# SER analysis of the MRC-OFDM receiver with pulse blanking over frequency selective fading channel

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## Abstract

Pulse blanking is a widely used method to eliminate impulsive interference in an orthogonal frequency division multiplexing (OFDM) receiver. To analyze the effect of the inter-carrier interference caused by pulse blanking on the symbol error rate (SER) performance of OFDM employing maximum ratio combining (MRC) receiver, the analytical expression of the signal-to-interference-noise ratio (SINR) of the OFDM employing MRC receiver is derived. Based on the SINR expression, furthermore, the SER performances over both Rayleigh and Ricean fading channels are also analyzed quantitatively. Simulation results validate the correctness of the derived formulas.

## Keywords

OFDM MRC Impulse interference Pulse blanking SINR SER Rayleigh Ricean## 1 Introduction

Orthogonal frequency division multiplexing (OFDM) is a multicarrier modulation technique which converts a frequency-selective channel into a parallel collection of frequency flat subchannels. Compared with single-carrier communication systems, OFDM has advantages, such as spectral efficiency, efficient implementation based on the fast Fourier transform (FFT), and simple channel equalization. For these reasons, the OFDM transmission scheme is widely employed in wireless and wired communications, e.g., digital subscriber lines (DSL), digital video broadcasting (DVB), digital audio broadcasting (DAB), wireless local area networks (WLAN), power line communications (PLC), long term evolution (LTE), and L-band digital aeronautical communication system (L-DACS) [1].

In practical applications, OFDM systems are often exposed to impulsive interference, e.g., ignition noise of passing vehicles, impulsive noise in power line or other systems operating in the same frequency range. Some studies have shown that the impulse interference with high power or frequent occurrence can significantly affect the performance of OFDM receivers [2, 3]. Thus, it is of great significance to eliminate the impulse interference in OFDM receivers.

To suppress the effect of impulsive noise on an OFDM receiver, an interference mitigation method based on pulse blanking is first proposed in [4, 5, 6]. However, when using this method to an actual system, two problems should be considered, the threshold of pulse blanking and the compensation of inter-carrier interference (ICI) caused by pulse blanking. To calculate the blanking threshold, an optimal threshold of pulse blanking is derived based on an expression for the signal-to-noise ratio (SNR) of the blanking nonlinearity [7]. With maximizing the signal-to-interference-and-noise ratio (SINR) criterion, an adaptive blanking threshold for the OFDM receiver is also proposed in [8]. To eliminate the ICI caused by pulse blanking, an iterative reconstructing and subtracting ICI method is proposed in [9, 10], and a frequency-domain ICI compensation scheme based on finite impulse response equalizer is proposed in [11].

Studies on the performance analysis of single-carrier systems in impulsive noise environment are presented in [12, 13, 14, 15, 16]. The performance of diversity combining technique over fading channels with impulsive noise is analyzed in [12]. Adopting the Middleton class A impulsive noise model, the performance of space-time coded systems over multiple-input multiple-output (MIMO) channels with impulsive noise is studied in [13, 14]. The effect of symmetric *α*-stable noise on space-time coded systems over MIMO fading channels is analyzed in [15]. A unified mathematical framework for the analysis of the asymptotic performance of amplify-and-forward cooperative diversity systems impaired by generic noise is provided in [16]. The influence of impulse noise on the symbol error rate (SER) performances of multicarrier and single-carrier communication systems is investigated and compared in [2]. With regard to the performance analysis of the OFDM receiver with blanking, the SNR expression of the OFDM receiver with blanking is obtained for the AWGN channel [17]. The SNR expression for the OFDM receiver with blanking is derived in frequency selective Rayleigh fading channel, as well as the SER performance on this SNR expression [18]. To the best of our knowledge, there is currently no literature regarding the SER performance of the maximum ratio combining (MRC)-OFDM receiver with pulse blanking.

In this paper, a closed-form expression for the SINR of the MRC-OFDM receiver with pulse blanking over frequency selective fading channel is derived. Furthermore, the SER performances of the MRC-OFDM receiver with pulse blanking over both Rayleigh and Ricean fading channels are also analyzed quantitatively based on this SINR expression. Finally, simulation results are presented that validate the correctness of the derived formulas.

This paper is arranged as follows. In Section 2, a system model comprising the OFDM transmitter and the MRC-OFDM receiver with pulse blanking is introduced. In Section 3, the closed-from expression for the SINR of the MRC-OFDM receiver with pulse blanking is derived, and the SER performances for both Rayleigh and Ricean fading channels are also analyzed. In Section 4, an overview of system and channel parameters is provided, and the analytically calculated and simulated symbol error performance curves are presented to validate our theoretical results. In Section 5, we draw the main conclusion.

## 2 System model

### 2.1 OFDM transmitter

*K*denotes the number of complex modulated symbols, and \(\left \{ {{S_{k}},k = 0,1,\ldots,K - 1} \right \}\) are assumed to be independent and identically distributed (i.i.d.) with \(E\left \{ {{S_{k}}} \right \} = 0\) and \(E\left \{ {{{\left | {{S_{k}}} \right |}^{2}}} \right \} = {\sigma _{S}^{2}}\).

**S**is then transformed into the time domain by the

*K*-point inverse fast Fourier transform (IFFT), and the output signal vector of IFFT is given by

**F**

^{ H }is defined by

with *k* being the subcarrier index in the frequency domain and *n* denoting the sample index in the time domain. As IFFT is a unitary transformation, the statistical property of **s** agrees with **S**, and \(\left \{ {{s_{n}},n = 0,1,\ldots,K - 1} \right \} \) are also i.i.d. with *E*{*s* _{ n }}=0 and \(E\left \{ {{{\left | {{s_{n}}} \right |}^{2}}} \right \} = {\sigma _{S}^{2}}\).

*K*

_{ g }-point cyclic prefix, the transmitted signal vector \(\mathbf {x} = {\left [{x_{0}},\ldots,{x_{n}},\ldots,{x_{K+{K_{g}}-1}}\right ]^{T}}\) can be expressed as

**P**

_{in}is the cyclic prefix insertion matrix denoted as

where \({\mathbf {0}}_{K \times {(K-K_{g})}}\) is a *K* _{ g }×(*K*−*K* _{ g }) zero matrix and **I** _{ K } is a *K*×*K* identity matrix.

The transmitted signal vector **x** is then converted to an analog signal *x*(*t*) by the D/A converter and *x*(*t*) is transformed into a RF signal by the RF front end. Finally, the RF signal is sent to a channel by the transmitter antenna.

### 2.2 MRC-OFDM receiver with pulse blanking

*v*th antenna can be represented as

where *x*(*t*) denotes the transmitted signal, ∗ the convolution operation, *h* ^{ v }(*t*) the channel impulse response of the *v*th channel, *n* ^{ v }(*t*) the complex Gaussian white noise signal from the *v*th channel, *i* ^{ v }(*t*) the impulsive noise from the *v*th channel, and *N* the number of receive antenna.

*r*

^{ v }(

*t*) is then converted to a sampled signal vector

**r**

^{ v }by the A/D converter. After removing the

*K*

_{ g }-point cyclic prefix, the received signal vector can be expressed as

where \(\mathbf {P}_{\text {out}} = \left [ {{{\mathbf {0}}_{K \times {K_{g}}}}} \quad {{\mathbf {I}_{K}}}\right ]\) denotes the cyclic prefix removal matrix.

**z**

^{ v }can be written as

**s**is given by Eq. (1), ⊗ denotes the circular convolution operator, and \({\mathbf {h}^{v}} = {\left [ {{h_{0}^{v}},\ldots,{h_{l}^{v}},\ldots,h_{{L_{v}} - 1}^{v}} \right ]^{T}}\) is the discrete-time channel impulse response of the

*v*th channel with

*L*

_{ v }paths, where \(\left \{{{h_{l}^{v}},l = 0,1\ldots,L_{v} - 1}\right \}\) are assumed to be i.i.d. and remain constant over one OFDM symbol interval, and the channel power is normalized to 1, i.e., \(\sum \limits _{l = 0}^{L - 1} {E\left \{ {{{\left | {{h_{l}^{v}}} \right |}^{2}}} \right \}} = 1\). In addition, \({\mathbf {n}^{v}} = {\left [{n_{0}^{v}},\ldots,{n_{n}^{v}},\ldots,n_{K - 1}^{v}\right ]^{T}}\) denotes the complex Gaussian white noise vector from the

*v*th channel where \(\left \{ {n_{n}^{v}},n = 0,1,\ldots,K - 1\right \} \) are the i.i.d. complex Gaussian random variables (RVs) of mean zeros and variance \({\delta _{n}^{2}}\) and \({\mathbf {i}^{v}} = {\left [ {{i_{0}^{v}},\ldots,{i_{n}^{v}},\ldots,i_{K - 1}^{v}} \right ]^{T}}\) denotes the impulsive noise vector where \({i_{n}^{v}}\) is modeled as a Bernoulli-Gaussian RV [2]

where \({b_{n}^{v}}\) is the Bernoulli process which is the arrival of impulsive noise with probability \({{\mathrm {P}}_{\mathrm {r}}}\left ({{b_{n}^{v}} = 1} \right) = p\) and \({g_{n}^{v}}\) is the complex Gaussian RV with mean zeros and variance \({{\delta _{g}^{v}}}^{2}\).

where \({\bar {\mathbf {B}}^{v}}\) denotes the impulse blanking matrix for the *v*th receive branch and where \({\bar {\mathbf {B}}^{v}} = \text {diag} \left (\bar {b}_{0}^{v},\ldots,\bar {b}_{n}^{v},\ldots, \bar b_{K - 1}^{v} \right)\) with \({\bar {b}_{n}^{v}} = 1 - {{b_{n}^{v}}}\).

**y**

^{ v }is then transformed into the frequency domain by the

*K*-point fast Fourier transform (FFT). Hence, the output signal vector of FFT \({\mathbf {Y}^{v}} = {\left [{Y_{0}^{v}},\ldots,{Y_{k}^{v}},\ldots,Y_{K - 1}^{v}\right ]^{T}}\) is given by

**F**is defined by

where \({\mathbf {H}^{v}} = \text {diag}\left ({{H_{0}^{v}},..,{H_{k}^{v}},\ldots,H_{K - 1}^{v}} \right)\) denotes the frequency domain channel transfer matrix of the *v*th channel and \({H_{k}^{v}}\) denotes the frequency response of the *k*th subchannel for the *v*th channel.

*k*th subchannel, which is given by

The signal vector \({\tilde {\mathbf {Y}}_{\text {MRC}}}\) is finally sent to the demodulator, where the output bit sequence \(\hat {\mathbf {I}}\) is sent to the sink.

## 3 Performance analysis

### 3.1 SINR for the MRC-OFDM demodulator

*n*th component represented as

*n*th component represented as

*n*th component \(\tilde {n}_{n}^{v}\) represented as

*k*th component of

**Y**

^{ v }can be expressed as

*k*th component of

**Y**

_{MRC}can be expressed as

*Y*

_{MRC,k }can be split into two parts, the first containing the desired signal of the

*k*th subchannel, denoted by

*E*

_{ k }, and the second containing the noise and the ICI caused by pulse blanking, denoted by

*W*

_{ k }.

*Y*

_{MRC,k }can be further expressed as

*E*

_{ k }is given as

*W*

_{ k }can be calculated as

*k*th subchannel at the output of MRC is given as

where \({\gamma _{k}^{v}} \buildrel \Delta \over = \frac {\rho }{{p \rho + \left ({1 - p} \right)}} {\left | {{H_{k}^{v}}} \right |^{2}}\) denotes the instantaneous SINR for the *k*th subchannel of the *v*th channel at the output of MRC. We assume the independence between different diversity reception branches, then \({{H_{k}^{v}}}\) is independent of \({{H_{k}^{j}}}\) if *j*≠*v*, and therefore \({\gamma _{k}^{v}}\) is independent of \({\gamma _{k}^{j}}\) if *j*≠*v*.

### 3.2 SER of the MRC-OFDM receiver with pulse blanking over Rayleigh fading channel

For the Rayleigh fading channel, the frequency response of the *k*th subchannel of the *v*th receive branch \({H_{k}^{v}} = \sum \limits _{l = 0}^{{L_{v}} - 1} {h_{_{l}}^{v}{e^{- j2\pi \frac {{kl}}{K}}}} \) is a complex Gaussian RV of mean zeros and variance one, thus \(\left |{H_{k}^{v}}\right |^{2}\) is *χ* ^{2} distributed with 2 degrees of freedom and \(E\left \{ {|{H_{k}^{v}}{|^{2}}} \right \} = 1\).

*p*is given, \({\gamma _{k}^{v}}\) is also

*χ*

^{2}distributed with 2 degrees of freedom and mean \(\bar \gamma = {\rho \left /{\left ({p\rho + \left ({1 - p} \right)} \right)}\right.}\). Therefore, the probability density function (PDF) of \({\gamma _{k}^{v}}\) is given by [20]

*k*th subchannel

*γ*

_{ k }becomes

*k*th subchannel for the OFDM receiver with

*M*-phase-shift keying (PSK) modulation over Rayleigh fading channel can be expressed as

*k*th subchannel for the OFDM receiver with

*M*-quadrature amplitude modulation (QAM) over Rayleigh fading channel can be expressed as

In the SER expressions given in Eqs. (32), (33), (35), and (36), the SER of the *k*th subchannel is independent of the subchannel index, which indicates that pulse blanking has the same effect on the error performance of each subchannel of the MRC-OFDM receiver.

where \(\mu = \sqrt {{{\bar {\gamma }} \left / {\left ({1 + \bar {\gamma }} \right)}\right.}} \).

The exact expression in Eq. (38) provides insight into a few special cases. (i) When there is no interference, i.e., *p*=0, \(\bar {\gamma }= \rho \), Eq. (38) reduces to the symbol error probability of a conventional *N*th-order space diversity receiver employing MRC. (ii) When there is interference, i.e., *p*≠0, and in the limiting case the input SNR is large, i.e., \(\rho \to \infty \), \(\mathop {\lim }\limits _{\rho \to \infty }\bar \gamma ={1\left / p\right.}\), thus, there will be an error floor for the SER performance curve. Considering a small *p* value (less than 0.1) fulfilling the condition \(\bar {\gamma }\gg 1\), the approximate theoretical expression of the error floor behaves as \({\left ({\frac {p}{4}}\right)^{N}}\left ({ \begin {array}{{c}} {2N-1}\\ N \end {array}} \right)\). Therefore, the error floor decreases proportionally with the *N*th power of the probability of impulsive noise occurrence *p* at an *N*th-order space diversity. For a given *p* value, the error floor can be efficiently reduced as the number of receive antenna increases. The same conclusion can be reached based on the approximate expressions in Eqs. (33) and (36).

### 3.3 SER of the MRC-OFDM receiver with pulse blanking over Ricean fading channel

For the Ricean fading channel, \({h_{0}^{v}}\) is assumed to be a non-zero mean complex Gaussian RV, i.e., \({h_{0}^{v}} \sim \mathrm {\mathcal {CN}}\left ({u_{v}},\delta _{0}^{{v}^{2}} \right)\), \(\left \{ {{h_{l}^{v}},l = 1,\ldots,{L_{v}} - 1} \right \}\) are assumed to be complex Gaussian RVs, i.e., \({h_{l}^{v}}\sim {\mathrm {\mathcal {CN}}}\left (0,\delta _{l}^{{v}^{2}} \right)\). The channel power is normalized to 1, i.e., \({\left | {{u_{v}}} \right |^{2}} + {\sum \nolimits }_{l = 0}^{L - 1} {{{\left | {{\delta _{l}^{v}}} \right |}^{2}}} = 1\), and the Ricean factor of the *v*th channel is defined as \({K_{v}} \buildrel \Delta \over = {{{{\left | {{u^{v}}} \right |}^{2}}} \left / {{\sum \nolimits }_{l = 0}^{L - 1} {{{\left | {{\delta _{l}^{v}}} \right |}^{2}}} }\right.}\). Hence, the frequency response of the *k*th subchannel \({H_{k}^{v}}\) is distributed according to complex Gaussian with mean *u* ^{ v } and variance \({\sum \nolimits }_{l = 0}^{L - 1} {{{\left | {{\delta _{l}^{v}}} \right |}^{2}}} - {\left | {{u^{v}}} \right |^{2}}\). Furthermore, \({\left | {{H_{k}^{v}}} \right |^{2}}\) is noncentral *χ* ^{2} distributed with 2 degrees of freedom and \(E\left \{ {|{H_{k}^{v}}{|^{2}}} \right \} = 1\).

*p*is given, \({\gamma _{k}^{v}}\) is also according to noncentral

*χ*

^{2}distributed with 2 degrees of freedom with mean \(\bar {\gamma } = {\rho \left / {\left ({p\rho + \left ({1 - p} \right)} \right)}\right.}\). Therefore, the PDF of \({\gamma _{k}^{v}}\) is given by [15]

*k*th subchannel for the OFDM receiver with

*M*-PSK modulation over Ricean fading channel can be expressed as

*k*th subchannel for the OFDM receiver with

*M*-QAM modulation over Ricean fading channel can be expressed as

Also, the SER derived above is only for the *k*th subcarrier. It has to be averaged over all the subcarriers. Comparing Eq. (44) with Eq. (33), Eq. (45), and Eq. (36), we find that for the specific Ricean factors, the same conclusion presented in Section 3.2 can be reached based on the approximate expressions in Eqs. (45) and (46).

## 4 Numerical results

### 4.1 System and channel parameters

System and channel parameters

Parameters | Value |
---|---|

System parameters | |

Modulator | BPSK, 8PSK, 16QAM |

Subcarrier number | 512 |

Data subcarrier number | 512 |

Cyclic prefix | 16 |

Channel parameters | |

Channel models | Rayleigh (10 paths) |

Ricean (10 paths, \(\left \{ {{K_{v}}} \right \} = 10\) dB) | |

Impulsive noise model | Bernoulli-Gaussian |

Probability of interference | |

occurrence | |

Receiver parameters | |

Number of receive antenna | |

Channel estimation | Ideal channel estimation |

Channel equalization | ZF equalization |

### 4.2 Symbol error performance curves

Figures 3 and 4 show the SER performances of the MRC-OFDM receiver with pulse blanking for BPSK modulation over Rayleigh fading channel for *N*=2 and *N*=4 receive antennas. Each figure contains four pairs of curves, which shows the theoretical and simulated SER of the OFDM receiver with pulse blanking at different *p* values. In both figures, it can be seen that theoretical results correspond well with the simulation results.

*p*are listed in Table 2. From the table, the simulated observations of the error floor achieve a good agreement with the theoretical calculations. We see that the error floor decreases as the probability of impulsive noise occurrence decreases. Further, the error floor decreases proportionally with the

*N*th power of the probability of impulsive noise occurrence

*p*. Comparing the two figures, it is shown that the error floor is efficiently reduced for

*N*=4 receive antennas compared with

*N*=2 for the same

*p*value.

Impact of probability of impulsive noise occurrence on the error floor

| Error floor (theory) | Error floor (simulation) |
---|---|---|

0.02 | 7.50e −005 | 7.0e −005 |

0.03 | 1.69e −004 | 1.5e −004 |

0.05 | 4.69e −004 | 4.5e −004 |

Similarly, for the same receiver and channel, Figs. 5 and 6 show the SER performances for 16QAM modulation with *N*=2 and *N*=4 receive antennas, respectively. From the four pairs of curves in each figure, the theoretical results correspond well with the simulation results and derive the same analysis as that in Figs. 3 and 4.

Figures 7 and 8 show the SER performances of the MRC-OFDM receiver with pulse blanking for 8PSK modulation over Ricean fading channel with Ricean factors \(\left \{ {{K_{v}}} \right \} = 10\) dB for *N*=2 and *N*=4 receive antennas, respectively. Similar to the previous paragraph, from the four pairs of curves in each figure, the theoretical results agree well with the simulation results. We arrive then with the same analysis as for Figs. 3 and 4.

Similarly, for the same receiver and channel, Figs. 9 and 10 show the SER performances for 16QAM modulation with *N*=2 and *N*=4 receive antennas, respectively. Again, from the four pairs of curves of in each figure, the theoretical results correspond well with the simulation results and yield the same analysis as for Figs. 3 and 4.

## 5 Conclusions

In this paper, we studied the symbol error performance of the MRC-OFDM receiver with pulse blanking over frequency selective fading channel. The closed-form expression of the SINR for the MRC-OFDM receiver with pulse blanking is derived. The SER of the MRC-OFDM receiver with pulse blanking over both Rayleigh and Ricean fading channels are also given. The simulation results validate the correctness of our derived formulas. The following conclusions are obtained: (i) the pulse blanking has same effect on the error performance of each subchannel of the MRC-OFDM receiver; (ii) the error floor for the SER performance is observed for the MRC-OFDM receiver with pulse blanking and the error floor depends on the probability of impulsive noise occurrence and the number of receive antenna; and (ii) the developed analysis method in this paper can be extended to the case of the Middleton class A noise environment and the error floor for the SER performance is determined by the probability of the Middleton class A impulsive noise occurrence.

## 6 \thelikesection Appendix

*s*

_{ m }and \({h_{l}^{v}}\) are statistically independent, \(E\left \{ {{{\left | {\tilde {i}_{n}^{v}} \right |}^{2}}} \right \}\) can be calculated as

*W*

_{ k }can be obtained as

## Notes

### Acknowledgements

This work was supported in part by National Natural Science Foundation of China under Grants No. U1233117 and No. 61271404.

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