research open access spectral broadening effects of ......ofdm relay links in the presence of hpa...

Ahmad et al. EURASIP Journal on Wireless Communications and Networking 2013, 2013:32http://jwcn.eurasipjournals.com/content/2013/1/32

RESEARCH Open Access

Spectral broadening effects of high-poweramplifiers in MIMO–OFDM relaying channelsIshtiaq Ahmad1, Ahmed Iyanda Sulyman1*, Abdulhameed Alsanie1, Awad Kh Alasmari1 and Saleh A Alshebeili1,2

Abstract

The combination of MIMO–OFDM is a very attractive solution for broadband wireless services. Thus, the twoprominent fourth-generation (4G) cellular systems, WiMAX and LTE-advanced, have both adopted MIMO–OFDMtransmission at the physical layer. OFDM signal however suffers from nonlinear distortions when passed throughhigh-power amplifier (HPA) at the RF stage. This nonlinear distortion introduces out-of-band spectral broadeningand in-band distortions on the transmitted signals. 4G cellular standards have placed strict limits on the allowablespectral broadening in their spectrum mask specifications, to insure that data transmission on a given channel isnot interfering significantly with an adjacent channel user. In this article, we characterize the out-of-band spectralbroadening 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 thecumulative effects of HPA on the spectrum of the transmitted signals are estimated. It is shown that depending onthe number of relays and the relaying configuration employed, it may happen that a transmitted MIMO–OFDMsignal with the transmit spectrum mask initially within the allowable set limit at the source node arrives at thedestination 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

IntroductionFourth-generation (4G) broadband communication sys-tems need to provide ultra-high data rate services inorder to meet the requirements of future high-bandwidth multimedia applications over cellular systemssuch as the digital TV distributions and interactive vid-eos planned in the WiMAX and LTE-advanced. Amongthe candidate physical layer technologies that can bedeployed to achieve these goals, MIMO–OFDM is themost potent solution that can provide such high datarate at high spectral efficiencies. Compared to thesingle-carrier systems however, OFDM has a large peak-to-average-power ratio [1,2] which makes it very sensi-tive to high-power amplifier (HPA) nonlinearities at theRF stage of the transmission chain [3]. There are twoimportant effects of the HPA nonlinearities introducedin the transmitted OFDM signals: in-band and out-of-band distortions. The in-band distortion degrades bit

* Correspondence: [email protected] of Electrical Engineering, King Saud University, Riyadh, SaudiArabiaFull list of author information is available at the end of the article

© 2013 Ahmad et al.; licensee Springer. This isAttribution License (http://creativecommons.orin any medium, provided the original work is p

error rate (BER) performance and capacity of the cellularoperator [4-10], whereas the out-of-band distortion aris-ing from the spectral broadening effect of the HPAaffects other users operating in the adjacent frequencybands [11-15]. While the BER and capacity analysis ofOFDM relay links in the presence of HPA nonlinearityare fairly well understood [5 and references there in], theeffects of HPA on out-of-band emissions for OFDMrelay link are not yet studied to the best of the authors’knowledge. Out-of-band emissions are strictly monitoredby the cellular regulators using the concept of transmitspectrum mask. Transmit spectrum mask is the powercontained in a specified frequency bandwidth at certainoffsets, relative to the total carrier power. In the 4G sys-tem such as WiMAX and LTE-advanced, strict limitshave been specified for the spectrum masks. Figure 1displays the spectrum mask of the IEEE 802.16 signalspecified in the WiMAX standard [16], where about25-dB attenuation is required between the reference car-rier power and all unwanted spurious emissions at cer-tain frequency offsets from the operating bandwidth.

an Open Access article distributed under the terms of the Creative Commonsg/licenses/by/2.0), which permits unrestricted use, distribution, and reproductionroperly cited.

mailto:[email protected]

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Figure 1 Spectrum mask of IEEE 802.16 WiMAX signal [16,Section 8.5.2, pp. 1126]. where A denotes the operatingbandwidth. and B is the maximum spectral broadening allowed.

Ahmad et al. EURASIP Journal on Wireless Communications and Networking 2013, 2013:32 of 14http://jwcn.eurasipjournals.com/content/2013/1/32

Banelli and Cacopardi [11] derived analytical expressionsfor the correlation function of the output of nonlinearHPA when the input to the amplifier is an OFDM signal.The power spectral density (PSD) of the signal is thencalculated using the Fourier transform of the correlationfunction. Grad et al. [12] studied spectral re-growth dueto HPA nonlinearity in code-division multiple access(CDMA) systems. They obtained analytical expressions forthe power spectrum of the CDMA signal at the output ofthe HPA, using a complex power-series model for the HPAcharacteristics. The out-of-band emission for the time div-ision synchronous CDMA system is presented in [13], interms of third-order intercept point (IP3). Cottais et al.[14] derived expressions for the PSD of a generalmulticarrier signal at the output of a memoryless HPA.They also obtained a closed-form expression for the PSDof the special case of single-carrier signals. Helaly et al. [15]examined the effects of the characteristics of the inputCDMA signal on the resulting out-of-band spectral re-growth at the output of the HPA. They pointed out that, inaddition to the HPA saturation level, the input signal’sthreshold crossing rate and the variance of the clipped sig-nal also contribute to the spectral re-growth. It is import-ant to note that OFDM signals share some similarities withCDMA signals in this regard. Recently also, Gregorio et al.[17] proposed a MIMO-predistortion (MIMO-PD) systemthat tries to compensate crosstalk and IQ imbalance insingle-hop MIMO–OFDM communication systems, wherethey have shown that some reduction in the spectral re-growth can be achieved using the proposed MIMO-PDsystem. The effectiveness of such a compensation schemein a multihop environment is however not yet known.

All the above-cited studies, and several others in theliterature however, focused on the spectral re-growthdue to HPA nonlinearity in a single-hop communicationsystem. Recently, the two prominent 4G cellularsystems, WiMAX and LTE-advanced, have definedrelaying as an integral part of the network design[18,19]. Thus, MIMO–OFDM signals transmitted in the4G systems will frequently pass through one or morerelay hops from source node to the destination node. In-vestigating the level of adherence to set limits on spec-tral broadening in cellular systems employing relayingtechnologies is therefore a deployment imperative. Tothe best of the authors’ knowledge, no work haspresented a detailed study of the broadening effects ofHPA nonlinearity on the spectrum of MIMO–OFDMsignals in multihop relaying channels.In this article, we characterize for the first time in the

literature, the cumulative spectral broadening effects ofmultiple HPAs when MIMO–OFDM signals are trans-mitted over multihop relaying channels. Expressions forthe PSD of a MIMO–OFDM signal are presented overmultihop relay channels, each equipped with nonlinearHPA’s. It is shown that due to the cumulative effect ofthe multiple HPA’s in a MIMO link, and the cascade ef-fect of many relaying channels in a multihop relay link,significant broadening effect occur which is much morethan what would be observed in a single-hop transmis-sion such as those characterized in [11-15]. We alsoshow that for the amplify-and-forward (AF) anddemodulate-and-forward (DemF) relaying options, theresulting cumulative re-growth may lead to spectralmask violations after a few relaying hops is traversed bythe transmitted OFDM signal, even though the set limitswere 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-for-ward (DF) relaying option, it is observed that less severespectral broadening are observed. However, due to thelatency problems associated with the DF relaying optionwhich degrades quality of broadband signals, AF orDemF are the preferred candidates for broadbandtransmissions over relaying channels and therefore thespectral broadening issues observed here must be givenconsiderable attentions in the design of broadbandmultihop relaying systems.

HPA nonlinearity model for MIMO–OFDM relayingchannelConsider a MIMO–OFDM relaying system employing nsubcarriers per OFDM symbol, M transmitting and L re-ceiving antennas at each hop in the transmission chain.We assume that all the transmitting antennas at eachnode simultaneously transmit different symbols (MIMO-multiplexing system), and that all L receiving antennas


at each receiving node are expended in separating eachof the M transmitted streams. For simplicity, we con-sider the case of N transmitting and N receiving anten-nas (where N ≤ M, L). Thus, in the ensuing analysis wefocus on the N × N MIMO–OFDM multiplexingrelaying system. We consider that the transmitted signalfrom a source node passes through a single-hopMIMO channel H0, and R multihop MIMO relayingchannels H1,. . .,HR, associated with R fixed or mobilerelaying nodes, to the destination node as shown inFigure 2.The MIMO channel matrix for each ith hop transmis-

sion Hi, i = 0, 1,. . ., R, is an nN × nN block diagonalmatrix, with the kth block diagonal entries H[K]icorresponding to the fading on the kth OFDM sub-carrier, k = 0, 1,. . ., n − 1, modeled as independent andidentically distributed (iid) random variables taken fromzero mean complex Gaussian distribution, with unitvariance. We assume that the set of random matrices{H0,. . ., HR} are independent, and that AF, DemF, or DFrelaying options could be employed at the relay nodes[5,20]. We also assume that the BS and all the relaystations (RS’s) employ similar nonlinear HPA’s whichintroduce similar nonlinear distortions per hop, in thetransmitted signal. A consequence of this assumption isthat if the BS or any of the RS employ a nonlinear HPAwith nonlinearity level more or less than the otherdevices, then the contribution of that device to the over-all spectral broadening will be more or less than

(a)Nonlinear Multi Hop MIMO OFDM

S/P

CP

rem

oval

DF

T

ampl

ifica

tion

S/P

DF

T

CP

rem

oval

ampl

ifica

tion

(b) Signal Proces

MIMOOFDM

Encoder

HPA

HPA

RelStati

Nx

1x

Nx

11y

Ny1

0H

1x

Figure 2 HPA nonlinearity model for MIMO–OFDM relaying channel.

estimated in this analysis. We can express the transmit-ted symbol in polar coordinate as

x tð Þ ¼ r tð Þejθ tð Þ ð1Þ

where r(t) is the amplitude and θ(t) is the phase of theinput signal into the HPA. The signal at the output ofthe HPA can then be expressed as

x tð Þ ¼ g r tð Þ½ �ejθ tð Þ ð2Þ

where g[r(t)] is a complex nonlinear distortion function,which only depends on the envelope of the transmittedsymbols. The nonlinear distortion function can beexpressed as

g r tð Þ½ � ¼ gA r tð Þ½ �ejgP r tð Þ½ � ð3Þ

where gA[r(t)] is the amplitude-to-amplitude (AM–AM)and gP[r(t)] is the amplitude-to-phase (AM–PM)conversions of the HPA. The AM–AM conversions fordifferent memoryless HPA models [Saleh model, Solid-State Power Amplifier (SSPA) model, and soft enveloplimiter (SEL) model] used in communication systems areplotted in Figure 3.

Communication Channel

Inve

rse

DF

T

CP

inse

rtio

n

P/S HPA

Inve

rse

DF

T

CP

inse

rtio

n

P/S HPA

sing at Relay Station (RS)

MIMOOFDM

Decoder

ayon Ny1ˆ

1ˆDy

NDy

RH1H11y

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Input Voltage (Normalized)

Out

put V

olta

ge(N

orm

aliz

ed)

AM-AM Conversion

Saleh Model

SEL Model

SSPA Model (β = .5)SSPA Model (β = 5)

Figure 3 AM–AM conversions for different amplifier models.


It is easily observed from this figure that Saleh modeldepicts the severest nonlinear distortions followed bythe SSPA model. The SEL model on the other handdepicts more closely the current state-of-the-art in HPAdesigns for cellular systems where approximately linearbehavior is obtained in the small signal conditions,followed by clipping (soft envelope limiting) at satur-ation. We focus on the SEL HPA model in this article.For the SEL model, the AM–AM and AM–PMconversions are given by [21]

gA r tð Þ½ � ¼ αAr tð Þ; 0 ≤ r tð Þ ≤ Ais

Ais; r tð Þ > Aisgp r tð Þ½ � ¼ 0

�ð4Þ

where αA is the small-signal gain, and Ais is the inputsaturation voltage of the HPA. Let x(t) denote the inputsignal into the HPA, and we assume that x(t) is Gaussiandistributed. Therefore, according to the Bussgang’s the-orem, the output of the HPA when the input is a Gauss-ian process is given as [22,23]

x tð Þ ¼ kx tð Þ þ w tð Þ ð5Þ

where k (i.e., 0 ≤ k ≤ 1) is an attenuation factor for thelinear part which represents the in-band distortion, andw(t) is a nonlinear additive noise which represents theout-of-band distortion. w(t) is a zero-mean complexGaussian random variable (r.v), with the in-phase andquadrature components mutually iid, and with variance

σw2 . The in-band and out-of-band distortion terms can becalculated as [24]

k ¼ E x tð Þx� tð Þf gE x tð Þx� tð Þf g ¼ 1

Pavg

Z10

rx∗ rð Þf rð Þdr0@

1A

∗

ð6Þ

σ2w ¼ Efjx� rð Þ 2j g � jkj2Efjx 2j g ¼ 1

Pavg

Z10

��x� rð Þ 2f rð Þdr�� k��2!

ð7Þwhere ‘*’ denotes complex conjugation, Pavg = E[|x|2] isthe average input energy per symbol, and f (r) is the prob-ability density function (PDF) of the envelope of the inputsignal into the HPA (i.e., r = |x|). The PDF of ‘r’ is Rayleighdistribution (since x(t) is assumed Gaussian). The closed-form expressions for the in-band and out-of-band distor-tion parameters for the SEL HPA model are calculated bysubstituting f (r) into Equations (6) and (7) to obtain

kj ¼ 1� e�γ2j þ 12

ffiffiffiπ

pγ jerfc γ j

� �ð8Þ

σ2wj¼ Pj 1� e�γ2j � k2j

� �ð9Þ

γ j ¼ffiffiffiffiffiffiffiffiffiIBOj

p ¼ Ais;jffiffiffiffiPj

p ð10Þ

where j = 0, 1, . . ., R denotes the number of relay hops, γjrepresents the clipping ratio (CR), Ais,j represents the input


saturation voltage of each and every HPA employed at thejth relay hop, and Pj represents the average input powerinto the HPA’s at the jth hop. The frequency domain (FD)expression of the signal above is then obtained by takingits discrete Fourier Transform (DFT).

Effect of HPA nonlinearity on the spectrum ofMIMO–OFDM relaying systemPSD of the nonlinearly amplified OFDM signal at the BSOFDM signal is well approximated by zero-mean com-plex Gaussian signals for the large number of subcarriers[3]. Using the Bussgang theorem, the output of the HPAat the MIMO–OFDM BS transmitter is given by

^x tð Þ ¼ Κ0x tð Þ þ w0 tð Þ ð11Þ

where the vector ^x tð Þ ¼ x1 tð Þ; . . . ; xN tð Þ� �and xj tð Þ are

the output of the HPA at the jth transmitting antenna ofthe BS when the input to the amplifier is xj(t). Κ0 is anN × N diagonal matrix, where k0

j represents the scalingfactor of the linear part (in-band distortion) at the jthtransmit antenna of the BS. w0(t) = [w0

1(t), . . ., w0N(t)],

and w0j (t) is the nonlinear distortions noise (out-of-band

distortion) due to HPA at the jth transmitting antennaof the BS. The PSD of a signal is commonly estimatedby computing the autocorrelation function of the signalfollowed by a Fourier transform, using the well-knownWiener–Khintchin theorem [25]. The PSD of the OFDMsignal at the output of the nonlinear HPA at the jth trans-mit antenna of a MIMO–OFDM transmitter is given by

Pjxjx j fð Þ ¼

Zþ1

�1Φ j

x j x j τð Þe�j2πf τdτ ð12Þ

where Pxj x jj fð Þ and Φx j x j

j τð Þ represent the PSD and auto-

correlation function of the OFDM signal at the output ofthe HPA at the jth transmitting antenna. The autocorrel-ation function for the output of the HPA at the jth trans-mitting antenna can be computed as

Φx j xj τð Þ¼ lim

T→112T

ZT�T

xj tð Þ xj t þ τð Þ� �∗dt

¼ E xj tð Þ xj t þ τð Þ� ��n o¼ E k j

0xj tð Þ þ wj

0 tð Þh i

k j0x

j t þ τð Þ þ wj0 t þ τð Þ

h i∗�

¼ jk j0j2E xj tð Þ xj t þ τð Þ� �∗n o

þ E wj0 tð Þ wj

0 t þ τð Þh i∗�

¼ kj02Φxjxj τð Þ þΦw j

0wj0τð Þ

��

ð13Þ

where k0j is the attenuation factor of the in-band part,

Φxjxj τð Þ represents the autocorrelation function of theOFDM signal at the input of the HPA, i.e., xj(t), and

Φw j0w

j0τð Þ represents the autocorrelation function of the

out-of-band nonlinear noise (i.e., wj(t)), contributed by theHPA at the jth transmitting antenna of the source node(BS or MS). Using Equations (12) and (13), the PSD of theOFDM signal at the output of the HPA at the jth transmit-ting antenna of the source node can be written as

Pjxjx j fð Þ ¼ jkj0j2Pj

xjxj fð Þ þ Pj

wj0w

j0

fð Þ ð14Þ

From Equation (14), it is easily observed that the PSDof the OFDM signal at the output of the HPA at the BSis attenuated by a factor of |k0

j|2, since 0 ≤ kij ≤ 1. It is

also observed from this equation that the PSD of theOFDM signal at the output of the HPA is broadened,compared to the PSD at the HPA input, by a factor of

P j

wj0w

j0

fð Þ . This spectral broadening term contributed by

the jth transmit antenna, P j

wj0w

j0

fð Þ, is related to the corres-

ponding nonlinear noise variance σ j

w j0

2 as

σ jwj0

2 ¼Zþ1

�1Pj

wj0w

j0

fð Þdf ð15Þ

Using the result from [26, Equation (16)], Equation

(15) can be approximated as σ j

w j0

2 ≈ Pj

wj0w

j0

fð ÞB , where B

denotes the bandwidth of the signal. This approximationis valid if the spectrum of w0

j is flat across the bandwidth,which is true for the subcarrier-based analysis consideredhere. As the variance of the nonlinear noise due to HPAincreases, the spectral re-growth of the transmittedOFDM signal increases as can be observed from Equation(15). Since the analysis above gives estimate of the spectralre-growth due to HPA per transmitting antenna, then forMIMO–OFDM system with N transmitting antennas, theoverall spectral re-growth that occur due to HPA

nonlinearity is given by PMIMOw0w0

¼XNj¼1

Pj

w j0w

j0

fð Þ.

PSD of the nonlinearly amplified OFDM signals at RS’sNext we calculate the PSD of the nonlinearly amplifiedOFDM signals at RS’s. For this analysis, we consider twocases. In the first case, we consider that each RS hasability to perform MIMO signal processing on thereceived signal before amplifying and forwarding it ontothe next hop, and thus we could examine the spectrumof the OFDM symbols obtained at each receiving anten-nas of the RS. This case corresponds to the DemFrelaying option. In the second case, we consider that theRS does not have ability to perform MIMO signalprocessing on the received signal before forwarding ontothe next hop. The RS simply AF the received OFDM


signals at the RF stage without demodulating the OFDMsignal. This case corresponds to the AF relaying option.AF has the best latency performance, and is attractivefor broadband transmissions over relaying channels. Thecase of DF relaying option where RS actually decodedata transmitted on each OFDM subcarriers before re-encoding/forwarding it is not analyzed here because theexisting single-hop analyses in [27] and other referencesare valid for that case using hop-by-hop analysis. How-ever, we later include the DF case in the simulationresults as a reference.

Case I: DemF relaying systemIn DemF relaying system, the received signal at the firstRS for the lth subcarrier is given by

YR1 l½ � ¼ H0 l½ �X l½ � þ N0 l½ �¼ K0H0 l½ �X l½ � þ H0 l½ �W0 l½ � þ N0 l½ �

ð16Þ

where X[l] = [X1[l]X2[l] ⋯ XN[l]]T is an N × 1 inputOFDM signal in FD at the BS, and Xj[l] is the input intothe HPA at the jth transmitting antenna on the lth sub-carrier. Ki = diag {ki

j}j=1N , i = 0, . . ., R is an N × N diagonal

matrix, where kij represents the scaling factor of the linear

part (in-band distortion) at the jth transmit antenna in theith hop. W[l] = [Wi

1[l]Wi2[l] ⋯ Wi

N[l]]T is an N × 1nonlinear distortion noise vector (out-of-band distortion)due to HPA and it is obtained by taking the DFT ofw(t). Wi

j[l] represents the nonlinear distortion noiseon the jth transmit antenna in the ith hop for the lthsubcarrier. N[l] = [Ni

1[l]Ni2[l] ⋯ Ni

N[l]]T is an N × 1complex additive noise vector and Ni

j[l] represents theiid zero-mean complex AWGN noise on the jthtransmit antenna in the ith hop for the lth subcarrier.

YR1 l½ � ¼ Y 1R1

l½ �Y 2R1

l½ � . . .YNR1

l½ �h iT

is an N × 1 received

OFDM symbol vector in FD at the first RS and Y jR1

l½ � ¼XN

m¼1H0

jm l½ �Xm l½ � þ Nj0 l½ � is the received OFDM symbol

at the jth received antenna of the first RS for the lth sub-carrier. We assume that the channel between the BS andthe RS is known at the RS. Thus, we can employ MIMOZero-Forcing or MIMO minimum mean square errortechnique for the MIMO signal processing at the RS forthe DemF relaying option as

YR1;DemF l½ � ¼ G0 l½ �YR1 l½ � ð17Þwhere

G0 l½ �¼ HH0 l½ �H0 l½ �

��1HH

0 l½ �HH

0 l½ �H0 l½ � þQ0 l½ �I ��1

HH0 l½ �

: ZF: MMSE

(ð18Þ

and Q0[l] = E[(H0[l]W0[l] + N0[l])(H0[l]W0[l] + N0[l])H].

Here, we used the ZF technique for simplicity. The

received symbol at the input of the first RS’s HPA can thenbe expressed as

YR1;DemF l½ � ¼ K0X l½ � þW0 l½ � þ GZF l½ �N0 l½ �

¼ K0X l½ � þW0 l½ � þ N 00 l½ �

ð19Þ

where YR1;DemF l½ � ¼ Y 1R1;DemF l½ �Y 2

R1;DemF l½ � . . .YNR1;DemF l½ �

h iTis an

N × 1 received OFDM symbol vector in FD at the input of

the RS’s HPA, and Y jR1;DemF l½ �; l = 1, 2, . . ., n is the input

symbol at the lth subcarrier and jth received antenna ofthe RS. Also, we use N 0

0[l] = GZF[l]N0[l] for notationalconvenience. The received OFDM signals are then ampli-fied (normalized) separately on each subcarrier with anamplification factor α, where α is an nN × nN block diag-onal matrix whose diagonal entries for the lth subcarrierand ith hop transmission is given by diag{αi

1[l], αi2[l], . . .,

αiN[l]}, l = 1, . . ., n. There are many choices for the selec-

tion of the amplification parameter α in relaying systems[5,20]. Sulyman et al. [20] amplify all the subcarriers in anOFDM symbol with the same parameter α without decod-ing the symbols. Riihonen et al. [5] demodulate theOFDM symbols at the RS to amplify each subcarrier sep-arately with the amplification parameter α. The amplifica-tion factor for the kth subcarrier is selected to satisfy thetransmit power constraint at the RS as

αjl k½ � ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

PR

E y j

Rl ;DemF k½ �� 2�

vuuut ð20Þ

where PR is the total transmit power available at the RS.Before transmitting onto the next hop, the OFDMsymbols are passed through the HPA at the RS as shownin Figure 2. The output of the HPA at the jth trans-mit antenna of the first RS (R1) in time domain (TD)is given by

y^jR1;DemF tð Þ ¼ α j

1kj0k

j1x

j tð Þ þ αj1kj1w

j0 tð Þ

þ αj1kj1n

00j tð Þ þ wj

1 tð Þ

ð21Þ

where n00j(t) and w0

j(t) are, respectively, the thermalnoise and the nonlinear noise of the HPA, propagatedfrom the first hop transmission. w1

j(t) and k1j are the

HPA distortion terms introduced by the first RS. Tocalculate the PSD of the output of the HPA at the jthtransmit antenna of the first RS, we calculate theautocorrelation function of the output of the HPA atthe jth transmit antenna of the RS as

ΦjR1R1

τð Þ ¼ E y jR1;DemF tð Þ y jR1;DemF t þ τð Þh iH�

¼ Eαj1k

j0k

j1x

j tð Þ þ αj1k

j1w

j0 tð Þ þ α j

1kj1n′0 tð Þ þ w j

1 tð Þh i

� α j1k

j0k

j1x

j t þ τð Þ þ α j1k

j1w

j0 t þ τð Þ þ α j

1kj1n′0 t þ τð Þ þ w j

1 t þ τð Þh iH

8><>:

9>=>;

¼ α j1

� �2k j0

�� 2 k j1

�� 2Φxjx j τð Þ þ α j1

� �2k j1

�� 2Φw j0w

j0τð Þ þ

α j1

� �2k j1

�� 2Φnj0n

j0τð Þ

NþΦw j

1wj1τð Þ

ð22Þ


where

E GiGHi

� � ¼ E HHi Hi

��1HH

i

h iHH

i Hi ��1

HHi

h iH� ¼ E HH

i Hi� ��1n o

¼ 1NIN

Thus, the PSD of the OFDM signal at the jth antenna ofthe RS is computed from the autocorrelation functionusing Wiener–Khintchin theorem and is given as

PjR1R1 ;DemF fð Þ ¼ α j

1

� �2k j0

�� 2 k j1

�� 2Pxjx j fð Þþ α j

1

� �2k j1

�� 2Pw j0w

j0fð Þ

þα j1

� �2k j1

�� 2Pnj0n

j0fð Þ

Nþ Pw j

1wj1fð Þ ð23Þ

In general, for R relay hops, we can repeat the steps inEquations (17)–(23) to obtain the spectrum of the trans-mitted OFDM signal at the jth transmit antenna of theRth RS as

PjRR;DemF fð Þ¼ α j

1

� �2⋯ α j

R

� �2� YRl¼0

jk jl j2 Px j

xj fð Þ

þ α j1

� �2⋯ α j

R

� �2� XR�1

m¼0

YR�m

n¼1

jk jRþ1�nj2 Pw

jmw

jmfð Þ

þ α j1

� �2⋯ α j

R

� �2� 1N

XR�1

p¼0

YR�p

q¼1

jk jRþ1�qj2 Pn j

pnjpfð Þ

þ Pw jRw

jRfð Þ

ð24Þ

From the above expression, it can be seen that thespectral re-growth at the RS depends both on thenonlinear distortion noise due to HPA and on the

thermal AWGN noise. For the case when we assume thesame HPA characteristic at the BS and R RS, then K0 =K1 =⋅⋅⋅⋅= KR = K, and if we assume the same amplifica-tion factor at all R RS so that α1 = α2 =⋅⋅⋅=αR = α, then Equation (24) can be expressed as

PjRR;DemF fð Þ ¼ αjð Þ2R kjj jð Þ2 Rþ1ð ÞPxjxj fð Þ

þ αjð Þ2RXR�1

m¼0

kj jð Þ2 R�mð ÞPw jmw

jmfð Þ

ð25Þ

þ 1N

αj �2RXR�1

p¼0

kj jð Þ2 R�pð ÞPn jpn

jpfð Þ

þ Pw

jRw

jRfð Þ

Case II: AF relaying systemIn AF relaying system, the received symbol at the inputof the HPA of the first RS (R1) in FD is given by

YR1 l½ � ¼ H0 l½ �X l½ � þ N0 l½ �¼ K0H0 l½ �X l½ � þH0 l½ �W0 l½ � þ N0 l½ �

ð26Þ

where YR1 l½ � ¼ Y 1R1

l½ �Y 2R1

l½ � . . .YNR1

l½ �h iT

is an N × 1

received OFDM symbol vector, and Y jR1

l½ � is the receivedOFDM symbol at the jth received antenna of R1 for thejth subcarrier and is given by

Y jR1

l½ � ¼XNm¼1

H0j;m l½ �Xm

m½ � þ Nj0 l½ �

¼XNm¼1

km0 H0j;m l½ �X m½ �

þXNm¼1

H0j;m l½ �Wm l½ � þ Nj

0 l½ � ð27Þ

The received OFDM signal at the jth receive an-tenna of R1 is then amplified with an amplificationfactor α1

j as shown in Figure 2b. The amplification

Table 1 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, Δf 19.53125 KHz (= 20 MHz/1024)

IFFT/FFT period, TFFT 51.2 μs

Cyclic prefix duration, TCP 6.4 μs (TFFT/8)

Total OFDM symbol duration, TS 57.6 μs


parameter αj1 l½ � ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

PR1E½jY j

R1l½ � 2�j

rfor the jth antenna on the

lth subcarrier is again selected such that the totaltransmitted power at R1 is PR1 . Before transmittingthe OFDM signal onto the next hop, it is passedthrough the HPA at the RS. In general, the transmit-ted OFDM signal at the mth RS (Rm) in TD is givenby

yRm;AF tð Þ ¼ αmKmHm�1yRm�1;AF tð Þþ αmKmnm�1 tð Þ þ wm tð Þ

ð28Þ

where yRm�1;AF tð Þ; m ¼ 1; . . . ;R is the transmittedOFDM symbol at the (m − 1)th RS, and the transmit-ted OFDM symbol at the output of the HPA at thejth transmit antenna of the mth RS is given by

y jRm;AFtð Þ ¼ αjmk

jm ∑

N

l¼1hm−1jl y jRm�1;AF

tð Þþ αjmk

jmn

jm−1 tð Þ þ wj

m tð Þ ð29ÞFor the case of m = 1, i.e., one RS, Equation (29) canbe written as

yRm;AFj tð Þ ¼ αj1k

j1 ∑N

l¼1kl0h

0jlx

l tð Þ þ αj1kj1 ∑N

l¼1hljlw

l0 tð Þ

þ αj1kj1n

j0 tð Þ þ wj

1 tð Þ ð30ÞThen the PSD of the transmitted OFDM symbol at the

output of the HPA at the jth transmit antenna for thecase m = 1 is given as

PjR1R1 ;AF

fð Þ ¼ α j1

� �2j k j1j2XNl¼1

jkl0j2Pxlx l fð Þ

þ α j1

� �2jk j

1j2XNl¼1

Pwl0w

l0fð Þ

þ α j1

� �2jk j

1j2Pnj0n

j0fð Þ þ Pwl

1wl1fð Þ

ð31Þ

Similarly, the general expression for the PSD of thetransmitted OFDM symbol at the output of the HPA atthe jth transmitting antenna of the Pth RS is derived forAF relaying system as

PjRPRP ;AF

fð Þ ¼ α jP

� �2jk j

Pj2XNl¼1

jklP�1j2PRlP�1R

lP�1

fð Þ

þ α jP

� �2jk j

Pj2XNl¼1

PwlP�1w

lP�1

fð Þ

þ α jP

� �2jk j

Pj2PnjP�1n

jP�1

fð Þ þ PwlPw

lPfð Þð32Þ

where PRlP�1R

lP�1

fð Þ is the PSD of the transmitted OFDM

symbol at the output of the HPA at the lth transmittingantenna of the (P − 1)th RS.

Simulation results and discussionsIn our simulation setup, we consider multihop MIMO–OFDM system for n = 1024 subcarriers and differentnumber of transmit and receive antennas. We adopt1024-FFT downlink sub-carrier allocation schemedefined in the WiMAX standard [16], as shown inTable 1. The HPA model employed in the simulationstudies is the SEL model.Figure 4 presents the time samples of the OFDM

symbol we passed through the HPA in our simulationstudies. It is evident from this figure that the OFDM sig-nal has high amplitude variations (or high PAPR), thussome part of the signal will fall into the nonlinear regimeof the HPA characteristics except if conservatively largeback-off is used, which results in severe power penalty.A popular approach is to clip the high peaks in theOFDM signal to enable linear amplifications [27]. How-ever, clipping operation itself is a nonlinear process andit results in in-band and out-of-band distortions. TheSEL model employed in our simulation captures thisHPA model. Figure 5 depicts the output of the HPAwhen the OFDM signal in Figure 4 is passed throughHPA for different values of the CR. It is easy to observefrom this figure that as the CR increases, the severity ofthe nonlinearity decreases and the time waveform of theoutput signal from the HPA approaches the input signalinto the HPA. However, large clipping ratio incurs severepower penalties; therefore, there is a trade-off betweenHPA power efficiency and nonlinearities introduced.Figure 6 shows the effect of HPA nonlinearity on the

spectrum of OFDM signal for different CRs. For thissimulation, we used the Welch’s modified periodogrammethod for the estimation of the PSDs with thefollowing parameters: 50% overlap between the segmentsand Hamming window. Welch function take into ac-count the abrupt changes of the phase between the twosuccessive symbols. It is observed from this figure thatthere is an in-band amplitude attenuation and an out-of-band spectral re-growth observed in the spectrum of

0 10 20 30 40 50 60 70 80 90 100-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

Time

Sig

nal a

mpl

itude

Figure 4 4-QAM, 1024 OFDM signal at the input of the HPA before clipping.


the output signal from the HPA, compared to its input.This confirms our earlier observations from the analysis.It is also observed from the figure that as we back-off theamplifier from the saturation region by increasing the CR,the out-of-band distortion decreases; however, the in-bandamplitude attenuation increases resulting in more in-bandsignal power attenuation. This results in significant powerpenalty, making the use of large CR as a means of avoidingHPA nonlinearity unaffordable in practice in the cellularsystems due to the need to transmit certain power levelsin order to meet signal quality requirements at cell edges.

0 10 20 30 40-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

T

Sig

nal a

mpl

itude

Figure 5 4-QAM, 1024 OFDM signal at the output of the HPA after cli

In [27], it is shown that at CR = 1.4 dB, the distortionintroduced by the HPA when transmitting OFDM signalsthrough it compares favorably with the distortionsencountered in a single-carrier QPSK signal using raised-cosine pulse shaping and a roll-off factor of 0.5. This levelof distortion is typically tolerable in many cellular systems,and hence CR = 1.4 is practically reasonable. We have thusused CR = 1.4 as a major reference in our simulationresults presented next.Figure 7 presents the spectrum of the OFDM signal

obtained by simulation versus the spectrum obtained

50 60 70 80 90 100ime

CR = 0dB

CR = 6dB

pping with CR of 0 and 6 dB.

-10 -8 -6 -4 -2 0 2 4 6 8 10-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

frequency, MHz

pow

er s

pect

ral d

ensi

ty [d

B]

HPA Input Spectrum

HPA Output Spectrum (CR = 6dB)HPA Output Spectrum (CR = 4dB)

HPA Output Spectrum (CR = 0dB)

-30.9 dB minimum, CR = 0dB

-36.1 dB minimum, CR = 4dB

-42.6 minimum, CR = 6dB

Figure 6 Effect of HPA nonlinearity on the spectrum of OFDM signals for different CRs at the BS.


from Equations (14) and (25). From Figure 7, it can beobserved that both the analytical and simulation resultsagree very well as expected, and that both results indi-cate that significant spectral re-growth occur when theOFDM symbol is nonlinearly amplified at the BS andthe RS. It is also observed that for CR = 1.4 dB, the gapbetween the main transmitted signal power and the out-of-band spectral re-growth at 1.4 MHz offset is roughly

-10 -8 -6 -4 -2-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

freque

pow

er s

pect

ral d

ensi

ty [d

B]

Transmit Sp

HPA Input Spectrum

HPA Output Spectrum

HPA Output SpectrumHPA Output Spectrum

HPA Output Spectrum

-25.1 dB m

-33.0 dB m

Figure 7 Comparison between analytical and simulation results for thand at the RS, SNR = 10 dB at RS.

(25–10) = 15 dB at the RS and (33–10) = 23 dB at theBS. This implies that while the WiMAX spectral mask isfairly satisfied at the output of HPA with CR = 1.4 dB atthe BS, it is significantly violated at the RS, even for the2-hop transmission shown in this figure.Figure 8 presents the spectrum of the nonlinearly

amplified OFDM signal at RSs, for different number ofrelay hops, using DemF relaying option and CR = 1.4

0 2 4 6 8 10ncy, MHz

ectrum at RS

at TS (Simulation, CR=1.4dB)

at TS (Analysis, CR = 1.4dB) at RS (Simulation, CR=1.4dB)

at RS (Analysis, CR = 1.4dB)

inimum, at RS

inimum, at TS

e spectrum of OFDM signals at the output of the HPA at the BS

-10 -8 -6 -4 -2 0 2 4 6 8 10-45

-40

-35

-30

-25

-20

-15

-10

-5

fequency, MHz

pow

er s

pect

ral d

ensi

ty [d

B]

HPA Input Spectrum

HPA Output Spectrum at RS(1 hop)HPA Output Spectrum at RS(3 hop)

HPA Output Spectrum at RS(5 hop)

Figure 8 Effect of HPA nonlinearity on the spectrum of OFDM signals for different number of relay hops (DemF, CR = 1.4 dB, SNR = 15 dB).


dB. It is observed from these results that the out-of-banddistortion (i.e., spectrum re-growth in adjacent band)increases as the OFDM signal is relayed across multipleRSs, as observed earlier from the analytical results inEquation (25). Thus, the WiMAX spectral maskdisplayed in Figure 1 would easily be violated when largenumbers of relay hops are involved. Figure 9 comparesthe effect of HPA nonlinearity on the spectrum of the

-10 -8 -6 -4 -2-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

freque

pow

er s

pect

ral d

ensi

ty [d

B]

HPA Inpu

HPA OutpHPA Outp

HPA Outp

-22.7 dB m

-25.3 dB m

-30.3 dB m

Figure 9 Effect of HPA nonlinearity on the spectrum of OFDM signals fo

relayed OFDM signals using AF and DemF relayingconfigurations. The spectrum of the relayed OFDMsignals using DF relaying option is also included in thefigure for reference. As expected, it can be observedfrom this figure that there is a higher spectral re-growthin AF relay option than in the DemF relay option. Alsocomparison between the spectral re-growth in DF relayoption with DemF and AF relay options, for different

0 2 4 6 8 10ncy, MHz

t Spectrum

ut Spectrum, AFut Spectrum, DemF

ut Spectrum, DF

inimum, AF

inimum, DemF

inimum, DF

r AF, DemF, and DF configurations at RSs (CR = 1.4 dB, SNR = 10 dB).


values of SNR at the RS, shows that less severe spectralbroadening is observed for the DF relaying option thanthe other two. It should be noted that these results wereevaluated from the analytical expressions derived above,which hold for any CR. Therefore, the relative perform-ance of DF, DemF, and AF relay options depicted in thisfigure hold also for other values of CR.Next we present results for the case when we window

the OFDM signal using linear-phase finite-impulse-re-sponse (FIR) digital filter before passing it through thenonlinear HPA, similar to the set up used for the worksin [15,28-30]. Since the analysis did not model the effectof filtering, this part of the study is conducted solely bysimulations. We use band-pass equi-ripple FIR filterwith 110 coefficients and pass-band 0.2 ≤ ω ≤ 0.6, whereω = 1 corresponds to the Nyquist frequency, for oursimulation. Figure 10 shows the effect of HPAnonlinearity on the spectrum of filtered OFDM signalfor different values of CR. It is observed that because ofthe attenuation of the filter in this case, the spectral re-growth decreases in filtered MIMO–OFDM systemcompared to the unfiltered case in Figure 6. However,the cumulative remnant re-growth is still significantconsidering the fact that the results in Figure 10 showwhat happens per MIMO antenna of the transmittingstation, and there are total of N antennas per node. Thespectrum of the nonlinear OFDM signal for differentnumber of relaying hops for DemF relaying is presentedin Figure 11. It is observed from these results that theout-of-band distortion (i.e., spectrum re-growth in adja-cent band) increases as the OFDM signal is relayedacross multiple RSs. For example, for N × N MIMO–

-10 -8 -6 -4 -2-80

-70

-60

-50

-40

-30

-20

-10

0

freque

pow

er s

pect

ral d

ensi

ty [d

B]

Filtererd, HPA I

Filtered, HPA O

Filtered, HPA O

Filtered, HPA O

Figure 10 Effect of HPA nonlinearity on the spectrum of filtered OFD

OFDM system with 2-hop relaying, the observed spec-tral re-growth from Figure 11 will be

PMIMOw2w2

¼ 10 log10 Nð Þ þ 32� 10ð Þ dB� � ð33Þ

which can result in significant spectral mask violation asthe number of relay hops increases. The result in thisfigure suggests that the WiMAX spectral mask will beviolated once the transmitted signal is relayed beyondtwo hops since according to Equation (33), PMIMO

w2w2¼

10log10 2ð Þ þ 32� 10ð Þ dB� �> 25 dB, which means that

a 25-dB gap between the main carrier power and the un-wanted spectral re-growth cannot be achieved once thenumber of relay hops involved are more than two.Hence, we can draw a conclusion that even though thegeneral mobile multihop relaying (MMR) system wheredata could traverse any number of relay hops fromsource to destination has widely been studied theoretic-ally [20,31,32], only the 2-hop version such as thoseproposed by the IEEE 802.16j group [18] may be advisedfor any practical deployments in MIMO–OFDMtransmissions because of the potentials for spectral maskviolations when going beyond 2-hop transmissions asshown in these results. In Figure 12, the effect of HPAnonlinearity on DF-relayed OFDM signal is presented. Itcan be observed from this figure that less severe spectralbroadening is observed for the DF relaying option. How-ever, due to the latency problems associated with the DFrelaying option which degrades quality of broadbandsignals, AF or DemF are the preferred candidates forbroadband transmissions over relaying channels andtherefore the spectral broadening issues observed here

0 2 4 6 8 10ncy, MHz

nput Spectrum

utput Spectrum(CR=1.4dB)

utput Spectrum (CR=3dB)

utput Spectrum (CR=5dB)

M signals at the BS for different CRs.

-10 -8 -6 -4 -2 0 2 4 6 8 10-80

-70

-60

-50

-40

-30

-20

-10

0

frequency, MHz

pow

er s

pect

ral d

ensi

ty [d

B]

Filtererd, HPA Input Spectrum

Filtered, HPA Output Spectrum at TS

Filtered, HPA Output Spectrum at RS, DemF(1 hop)

Filtered, HPA Output Spectrum at RS, DemF(2 hops)

Figure 11 Effects of HPA nonlinearity on the spectrum of filtered OFDM signals in multihop relay channel (CR = 1.4 dB, SNR = 10 dB ateach RS), DemF relaying at RSs.


must be given considerable attentions in the design ofbroadband multihop-relaying systems.

ConclusionsThis article presents new insights on the out-of-bandspectral re-growth due to HPA nonlinearities whenMIMO–OFDM signals are transmitted over multiplerelay channels. Expressions for the PSD of a MIMO–

-10 -8 -6 -4 -2-80

-70

-60

-50

-40

-30

-20

-10

0

freque

pow

er s

pect

ral d

ensi

ty [d

B]

Filtererd, HPA Input

Filtered, HPA Outpu

Filtered, HPA Outpu

Figure 12 Effects of HPA nonlinearity on the spectrum of filtered OFDeach RS), DF relaying at RSs.

OFDM signal are presented when it is transmitted froma BS to the MS via multiple RSs, all equipped withnonlinear HPAs. It is shown that significant spectral re-growth occurs for the AF and DemF relaying options asthe OFDM signal traverse one or more relay hops. ForMIMO–OFDM systems with large number of relay hopsand MIMO antennas, the cumulative effects of this re-growth can result in significant spectral broadening

0 2 4 6 8 10ncy, MHz

Spectrum

t Spectrum at TS

t Spectrum at RS, DF, (1 hop)

M signals in multihop relay channel (CR = 1.4 dB, SNR = 10 dB at


exceeding specified limits on the spectral masks. Hence,it is concluded from our results that even though thegeneral MMR system where data could traverse anynumber of relay hops from source to destination havewidely been studied theoretically, only the 2-hop versionproposed by the IEEE 802.16j group [18] may be advisedfor any practical deployments in MIMO–OFDM trans-missions because of the potentials for spectral maskviolations when going beyond 2-hop transmissions.

Competing interestsThe authors declare that they have no competing interests.

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

Author details1Department of Electrical Engineering, King Saud University, Riyadh, SaudiArabia. 2KACST Technology Innovation Center RFTONICS, King SaudUniversity, Riyadh, Saudi Arabia.

Received: 14 June 2012 Accepted: 10 November 2012Published: 15 February 2013

References1. S-H Wang, C-P Li, A low-complexity PAPR reduction scheme for SFBC

MIMO-OFDM systems. IEEE Signal Process. Lett. 16(11), 941–944 (2009)2. Y Wang, W Chen, C Tellembura, A PARP reduction method based on artificial

bee colony algorithm for OFDM signals. IEEE Trans. Wirel. Commun. 9(10) (2010)3. E Costa, S Pupolin, M-QAM-OFDM system performance in the presence of a

nonlinear amplifier and phase noise. IEEE Trans. Commun. 50(3), 462–472 (2002)4. AI Sulyman, M Ibnkahla, Performance analysis of nonlinearly amplified M-

QAM signals in MIMO channels. Eur. Trans. Telecommun. 19(1), 15–22 (2008)5. T Riihonen, S Werner, F Gregorio, R Wichman, J Hamalainen, BEP analysis of

OFDM relay links with nonlinear power amplifiers, in Proceedings of the 2010IEEE Wireless Communications and Networking Conference (WCNC), Sydney,Australia, 2010, p. 6

6. J Qi, S Aissa, On the power amplifier nonlinearity in MIMO transmitbeamforming systems. IEEE Trans. Commun. 60(3), 876–887 (2012)

7. PM Castro, JP Gonzalez-Coma, JA Garcia-Naya, L Castedo, Performance ofMIMO systems in measured indoor channels with transmitter noise. EURASIP J.Wirel. Commun. Netw. no. 1, 19 (2012). doi:10.1186/1687-1499-2012-109

8. J Qi, S Aissa, Analysis and compensation of power amplifier nonlinearity in MIMOtransmit diversity systems. IEEE Trans. Veh. Technol. 59(6), 2921–2931 (2010)

9. R Heung-Gyoon, System design and analysis of MIMO SFBC CI-OFDMsystem against the nonlinear distortion and narrowband interference. IEEETrans. Consum. Electron. 54(2), 368–375 (2008)

10. I Ahmad, AI Sulyman, A Alsanie, A Alasmari, On the effect of amplifier non-linearity on the capacity of MIMO systems, in Proceedings of the IEEE GCCConference (GCC), Dubai, UAE, 2011, pp. 108–111

11. P Banelli, S Cacopardi, Theoretical analysis and performance of OFDM signals innonlinear AWGN channels. IEEE Trans. Commun. 48(3), 430–441 (2000)

12. KG Grad, LE Larson, MB Steer, The impact of RF front-end characteristics onthe spectral regrowth of communications signals. IEEE Trans. Microw.Theory 53(6), 2179–2186 (2005)

13. X Li, H Xiao, C Liu, F Li, Nonlinearity analysis of RF power amplified TD-SCDMA signals, in Proceedings of the 10th IEEE International Conference onSignal Processing (ICSP), Beijing, China, 2010, pp. 1636–1641

14. E Cottais, Y Wang, S Toutain, Spectral regrowth analysis at the output of amemoryless power amplifier with multicarrier signals. IEEE Trans. Commun.56(7), 1111–1118 (2008)

15. TK Helaly, RM Dansereau, M El-Tanany, An efficient measure for nonlineardistortion severity due to HPA in downlink DS-CDMA signals. EURASIP J.Wirel. Commun. Netw. no. 1, 9 (2010). doi:10.1155/2010/945427. article ID945427

16. IEEE Standard for Local and Metropolitan Area Networks- Parts 16: Air Interfacefor Fixed and Mobile Broadband Wireless Access System, 2009

17. F Gregorio, J Cousseau, S Werner, T Riihonen, R Wichman, Power amplifierlinearization technique with IQ imbalance and crosstalk compensation forbroadband MIMO-OFDM transmitters. EURASIP J. Adv. Signal Process. no. 1,15 (2011). doi:10.1186/1687-6180-2011-19

18. SW Peters, RW Heath, The future of WiMAX: multihop relaying with IEEE802.16j. IEEE Commun. Mag. 47, 104–111 (2009)

19. A Ghosh, R Ratasuk, B Mondal, N Mangalvedhe, T Thomas, LTE-advanced:next-generation wireless broadband technology [Invited paper]. IEEE Trans.Wirel. Commun. 17(3), 10–22 (2010)

20. AI Sulyman, G Takahara, H Hassanein, M Kousa, Multi-hop capacity of MIMO-multiplexing relaying systems. IEEE Trans. Wirel. Commun. 8, 3095–3103 (2009)

21. FH Gregorio, Analysis and compensation of nonlinear power amplifier effectsin multi-antenna OFDM systems (Ph.D. dissertation, Helsinki University ofTechnology, Espoo, Finland, 2007)

22. JJ Bussgang, Crosscorrelation functions of amplitude-distorted Gaussian signals, inResearch Lab. Electron, M.I.T, Technical Report, Cambridge, MA, USA, 1952, p. 216

23. NY Ermolova, SG Haggman, An extension of Bussgang’s theory to complex-valued signals, in Proceedings of the 6th Nordic Signal Processing Symposium(NORSIG’04), Espoo, Finland, 2004

24. F Gregorio, S Werner, TI Laakso, J Cousseau, Receiver cancellation techniquefor nonlinear power amplifier distortion in SDMA–OFDM systems. IEEETrans. Veh. Technol. 56(5), 2499–2516 (2007)

25. JG Proakis, Digital Communications, 4th edn. (McGraw-Hill, New York, 2001)26. DW Chi, P Das, Effects of nonlinear amplifiers and narrowband interference

in MIMO-OFDM with application to 802.11n WLAN, in Proceedings of the2nd International Conference on Signal Processing and CommunicationSystems (ICSPCS), Gold Coast, Australia, 2008, pp. 1–7

27. X Li, LJ Cimini Jr, Effects of clipping and filtering on the performance ofOFDM. IEEE Commun. Lett. 2(5), 131–133 (1998)

28. GR Tsouri, D Wulich, Capacity analysis and optimization of OFDM withdistortionless PAPR reduction. Eur. Trans. Telecommun. 19, 781–790 (2008)

29. S Woo, H Yu, J Lee, C Lee, J Laskar, Effects of RF impairments in transmitterfor the future beyond-3 G communications systems, in Proceedings of the2006 IEEE International Symposium on Circuits and Systems (ISCAS), Island ofKos, Greece, 2006, pp. 5668–5671

30. PB Kenington, RF and Baseband Techniques for Software Defined Radio, 1stedn. (Norwood, MA, Artech House, 2005), p. 8

31. Y Rong, Y Xiang, Multiuser multi-hop MIMO relay systems with correlatedfading channels. IEEE Trans. Wirel. Commun. 10(9), 2835–2840 (2011)

32. Q Li, KH Li, KC Teh, Achieving optimal diversity-multiplexing tradeoff for full-duplexMIMO multihop relay networks. IEEE Trans. Inf. Theory 57(1), 303–316 (2011)

doi:10.1186/1687-1499-2013-32Cite this article as: Ahmad et al.: Spectral broadening effects of high-power amplifiers in MIMO–OFDM relaying channels. EURASIP Journal onWireless Communications and Networking 2013 2013:32.

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