 (3) √ √ 2Eb 2Eb where B1 = LN0 and B2 = − LN0 . zl represents the real-valued noise samples at the output of the matched filter in the lth diversity channel. When the mean duration of the interference waveforms comprising wl (t) is comparable to a bit duration Tb , the noise samples zl can be modeled by an MCA density as [2] ∞ z2 ∑ βm − 2σl2 √ p(zl ) = (4) e m , 2 2πσ m m=0 where βm =
e−A Am , m!
(5)
and
m/A + Γ . (6) 1+Γ This model is defined using two parameters A and Γ. The parameter A reflects the impulsiveness of the noise process, e.g., at small values of A, the statistics of the noise samples are characterized as a summation of a few interfering waveforms and the interference has an impulsive appearance. For a large number of interferers, i.e., A ≫ 1, the noise statistic is almost Gaussian. The parameter Γ represents the power ratio of a Gaussian part to an impulsive part of noise. Regarding (4), the MCA model can be seen as a noise process of an infinite number of noise states [8]. The first state, m = 0, represents AWGN samples, whereas the remaining states, m ≥ 1, model the noise samples contaminated by impulses. The state m has a 2 Gaussian probability density function (pdf) with variance σm , 2 2 2 such that σ0 ≤ σ0 ≤ · · · ≤ σ∞ . m is a Poisson-distributed random variable such that the probability of being in a given state m is equal to βm . [Using the variance ] this ∑∞description, 2 of the noise process is E zl2 = m=0 βm σm , which can be easily shown to equal 1. Similar to the conventional model [2], we assume that the interfering waveforms creating the impulse are independent over time. Then the noise observations zl , ∀l = 1, · · · , L, are statistically independent. Therefore, the pdf of the received noise vector z = [z1 · · · zL ] can be expressed as
Fig. 1.
Model of binary digital communications with diversity
where hl = αl e−jθl is a complex-valued channel gain with Rayleigh distributed envelope and uniformly distributed phase. In this model, zl is a complex-valued noise process. Since the in-phase and quadrature (IQ) components of zl are generated by the same physical process creating the impulse, the noise samples zl = zI,l + jzQ,l can be modeled by a bivariate MCA density as [8], [10] ∗ ∞ ∑ βm − z2σl 2zl p(zl ) = e m . 2 2πσ m m=0
2 σm =
p(z) =
L ∑ ∞ ∏
β √ m e 2 2πσm l=1 m=0
z2 − 2σl2 m
l = 1, 2, · · · , L ,
III. O PTIMUM C OMBINER The optimum detector in real-valued MCA channels is quantitatively studied in [3]. Given the received vector r = [r1 · · · rL ], assuming equally likely transmitted signals, the optimum detector computes the following statistics {∏ } L H1 p(r |B ) l 1 ≥ ΛM L = log ∏l=1 (10)
