# The Capacity Performance of OFDM Systems with Nonlinear Pulse Blanking in Frequency Selective Fading Channels

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## Abstract

Nonlinear pulse blanking is a widely applied method to mitigate the impulse interference of orthogonal frequency division multiplexing (OFDM) systems. To quantitatively analyze the capacity of the OFDM system with nonlinear pulse blanking, the analytical expressions of the instantaneous output signal-to-interference-plus-noise ratio of the OFDM system with ideal blanking, peak value blanking, and optimum threshold blanking are first derived. The probability density function of the instantaneous capacity of the OFDM system with nonlinear pulse blanking is then estimated by the finite mixture with the expectation–maximization algorithm. The ergodic capacity and the probability of the outage capacity of the OFDM system with nonlinear pulse blanking are obtained based on the estimated probability density function. Finally, the simulation results are provided to indicate its good agreement with the theoretical results.

## Keywords

OFDM system Capacity Impulsive noise Nonlinear pulse blanking FM-EM algorithm## 1 Introduction

The orthogonal frequency division multiplexing (OFDM) scheme has several advantages such as its high spectral efficiency, robustness to channel fading, and easy implementation as compared with the single carrier transmission scheme. Hence, the OFDM transmission scheme has been widely adopted in many modern communication systems [1]. The received signal of an OFDM system in practical applications is not only affected by additive Gaussian white noise (AWGN) but also by impulsive noise generated by vehicle ignition devices, adjacent channel electromagnetic radiation, and electrical devices of power lines [2, 3, 4, 5]. Since the statistical properties of impulse noise are quite different than that of AWGN, the transmission reliability of an OFDM system, which is designed and optimized under the assumption of AWGN, severely decreases due to impulse noise interference. Hence, it is of great significance to conduct research on the impulse interference mitigation of OFDM systems.

To improve the transmission reliability of an OFDM system with impulse interferences, a great deal of research has been conducted on the mitigation of impulse noise. These research studies can be classified into three categories: nonlinear impulse interference suppression such as pulse blanking, pulse clipping, and joint pulse blanking and pulse clipping [6, 7, 8, 9, 10]; interference mitigation based on impulse signal reconstruction such as decision-directed and compressed sensing [11, 12, 13, 14, 15, 16]; and impulse mitigation based on coding and iterative decoding [17, 18]. Although the transmission reliability of an OFDM system with the nonlinear impulse interference suppression method is inferior to the other two methods, the nonlinear impulse interference suppression method has many advantages such as its low computational complexity, easy engineering implementation, and strong adaptability [19]. Therefore, the nonlinear impulse interference suppression method is widely used in OFDM systems.

In the field of the nonlinear impulse interference suppression for OFDM systems, the related research is given as follows. Haffenden [6] and Cowley [7] first proposed an impulse mitigation method based on pulse blanking and pulse clipping. To apply this method to an OFDM receiver, two key issues should be addressed: (1) the optimal threshold of pulse blanking and pulse clipping and (2) the elimination of inter-carrier interference (ICI) caused by nonlinear pulse blanking. To address the optimal threshold setting problem of pulse blanking, Zhidkov [20] proposed a method based on maximizing the output SNR criterion in AWGN channels. Epple [21] further proposed an adaptive threshold optimization method based on maximizing the SINR criterion. Recently, a threshold setting method based on the peak value amplitude of received OFDM signals was presented by Alsusa [22, 23]. To mitigate the ICI caused by pulse blanking, Yih [24] proposed an iterative ICI cancellation scheme, while Epple [25] presented a signal combination method based on the maximization of the SINR criterion. Furthermore, Darsena et al. [26] proposed a frequency-domain linear FIR equalizer.

In the performance analysis of an OFDM system with impulse interference, Ghosh [27] first compared the influence of impulse noise on the transmission reliability of a single carrier and a multi-carrier system. Ma [28] derived the bit error rate performance of an OFDM system over a multipath and impulse noise environment. Amirshahi [29] further extended this result to the coded OFDM system. Liu [30] investigated the symbol error probability of an OFDM system with ideal pulse blanking over the frequency-selective fading channel, and this result was further extended to an OFDM system with peak value blanking in [31]. In addition, Zhang et al. [32] investigated the symbol error ratio performance of an OFDM system in the hidden semi-Markov impulse channel. Although many studies on nonlinear impulse interference have been presented, the effect of nonlinear impulse interference suppression on the capacity performance of an OFDM system has not been fully understood. These are the primary objectives of our study: (1) to generate the analytical expressions of the capacity of an OFDM system with nonlinear pulse blanking and (2) to compare the performance of an OFDM system with different pulse blanking schemes from the perspective of the channel capacity.

In this paper, the probability density function (PDF) of the instantaneous channel capacity of an OFDM system is derived by the finite mixture with expectation–maximization (FM-EM) algorithm over the frequency-selective Rayleigh and Rician fading channels. The ergodic and the probability of the outage capacity of an OFDM system with nonlinear pulse blanking are obtained based on the PDF of the instantaneous channel capacity of an OFDM system. At the end, theoretical results are validated by the computer simulation results.

The contributions of this paper include the following two aspects: (1) the ergodic capacity and the probability of the outage capacity of an OFDM system with pulse blanking are derived based on the FM-EM algorithm and (2) from the perspective of the channel capacity, three nonlinear blanking schemes are compared, specifically ideal pulse blanking, peak value blanking, and optimum threshold blanking.

The rest of this paper is organized as follows. In Sect. 2, the model of OFDM system with nonlinear pulse blanking is introduced. In Sect. 3, the instantaneous SINR expressions of an OFDM system with nonlinear pulse blanking are derived. In addition, the ergodic capacity and the probability of the outage capacity of a nonlinear OFDM system are presented in Sect. 4. The computer simulation results and analysis are presented in Sect. 5. Finally, Sect. 6 presents the conclusions.

## 2 System Model

*N*is the number of the modulated symbols. The

*k*-th symbol \(X_k\) of \({\mathbf {X}}\) is assumed to be independent and identically distributed (i.i.d.) with \(E\left[ {X_k } \right] = 0\) and \(E\left[ {\left| {X_k } \right| ^2 } \right] = \sigma _s^2\). The modulated symbol vector \({\mathbf {X}}\) is then transformed into \({\mathbf {x}}\) by the N-points inverse discrete Fourier transform (IDFT) operation, of which the output signal vector of the IDFT is presented as follows:

*k*represents the subcarrier index in the frequency domain and

*n*denotes the samples index in the time domain. As the IDFT operation is a unitary transformation, the statistical property of \({\mathbf {x}}\) agrees with \({\mathbf {X}}\). Thus, \(x_n\) is also i.i.d such that \(E\left[ {x_n } \right] = 0\) and \(E\left[ {\left| {x_n } \right| ^2 } \right] = \sigma _s^2\). Finally, the transmitted signal \({\mathbf {x}}\) is inserted the cycle prefix (CP) and sent to a multipath fading channel.

*L*paths and the discrete-time channel power is normalized to one, i.e., \(\sum \nolimits _{l = 0}^{L - 1} {E\left[ {\left| {h_l } \right| ^2 } \right] = 1}\), where \(h_l\) is an i.i.d. complex Gaussian random variable. \({{\mathbf {n}}} = \left[ {n_0 , \ldots ,n_n , \ldots ,n_{N - 1} } \right] ^T\) denotes the complex AWGN vector, where \(n_n\) is an i.i.d complex Gaussian random variable with a mean of zero and a variance \(\sigma _n^2\). \({\mathbf {i}} = \left[ {i_0 ,i_1 , \ldots ,i_n , \ldots ,i_{N - 1} } \right] ^T\) represents the impulse noise vector coming from the channel, where \(i_n\) is modeled as a Bernoulli–Gaussian random variable [27, 28], which can be expressed as:

*p*and \(1-p\), respectively. The input signal-to-noise ratio is defined as SNR \(\buildrel {\Delta } \over = {{\sigma _s^2 } \big / {\sigma _n^2 }}\), and the input signal-to-interference ratio is defined as SIR \(\buildrel {\Delta } \over = {{\sigma _s^2 } \big / {\sigma _g^2 }}\).

*k*-th sub-channel is given by:

*k*-th component of \({\mathbf {H}}\) and \(( \cdot )^*\) denotes the complex conjugate operator. Finally, the output signal vector \({\hat{{\mathbf {Y}}}} = [{\hat{Y}}_0 , \ldots ,{\hat{Y}}_k , \ldots ,{\hat{Y}}_{N - 1} ]^T\) of the equalizer is sent to the demodulator, wherein the output of the demodulator \({\hat{{\mathbf {I}}}}\) is the estimation of the transmitted information bit vector \({\mathbf {I}}\).

## 3 SINR of the OFDM Receiver

In this section, the output SINR expression of a conventional OFDM receiver is first presented in Sect. 3.1. The output SINR expressions of the nonlinear OFDM receivers with ideal blanking, peak value blanking, and optimum threshold blanking are then derived in Sects. 3.2, 3.3, and 3.4, respectively.

### 3.1 SINR of the Conventional OFDM Receiver

*k*-th component of \({\mathbf {Y}}\) is presented as follows:

*k*-th components of the vectors \({\mathbf {N}}\) and \({\mathbf {I}}\), respectively. Since \({\mathbf {F}}\) is a unitary matrix, the statistical property of \({\mathbf {N}}\) agrees with \({\mathbf {n}}\). Therefore, \(N_k\) is also an i.i.d. complex Gaussian random variable with a mean of zero and a variance \(\sigma _n^2\).

*k*-th sub-channel, and the rest of the two terms represent the noise signal, which is contributed by the AWGN and the impulse noise. Therefore, Eq. (12) can be rewritten as follows:

*k*-th sub-channel, \({\tilde{N}}_k\) represents the equivalent frequency domain noise, which is defined as follows:

*k*-th component of \({\mathbf{G = }}{{\mathbf {F}}} \cdot {\mathbf{g}}\), such that \(G_k\sim {{{{\mathcal {C}}}\mathcal{N}}}\left( {0,\sigma _{\mathrm{g}}^2} \right)\). Considering that the random variable \(N_k\) and \(G_k\) are statistically independent, the variance of \({\tilde{N}}_k\) can be calculated as follows:

*k*-th sub-channel is given by:

*k*-th sub-channel is presented as follows:

In the following situation, if no impulse noise is presented at the input of the OFDM receiver, i.e., \(p = 0\), the instantaneous SINR expression given in Eq. (17) is then reduced to \(r_{k,\mathrm{out}} = \left( {{{\sigma _s^2 } \big / {\sigma _n^2 }}} \right) \cdot \left| {H_k } \right| ^2\), which was reported in [28].

### 3.2 SINR of the OFDM Receiver with Ideal Blanking

*n*-th diagonal element of the nonlinear pulse blanking matrix \({\mathbf {D}}\) is presented as follows:

*n*-th component of the signal vector \({\mathbf{y}}\) can be expressed as follows:

*k*-th component of \({\mathbf {Y}}\) is defined as follows:

*k*-th sub-channel is defined as follows:

### 3.3 SINR of the OFDM Receiver with Peak Value Blanking

*n*-th diagonal element of the nonlinear pulse blanking matrix \({\mathbf {D}}\) is defined as follows:

*n*-th component of signal vector \(\mathbf{y}\) can be rewritten as follows:

*k*-th component of \({\mathbf {Y}}\) is defined as follows:

*k*-th sub-channel is defined as follows:

### 3.4 SINR of the OFDM Receiver with Optimum Threshold Blanking

*n*-th diagonal element \(d_n\) of the nonlinear pulse blanking matrix \({\mathbf {D}}\) is defined as follows:

*n*-th component of the signal vector \({\mathbf {y}}\) is then defined as follows:

*k*-th component of \({\mathbf {Y}}\) is defined as follows:

*k*-th sub-channel is defined as follows:

*I*denotes an event wherein the impulse noise occurs at a probability of \(p\left( I \right) = p\); \({\bar{I}}\) is the complement of event

*I*with a probability of \(P\left( {{\bar{I}}} \right) = 1 - p\); \(\sigma _I^2 = \sigma _s^2 + \sigma _n^2 + \sigma _g^2\); and \(\sigma _{{\bar{I}}}^2 = \sigma _s^2 + \sigma _n^2\).

## 4 Capacity of the OFDM System with Nonlinear Pulse Blanking

Note that it is difficult to directly calculate the ergodic capacity based on Eq. (43). The present study first utilize the FM-EM algorithm to estimate the PDF of the instantaneous capacity of the OFDM system. The ergodic capacity and the probability of outage capacity of the OFDM system are then obtained based on the estimated PDF.

*M*is the number of weighted densities, \(\pi _m\) represents the weighting coefficient of the m-th component and satisfies \(\sum \nolimits _{m = 1}^M {\pi _m } = 1\left( {\pi _m > 0} \right)\), and \(\mu _m\) and \(\sigma _m^2\) denote the mean and the variance of the

*m*-th component, respectively. To determine the values of \(\pi _m\), \(\mu _m\), and \(\sigma _m^2\), the famous EM algorithm is used to estimate the \(3M - 1\) unknown parameters in Eq. (44). Using the PDF given in Eq. (43), the ergodic capacity of the OFDM system can be obtained as follows [34]:

*Q*-function, which is defined as follows:

## 5 Numerical Simulations

In this section, we provide computer simulation results to verify the accuracy of the analytical results derived by Eqs. (45) and (47).

### 5.1 Simulation Settings

The present study simulated an OFDM system with 512 subcarriers, a cyclic prefix length of 16, and QPSK and 16QAM modulations. The following channel models were employed: frequency-selective Rayleigh fading channels with 9 paths, frequency-selective Rician fading channels with 9 paths, and a Rician factor of 10 dB. In addition, the present study employed a Bernoulli–Gaussian impulse noise model. Four impulse mitigation schemes, specifically the conventional method, ideal pulse blanking, peak value blanking, and optimum threshold blanking, were adopted in the OFDM receivers to eliminate the impulse noise. An ideal channel estimation was assumed, and the linear zero-forcing equalizer was used to compensate for the channel distortion. In addition, the theoretical ergodic capacity and the probability of the outage capacity of the OFDM system with nonlinear pulse blanking were obtained as follows. First, the given parameter and input SNR were applied to calculate the instantaneous SINR by Eqs. (17), (25), (32), and (39). The instantaneous capacity was then obtained by Eq. (44). Afterwards, the weighting coefficients \(\pi _m\), means \(\mu _m\), and variances \(\sigma _m^2\) were calculated by Eqs. (51), (53), and (54). Finally, the ergodic capacity and the probability of the outage capacity were calculated by Eqs. (45) and (47), respectively.

### 5.2 Frequency-Selective Rayleigh Fading Channels

Figure 3 presents the ergodic capacity of the OFDM system with ideal pulse blanking over the frequency-selective Rayleigh fading channels (\({\mathrm{SIR}}=-10\) dB), wherein the impulse occurrence probability *p* was set to 0, \(10^{-4}\), \(10^{-3}\), and \(10^{-2}\). Two kinds of curves are presented in Fig. 3, wherein the curves marked as “MO” were generated by the Monte Carlo simulations and the curves marked as “FM” were generated by the FM-EM algorithm. Based on these curves, the following results were observed: (1) the theoretical ergodic capacity obtained by the FM-EM algorithm agrees with the Monte Carlo simulation results; and (2) the ergodic capacity of the OFDM system with ideal pulse blanking tended to decrease following an increase in the impulse occurrence probability because the ideal pulse blanking method completely mitigated the impulse noise and resulted in the mitigation of useful signals. Hence, an increase in the occurrence probability of the impulse noise increased the loss of useful signal energy, thereby resulting in an increased loss of channel capacity.

*p*is also set to 0, \(10^{-3}\), \(10^{-4}\), and \(10^{-2}\). The dashed curves were obtained by the Monte Carlo simulations and the solid curves were obtained by the FM-EM algorithm. The following results were observed: (1) the ergodic capacity of the OFDM system with peak value blanking tended to decrease following an increase in the impulse occurrence probability, which is similar to the ideal pulse blanking method; and (2) under the same interference condition, the ergodic capacity of the OFDM system with QPSK modulation was higher than that of the OFDM system with 16QAM modulation because the peak value of the OFDM symbol in the QPSK modulation was lower than the 16QAM modulation. As a result, the energy of the residual impulse noise in the nonlinear OFDM system with 16QAM modulation was more than that of the QPSK modulation.

### 5.3 Frequency-Selective Rician Fading Channels

Figures 7 and 8 present the ergodic capacity and the probability of outage capacity of the OFDM system with different blanking schemes over frequency-selective Rician fading channels (QPSK, \({\mathrm{SIR}}=-10\) dB, \(K_{Rice}=10\) dB, and \(p=10^{-2}\)). The dashed curves were derived from the Monte Carlo simulation and the solid curves were obtained by the FM-EM algorithm. The following two results were observed based on the presented curves: (1) the theoretical results were well in agreement with the simulation results; and (2) from the aspect of the ergodic capacity and the outage capacity probability, ideal pulse blanking was deemed the best impulse mitigation scheme. Optimum threshold blanking and peak value blanking were characterized as the second and third best schemes, respectively.

## 6 Conclusions

The present study derived the ergodic capacity and the outage capacity probability of an OFDM system with nonlinear pulse blanking by the FM-EM algorithm. Computer simulation results were generated to verify the correctness of the theoretical formulas. From our research, the following conclusions were obtained: (1) nonlinear pulse blanking schemes resulted in a loss of channel capacity compared with the conventional OFDM receiver and (2) in terms of the ergodic and outage capacity, ideal pulse blanking was characterized as the best impulse blanking scheme. In addition, optimum threshold blanking was deemed the second scheme, whereas peak value blanking was characterized as the worst scheme.

## Notes

### Acknowledgements

This work was supported by National Natural Science Foundation of China under Grants Nos. U1633108 and U1733120, National Key Research and Development Program of China under Grant No. 2016YFB0502402.

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