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# Theoretical analysis and compensation for the joint effects of HPA nonlinearity and RF crosstalk in VBLAST MIMO-OFDM systems over Rayleigh fading channel

- Maha Cherif Dakhli
^{1}Email author, - Rafik Zayani
^{1}, - Oussama B Belkacem
^{1}and - Ridha Bouallegue
^{1}

**2014**:61

https://doi.org/10.1186/1687-1499-2014-61

© Dakhli et al.; licensee Springer. 2014

**Received:**29 June 2013**Accepted:**13 March 2014**Published:**22 April 2014

## Abstract

In this paper, we investigate the joint effects of nonlinear distortions due to high-power amplifier (HPA) and radio-frequency (RF) crosstalk on the performance of a multiple-input multiple-output-orthogonal frequency division multiplexing (MIMO-OFDM) system. The performance of the vertical bell laboratories layered space-time (VBLAST) MIMO-OFDM system, in terms of bit error rate (BER) and system capacity, is derived and evaluated for a Rayleigh flat fading channel. In the performance analysis, M-ary quadrature amplitude modulation (M-QAM) is considered and the joint effects of nonlinear HPA distortions and RF crosstalk are studied. Theoretical results and comparisons are provided for several system parameters, such as the input back-off (IBO) and crosstalk intensity. It is shown that the results obtained from the theoretical study are very close to the ones obtained from the simulation. Indeed, we introduce a nonlinear distortion cancellation (NDC) technique to estimate and mitigate nonlinear distortions and crosstalk effects jointly.

## Keywords

- MIMO
- OFDM
- HPA
- Crosstalk
- Nonlinear distortion cancellation
- MMSE

## 1. Introduction

The combination of multiple-input multiple-output (MIMO) and orthogonal frequency division multiplexing (OFDM) guarantees high performance with high data rates in 4G broadband wireless communications [1]. Indeed, OFDM combats frequency-selective fading effectively by dividing a wideband frequency-selective fading channel into parallel narrowband flat fading subchannels, while MIMO systems increase the data rate significantly over single-input single-output (SISO) systems by employing multiple transmit and receive antennas.

Like in classical SISO-OFDM, MIMO-OFDM exhibits large peak-to-average power ratios (PAPR), i.e., large fluctuations in their signal envelopes. Moreover, the performance of the transceiver is very sensitive to nonlinear distortions caused by high-power amplifiers (HPAs). On the other hand, MIMO transceivers use multiple transmission/reception paths implemented in the same chipset. Crosstalk or coupling effect is the result of the interference between the signals of different paths, and it affects the quality of the transmitted signal and degrades the system performance. This type of crosstalk can be linear or nonlinear as explained in [2], where the authors discussed the difference between linear MIMO crosstalk, which is relatively benign since it is being corrected by the MIMO equalizer, and nonlinear crosstalk that is generated before the HPA. This nonlinear crosstalk is not corrected by the MIMO receiver, and it is thus much more harmful. HPA nonlinearity in SISO-OFDM systems was investigated in [3], where the output at the HPA is expressed as the summation of the input signal multiplied by a complex scale factor, and an additive Gaussian noise which is uncorrelated with the input signal.

Recent research effort has dealt with the issue of HPA nonlinearity in MIMO systems. For instance, based on this study, the effect of HPA nonlinearity on the bit error rate of quadrature amplitude modulation (QAM) was analyzed in [4] for MIMO systems. In addition, the impact of HPA nonlinearity on the bit error rate was studied in [5] for SDMA-OFDM systems and was analyzed in [3] in order to determine the parameters of amplification such as the complex attenuation coefficient *K*_{0} and the variance of the nonlinear distortion $\phantom{\rule{0.25em}{0ex}}{\mathit{\sigma}}_{\mathit{d}}^{2}$.

Up to now, in the open literature, these terms have been investigated analytically only for the soft envelope limiter (SEL) model [3], and a similar framework was proposed for the traveling wave tube amplifier (TWTA) and the solid-state power amplifier (SSPA) models, but limited actually to a semi-analytic study [3, 5].

The compensation methods and comprehensive theoretical analysis of several radio-frequency (RF) impairments together, in MIMO communication systems, are extremely challenging. A crossover digital predistorter was proposed for the compensation of HPA nonlinearity and crosstalk in MIMO systems and was evaluated through simulations [6, 7].

In this paper, we investigate the joint effects of HPA nonlinearity and RF crosstalk in vertical bell laboratories layered space-time (VBLAST) MIMO-OFDM systems, and we derive the new closed-form probability density function (pdf) signal-to-noise ratio (SNR) in different scenarios: (1) nonlinear MIMO-OFDM system and (2) nonlinear MIMO-OFDM system with crosstalk.

In addition, we introduce a new approach to derive analytically *K*_{0} and $\phantom{\rule{0.25em}{0ex}}{\mathit{\sigma}}_{\mathit{d}}^{2}$, and this approach is valid for any HPA model, such as TWTA, SSPA, and the real measured one. Thus, a polynomial approximation is considered in order to calculate the values of these parameters. Based on this approximation, we provide an analytical performance evaluation, in terms of bit error rate (BER) and channel capacity, of a VBLAST MIMO-OFDM system under a Rayleigh fading channel and memoryless nonlinear HPA model. Finally, we introduce a compensation method that aims to estimate and mitigate simultaneously the HPA nonlinearity and RF crosstalk at the receiver side. This proposed method is based on the theoretical analysis done in this paper.

Results were supported with MATLAB simulation of a VBLAST MIMO-OFDM system under a Rayleigh fading channel. The remainder of this paper is organized as follows: In Section 2, the system model is introduced. The approximation of the HPA model is studied in Section 3. The performance of the VBLAST MIMO-OFDM system in terms of BER and capacity is evaluated in Section 4. Section 5 presents a nonlinear distortion cancellation (NDC) technique for compensating the nonlinear crosstalk. Numerical results and discussions are presented in Section 6, while the conclusion is given in Section 7.

## 2. System model

*Nt*transmit antennas and

*Nr*receive antennas shown in Figure 1. It is assumed that the system employs

*Nc*subcarriers and M-QAM, and the fast Fourier transform (FFT) matrix is denoted by V ∈ ℂ

^{Nc × Nc}.

_{ k }is assumed to contain the source information, is mapped by using a VBLAST encoder, and is transmitted over a Rayleigh fading channel. The transmitted signal x

_{ k }(

*n*) from

*k*antenna at time instant

*n*is obtained by means of an inverse fast Fourier transform (IFFT) of the X

_{ k }(

*n*) and is given by

### 2.1 Linear MIMO-OFDM system

*Nr*receiving antennas, the transmitted signals are affected by the propagation channel, which can be modeled by the propagation matrix H

_{m,k}. The received signal, in a linear HPA case, is represented by

_{ m }(

*n*) is the additive white Gaussian noise (AWGN) and H

_{m,k}(

*n*) is an

*Nc*×

*Nc*circulant time domain (TD) channel matrix at time instant

*n*, which is formed by the channel response vector h

_{m,k}(

*n*) for the link between transmit antenna

*k*and receive antenna

*m*. The FD expression of the received signal is obtained by taking the FFT of Equation 2 giving

^{ L }(

*n*,

*i*) = Vy

^{ L }(

*n*,

*i*) = [

*Y*

_{1}(

*n*,

*i*),…,

*Y*

_{ Nr }(

*n*,

*i*)]

^{ T }denote the vector of received signals in the linear case for all antenna on subcarrier

*i*. Then, the received signal vector for each subcarrier can be expressed as

*n*,

*i*) = [

*X*

_{1}(

*n*,

*i*),…,

*X*

_{ Nt }(

*n*,

*i*)]

^{ T }is the vector that contains transmitted signals from each antenna and the Gaussian noise term $\mathcal{N}\left(\mathit{n},\mathit{i}\right)$ as obtained by FFT of the thermal Gaussian noise of the receiver has zero mean and variance ${\mathit{\sigma}}_{\mathit{n}}^{2}$. The frequency domain channel transfer matrix

**H**(

*n*,

*i*) in Equation 4 is given by

where *h*_{m,k}(*n*,*i*) denotes the *Nr* × *Nt* channel coefficient on subcarrier *i* at time *n* between the *k* th transmit and *m* th receive antennas.

### 2.2 Nonlinear MIMO-OFDM system

#### 2.2.1 Nonlinear HPA model

*ρ*

_{ k }and

*θ*

_{ k }denote, respectively, the amplitude and the phase of x

_{ k }, the complex envelope of a memoryless HPA output z

_{ k }(

*n*) can be modeled as follows:

*g*[.] is the HPA transfer function and

*A*(.) and

*P*(.) denote the HPA AM/AM and AM/PM conversions, respectively. For a nonlinear HPA model, we consider Saleh's model [8], which has the advantage of good accuracy and reasonable complexity than other models. The AM/AM and AM/PM conversions can be represented as follows [5, 9]:

*α*

_{ a }and

*β*

_{ a }are the parameters that decide the nonlinear level, and

*α*

_{ p }and

*β*

_{ p }are phase displacements. For validation of our theoretical results, the operating point of the amplifier is usually identified by the ‘back-off’. In the simulations, we define the input back-off as [9]

*A*

_{0}is the saturation amplitude at the input of HPA and

*P*

_{0}is the input average power. From the Bussgang theorem and by extending that to complex Gaussian processes, the output signal at the HPA can be expressed as [3]

*K*

_{0}is the attenuation coefficient and d

_{ k }(

*n*) is the nonlinear distortion term uncorrelated with the input signal x

_{ k }(

*n*). The value of

*K*

_{0}is given by Dardari et al. in ([3], eq. (19)) by

*S′*(

*ρ*

_{ k }) denotes the differential of

*S*(

*ρ*

_{ k }). Furthermore, the variance of the nonlinear distortion term (d

_{ k }(

*n*)) is given by ([3], eq. (37))

*Nt*antennas sharing the same space-frequency resource. The received signal is assumed to be corrupted by a Gaussian noise at the array elements, and it is given by

*i*) can be written as [5, 10]

#### 2.2.2 Nonlinear MIMO-OFDM with RF crosstalk

_{ l }(

*n*) is the output of the FFT bloc of the

*l*path of the transceiver, and

*α*

_{ lk }(

*n*) is the filter representing the crosstalk and modeling the coupling between path

*l*to path

*k*[7]. Since the signals in different paths use the same operating frequency and have equal transmission power, crosstalk is more likely between the paths. In fact, the received signal of a nonlinear crosstalk is represented by

*α*

_{21}=

*α*

_{12}=

*α*) [6]. The frequency domain expression of the received signal vector for each subcarrier can be written as

where $\mathbf{\chi}\left(\mathit{n},\mathit{i}\right)=\left[\begin{array}{c}\hfill {\mathit{x}}_{2}\left(\mathit{n},\mathit{i}\right)\hfill \\ \hfill {\mathit{x}}_{1}\left(\mathit{n},\mathit{i}\right)\hfill \end{array}\right].$

### 2.3 MMSE receiver

*i*can be estimated with the aid of a suitable linear combiner W ∈ ℂ

^{Nr × Nt}. The minimum mean square error (MMSE) detection minimizes the mean square error between the transmit signal and the estimated signal [13–15]. The optimal MMSE weight is obtained as follows:

## 3. Nonlinear distortion characterization

To evaluate the nonlinear distortion effects in MIMO-OFDM systems, in this section, we propose an analytical study to calculate the parameters of a power amplifier. In [3, 5, 16], the terms of the nonlinear distortion are evaluated with semi-analytic expressions. Motivated by these results, we will expand this study in order to introduce a new approach to derivate analytically the complex attenuation coefficient *K*_{0} and the variance of the nonlinear distortion $\phantom{\rule{0.25em}{0ex}}{\mathit{\sigma}}_{\mathit{d}}^{2}$, and this approach is valid for any HPA model even a real measured one. Thus, we will consider a polynomial approximation in order to calculate the values of these parameters.

## 3.1 Polynomial approximation of the nonlinear distortion

*K*

_{0}and the variance $\phantom{\rule{0.25em}{0ex}}{\mathit{\sigma}}_{\mathit{d}}^{2}$ is not feasible because we have to calculate the expectation (

*E*[.]) of the complicated function. Therefore, we propose a new approach, which is based on polynomial approximation in order to estimate the nonlinear HPA. In this case, the output of HPA can be represented by

*L*is the order of the polynomial and

*p*

_{2l + 1}are the complex coefficients. Therefore, the new expression of

*S*(

*ρ*) is given by:

## 3.2 Estimation of *p*_{2l + 1}

*p*

_{2l + 1}, we use the least square (LS) approach [17]:

*L*= 10. The considered HPA is Saleh's model with

*α*

_{ a }= 1,

*β*

_{ a }= 1, ${\mathit{\alpha}}_{\mathit{p}}=\frac{\mathit{\pi}}{3}$, and

*β*

_{ p }= 1 and input back-off (IBO) of 8 dB. We note from these results the AM/AM and AM/PM characteristics given by Saleh's model and those obtained by polynomial approximation are well matched.

## 3.3 Analytical computation of *K*_{0}and $\phantom{\rule{0.25em}{0ex}}{\mathit{\sigma}}_{\mathit{d}}^{2}$

*K*

_{0}is obtained by substituting (22) and (23) in (10) and we obtain

*l*-order moment (

*E*[

*ρ*

^{2l}]), where

*ρ*can be written as

*a*and

*b*represent the real and imaginary parts, respectively. Then,

*E*[

*ρ*

^{2l}] is given by

*Γ*is the gamma function, which is defined as

*K*

_{0}:

*K*

_{0}and $\phantom{\rule{0.25em}{0ex}}{\mathit{\sigma}}_{\mathit{d}}^{2}$ obtained, respectively, by ((10) and (11)) and ((33) and (38)) for different values of IBO. According to this table, we observe a good agreement between simulation and theoretical results of the both

*K*

_{0}and $\phantom{\rule{0.25em}{0ex}}{\mathit{\sigma}}_{\mathit{d}}^{2}$, which validate our analysis.

**Comparison of analytical and simulation results of** K_{
0
}**and**${\mathit{\sigma}}_{\mathit{d}}^{2}$ **for different values of IBO,** L **= 10, and Saleh's model**

IBO (dB) | Simulation | Analytical method | ||
---|---|---|---|---|

K
| $\phantom{\rule{0.25em}{0ex}}{\mathit{\sigma}}_{\mathit{d}}^{2}$ |
K
| $\phantom{\rule{0.25em}{0ex}}{\mathit{\sigma}}_{\mathit{d}}^{2}$ | |

4 | 0.5556 + 0.2152 | 0.0142 | 0.5560 + 0.2152 | 0.0142 |

6 | 0.6625 + 0.1953 | 0.0074 | 0.6627 + 0.1953 | 0.0074 |

8 | 0.7556 + 0.1643 | 0.0034 | 0.7560 + 0.1641 | 0.0033 |

10 | 0.8304 + 0.1287 | 0.0013 | 0.8302 + 0.1288 | 0.0013 |

## 4. VBLAST MIMO-OFDM performance analysis

In this section, we evaluate analytically the performance of VBLAST MIMO-OFDM systems in terms of BER and the system capacity. The considered system uses M-QAM under a Rayleigh flat fading channel.

## 4.1 BER performance

where *p*_{
γ
}(*γ*) is the pdf of the instantaneous SNR (*γ*).

*M*denotes the modulation order and $\phantom{\rule{0.25em}{0ex}}{\mathit{\gamma}}_{\mathit{b}}=\frac{{\mathit{\sigma}}_{\mathit{x}}^{2}}{{\mathit{\sigma}}_{\mathit{n}}^{2}}$ while

*Q*is the standard

*Q*-function defined as [18]

In this investigation, we calculate, theoretically, the instantaneous SNR (*γ*) and its pdf (*p*_{
γ
}(*γ*)) in different scenarios such as linear MIMO-OFDM, nonlinear MIMO-OFDM, and nonlinear MIMO-OFDM with crosstalk. This analysis will be used to calculate the average BER.

### 4.1.1 Linear MIMO-OFDM system

**H**(

*n*,

*i*) is assumed to be a flat fading one, and it is exponentially distributed with a pdf given by

where *Ω* = *E*[|**H**(*n*,*i*)|^{2}].

### 4.1.2 Nonlinear MIMO-OFDM system

^{NL}(

*n*,

*i*), the following estimate of the transmitted signal is obtained:

*ℵ*= |

**H**(

*n*,

*i*)|

^{2}, and Equation 48 then becomes

*g*(

*γ*) =

*ℵ*and

*g*(

*f*(

*ℵ*)) =

*ℵ*. Moreover, the instantaneous SNR in the nonlinear case can be expressed by

*g*(

*γ*) and then determine

*g′*(

*γ*), which are given by (52) and (53), respectively:

### 4.1.3 Nonlinear MIMO-OFDM system with RF crosstalk

*g*(

*γ*) and

*g′*(

*γ*):

The integrals presented in Equations 55 and 63 are computed using MATLAB in order to determine the average BER, because the calculation of these integrals is intractable. An approximation for this integral is suggested in full perspective.

### 4.2 System capacity

### 5. Compensation of the joint effects of HPA nonlinearity and crosstalk

In this section, we introduce an iterative estimation and cancellation technique of nonlinearly distorted signals received by the VBLAST MIMO-OFDM transmission system. Indeed, we also propose two compensation methods for two scenarios such as (1) the nonlinear distortion due to HPA without crosstalk and (2) the nonlinear distortion with crosstalk. Accordingly, in scenario 1, we attempt to estimate and mitigate the nonlinear distortion term in Equation 9, and in scenario 2, we try to eliminate jointly the nonlinear distortion due to HPA and the crosstalk between the different branches present in Equation 15.

### 5.1 NDC in nonlinear MIMO-OFDM system

The NDC technique requires knowledge of the HPA parameters. Indeed, the aim of this paper is to estimate and mitigate simultaneously HPA nonlinearity and RF crosstalk basing on the theoretical analysis done in this paper, where we used a polynomial approximation in order to estimate the nonlinear HPA (*g*[.]), which is described in Section 3.

The proposed method is applied at the receiver side, independently for each antenna using its own HPA parameters, and it consists of the following steps [5, 20–22]:

*Step 1*: Using ${\widehat{\mathit{X}}}_{\mathit{k}}$, an estimate r

_{ k }of the original transmitted constellation X

_{ k }is obtained by applying hard decoding. This process is carried out for all active carriers. Using the recovered symbols, the time domain signal is reproduced via IFFT as

*Step 2*: Using the nonlinear model of HPA that is approximated by a polynomial model, which is described in the previous section, we compute the estimation ${\mathit{d}}_{\mathit{k}}^{\left(\mathit{q}\right)}\left(\mathit{n}\right)$ of the nonlinear distortion terms using Equation 9:

where ${\widehat{\mathit{x}}}_{\mathit{k}}^{\left(\mathit{q}\right)}\left(\mathit{n}\right)$ is the time domain representation of the recovered signal at iteration *q* and $\widehat{\mathrm{g}}\left[.\right]$ is the estimated transfer function.

*Step 3*: The distortion ${\widehat{\mathit{D}}}_{\mathit{k}}^{\left(\mathit{q}\right)}\left(\mathit{n}\right)$ is subtracted from the estimated signal ${\widehat{\mathit{x}}}_{\mathit{k}}^{\left(\mathit{q}\right)}$. Using this result, the transmitted constellation is re-estimated in a new decoding/distortion cancellation step, and the process can be carried out iteratively.

### 5.2 NDC in nonlinear MIMO-OFDM system with crosstalk

This process can also be performed iteratively.

### 6. Numerical results

In this section, we present theoretical and simulation results illustrating the performance of VBLAST MIMO-OFDM systems with the proposed compensation techniques for HPA nonlinearity and crosstalk, specifically on the average BER and system capacity with 64 subcarriers and 16-QAM. Herein, we consider a Rayleigh flat fading channel that is quasi-stationary on two successive OFDM symbols. The values of the parameters considered for HPA are assumed to be $\phantom{\rule{0.25em}{0ex}}{\mathit{\alpha}}_{\mathit{a}}=1,\phantom{\rule{0.5em}{0ex}}{\mathit{\beta}}_{\mathit{a}}=1,\phantom{\rule{0.5em}{0ex}}{\mathit{\alpha}}_{\mathit{p}}=\frac{\mathit{\pi}}{3},\phantom{\rule{0.5em}{0ex}}\mathit{and}\phantom{\rule{0.5em}{0ex}}{\mathit{\beta}}_{\mathit{p}}=1$.

Figure 7 illustrates BER versus *Eb*/*N* 0 for IBO = {4*-* 8}. As illustrated, *Eb*/*N* 0 imposes a great impact on the BER performance. When IBO is decreasing (e.g., 4 dB), the modulated input signal can be higher than *A*_{0} with a strong probability. This gives an irreducible error at higher *Eb*/*N* 0.

To further illustrate the effect of the crosstalk in the nonlinear MIMO-OFDM system, Figure 8 shows the average BER for different values of the crosstalk parameters, fixing IBO = 8 dB. The residual degradation on the performance of the average BER, due to the nonlinear crosstalk, increases as the parameters (*α*) become larger.

Figure 9 show the effect of the nonlinearities due to HPA on the system capacity with different values of IBO. As noticed, the nonlinear distortion result causes a decrease in the capacity of MIMO-OFDM systems. On the other hand, the capacity of the system improves with the increase in the value of the IBO.

In Figure 10, we show the effect of the crosstalk on the system capacity as a function of *Eb*/*N* 0, taking into account the HPA nonlinearity, for IBO = 8 dB. As observed, the system capacity increases as the crosstalk parameters (*α*) become larger. However, profiting of the RF crosstalk can improve the system capacity.

**7**. Conclusions

In this paper, we have investigated and analyzed the joint effects of nonlinear HPA and crosstalk on the performance of the M-QAM VBLAST MIMO-OFDM system, in terms of BER and system capacity over a Rayleigh fading channel. In addition, new closed-form pdf signal-to-noise ratios are derived in order to develop theoretical expressions of the average BER in different scenarios: nonlinear MIMO-OFDM system and nonlinear MIMO-OFDM system with crosstalk.

On the other hand, we have introduced a compensation method (NDC) that aims to estimate and mitigate simultaneously the HPA nonlinearity and RF crosstalk at the receiver side. This proposed method is based on the theoretical analysis done in this work.

Numerical results showing the joint effects of HPA nonlinearity and crosstalk on the performances of MIMO-OFDM were evaluated and compared. In fact, the results obtained using the theoretical expressions of BER are very close to the simulation results, and it was shown that the NDC method can efficiently compensate these two impairments even after one iteration.

## Declarations

## Authors’ Affiliations

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