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# Spectral broadening effects of high-power amplifiers in MIMO–OFDM relaying channels

- Ishtiaq Ahmad
^{1}, - Ahmed Iyanda Sulyman
^{1}Email author, - Abdulhameed Alsanie
^{1}, - Awad Kh Alasmari
^{1}and - Saleh A Alshebeili
^{1, 2}

**2013**:32

https://doi.org/10.1186/1687-1499-2013-32

© Ahmad et al.; licensee Springer. 2013

**Received: **14 June 2012

**Accepted: **10 November 2012

**Published: **15 February 2013

## Abstract

The combination of MIMO–OFDM is a very attractive solution for broadband wireless services. Thus, the two prominent fourth-generation (4G) cellular systems, WiMAX and LTE-advanced, have both adopted MIMO–OFDM transmission at the physical layer. OFDM signal however suffers from nonlinear distortions when passed through high-power amplifier (HPA) at the RF stage. This nonlinear distortion introduces out-of-band spectral broadening and in-band distortions on the transmitted signals. 4G cellular standards have placed strict limits on the allowable spectral broadening in their spectrum mask specifications, to insure that data transmission on a given channel is not interfering significantly with an adjacent channel user. In this article, we characterize the out-of-band spectral broadening introduced by HPA when MIMO–OFDM signals are transmitted over multiple relaying channels. Expressions for the power spectral density of MIMO–OFDM signals are derived over multiple relay channels, and the cumulative effects of HPA on the spectrum of the transmitted signals are estimated. It is shown that depending on the number of relays and the relaying configuration employed, it may happen that a transmitted MIMO–OFDM signal with the transmit spectrum mask initially within the allowable set limit at the source node arrives at the destination violating this limit due to the cumulative effects of the multiple HPA’s in a multihop relaying channel.

## Keywords

- Spectral re-growth
- Amplifier nonlinearity
- Spectral mask
- MIMO–OFDM
- Relaying channels

## Introduction

Banelli and Cacopardi[11] derived analytical expressions for the correlation function of the output of nonlinear HPA when the input to the amplifier is an OFDM signal. The power spectral density (PSD) of the signal is then calculated using the Fourier transform of the correlation function. Grad et al.[12] studied spectral re-growth due to HPA nonlinearity in code-division multiple access (CDMA) systems. They obtained analytical expressions for the power spectrum of the CDMA signal at the output of the HPA, using a complex power-series model for the HPA characteristics. The out-of-band emission for the time division synchronous CDMA system is presented in[13], in terms of third-order intercept point (IP3). Cottais et al.[14] derived expressions for the PSD of a general multicarrier signal at the output of a memoryless HPA. They also obtained a closed-form expression for the PSD of the special case of single-carrier signals. Helaly et al.[15] examined the effects of the characteristics of the input CDMA signal on the resulting out-of-band spectral re-growth at the output of the HPA. They pointed out that, in addition to the HPA saturation level, the input signal’s threshold crossing rate and the variance of the clipped signal also contribute to the spectral re-growth. It is important to note that OFDM signals share some similarities with CDMA signals in this regard. Recently also, Gregorio et al.[17] proposed a MIMO-predistortion (MIMO-PD) system that tries to compensate crosstalk and IQ imbalance in single-hop MIMO–OFDM communication systems, where they have shown that some reduction in the spectral re-growth can be achieved using the proposed MIMO-PD system. The effectiveness of such a compensation scheme in a multihop environment is however not yet known.

All the above-cited studies, and several others in the literature however, focused on the spectral re-growth due to HPA nonlinearity in a single-hop communication system. Recently, the two prominent 4G cellular systems, WiMAX and LTE-advanced, have defined relaying as an integral part of the network design[18, 19]. Thus, MIMO–OFDM signals transmitted in the 4G systems will frequently pass through one or more relay hops from source node to the destination node. Investigating the level of adherence to set limits on spectral broadening in cellular systems employing relaying technologies is therefore a deployment imperative. To the best of the authors’ knowledge, no work has presented a detailed study of the broadening effects of HPA nonlinearity on the spectrum of MIMO–OFDM signals in multihop relaying channels.

In this article, we characterize for the first time in the literature, the cumulative spectral broadening effects of multiple HPAs when MIMO–OFDM signals are transmitted over multihop relaying channels. Expressions for the PSD of a MIMO–OFDM signal are presented over multihop relay channels, each equipped with nonlinear HPA’s. It is shown that due to the cumulative effect of the multiple HPA’s in a MIMO link, and the cascade effect of many relaying channels in a multihop relay link, significant broadening effect occur which is much more than what would be observed in a single-hop transmission such as those characterized in[11–15]. We also show that for the amplify-and-forward (AF) and demodulate-and-forward (DemF) relaying options, the resulting cumulative re-growth may lead to spectral mask violations after a few relaying hops is traversed by the transmitted OFDM signal, even though the set limits were initially met at the source node [at the base station (BS) for downlink transmission, or at the mobile station (MS) for uplink transmission]. For the decode-and-forward (DF) relaying option, it is observed that less severe spectral broadening are observed. However, due to the latency problems associated with the DF relaying option which degrades quality of broadband signals, AF or DemF are the preferred candidates for broadband transmissions over relaying channels and therefore the spectral broadening issues observed here must be given considerable attentions in the design of broadband multihop relaying systems.

## HPA nonlinearity model for MIMO–OFDM relaying channel

*n*subcarriers per OFDM symbol,

*M*transmitting and

*L*receiving antennas at each hop in the transmission chain. We assume that all the transmitting antennas at each node simultaneously transmit different symbols (MIMO-multiplexing system), and that all

*L*receiving antennas at each receiving node are expended in separating each of the

*M*transmitted streams. For simplicity, we consider the case of

*N*transmitting and

*N*receiving antennas (where

*N*≤

*M*,

*L*). Thus, in the ensuing analysis we focus on the

*N*×

*N*MIMO–OFDM multiplexing relaying system. We consider that the transmitted signal from a source node passes through a single-hop MIMO channel

*H*

_{0}, and

*R*multihop MIMO relaying channels

*H*

_{1},…,

*H*

_{ R }, associated with

*R*fixed or mobile relaying nodes, to the destination node as shown in Figure 2.

*i*th hop transmission H

_{ i },

*i*= 0, 1,…,

*R*, is an

*nN*×

*nN*block diagonal matrix, with the

*k*th block diagonal entries

*H*[

*K*]

_{ i }corresponding to the fading on the

*k*th OFDM subcarrier,

*k*= 0, 1,…,

*n*- 1, modeled as independent and identically distributed (iid) random variables taken from zero mean complex Gaussian distribution, with unit variance. We assume that the set of random matrices {

*H*

_{0},…,

*H*

_{ R }} are independent, and that AF, DemF, or DF relaying options could be employed at the relay nodes[5, 20]. We also assume that the BS and all the relay stations (RS’s) employ similar nonlinear HPA’s which introduce similar nonlinear distortions per hop, in the transmitted signal. A consequence of this assumption is that if the BS or any of the RS employ a nonlinear HPA with nonlinearity level more or less than the other devices, then the contribution of that device to the overall spectral broadening will be more or less than estimated in this analysis. We can express the transmitted symbol in polar coordinate as

*r*(

*t*) is the amplitude and

*θ*(

*t*) is the phase of the input signal into the HPA. The signal at the output of the HPA can then be expressed as

*g*[

*r*(

*t*)] is a complex nonlinear distortion function, which only depends on the envelope of the transmitted symbols. The nonlinear distortion function can be expressed as

*g*

_{ A }[

*r*(

*t*)] is the amplitude-to-amplitude (AM–AM) and

*g*

_{ P }[

*r*(

*t*)] is the amplitude-to-phase (AM–PM) conversions of the HPA. The AM–AM conversions for different memoryless HPA models [Saleh model, Solid-State Power Amplifier (SSPA) model, and soft envelop limiter (SEL) model] used in communication systems are plotted in Figure 3.

*α*

_{ A }is the small-signal gain, and

*A*

_{is}is the input saturation voltage of the HPA. Let

*x*(

*t*) denote the input signal into the HPA, and we assume that

*x*(

*t*) is Gaussian distributed. Therefore, according to the Bussgang’s theorem, the output of the HPA when the input is a Gaussian process is given as[22, 23]

*k*(i.e., 0 ≤

*k*≤ 1) is an attenuation factor for the linear part which represents the in-band distortion, and

*w*(

*t*) is a nonlinear additive noise which represents the out-of-band distortion.

*w*(

*t*) is a zero-mean complex Gaussian random variable (r.v), with the in-phase and quadrature components mutually iid, and with variance

*σ*

_{ w }

^{2}. The in-band and out-of-band distortion terms can be calculated as[24]

*P*

_{avg}=

*E*[|

*x*|

^{2}] is the average input energy per symbol, and

*f*(

*r*) is the probability density function (PDF) of the envelope of the input signal into the HPA (i.e.,

*r*= |

*x*|). The PDF of ‘

*r*’ is Rayleigh distribution (since

*x*(

*t*) is assumed Gaussian). The closed-form expressions for the in-band and out-of-band distortion parameters for the SEL HPA model are calculated by substituting

*f*(

*r*) into Equations (6) and (7) to obtain

where *j* = 0, 1, …, *R* denotes the number of relay hops, *γ*_{
j
} represents the clipping ratio (CR), *A*_{is,j} represents the input saturation voltage of each and every HPA employed at the *j* th relay hop, and *P*_{
j
} represents the average input power into the HPA’s at the *j* th hop. The frequency domain (FD) expression of the signal above is then obtained by taking its discrete Fourier Transform (DFT).

## Effect of HPA nonlinearity on the spectrum of MIMO–OFDM relaying system

### PSD of the nonlinearly amplified OFDM signal at the BS

*j*th transmitting antenna of the BS when the input to the amplifier is

*x*

^{ j }(

*t*).

*Κ*

_{0}is an

*N*×

*N*diagonal matrix, where

*k*

_{0}

^{ j }represents the scaling factor of the linear part (in-band distortion) at the

*j*th transmit antenna of the BS.

**w**

_{0}(

*t*) = [

*w*

_{0}

^{1}(

*t*), …,

*w*

_{0}

^{ N }(

*t*)], and

*w*

_{0}

^{ j }(

*t*) is the nonlinear distortions noise (out-of-band distortion) due to HPA at the

*j*th transmitting antenna of the BS. The PSD of a signal is commonly estimated by computing the autocorrelation function of the signal followed by a Fourier transform, using the well-known Wiener–Khintchin theorem[25]. The PSD of the OFDM signal at the output of the nonlinear HPA at the

*j*th transmit antenna of a MIMO–OFDM transmitter is given by

*j*th transmitting antenna. The autocorrelation function for the output of the HPA at the

*j*th transmitting antenna can be computed as

*k*

_{0}

^{ j }is the attenuation factor of the in-band part,${\Phi}_{{x}^{j}{x}^{j}}\left(\tau \right)$ represents the autocorrelation function of the OFDM signal at the input of the HPA, i.e.,

*x*

^{ j }(

*t*), and${\Phi}_{{\mathbf{w}}_{0}^{j}{\mathbf{w}}_{0}^{j}}\left(\tau \right)$ represents the autocorrelation function of the out-of-band nonlinear noise (i.e.,

*w*

^{ j }(

*t*)), contributed by the HPA at the

*j*th transmitting antenna of the source node (BS or MS). Using Equations (12) and (13), the PSD of the OFDM signal at the output of the HPA at the

*j*th transmitting antenna of the source node can be written as

*k*

_{0}

^{ j }|

^{2}, since 0 ≤

*k*

_{ i }

^{ j }≤ 1. It is also observed from this equation that the PSD of the OFDM signal at the output of the HPA is broadened, compared to the PSD at the HPA input, by a factor of${P}_{{w}_{0}^{j}{w}_{0}^{j}}^{\mathit{j}}\left(f\right)$. This spectral broadening term contributed by the

*j*th transmit antenna,${P}_{{w}_{0}^{j}{w}_{0}^{j}}^{\mathit{j}}\left(f\right)$, is${\sigma}_{{w}_{0}^{j}}^{j}{}^{2}$ as

Using the result from[26], Equation (16)], Equation (15) can be approximated as${\sigma}_{{w}_{0}^{j}}^{j}{}^{2}\approx {P}_{{w}_{0}^{j}{w}_{0}^{j}}^{j}\left(f\right)B$, where *B* denotes the bandwidth of the signal. This approximation is valid if the spectrum of *w*_{0}^{
j
} is flat across the bandwidth, which is true for the subcarrier-based analysis considered here. As the variance of the nonlinear noise due to HPA increases, the spectral re-growth of the transmitted OFDM signal increases as can be observed from Equation (15). Since the analysis above gives estimate of the spectral re-growth due to HPA per transmitting antenna, then for MIMO–OFDM system with *N* transmitting antennas, the overall spectral re-growth that occur due to HPA nonlinearity is given by${P}_{{w}_{0}{w}_{0}}^{\mathrm{MIMO}}={\displaystyle \sum _{j=1}^{N}}{P}_{{w}_{0}^{j}{w}_{0}^{j}}^{j}\left(f\right)$.

### PSD of the nonlinearly amplified OFDM signals at RS’s

Next we calculate the PSD of the nonlinearly amplified OFDM signals at RS’s. For this analysis, we consider two cases. In the first case, we consider that each RS has ability to perform MIMO signal processing on the received signal before amplifying and forwarding it onto the next hop, and thus we could examine the spectrum of the OFDM symbols obtained at each receiving antennas of the RS. This case corresponds to the DemF relaying option. In the second case, we consider that the RS does not have ability to perform MIMO signal processing on the received signal before forwarding onto the next hop. The RS simply AF the received OFDM signals at the RF stage without demodulating the OFDM signal. This case corresponds to the AF relaying option. AF has the best latency performance, and is attractive for broadband transmissions over relaying channels. The case of DF relaying option where RS actually decode data transmitted on each OFDM subcarriers before re-encoding/forwarding it is not analyzed here because the existing single-hop analyses in[27] and other references are valid for that case using hop-by-hop analysis. However, we later include the DF case in the simulation results as a reference.

#### Case I: DemF relaying system

*l*th subcarrier is given by

*X*[

*l*] = [

*X*

^{1}[

*l*]

*X*

^{2}[

*l*] ⋯

*X*

^{ N }[

*l*]]

*T*is an

*N*× 1 input OFDM signal in FD at the BS, and

*X*

^{ j }[

*l*] is the input into the HPA at the

*j*th transmitting antenna on the

*l*th subcarrier.

**K**

_{ i }= diag {

*k*

_{ i }

^{ j }}

_{j=1}

^{ N },

*i*= 0, …,

*R*is an

*N*×

*N*diagonal matrix, where

*k*

_{ i }

^{ j }represents the scaling factor of the linear part (in-band distortion) at the

*j*th transmit antenna in the

*i*th hop.

*W*[

*l*] = [

*W*

_{ i }

^{1}[

*l*]

*W*

_{ i }

^{2}[

*l*] ⋯

*W*

_{ i }

^{ N }[

*l*]]

^{ T }is an

*N*× 1 nonlinear distortion noise vector (out-of-band distortion) due to HPA and it is obtained by taking the DFT of

**w**(

*t*).

*W*

_{ i }

^{ j }[

*l*] represents the nonlinear distortion noise on the

*j*th transmit antenna in the

*i*th hop for the

*l*th subcarrier.

*N*[

*l*] = [

*N*

_{ i }

^{1}[

*l*]

*N*

_{ i }

^{2}[

*l*] ⋯

*N*

_{ i }

^{ N }[

*l*]]

^{ T }is an

*N*× 1 complex additive noise vector and

*N*

_{ i }

^{ j }[

*l*] represents the iid zero-mean complex AWGN noise on the

*j*th transmit antenna in the

*i*th hop for the

*l*th subcarrier.${Y}_{{R}_{1}}\left[l\right]={\left[{Y}_{{R}_{1}}^{1}\left[l\right]{Y}_{{R}_{1}}^{2}\left[l\right]\dots {Y}_{{R}_{1}}^{N}\left[l\right]\right]}^{T}$ is an

*N*× 1 received OFDM symbol vector in FD at the first RS and${Y}_{{R}_{1}}^{j}\left[l\right]={\displaystyle {\sum}_{m=1}^{N}{H}_{\mathit{jm}}^{0}\left[l\right]{\widehat{X}}_{m}\left[l\right]+{N}_{0}^{j}\left[l\right]}$ is the received OFDM symbol at the

*j*th received antenna of the first RS for the

*l*th subcarrier. We assume that the channel between the BS and the RS is known at the RS. Thus, we can employ MIMO Zero-Forcing or MIMO minimum mean square error technique for the MIMO signal processing at the RS for the DemF relaying option as

**Q**

_{0}[

*l*] =

*E*[(

**H**

_{0}[

*l*]

*W*

_{0}[

*l*] +

*N*

_{0}[

*l*])(

**H**

_{0}[

*l*]

*W*

_{0}[

*l*] +

*N*

_{0}[

*l*])

^{ H }]. Here, we used the ZF technique for simplicity. The received symbol at the input of the first RS’s HPA can then be expressed as

*N*× 1 received OFDM symbol vector in FD at the input of the RS’s HPA, and${Y}_{{R}_{1},\mathrm{DemF}}^{j}\left[l\right],$

*l*= 1, 2, …,

*n*is the input symbol at the

*l*th subcarrier and

*j*th received antenna of the RS. Also, we use

*N*

^{′}

_{0}[

*l*] =

**G**

_{ ZF }[

*l*]

*N*

_{0}[

*l*] for notational convenience. The received OFDM signals are then amplified (normalized) separately on each subcarrier with an amplification factor α, where α is an

*nN*×

*nN*block diagonal matrix whose diagonal entries for the

*l*th subcarrier and

*i*th hop transmission is given by diag{

*α*

_{ i }

^{1}[

*l*],

*α*

_{ i }

^{2}[

*l*], …,

*α*

_{ i }

^{ N }[

*l*]},

*l*= 1, …,

*n*. There are many choices for the selection of the amplification parameter α in relaying systems[5, 20]. Sulyman et al.[20] amplify all the subcarriers in an OFDM symbol with the same parameter α without decoding the symbols. Riihonen et al.[5] demodulate the OFDM symbols at the RS to amplify each subcarrier separately with the amplification parameter α. The amplification factor for the

*k*th subcarrier is selected to satisfy the transmit power constraint at the RS as

*P*

_{ R }is the total transmit power available at the RS. Before transmitting onto the next hop, the OFDM symbols are passed through the HPA at the RS as shown in Figure 2. The output of the HPA at the

*j*th transmit antenna of the first RS (

*R*

_{1}) in time domain (TD) is given by

*n*

^{′}

_{0}

^{ j }(

*t*) and

*w*

_{0}

^{ j }(

*t*) are, respectively, the thermal noise and the nonlinear noise of the HPA, propagated from the first hop transmission.

*w*

_{1}

^{ j }(

*t*) and

*k*

_{1}

^{ j }are the HPA distortion terms introduced by the first RS. To calculate the PSD of the output of the HPA at the

*j*th transmit antenna of the first RS, we calculate the autocorrelation function of the output of the HPA at the

*j*th transmit antenna of the RS as

*j*th antenna of the RS is computed from the autocorrelation function using Wiener–Khintchin theorem and is given as

*R*relay hops, we can repeat the steps in Equations (17)–(23) to obtain the spectrum of the transmitted OFDM signal at the

*j*th transmit antenna of the

*R*th RS as

*R*RS, then

**K**

_{0}=

**K**

_{1}=⋅⋅⋅⋅=

**K**

_{ R }=

**K**, and if we assume the same amplification factor at all

*R*RS so that

**α**

_{1}=

**α**

_{2}=⋅⋅⋅=

**α**

_{ R }=

**α**, then Equation (24) can be expressed as

#### Case II: AF relaying system

*R*

_{1}) in FD is given by

*N*× 1 received OFDM symbol vector, and${Y}_{{R}_{1}}^{j}\left[l\right]$ is the received OFDM symbol at the

*j*th received antenna of

*R*

_{1}for the

*j*th subcarrier and is given by

*j*th receive antenna of

*R*

_{1}is then amplified with an amplification factor

*α*

_{1}

^{ j }as shown in Figure 2b. The amplification parameter${\alpha}_{1}^{j}\left[l\right]=\sqrt{\frac{{P}_{{R}_{1}}}{E\left[|{Y}_{{R}_{1}}^{j}\left[l\right]{|}^{2}\right]}}$ for the

*j*th antenna on the

*l*th subcarrier is again selected such that the total transmitted power at

*R*

_{1}is${P}_{{R}_{1}}$. Before transmitting the OFDM signal onto the next hop, it is passed through the HPA at the RS. In general, the transmitted OFDM signal at the

*m*th RS (

*R*

_{ m }) in TD is given by

*m*- 1)th RS, and the transmitted OFDM symbol at the output of the HPA at the

*j*th transmit antenna of the

*m*th RS is given by

*m*= 1, i.e., one RS, Equation (29) can be written as

*j*th transmit antenna for the case

*m*= 1 is given as

*j*th transmitting antenna of the

*P*th RS is derived for AF relaying system as

where${P}_{{R}_{P-1}^{l}{R}_{P-1}^{l}}\left(f\right)$ is the PSD of the transmitted OFDM symbol at the output of the HPA at the *l* th transmitting antenna of the (*P* - 1)th RS.

## Simulation results and discussions

*n*= 1024 subcarriers and different number of transmit and receive antennas. We adopt 1024-FFT downlink sub-carrier allocation scheme defined in the WiMAX standard[16], as shown in Table 1. The HPA model employed in the simulation studies is the SEL model.

**1024-FFT/IFFT parameters in 20 MHz bandwidth**

Parameter | Value |
---|---|

Channel bandwidth | 20 MHz |

Modulation | 4-QAM |

Number of DC subcarriers | 1 |

Number of guard subcarriers, left | 80 |

Number of guard subcarriers, right | 79 |

Number of data subcarriers | 864 |

Subcarrier frequency spacing, Δ | 19.53125 KHz (= 20 MHz/1024) |

IFFT/FFT period, | 51.2 μs |

Cyclic prefix duration, | 6.4 μs ( |

Total OFDM symbol duration, | 57.6 μs |

*ω*≤ 0.6, where

*ω*= 1 corresponds to the Nyquist frequency, for our simulation. Figure 10 shows the effect of HPA nonlinearity on the spectrum of filtered OFDM signal for different values of CR. It is observed that because of the attenuation of the filter in this case, the spectral re-growth decreases in filtered MIMO–OFDM system compared to the unfiltered case in Figure 6. However, the cumulative remnant re-growth is still significant considering the fact that the results in Figure 10 show what happens per MIMO antenna of the transmitting station, and there are total of

*N*antennas per node. The spectrum of the nonlinear OFDM signal for different number of relaying hops for DemF relaying is presented in Figure 11. It is observed from these results that the out-of-band distortion (i.e., spectrum re-growth in adjacent band) increases as the OFDM signal is relayed across multiple RSs. For example, for

*N*×

*N*MIMO–OFDM system with 2-hop relaying, the observed spectral re-growth from Figure 11 will be

## Conclusions

This article presents new insights on the out-of-band spectral re-growth due to HPA nonlinearities when MIMO–OFDM signals are transmitted over multiple relay channels. Expressions for the PSD of a MIMO–OFDM signal are presented when it is transmitted from a BS to the MS via multiple RSs, all equipped with nonlinear HPAs. It is shown that significant spectral re-growth occurs for the AF and DemF relaying options as the OFDM signal traverse one or more relay hops. For MIMO–OFDM systems with large number of relay hops and MIMO antennas, the cumulative effects of this re-growth can result in significant spectral broadening exceeding specified limits on the spectral masks. Hence, it is concluded from our results that even though the general MMR system where data could traverse any number of relay hops from source to destination have widely been studied theoretically, only the 2-hop version proposed by the IEEE 802.16j group[18] may be advised for any practical deployments in MIMO–OFDM transmissions because of the potentials for spectral mask violations when going beyond 2-hop transmissions.

## Declarations

### Acknowledgments

This study was sponsored by a grant (No. 09-ELE928-02) from The National Plan for Science and Technology (NPST), King Saud University, Saudi Arabia.

## Authors’ Affiliations

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