BROADBAND communications based on multiple carriers

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Clipping Distortion Performance of Non-square M -QAM-OFDM Systems on Nonlinear

Time-variant Channels

Luciano Leonel Mendes, Renato Baldini Filho

Abstract—OFDM systems usually make use of a set of square M -QAM constellations to obtain a good trade-off between throughput and symbol-error robustness. However, the switch to the next constellation increases the number of bits per modulation symbol by two. The introduction of non-square M -QAM constel- lations in such systems brings extra advantages such as smoother transition among bit rates and a reduction of the peak-to- average ratio of the OFDM signal. Therefore, this paper unfolds analytical expressions to evaluate the symbol-error performance of non-square M -QAM-OFDM on nonlinear time-variant AWGN channels taking into account clipping distortion. Cross and overlaid M -QAM are considered. Analytical performances are evaluated and confronted to computational simulations that show good agreement.

Index Terms—Non-square M -QAM, Nonlinear channels, OFDM, clipping, time-variant channels.

I. INTRODUCTION

B

ROADBAND communications based on multiple carriers have attracted a great amount of attention in recent years. High data rate transmission standards, such as DVB (Digital Video Broadcasting) [1], ISDB (Integrated Service Digital Broadcasting) [2], Wi-Fi (Wireless Fidelity) [3] and WiMAX (Worldwide Interoperability for Microwave Access) [4], utilize OFDM (Orthogonal Frequency Division Multi- plexing). OFDM offers high data rate transmission with high spectral efficiency, immunity to multipath fading, and simple implementation using FFT (fast Fourier transform). However, OFDM systems present high PAPR (peak-to-average power ratio) that can result in poor power efficiency, degradation in BER (bit-error-rate) performance, and spectral regrowth.

High PAPR means that an OFDM signal presents much higher instantaneous amplitude than its average amplitude. This is a significant restriction in power-limited transmission systems.

High efficient power amplifiers overcome this restriction by operating nearby the saturation region of their power-transfer functions. However, the saturation of the amplifier output introduces nonlinear interferences in the transmitted OFDM signal, resulting on ICI (Intra-Carrier Interference) that de- creases the system performance.

However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to [email protected].

L. L. Mendes is with Instituto Nacional de Telecomunicac¸˜oes, Santa Rita do Sapucai, MG, 37540-000 Brazil, e-mail: [email protected].

R. Baldini Filho is with Departamento de Comunicações, Faculdade de Engenharia Elétrica e de Computação, Universidade Estadual de Campinas, Campinas, SP, 13083-852, Brazil, e-mail: [email protected].

There are several technical papers [5] [6] [7] with proposals to reduce the nonlinear degradation introduced by the power amplifier in OFDM signals. For instance, the back-off approach [5] reduces the average power of the incoming signal to certify that high peaks of the OFDM signal will lay down in the linear region of the amplifier power-transfer function. The companding approach uses an amplitude compressor at the transmitter and an expander at the receiver to eliminate high amplitude peaks [6]. Error control codes [7] are used to remove symbol combinations at the input of the IFFT (Inverse Fast Fourier Transform) device that produce high amplitude peaks.

The selective mapping technique [8] multiplies the data stream by a set of distinct sequences, the product that leads to the smaller PAPR are employed in the transmission. The partial transmission sequence scheme [9] splits the data into sub- blocks which are optimally combined to minimize the PAPR.

Each block is multiplied by a phase rotation factor that leads to minimal PAPR for a specific sub-block combination. The search for the best sub-block combination and phase rotation factors become a hard task for large number of sub-blocks.

There are other schemes to reduce the PAPR, such as the SOCP (Second Order Conic Program) optimization algorithm [10], tone injection scheme [11], etc.

Moreover, appropriate modeling of nonlinear channel is of great practical and academic interest. For instance, Mestdagh et al [12] have modeled the nonlinear interferences as AWGN (Additive White Gaussian Noise), with overall power equal to the power of the clipped part of the signal. However, it is an optimistic approach to consider the nonlinear effects of the clipping as linear interferences. Actually, the signal degradation is much higher than that estimation. On the other hand, Bahai et al. [13] [14] have considered the clipping effect as an impulsive parabolic pulse, achieving a better estimation for the symbol error probability. This analysis has also been extended to nonlinear AWGN and time-variant channels for square M -QAM-OFDM systems.

Technical standards related to OFDM systems make use of high-order square M -QAM modulation to obtain a good trade- off between throughput and bit-error robustness. However, there are no technical reasons to avoid the use of non-square M -QAM constellations in such systems. Non-square M -QAM constellation may bring some extra advantages to OFDM systems, such as a further reduction of the overall PAPR [15] [16] [17]. Non-square M -QAM carries an odd number of bits per modulation symbol, therefore, adaptive multi-rate OFDM systems can use both square and non-square M -QAM

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allowing smoother transitions of bit rates.

In addition, Demjanenko et al. [18] propose the introduction of non-square M -QAM constellations in PAN IEEE 802.15 standard. DVB-C (Digital Video Broadcasting - Cable) [19]

systems, ADSL (Asymmetric Digital Subscriber Line) and VDSL (Very High Speed Digital Subscriber Line) systems [20]

also use non-square M -QAM constellations. The combination of Trellis coded modulation (TCM) with non-square QAM modulation is also a promising technique to support high- spectral-efficiency and high performance COFDM (Coded OFDM) transmission [21].

Therefore, the importance to extend the theoretical approach presented in [13] [14] to non-square M -QAM constellations is evident and provides a complementary tool to evaluate the performance of OFDM systems. Hence, the aim of this paper is to provide analytical expressions to evaluate the perfor- mance of non-square M -QAM-OFDM systems on nonlinear time-variant AWGN channels with clipping distortion. More specifically two schemes are analyzed: cross and overlaid M - QAM-OFDM. The analytical results are compared to those obtained by computational simulations.

This paper is organized as follows. Section II unveils analytical expressions to estimate the symbol error probability for non-square M -QAM OFDM systems on frequency-selective fading LTI (Linear Time-Invariant) channel. In Section III, the analysis performed in [14] is extended to non-square M -QAM constellations on nonlinear channels. Section IV presents a generalized performance expression for M -QAM OFDM taking into account the effects of the AWGN and clipping. Section V devises a generalized expression to es- timate the symbol error probability of M -QAM-OFDM in presence of clipping, AWGN and Rayleigh fading. Finally, some conclusions are drawn in Section VI.

II. PERFORMANCE OFM -QAM-OFDM SYSTEM ON

LINEARTIME-INVARIANTCHANNEL

An OFDM system with large number of subcarriers and a single carrier system both with same overall bandwidth have in practice equivalent performance on AWGN channel [13].

Performance expressions for square M -QAM constellations can be easily found in basic telecommunications literature [22]. However, performance expressions for non-square M - QAM are approximations of those for square constellations and, therefore, they do not take appropriately into account the geometry of the non-square constellations. Then, precise expressions to estimate the performance of cross and overlaid M -QAM constellations are derived as follows.

A. Cross M -QAM constellations

A cross M -QAM constellation has M = 2^k symbols, where k is an odd integer. This constellation can be seen as a composite of a square sub-constellation with M/2 symbols and 4 branches with M/8 symbols each as depicted in Fig.

1. Notice that adjacent symbols are equally spaced by 2ν, i.

e., by the minimum distance between neighbor symbols.

The square sub-constellation has p

M/2 projections on each axis. And each branch has ¹₄p

M/2 projections. Then,

Fig. 1. Cross 128-QAM constellation.

the number of projections for the cross M -QAM on the I (or Q) axis is L = ³₄√

2M . The average energy of the cross M -QAM constellation, as a function of ν, can be evaluated by

E =¯

L2X−1 i=0

L2−1

X

j=0

[ν(2i + 1)]²+ [ν(2j + 1)]²

¡_L

2

¢₂

−¡_L

2 −^L₃¢₂ − +

L2−1

X

i=^L₃

L2X−1 j=^L₃

[ν(2i + 1)]²+ [ν(2j + 1)]²

¡_L

2

¢2

−¡_L

2 −^L₃¢2

=ν² 6

µ31 9 L²− 4

¶ .

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Therefore, ν can be written as ν =

s 54 ¯E

31L²− 36. (2)

From Fig. 1, the average number of nearest neighbors is given by

¯

µ = 8L − 9

2L . (3)

For AWGN channel, with bilateral power spectrum density function equal to σ_w² = N0/2 and N0= constant, the symbol error rate can be approximated to [22]

pe≈ ¯µQ µ ν

σw

¶

, (4)

where Q(.) is the gaussian error integral. Applying (2) and (3) into (4) leads to the symbol error probability

pe≈8L − 9 2L Q



 s

108 31L²− 36

E¯ N0



 . (5)

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B. Overlaid M -QAM constellations

An overlaid M -QAM constellation, with M = 2^k symbols, k an odd integer, can be seen as two superposed offset square (M/2)-QAM constellations, as shown in Fig. 2.

-7 -5 -3 -1 1 3 5 7

Re{c_n} Imag{c_n}

2v 2v

Fig. 2. Overlaid 32-QAM constellation.

The average energy of this constellation is evaluated by

E =¯

√2M 2 −1

X

k=−^√^2M₂

√2M 2 −1

X

i=−^√^2M₂

h

(2i + 1)ν^√₂² i₂

+ h

(2k + 1)ν^√₂² i₂ 2M

= 1

3ν²(2M − 1).

(6) Therefore, ν is given by

ν = s

3 ¯E

2M − 1. (7)

The average number of nearest neighbors for overlaid M - QAM constellations can be evaluated by

¯

µ = 4M − 4√ 2M + 2

M . (8)

Applying (7) and (8) into (4) leads to the symbol error probability

pe≈ 4M − 4√ 2M + 2

M Q



 s

6 2M − 1

E¯ N0



 . (9)

Fig. 3 presents a performance comparison of (5) and (9) with the results obtained by simulation for cross 32-QAM and overlaid 8-QAM constellations, respectively. The OFDM uses 256 subcarriers for the computer simulations. Notice that theoretical and simulation curves match perfectly for high values of SNR.

Fig. 3. Performance of overlaid 8-QAM and cross 32-QAM on AWGN and frequency-selective channels.

C. Performance of M-QAM OFDM Systems on Linear Time- Invariant Frequency-Selective Channels

For frequency-selective fading channel, the M -QAM OFDM system performance is an average of the performance of each subcarrier, weighted by the channel frequency response. From (5) and (9), the symbol error rate approximation of M -QAM OFDM systems is

pe≈ ¯µQ



 s

ξ E¯ N0



 , (10)

where ξ is a parameter that depends on the geometry of the modulation. For instance, for cross constellations ξ = _31L¹⁰⁸2−36

and for overlaid constellations ξ = ³₄√ 2M .

The channel frequency response changes the SNR for each subcarrier, therefore the overall symbol error rate for an M - QAM OFDM system is evaluated by

pes ≈ µ¯ N

N −1X

i=0

Q



 s

|Hi|²ξ E¯ N0



 , (11)

where Hi is the channel frequency response for the i^th sub- carrier and N is the number of subcarriers. The Q-f unction can be tightly approximated for high values of its argument by

Q(x) ≈ x

√2π (1 + x²)e⁻^x2² . (12)

If γi= q

|Hi|²ξ_N^E^¯

0 and applying (12) in (11) leads to pes ≈ µ¯

√2π N

N −1X

i=0

γi

1 + γ_i²e⁻^γ2²ⁱ. (13) For simulation, the multipath channel impulse response is set to

hm= 0.873δm+ 0.436δm−4− 0.218δm−13. (14)

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Fig. 3 shows the performance evaluated by (13) compared to that obtained by simulation for overlaid 8-QAM and cross 32-QAM OFDM. The number of subcarriers is 256 and the time guard interval is equal to T /16, where T is the duration of an OFDM symbol. Notice that the comparisons show that (13) is tight and, therefore, it is a good tool to estimate the symbol error probability of an M -QAM OFDM system on frequency selective channels.

III. M -QAM-OFDMONNONLINEARCHANNELS

An OFDM symbol is represented by a sum of N complex random variables, where N is large. Therefore, its amplitude can be modeled as a Gaussian variable with standard deviation σs [23]. The Gaussian nature of the OFDM signal produces high amplitude peaks that drive the high-power amplifier to its saturation, clipping the signal. In this case, the amplitude range of the output signal is limited to the clipping thresholds ± l Volts. Thus, the probability of clipping in one OFDM sample can be defined as

p_c= P [|s(t)| > l] = 2Q (l) . (15) The number ϕ of times that the amplitude of the Gaussian signal exceeds the thresholds ± l can be modeled as Poisson- distributed random variable [23] with PDF (Probability Den- sity Function) given by

p(ϕ) =λ^ϕ_l e^−λ^l

ϕ! ϕ ≥ 0, (16)

where λl is the crossing rate given by [23]

λl= 2N

√3 T exp µ

−l² 2

¶

, (17)

for an unitary-power OFDM signal.

The clipping duration τc is a Rayleigh-distributed variable [23] with PDF given by

p(τc) = π 2

τc

τ_m² e⁻^π⁴(_τm^τc)² τc> 0. (18) where the average clipping duration τmis [23]

τm=2Q(l) λl

. (19)

The Q-f unction can be expanded in a series format as Q(l) = 1

√2πe⁻^l2²

·1 l − 1

l³ + 3 l⁵ − . . .

¸

. (20) In practical systems, such as DTV transmitters and base stations, it is common to use pre-distortion algorithms for linearization of power amplifiers. High back-off for the amplifier is also used for low cost CPEs (Customer Provided Equipments). These algorithms generally result in relatively high clipping level. Thus, it is possible to consider, without loss of generality, that the clipping level in the majority of the cases is higher than 3 times the standard deviation σs of the OFDM signal. Therefore, (20) can be truncated at its first term.

Notice that the clipping probability of an OFDM-signal sample is low for l ≥ 3σs, but the clipping probability of the OFDM signal may not be low. In fact, if the clipping probability of

one specific sample of the OFDM symbol is pc, probability of a clipping in an OFDM signal with N samples is

pcOF DM = N pc(1 − pc)^{N −1}. (21) For instance, if the threshold is set to l = 3σs, the OFDM symbol clipping probability is 14.57% for a Wi-FI system with N = 64.

Applying (17) and (20) approximated by its first term into (19) leads to

τm≈ r 3

2π T

N · l . (22)

The clipped portion of the signal can be modeled as a parabolic-shape pulse expressed by [23]

pτ(t) =2π²N²l 3T²

µ

−t²+1 4τ_c²

¶ rect

µt τc

¶

. (23) This pulse can be seen as an impulsive noise added to the OFDM signal. To evaluate this interference in the transmitted data, let’s analyze it in the frequency domain. The Fourier transform of the sampled version pτ(m) of the pulse pτ(t) is [13] [14]

Pτ(k) ≈lτ_c³π²N⁵² 9T³ exp

·

−j2π k T

³ t0+τc

2

´¸

, (24) that is the influence of the clipping on the k^th subcarrier.

Assuming that the phase rotation introduced by the clipping can be mitigated by an equalizer in the receiver end, then the only relevant term is the magnitude of Pτ(k), now on stated as η.

If an error is introduced when the magnitude of the inter- ference is higher than a given threshold, ϑ, then the maximum duration of the clipping without introducing error is

τ_max=

µ 9T³ϑ lπ²N⁵²

¶¹

3

. (25)

Therefore, the probability of introducing an error in the OFDM symbol by a clipping can be written in terms of the its duration as

P [η > ϑ] = P

"

τc>

µ 9T³ϑ lπ²N⁵²

¶¹

3

#

. (26)

Since τc is a random variable with PDF given by (18), then it is easy to show that the error probability given that a clipping have occurred is

P [error/clipping] ≈ ¯µQ



 Ã

ν

√ 2 3N l²π

!¹

3



 . (27)

Applying (15) in (27) leads to the symbol error probability

pec ≈ 2¯µQ(l)Q



 Ã

ν√ 3N l²π

2

!¹

3



 . (28)

This probability is a function of the average number of symbols, ¯µ, and the distance parameter ν for a given modulation.

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Table I shows the values of ¯µ and ν for M -QAM constellations depicted in Section II.

TABLE I

VALUES OFνANDµ¯FORM -QAMCONSTELLATIONS.

Constellations ν (normalized) µ¯ L

Cross q

54

(31L²−36)N 8L−9

2L 3

4

√2M

Overlaid q

3

(2M −1)N 4M −4√ 2M +2

M —

Notice that the value of ν on this table has been normalized by 1/√

2N to keep the standard deviation of the OFDM symbol equal to unity [13] [14].

Fig. 4 presents theoretical and simulation curves for the symbol error probability due to clipping for cross and overlaid 32-QAM-OFDM systems. Notice that (28) can estimate the symbol error rate of a non-square M -QAM-OFDM system on a nonlinear channel for clipping level higher than 3σs.

2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2

10^-7 10^-6 10^-5 10^-4 10^-3 10^-2 10^-1 10⁰

Normalized Clipping Threshold ( l)

SER

* *

*

Theoretical curve Cross 32-QAM

***

Simulation curve Cross 32-QAM Theoretical curve Overlaid 32-QAM Simulation curve Overlaid 32-QAM

Fig. 4. Performance of Cross and Overlaid 32-QAM-OFDM on nonlinear channel.

IV. M -QAM OFDMONNONLINEAR

FREQUENCY-SELECTIVEFADINGCHANNEL

The symbol error probability of an OFDM system in a nonlinear frequency-selective channel in presence of AWGN can be evaluated by [13] [14]

pecw ≤ P [e/clip, w]pc+ pes, (29) where pes is the symbol error probability due to AWGN in a frequency-selective fading channel and P [e/clip, w] is the conditional symbol error probability due to clipping in the presence of noise, that is

P [e/clip, w] = pcw≈ ¯µ Z _∞

−∞

Q



 Ã

ν − w

√ 2 3N l²π

!¹

3



 p(w)dw, (30)

where p(w) is the Gaussian distribution with zero mean and variance σ_w². Therefore, the symbol error probability due to the noise and clipping can be estimated by

pcw ≈ µ¯

√2π σw

Z _∞

−∞

exp µ

− n² 2σ²_w

¶ Q





Ã ν − w

√ 2 3N πl²

!¹

3



dw.

(31) The symbol error probability for an M -QAM OFDM system on a frequency-selective fading channel with AWGN is given by (13). Thus, the symbol error probability considering both AWGN and clipping is given by

pe_cw ≈ 2Q(l)¯µ

√2π σw

Z _∞

−∞

exp µ

− w² 2σ²_w

¶ Q





Ã ν − w

√ 2 3N πl²

!¹

3



dw

+ µ¯

√2π N

N −1X

i=0

γi

1 + γ_i²e⁻^γ2²ⁱ.

(32) The symbol error probabilities for cross and overlaid M -QAM OFDM systems can be evaluated using the values of ¯µ and ν presented in Table I.

The first term on the right hand side of (32) is the error floor introduced by the clipping that prevails for high SNR. The second term is the performance of an M -QAM OFDM system for low SNR. Fig. 5 presents the symbol error probability for the cross 128-QAM OFDM evaluated by (32) for a clipping threshold of 3.6σs. The number of subcarriers is still equal 256. It has been assumed that the receiver knows the channel frequency response and that the equalizer is able to correct the phase rotation introduced by the clipping.

15 20 25 30 35 40 45 50 55

10^-4 10^-3 10^-2 10^-1 10⁰

SNR

SER

Theoretical curve Simulation

Fig. 5. Performance of cross 128-QAM-OFDM on nonlinear AWGN channel with clipping level equal to 3.6σs.

Notice from Fig. 5 that (32) produces a good performance estimate of non-square M -QAM OFDM systems on nonlinear frequency-selective fading channel with AWGN. The mismatch observed in the figures is caused by the approximation performed in (20).

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V. M -QAM-OFDM SYSTEMS ONNONLINEAR

TIME-VARIANTCHANNEL

The symbol error probability for M -QAM OFDM systems on nonlinear time-variant channel can be evaluated assuming that the coherence bandwidth is larger than the frequency spac- ing between two adjacent subcarriers and the coherence time is larger than an OFDM symbol duration. The i^th propagation path presents a time delay τi and an amplitude gain rk(t, τ ) modeled as a Rayleigh random process. The fading is flat for each subcarrier. Then the overall symbol error probability can be evaluated by [22]

pe≈ Z _∞

0

P [e/r]p(r)dr, (33) where r is a Rayleigh random variable and p(r) is its probabil- ity density function. The conditional symbol error probability P [e/r] can be obtained from (32) and it is given by

P [e/r] ≈ 2Q(l)¯µ

√2π σw

Z _∞

−∞

e⁻^2σ2^w2^wQ





"

r (ν − w)

√ 2 3N πl²

#¹

3



dw+

+ µ¯

√2π N σ_r²

N −1X

i=0

r γi

1 + r²γ_i²e⁻^r2γ2²ⁱ.

(34) Thus, the symbol error probability pefor an M -QAM OFDM system on a nonlinear time-variant channel is obtained by averaging P [e/r] in relation to the random variable r, that is

pe≈ 2Q(l)¯µ

√2π σwσ_r² Z _∞

0

Z _∞

−∞

r e⁻^2σ2^w2^w⁻^2σ2^r2^r Q(ζ)dwdr+

+ µ¯

√2πN σ²_r Z _∞

0 N −1X

i=0

r²γi

1 + r²γ_i²exp

·

−r²

µ1 + γ_i²σ_r² 2σ_r²

¶¸

dr.

(35) where

ζ =

"

r (ν − w)

√ 2 3N πl²

#¹

3

(36) Notice that the variance σ_r²in (35) depends on the impulse response of the channel. This equation can be particularized to cross and overlaid M -QAM constellations by applying the values of ¯µ and ν presented in Table I.

Fig. 6 compares the results obtained by computational simulation with the theoretical curves obtained from (35) for cross and overlaid 32-QAM OFDM systems with l = 3.6σson a time-variant frequency-selective fading channel. This figure shows that the results obtained by (35) are close to those obtained by the simulation.

The mismatch between the theoretical and the simulated error floors presented in this figure is not significant because the clipping threshold has been made equal to 3.6σs. Larger mismatches may be expected for lower clipping thresholds because of the approximations in (28). Therefore, (35) can be used the estimate the symbol error probability of non- square M -QAM OFDM systems on a nonlinear time-variant

20 30 40 50 60 70 80

10^-5 10^-4 10^-3 10^-2 10^-1

SNR [dB

SER

Theoretical curve Cross 32-QAM

***

Simulation curve Cross 32-QAM Theoretical curve Overlaid 32-QAM Simulation curve Overlaid 32-QAM

*

* * * * * * * *

Fig. 6. Performance of overlaid and cross 32-QAM-OFDM on time-varying frequency-selective fading nonlinear channel with clipping level equal to 3.6σs.

frequency-selective fading channel with AWGN. However, it is important to highlight that the numerical integration algorithm to solve (35) must be chosen carefully to avoid problems of instability and convergence. Two suitable algorithms to evaluate (35) are quad and dblquad from Matlab^r.

VI. CONCLUSION

A theoretical analysis of the effects of clipping distortion on linear and nonlinear frequency-selective fading channels with AWGN for non-square M -QAM OFDM systems has been presented. Performance expressions that provide good estimations for the symbol-error rate have been introduced.

Non-square M -QAM allows adaptive OFDM systems with smoother bit-rate transitions and also may further reduce the effects of the PAPR. Moreover, non-square M -QAM has inherent memory, that is, it can be by itself considered as a form of coded modulation scheme. Therefore, non-square M - QAM concatenated with a simple outer code allows iterative decoding. Finally, cross M -QAM constellations can also re- duce the PAPR in OFDM systems when compared with square M -QAM constellations.

REFERENCES

[1] ETSI EN 300 744 V1.4.1, European Broadcasting Union, Digital Video Broadcasting (DVB); Framing structure, channel coding and modulation for digital terrestrial television, 2001.

[2] ITU-R 205/11, International Telecommunication Union, Channel Cod- ing, Frame Structure and Modulation Scheme for Terrestrial Integrated Services Digital Broadcasting (ISDB-T), 1999.

[3] IEEE Standard 802.11, Institute of Electrical and Electronic Engineers, Wireless LAN media access control (MAC) and physical layer (PHY) Specifications, 1999.

[4] IEEE Standard 802.16, Institute of Electrical and Electronic Engineers, Air interface for fixed and mobile broadband wireless access systems - physical and medium control layers for combined fixed and mobile operation in lincensed bands, 2005.

[5] B. J. Dixon, R. D. Pollard and S. Iexekiel, “A discussion of the effects of amplifier back-off on OFDM”, in Proocedings IEEE High Frequency Postgraduate Student Colloquium, Leeds, UK, September, 1999.

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[6] Y. Guo and J. R. Cavallaro, “Reducing peak-to-average power ratio in OFDM systems by adaptive dynamic range companding”, in Proceed- ings of the World Wireless Congress, San Francisco, CA, May, 2002.

[7] K. G. Paterson, “Generalized Reed-Muller codes and power control in OFDM modulation”, Transaction on Information Theory, vol. 46, pp.

104-119, January, 2000.

[8] S. Yamg and Y. Shin, “A partitioned-SLM with low complexity for OFDM PAPR reduction”, IEICE Transactions on Communications, vol.

E91-B, no. 4, pp. 1198-1202, 2008.

[9] D. Park and H. Song, “PAPR reduction of an OFDM signal by use of PTS with minimum nonlinear distortion”, IEICE Electronics Express, vol. 4, no. 13, pp. 417-422, 2007.

[10] B. Rihawi and Y. Louet, “PAPR reduction scheme with SOCP for MIMO-OFDM systems”, International Journal of Communications, Network and System Sciences, vol. 1, pp. 29-35, February, 2008.

[11] S. Cioffi and J. Lee, “Tone injection with hexagonal constellation for peak-to-average power ratio reduction in OFDM”, IEEE Communica- tions Letters, vol. 10, no. 9, pp. 646-648, September, 2006.

[12] D. J. G. Mestdagh, P. Spruyt and B. Biran, “Analisys of clipping effect in DMT-based ADSL systems”, in Proceeding of the IEEE International Conference on Communication’94, vol. 1, pp. 293-300, New Orleans, LA, May, 1994.

[13] A. R. Bahai and B. R. Saltzberg, Multi-carrier digital communications - theory and applications of OFDM, Kluwer Academic Publisher, 1999.

[14] A. R. S. Bahai, M. Singh, A. J. Goldsmith and B. R. Saltzberg, A New Approach for Evaluating Clipping Distortion in Multicarrier Systems, IEEE Journal on Selected Areas in Communications, vol. 20, pages 1037-1046, June 2002.

[15] J. Seo, Power Efficient Multi-ary QAM Systems, PhD Thesis, Ottawa- Carleton Institute for Electrical Engineering, University of Ottawa, 1988.

[16] J. Seo and K. Feher, “SQAM: a new superposed QAM modem tech- nique”, IEEE Transactions on Communications, Vol. 33 no. 3, pp. 296- 300, 1985.

[17] J. Seo and K. Feher, “Bandwidth compressive 16-state SQAM Modems through saturated amplifiers”, IEEE Transactions on Communications, Vol. 35 no. 3, pp. 339-345, 1987.

[18] V. Demjanenko, P. Marzec and A. Torres, “Reason to use non squared QAM constellations with independent I&Q in PAN systems”, submitted to IEEE P802.15-15-03, 0311-00-003a, August, 2003.

[19] ETSI EN 300 439 V1.2.1, European Broadcasting Union, Digital video broadcasting (DVB); framing structure, channel coding and modulation for cable systems, 1998.

[20] D. Ginsburg, Implementing ADSL, Addison-Wesley Professional, Boston, MA, 1999.

[21] Q. Yang; X. Liu; S. Chandrasekhar and W. Shieh ”Experimental demon- stration of 44-Gb/s coherent optical OFDM with trellis-coded 32-QAM”, 15th OptoeElectronics and Communications Conference (OECC), pp.

474 - 475, 2010.

[22] S. Benedetto and E. Biglieri, Principles of digital transmission: with wireless applications, Plenum Pub Corp, 1999.

[23] S. O. Rice, “Distribution of the duration of FFDES in radio transmission:

Gaussian noise model”, Bell System Technology Journal, vol. 37, pp.

581-635, May, 1958.