Universidade Estadual de LondrinaCentro de Tecnologia e UrbanismoDepartamento de Engenharia Elétrica

Caio Henrique Azolini Tavares

Otimização de PAPR e Nível de IBO emSistemas OFDM

Londrina2016

Ficha Catalográfica

Caio Henrique Azolini TavaresOtimização de PAPR e Nível de IBO em Sistemas OFDM - Londrina, 2016 -42 p., 30 cm.Orientador: Prof. Dr. Taufik Abrão1. Multiplexagem por Divisão de Frequência Ortogonal (OFDM). 2. RelaçãoPotência de Pico por Potência Média (PAPR). 3. Recuo do Nível de Entrada(IBO). 4. Amplificador de Alta Potência (HPA). 5. OtimizaçãoI. Universidade Estadual de Londrina. Curso de Engenharia Elétrica. II.Otimização de PAPR e Nível de IBO em Sistemas OFDM.

Universidade Estadual de Londrina

Centro de Tecnologia e UrbanismoDepartamento de Engenharia Elétrica

Caio Henrique Azolini Tavares

Otimização de PAPR e Nível de IBO em SistemasOFDM

Trabalho de Conclusão de Curso intitulado “Otimização de PAPRe Nível de IBO em Sistemas OFDM”, apresentado à UniversidadeEstadual de Londrina como parte dos requisitos necessários à ob-tenção do título de Bacharel em Engenharia Elétrica.

Orientador: Prof. Dr. Taufik Abrão

Londrina2016

Caio Henrique Azolini Tavares

Otimização de PAPR e Nível de IBO emSistemas OFDM

Trabalho de Conclusão de Curso apresentado ao Curso deEngenharia Elétrica da Universidade Estadual de Londrina,como requisito parcial para a obtenção do título de Bacharelem Engenharia Elétrica.

Comissão Examinadora

Prof. Dr. Taufik AbrãoUniversidade Estadual de Londrina

Orientador

Prof. MSc Décio Luiz Gazzoni FilhoUniversidade Estadual de Londrina

Prof. MSc José Carlos Marinello FilhoUniversidade Estadual de Londrina

Londrina, 21 de fevereiro de 2016

Agradecimentos

Agradeço aos meus pais pela dedicação, sacrifícios e incentivos que me permitiramrealizar este curso.

Ao meu primo Paulo Rogério Scalassara pelos ensinamentos e influências duradouras.Aos amigos que conheci e convivi durante o curso, pela diversão, aprendizado e com-

panheirismo que me cercavam todos os dias.À minha namorada, Rebecca Carolline, pelo apoio, incentivo e café inabaláveis que

me forneceu durante toda esta etapa.Ao José Carlos Marinello Filho, pela grande ajuda durante a realização deste trabalho.Aos professores que se dedicaram e trabalharam duro durante minha formação profis-

sional e que, de alguma forma, me ajudaram a evoluir em conhecimento e sabedoria; emespecial ao professor Francisco Granziera Junior, por me mostrar o verdadeiro valor dequalquer projeto.

Ao meu orientador e mentor Taufik Abrão, pela oportunidade de aprendizado pessoale profissional, além das proveitosas discussões que ajudaram no desenvolvimento destetrabalho.

“Imagination is more important than knowledge.For knowledge is limited to all we know and understand, while imagination embraces the

entire world, and all there ever will be to know and understand.”Albert Einstein

Caio Henrique Azolini Tavares. 2016. 42 p. Trabalho de Conclusão de Curso emEngenharia Elétrica - Universidade Estadual de Londrina, Londrina.

ResumoNeste trabalho foram empregadas técnicas de otimização visando o aumento de desempe-nho em termos da relação sinal-ruído-mais-distorção (SNDR) em sistemas de multiple-xagem por divisão de frequência ortogonal (OFDM). Primeiramente, foram consideradosdois algoritmos de otimização para redução de complexidade computacional do métodode transmissão por sequência parcial (PTS) utilizado na redução da relação potência depico por potência média (PAPR) em sinais OFDM. Por meio de simulações computa-cionais Monte-Carlo, foi demonstrado que a partir do uso da otimização heurística porenxame de partículas (PSO) e busca pseudoaleatória (PRS), é possível reduzir o custocomputacional do algoritmo PTS para até um quarto do total de operações de pontoflutuante (FLOPS), mantendo uma considerável redução de PAPR. Além disso, foi pos-sível demonstrar que o algoritmo de busca pseudoaleatória possui um compromisso deredução de PAPR por operações de ponto flutuante superior ao PSO, devido à irregu-laridade da função custo de minimização. Na segunda parte deste trabalho, foi propostauma expressão analítica-iterativa para o ponto ótimo de recuo do nível de entrada (IBO)em amplificadores de potência de estado sólido (SSPA) para sinais OFDM, buscandomaximizar a relação sinal-ruído-mais-distorção (SNDR) do sinal de saída. Os resultadosencontrados, corroborados por simulações computacionais Monte-Carlo, demonstram queo nível ótimo de IBO depende apenas da potência de ruído branco aditivo gaussiano(AWGN) e da potência de saturação do amplificador.

Palavras-Chave: 1. Multiplexagem por Divisão de Frequência Ortogonal (OFDM). 2.Relação Potência de Pico por Potência Média (PAPR). 3. Recuo do Nível de Entrada(IBO). 4. Amplificador de Alta Potência (HPA). 5. Otimização

PAPR and IBO Optimization in OFDM Systems. 2016. 42 p. Monograph inElectrical Engineering - State University of Londrina, Londrina.

AbstractIn this work, optimization techniques were employed with the goal of increasing the per-formance in terms of signal-to-noise-plus-distortion ratio (SNDR) of orthogonal frequencydivision multiplexing systems (OFDM). At first, two optimization algorithms were em-ployed for computational complexity reduction in the partial transmit sequence methodfor peak-to-average power ratio (PAPR) reduction in OFDM signals. Through Monte-Carlo computational simulations, it was demonstrated that by using particle swarm op-timization (PSO) and pseudo-random search (PRS) heuristics, it is possible to decreasethe computational cost of the PTS algorithm to approximately a quarter of the totalfloating-point operations (FLOPS), while keeping a considerable PAPR reduction. Fur-thermore, it was possible to demonstrate that the pseudo-random search algorithm hasa more atractive trade-off between PAPR reduction and floating-point operations thanPSO, due to the irregularity of the minimization cost function. On the second part ofthis work, an iterative analytical expression was proposed for the optimum input back-off level (IBO) on solid-state power amplifiers (SSPA) for OFDM signals, aiming tomaximize the output signal-to-noise-plus-distortion (SNDR). The encountered results,corroborated by Monte-Carlo computational simulations, demonstrate that the optimumIBO level depends only on the channel additive white gaussian noise (AWGN) powerand the amplifier saturation power.

Keywords: 1. Orthogonal Frequency Division Multiplexing (OFDM). 2. Peak-to-average Power Ratio (PAPR). 3. Input back-off (IBO). 4. High-power Amplifiers (HPA).5. Optimization

Lista de ilustrações

Figura 1 – Transmissor OFDM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Figura 2 – Resposta de amplitude de um amplificador típico. . . . . . . . . . . . . 13Figura 3 – Exemplo de funcionamento do método de transmissão por sequência

parcial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Lista de Siglas e Abreviaturas

ACO Ant Colony OptimizationAWGN Additive White Gaussian NoiseBER Bit Error RateDSP Digital Signal ProcessorDVB-T Digital Video Broadcasting - TerrestrialFFT Fast Fourier TransformGA Genetic AlgorithmHPA High-power AmplifierIBO Input Back-offICI Inter-carrier InterferenceIFFT Inverse Fast Fourier TransformLTE Long-Term EvolutionOBO Output Back-offOFDM Orthogonal Frequency Division MultiplexingPAPR Peak-to-average Power RatioPSK Phase Shift KeyingPSO Particle Swarm OptimizationPTS Partial Transmit SequenceRF Radio FrequencySLM Selective MappingSNDR Signal-to-noise-plus-distortion RatioSNR Signal-to-noise RatioSNR𝑠𝑎𝑡 Saturation Signal-to-noise RatioTI Tone InjectionTR Tone ReservationWIFI Wireless FidelityWIMAX Worldwide Interoperability for Microwave Access

10

Sumário

Lista de ilustrações . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Sumário . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1 INTRODUÇÃO . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 PROPOSTA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.1 Redução de PAPR . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Otimização de Nível de IBO e SNDR . . . . . . . . . . . . . . . 162.3 Desenvolvimento . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 CONCLUSÕES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Anexo A - Abordagem Heurística no Algoritmo PTS para Redução de

PAPR em Sistemas OFDM SISO . . . . . . . . . . . . . . . . . . 19Anexo B - Otimização do Nível de IBO em Sistemas OFDM . . . . . . . 36

REFERÊNCIAS . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

11

1 IntroduçãoOs esquemas de transmissão multiplexados por divisão de frequência ortogonal (Ortho-

gonal Frequency Division Multiplexing - OFDM) estão presentes em muitas das tecno-logias recentes de telecomunicações, como IEEE 802.11 (WiFi) (GOLDSMITH, 2005),comunicações móveis de 4a geração (LTE) (GOUDA; HUSSIEN, 2013), IEEE 802.16(WiMAX) e o sistema de comunicação europeu de vídeo digital terrestre (DVB-T DigitalVideo Broadcasting - Terrestrial) (MAHAFENO; LOUëT; HéLARD, 2008).

O primeiro sistema OFDM foi desenvolvido em 1966 por Robert W. Chang (LA-SORTE; BARNES; REFAI, 2008), entretanto, seu uso só foi popularizado após o adventoda transformada rápida de Fourier (Fast Fourier Transform - FFT) e transformada rá-pida inversa de Fourier (Inverse Fast Fourier Transform - IFFT) via processadores di-gitais de sinal (Digital Signal Processor - DSP). Tais sistemas empregam modulaçãomultiportadora com sobreposição espectral parcial, o que gera robustez em canais comdesvanecimento seletivo em frequência ao usar subportadoras de banda estreita justapos-tas, assim como um ganho em eficiência espectral. A Figura 1 apresenta um transmissorOFDM, considerando uma entrada binária, seguindo pela transformação série-paralelopara posterior IFFT e adição de prefixo cíclico. O sinal já no domínio do tempo passapor um conversor paralelo-série, modula as portadoras em fase e quadratura e, por fim,é amplificado no bloco do amplificador de alta-potência (High-power amplifier - HPA)para transmissão no canal.

Figura 1 – Transmissor OFDM.

S/P

Mapeador

DadosIFFT

P/S

P/S

Real

Imaginário

HPA90°

Inserção de Prefixo

Cíclico

Fonte: Elaborada pelo autor

Apesar das vantagens citadas, uma das principais desvantagens da tecnologia OFDMé sua alta relação entre a potência de pico e potência média (Peak-to-Average Power Ratio- PAPR) devido à possível combinação de fase das subportadoras no domínio do tempo.Tal efeito tem impacto principalmente nos amplificadores de alta-potência presentes noscircuitos transmissores, que devem possuir uma elevada faixa de operação dinâmica afim de manter o sinal de saída livre de distorção, requisito que necessariamente impactanegativamente na eficiência energética do amplificador.

Capítulo 1. Introdução 12

Por meio de simulações é possível afirmar que a PAPR aumenta conforme o número desubportadoras do sinal. Considerando que o efeito da elevada PAPR é devido à possibi-lidade de ocorrer combinações coerentes das subportadoras no domínio do tempo, quantomaior o número de subportadoras, maior a amplitude que tais combinações podem atingir.Para um símbolo OFDM com modulação de amplitude constante e 𝑁 subportadoras, amáxima PAPR teórica deste símbolo é, desta forma, igual a 𝑁 em banda base.

Existem diversas formas de obter a redução da PAPR em sinais OFDM. De acordocom Molisch (2010), as mais promissoras podem ser divididas basicamente em quatrogrupos:

1) Codificação: Consiste em alterar o vetor de símbolos modulados no domínio dafrequência, de forma a encontrar a sequência que possua a menor PAPR no domíniodo tempo. Um exemplo de técnica de codificação é o mapeamento seletivo (SelectiveMapping - SLM), (WEN et al., 2008).

2) Ajuste de fases: possui como característica central a multiplicação do sinal OFDMpor um vetor de rotação, de forma a evitar a combinação construtiva de fases dassubportadoras. Um exemplo é a técnica de transmissão de sequência parcial (Par-tial Transmit Sequence - PTS), (INOUE; TSUTSUI; MIYANAGA, 2013), (PHET-SOMPHOU; YOSHIZAWA; MIYANAGA, 2010)

3) Correção por fator multiplicativo: sua forma mais simples é a técnica de clipping(SUDHA; BALAN; KUMAR, 2014), (RYU et al., 2004), que multiplica o sinalOFDM por um fator de redução de potência quando esta atinge um dado limiar.

4) Correção por fator aditivo: considera a soma do sinal OFDM por subportado-ras redundantes visando reduzir a PAPR. Exemplos são as técnicas de Tone Injec-tion (TI) e Tone Reservation (TR), (WATTANASUWAKULL; BENJAPOLAKUL,2005), (PHOOMCHUSAK; PIRAK, 2011), (JIAO; LIU; WANG, 2008).

Dentre estas técnicas, a PTS se destaca por atingir uma redução significativa dapotência de pico do sinal OFDM sem introduzir distorção, como na técnica de Clipping(SUDHA; BALAN; KUMAR, 2014). Não obstante, o esquema de PTS convencionalrequer um elevado custo computacional para obter bons resultados, o que pode torná-lo impraticável em cenários de aplicação realistas. Uma alternativa para este problemaé usar algoritmos heurísticos, tais como otimização por enxame de partículas (ParticleSwarm Optimization - PSO) (HUNG et al., 2008), otimização por colônia de formigas(Ant Colony Optimization - ACO) (ABRãO, 2013), (BAOCHENG; TIANE; ZENGHUI,2012) e algoritmo genético (Genetic Algorithm - GA) (LIANG et al., 2009), tendo emvista reduzir o tempo de execução do PTS e atingir resultados de redução de PAPRquase-ótimos.

Capítulo 1. Introdução 13

Outra característica importante para a manutenção da eficiência energética dos am-plificadores de rádio frequência (RF) de alta potência, assim como controle da distorçãodo sinal de saída, é o recuo do nível de entrada (input back-off - IBO), que mede o dis-tanciamento entre a potência de saturação do amplificador e a potência média do sinal deentrada (Figura 2). Por meio do ajuste do nível de IBO, é possível reduzir a distorção dosinal de saída do amplificador, evitando que amostras do sinal de entrada com potênciaselevadas caiam na região de saturação. Desta forma, evita-se que ocorra interferênciainter-portadora (inter-carrier interference - ICI) devido à perda de ortogonalidade dosinal OFDM, além de reduzir a radiação fora da banda (out-of-band radiation), efeitosque prejudicariam o desempenho geral do sistema em termos de taxa de erro de bit (biterror rate - BER) no receptor.

Na Figura 2 temos a resposta de amplitude de um amplificador típico com potência desaturação igual a 30 dBm, considerando três níveis diferentes para o fator de suavização desaturação. Um elevado fator de suavização confere ao amplificador uma resposta próximade um limitador; neste cenário, caso seja aplicado um sinal com potência média de 20dBm na entrada, teremos um IBO e um recuo de nível de saída (output back-off - OBO)de 10 dB, segundo o modelo de amplificador de ganho unitário adotado neste trabalho.Já para pequenos fatores de suavização, o IBO é medido 3 dB abaixo da potência desaturação do amplificador.

Figura 2 – Resposta de amplitude de um amplificador típico.

Pin

[dBm]

0 5 10 15 20 25 30 35 40 45 50

Po

ut

[d

Bm

]

0

5

10

15

20

25

30

35

40

45

50

p=10

p=1

p=0.5

IBO

OBO

Fonte: Elaborada pelo autor

Capítulo 1. Introdução 14

Este trabalho de conclusão de curso é composto por duas partes. A primeira parteexplora o uso de heurística para ajustes de fase do símbolo OFDM com a técnica PTS,visando obter a redução de sua complexidade computacional e manter seu desempenho emtermos de redução de PAPR a níveis quase-ótimos. A segunda parte deste trabalho temcomo objetivo obter uma expressão analítica-iterativa que descreva o ponto ótimo de IBOem amplificadores de estado sólido para sinais OFDM de acordo com a relação sinal-ruído(Signal-to-noise ratio - SNR) do canal de transmissão, visando obter a máxima relaçãosinal-ruído-mais-distorção (Signal-to-noise-plus-distortion ratio - SNDR).

15

2 Proposta

2.1 Redução de PAPR

O método de transmissão por sequência parcial representado na Figura 3 possui grandeapelo por seus resultados no que diz respeito à redução de PAPR sem distorção do sinal.O método consiste em particionar o sinal modulado em 𝑀 ramos, aplicar a IFFT emcada ramo separadamente e, então, multiplicá-los por um determinado fator de fase 𝜔𝑖,escolhido dentre 𝑊 possibilidades; o sinal é então recombinado e sua PAPR avaliada. Aofinal do processo, todas as possíveis combinações de fatores de fase terão sido testadas eo sinal de menor PAPR é escolhido para transmissão, juntamente com os dados relativosà sequência de fases utilizada, necessários em sua decodificação.

Contudo, uma de suas principais desvantagens é sua complexidade computacional,pois este precisa realizar uma busca exaustiva em 𝑊 𝑀 possíveis vetores de solução, oque torna o tempo de execução proibitivo para um número grande de ramos, além danecessidade de realizar 𝑀 operações de IFFT paralelas ao construir o sinal OFDM. Deve-se ter em mente também que os dados adicionais relativos à sequência de fase utilizada sãoimperativos para demodulação do sinal, exigindo menores taxas de codificação ou ordemde modulação reduzida para transmissão, evitando, portanto, que sejam corrompidos pelocanal.

Figura 3 – Exemplo de funcionamento do método de transmissão por sequência parcial.

Comprimento da FFT

M ramos

x[5] ... x[8] x[9] ... x[12]

x[5] ... x[8]

x[9] ... x[12]

x[13] ... x[16]

x[13] ... x[16]x[1] ... x[4]

x[1] ... x[4] ω[1]

ω[2]

ω[3]

ω[4]

Sinal OFDM de mínima

PAPR

IFFT

IFFT

IFFT

IFFT

+

Fonte: Elaborada pelo autor

Levando em conta a restrição de tempo de execução, este trabalho propõe um esquemade busca guiada, utilizando como base o algoritmo de otimização heurística por enxame departículas denominado PSO-PTS, de forma a encontrar uma solução quase-ótima paraa minimização da PAPR em um tempo reduzido de execução. Além disso, propõe-se

Capítulo 2. Proposta 16

e avalia-se comparativamente um método similar ao PSO-PTS, contudo, desprovido dequalquer inteligência artificial para auxiliar a busca, apenas sorteando soluções aleatóriasa cada iteração e armazenando a de melhor desempenho. Este método foi denominadoPseudo-Random Search, ou PRS-PTS. Ambos os métodos são comparados com o algo-ritmo PTS convencional disponível na literatura, utilizando sinais OFDM com modulaçãoQPSK, a fim de verificar os ganhos de desempenho conferidos pelas técnicas heurísticasem detrimento do método convencional exaustivo.

No âmbito da redução de PAPR, os objetivos deste trabalho são:

∙ Redução da complexidade computacional do algoritmo PTS.

∙ Comparar o desempenho dos algoritmos PSO-PTS e PRS-PTS.

2.2 Otimização de Nível de IBO e SNDR

Tendo em mente o esquema de transmissão OFDM da Figura 2, após a multiplicaçãopelos fatores de fase no processo de redução de PAPR, o sinal OFDM passa pelo ampli-ficador de alta potência, modelado neste trabalho como um amplificador de potência deestado sólido (Solid State Power Amplifier - SSPA). Do ponto de vista de desempenhodo sistema, a potência de saturação deve ser levada em consideração a fim de evitar adistorção do sinal de saída, o que acarretaria em um aumento nas emissões do sinal forada banda e aumento na taxa de erro de bit.

Nesse aspecto, usando como base as equações de processos não-lineares de entradasgaussianas (ROWE, 1982), assim como os trabalhos de Ochiai e Imai (2000) e Ochiai eImai (2002), foi possível escrever a equação da SNDR considerando o modelo de canal deruído branco aditivo gaussiano (Additive White Gaussian Noise - AWGN) de potência deruído constante e dependente da SNR𝑠𝑎𝑡, a qual é obtida quando a potência do sinal desaída do amplificador é igual à potência de saturação. A partir da equação da SNDR foipossível obter sua derivada, que, quando igualada a zero, fornece o ponto de IBO ótimopara o desempenho do amplificador. Por fim, os resultados analíticos obtidos foramcorroborados por simulações computacionais Monte Carlo.

No âmbito da otimização de nível de IBO e SNDR, os objetivos deste trabalho são:

∙ Caracterizar a SNDR em termos de IBO e SNR.

∙ Obter analiticamente o ponto ótimo de IBO a partir da equação da SNDR.

2.3 Desenvolvimento

O desenvolvimento da primeira parte deste trabalho é descrita no Anexo A. Nestetrabalho de investigação foram obtidas reduções significativas do tempo de execução do

Capítulo 2. Proposta 17

método PTS por meio dos dois algoritmos propostos, mantendo a redução de PAPR emníveis próximos aos obtidos pelo método PTS convencional.

A segunda parte deste trabalho é descrita no Anexo B. Neste trabalho foi possívelobter as equações que descrevem a SNDR em função de IBO, bem como a equação doponto ótimo de IBO, porém de forma semi-fechada (analítica-iterativa).

Os Anexos A e B são apresentados na forma de artigo científico, uma vez que ambosencontram-se submetidos para revistas científicas da área de Telecomunicações:

A - Caio Henrique Azolini Tavares, Taufik Abrão, A Heuristic Approach onPTS Algorithm for PAPR Reduction on SISO OFDM Systems, International Jour-nal of Communications Systems, Wiley (Submetido)

B - Caio Henrique Azolini Tavares, José Carlos Marinello Filho, Cris-tiano Magalhães Panazio, Taufik Abrão, Input Back-Off Optimization inOFDM Systems. IEEE Wireless Communications Letters (Submetido)

18

3 ConclusõesLevando em consideração os objetivos propostos na seção 2.1, os resultados obtidos nas

simulações realizadas no primeiro trabalho (Anexo A) mostram que ambos os algoritmospropostos para redução da complexidade computacional do método PTS, a saber, PSO-PTS e PRS-PTS, atingiram resultados quase-ótimos em relação ao método convencionalexaustivo no âmbito da redução da PAPR, porém o fizeram em frações da quantidade deoperações de ponto flutuante exigidas pelo método PTS convencional. Desta forma,o compromisso entre redução de PAPR e complexidade computacional dos algoritmospropostos se mostra superior.

Por meio de simulações computacionais, foi possível demonstrar que ao aumentar adimensão do problema, os algoritmos heurísticos demonstraram um aumento marginal decomplexidade computacional, ou até mesmo uma redução de complexidade, juntamentecom um melhor resultado de redução de PAPR, em detrimento do aumento proibitivo decomplexidade do algoritmo PTS convencional.

Por outro lado, comparando o compromisso dos algoritmos propostos, é possível dizerque a busca pseudo-aleatória possui vantagem sobre a otimização por enxame de partícu-las, resultado justificável pela forma extremamente irregular (elevado número de mínimoslocais) da função custo que os algoritmos buscam minimizar.

Em relação aos objetivos propostos para otimização do nível de IBO, os resultadosexibidos no Anexo B demonstram que foi possível obter a SNDR analítica para um dadonível de IBO e SNR de saturação. A partir da equação obtida, foi possível derivar arelação que caracteriza o ponto ótimo de IBO, equivalente à máxima SNDR alcançável.O ponto ótimo de IBO encontrado foi corroborado por meio de simulações computacionaisMonte Carlo. Por meio da análise realizada, é possível dizer que em determinados cenáriostorna-se favorável reduzir o ponto de IBO do amplificador, introduzindo distorção no sinal,para contra-intuitivamente obter uma melhor SNDR no receptor, por conta da potênciade ruído do canal.

Outro resultado relevante obtido com o segundo trabalho (Anexo B) diz respeito àPAPR do sinal OFDM na entrada do amplificador, que mostrou não influenciar o pontoótimo de IBO do sistema. Apesar de ter efeito negativo na SNDR do sistema, umaalta PAPR no sinal de entrada não altera o tipo da distribuição de potências do sinal,que, devido ao teorema do limite central, é aproximadamente exponencial e com precisãocrescente de acordo com o número de subportadoras (ARAúJO; DINIS, 2011), a partirda qual o resultado de IBO ótimo foi obtido.

Anexo AAbordagem Heurística no Algoritmo

PTS para Redução de PAPR emSistemas OFDM SISO

A Heuristic Approach on PTS Algorithm for PAPR

Reduction on SISO OFDM Systems

Caio Henrique Azolini Tavares & Taufik Abrão∗

February 21, 2016

Abstract

The main subject of this paper is the optimization of peak-to-average power ratio (PAPR) reduction in

Orthogonal Frequency Division Multiplexing (OFDM) systems. A high PAPR implies in power inefficiency

by requiring a lower operating point in the high-power amplifiers (HPA) present at the transmitter RF

electronics. Currently there are many ways to deal with this problem, but for the sake of this paper we have

analyzed and evaluated the partial transmit sequence (PTS) technique for reducing PAPR levels by means

of heuristic optimization, namely particle swarm optimization (PSO) and a pseudo-random search (PRS),

while comparing the results with the conventional PTS (C-PTS) algorithm regarding the trade-off between

computer complexity and PAPR decrease. Numerical results have demonstrated that by using the PSO-PTS

and PRS-PTS algorithm it is possible to achieve near optimal results in approximately a quarter of the time

used by conventional PTS. Also, PRS-PTS have shown a better performance trade-off when compared to

the PSO-PTS technique, due to the irregular format of the minimization problem cost function.

Keywords – Orthogonal frequency division multiplexing (OFDM); peak-to-average power ratio (PAPR);

partial transmit sequence (PTS); heuristic optimization; particle swarm optimization (PSO)

1 Introduction

Orthogonal frequency division multiplexing (OFDM) transmission schemes are present in many recent

wireless communication technologies and standards, such as IEEE 802.11 (WiFi) [1], 4th generation mobile

communications (LTE) [2], IEEE 802.16 (WiMAX) [3] and DVB-T. OFDM employs multicarrier modula-

tion through ifft/fft algorithms to provide robustness against frequency selective fading channels by using

narrowband subcarriers, as well as high spectral efficiency. On the other hand, one of the main disadvantages

of OFDM systems lies in its intrinsically high peak-to-average power ratio (PAPR) due to the time-domain

combination of the subcarriers. To deal with a high PAPR signal at the transmitter side, the operation point of

the high-power amplifier (HPA) must be lowered in order to reduce the output signal distortion, this is known

as input back-off (IBO). By doing this, the energy efficiency of the HPA is reduced. On the other hand, if the

output signal is distorted, inter-carrier interference (ICI) can happen due to loss of subcarrier orthogonality, as

well as out-of-band radiation, diminishing the overall performance of the system.

Through computational simulations it is possible to affirm that the PAPR increases on the OFDM symbol

according to the number of subcarriers. Besides, since some current communication technologies employ a∗C. Tavares and T. Abrão are with Department of Electrical Engineering, State University of Londrina, Parana, Brazil. E-mail:

caio.tavares11@gmail.com; taufik@uel.br

1

great number of subcarriers aiming to increase spectral efficiency in transmission, the PAPR problem becomes

a serious barrier for achieving an efficient high data-rate link.

There are several ways to obtain PAPR reduction in the context of OFDM transmission, including tone

reservation (TR), tone injection (TI) [4] [5] [6], partial transmit sequence (PTS) technique [7] [8], among

others. Among these techniques, PTS can achieve a significant reduction on the OFDM symbol peak power

without introducing distortion at the signal such as the clipping technique [9]. Nevertheless, conventional PTS

schemes uses a lot of signal processing resources in order to obtain good results, which may render it unusable

in scenarios of practical interest. An alternative consists in the use of heuristic algorithms, such as particle

swarm optimization [10], ant colony optimization [11] [12] and genetic algorithm [13] aiming to minimize PTS

execution time, while achieving quasi-optimal performance results.

This paper is organized as follows. In Section 2 the system model for OFDM including the PAPR effect

is presented. In Section 3 the partial transmit sequence technique for PAPR reduction is revisited. PSO and

PRS-based heuristics approaches for PAPR reduction purpose are explored in Section 3.1. Numerical results are

analyzed in Section 4. Conclusion remarks are offered in Section 5.

2 System Model

As the basis of the problem discussed on this paper, an OFDM symbol composed by N phase shift keying

(PSK) data symbols with modulation order M (M-PSK) can be described as:

x(n) =1√N

N−1∑i=0

siej2πni/N , 0 ≤ n ≤ N − 1 (1)

where si is the PSK data symbol at the i-th subcarrier taken from s = [s1 s2 . . . sN ]T , N is the FFT length

and x(n) is n-th sample of the OFDM symbol vector.

The peak-to-average power ratio (PAPR) of an OFDM symbol is defined as the ratio between the maximum

instantaneous power and the mean square power taken from a symbol period, and it can be written as:

PAPR = 10 · log10

max0≤n≤N−1

|x(n)|2

E {|x(n)|2}

[dB] (2)

where E[·] is the expectation operator.

For N higher than 64, x(n) can be accurately treated as a Gaussian distributed random variable [14], hence,

the PAPR is more suitably analyzed by means of its probability density function (PDF) and complementary

cumulative distribution function (CCDF). The CCDF is defined as [15,16]:

CCDF = Pr(PAPR > PAPR0) = 1 − Pr(PAPR ≤ PAPR0)

= 1 − (1 − e−PAPR0)N (3)

where PAPR0 is the PAPR threshold.

To better understand the behavior of an OFDM system regarding the distribution of PAPR values, a

numerical example of a BPSK OFDM signal with 8 subcarriers was simulated exhaustively throughout all

possible combinations of BPSK symbols along the subcarriers. Figure 1 depicts the variation of PAPR,

indicating three OFDM symbol combinations that achieved the theoretical maximum PAPR for 8 subcarriers,

2

i.e., PAPRmax ≈ 9 [dB]. Notice that there is a variation of around 5.1 dB between the maximum PAPR and

the average PAPR.

Therefore, the goal of the PAPR reduction methods consists in reducing the occurrence of these distinct

high peaks of signal power, as well as reducing the overall mean peak power.

Combinations0 50 100 150 200 250

PA

PR

[dB

]

0

1

2

3

4

5

6

7

8

9

10

BPSK 8-FFT SignalE [PAPR]

Figure 1: Possible PAPR values for a BPSK and 8-FFT OFDM Signal

3 Partial Transmit Sequence

The conventional partial transmit sequence (C-PTS) technique is a distortionless PAPR reduction method

employed at the transmitter side; the modulated signal s on the transmitter is divided into M branches of

length N/M , where N is the number of OFDM subcarriers and M is the number of branches. The resulting

vectors are zero-filled so that each branch has the same length as the original signal, N . Then, ifft operation

is applied to each branch along with a multiplication by a certain weight factor ω(m), known as the phase

factor, taken from the set of W uniformly distributed phases from:

ω(m) ∈{

ej 2πiW | i = 0, 1, . . . , W − 1

}, m = 1 . . . M (4)

The conventional PTS algorithm evaluates every combination of phase factors to find the one that results in

the lowest PAPR and returns both the weighted signal and the phase sequence used. It is worth noting that this

conventional PTS executes an exhaustive search through all possible phase sequences, thus looking into WM

combinations, making it impractical for M larger than 16 branches. After the application of proper weighting

factor, the branches are then summed together, composing the new OFDM symbol. By rotating a slice of the

signal by a certain phase factor and then recombining all the branches, the phase alignment between adjacent

subcarriers of the OFDM symbol that causes a high peak power can be mitigated. Also, the probability of very

high peak power is significantly decreased [10].

3

FFT Length

M branches

x[5] ... x[8] x[9] ... x[12]

x[5] ... x[8]

x[9] ... x[12]

x[13] ... x[16]

x[13] ... x[16]x[1] ... x[4]

x[1] ... x[4] �[1]

�[2]

�[3]

�[4]

Min PAPR

OFDM

Signal

IFFT

IFFT

IFFT

IFFT

Figure 2: PTS process illustration for M = 4 branches and N = 16 subcarriers

Besides the high computational complexity, one of the downsides of this type of PAPR reduction technique

is that it requires the transmission of M · log2(W ) bits of side information regarding the rotation sequence used,

which is imperative to the signal demodulation, therefore requiring reduced coding rate or a reduced modulation

order to decrease the chances of bit error, thus impacting the system spectral efficiency.

Figure 2 depicts the main process of a PTS implementation with M = 4 branches and fft length of N = 16.

In a more compact notation, the phase factors can be expressed as the vector ω = [ω(1) ω(2), . . . , ω(M)]T .

After the sum of all M branches, the OFDM symbol to be transmitted is formed by the elements:

x(n) =

M∑

m=1

ωm ·N−1∑

k=0

sm(k)ej2πnk/N , 0 ≤ n ≤ N − 1 (5)

where sm(k) is the PSK modulated symbol in the m-th branch.

3.1 Particle Swarm Optimization Based PTS (PSO-PTS) for PAPR Reduction

As an alternative to handling the optimization problem using deterministic methods, the particle swarm

optimization (PSO) technique was considered, which is well known due to its simple design and configuration

when compared to other heuristic methods available in the literature. It is based on the behavior of a flock

of birds during flight. The algorithm generates a population P of “particles” with random positions along the

solution space and iteratively update their positions based on their fitness value retrieved from the objective

function inspired in eq.(2) and subject to the constraint given by eq.(4); as a result, the optimization problem

is given by:

minimizex∈CN

PAPR(x)

s.t. ω(m) ∈{

ej 2πiW | i = 0, 1, . . . , W − 1

}, m = 1 . . .M (6)

For every i-th particle there are a few indicators: their current position θi = [θ1 θ2 . . . θM ], their local

best position vector so far or pbestiand their current speed vi, used to update their positions throughout the

iterations. There is also the global best position gbest which indicates the best position achieved by any particle

so far.

4

At the end of the iterations, the current global best fitness is compared to a predefined threshold level,

if gbest is either equal or inferior to the threshold, the iterative process stops and the algorithm returns gbest,

otherwise the speed and position of all particles are updated through Eq.(7) and Eq.(8), respectively, and the

iterative process continues.

vi(t + 1) = I · vi(t) + c1 [pbesti(t) − θi(t)] + c2 [gbest(t) − θi(t)] (7)

where I is the inertia factor, used to refer to the velocity from the previous iteration, c1 is the individual cognitive

factor, which gives the weight of pbestiof the i-th particle; c2 is the social cognitive factor, that weights the

effect of gbest and vi(t) is the i-th particle speed at iteration t. The positions of the particles in the t + 1

iteration are given by:

θi(t + 1) = θi(t) + vi(t + 1) (8)

At every iteration the particles are sorted according to their fitness values from the previous iteration and a

given percentage of them are replaced by new random particles. This assures a level of variability to the process

and minimizes the chances of the particles getting stuck on local minima. Algorithm 1 show the pseudo-code

of the PSO-PTS for the PAPR optimization problem.

Algorithm 1 PSO-PTSprocedure Initialization

Step 1: Initialize a swarm of P particles with random positions and zero velocity.

Step 2: Evaluate the fitness of all particles. Save personal and global best positions (pbesti); (gbest).

Step 3: Update particles velocities and positions according to current inertia, pbest and gbest, Eq.(7), (8).

procedure Iterative Process

Step 1: Sort particles according to their fitness and replace a predefined percentage of the worst by new

random particles with zero velocity and zero pbest.

Step 2: Evaluate the fitness level of entire population; if necessary, update pbesti and gbest.

Step 3: Compare gbest with the threshold level. If met, stop execution and return gbest as solution.

Step 3: Update particles positions based on Eq. (8).

procedure Convergence

By Threshold The process converges when the gbest fitness value is PAPR(gbest) ≤ PAPRth

By Time If the threshold value is not achieved during the iterative process, it will return the gbest particle

when the maximum number of iterations has been executed.

The Initialization procedure sets up the particles population with random positions in the solution space

and zero velocity. Then, their fitness are evaluated, the personal best and global best are saved and compared

to the threshold level; if it is achieved, the algorithm returns the best particle and terminates the procedure,

otherwise their velocity and positions are updated and the iterative process begins.

At the first step, the iterative procedure sorts the particles according to their fitness from the previous

iteration and replaces a given percentage of the worst particles by random particles. After the sorting and

replacement process, their new fitness values are evaluated; individual best and global best are updated if better

values were achieved and the global best is tested against the threshold value.

The algorithm terminates when the gbest particle has achieved the fitness threshold level or when the

maximum number of iterations has been executed.

5

3.2 Pseudo-Random Search PTS (PRS-PTS)

Keeping in mind that the objective function for the PSO-PTS method is evaluated as a discrete M -

dimensional function with input given by the phase factor vector ω, the complexity of the search space through

which the PSO algorithm must find the global minimum can become overwhelmingly complex.

Thus, to investigate the gains of the PSO, a pseudo-random search (PRS-PTS) algorithm was implemented

following the same design as the PSO-PTS, but free from any optimization parameters on phase factor selection

through iterations. The algorithm generates P random particles and evaluates their fitness at every iteration

based on PAPR objetive function, keeping track of the best result achieved so far, labeled gbest. The algorithm

returns the best solution when it meets a predefined threshold level or when the maximum number of iterations

has been executed.

To aid the performance analysis of the PSO-PTS algorithm, the PTS problem search space for a random

OFDM symbol was calculated, with M = 4 branches and W = 16, 8 and 4 possible phase factors, resulting in

164, 84 and 44 possible solutions, respectively. As reference, the numerical results for PAPR cost function are

depicted in Fig. 12, provided at Appendix A.

3.3 Convergence of the Cost Function: PSO × PRS

With the goal of comparing the performance of both heuristic methods in a minimization problem, the

PSO algorithm is performed against the pseudo-random search algorithm, described at Section 3.2, with various

population sizes in the Ackeley benchmark function, Fig. 3.

Iterations0 5 10 15 20 25 30

min

f(x,

y) a

chie

ved

0

2

4

6

8

10

12

14

16Ackeley Benchmark Function

PSO - Population = 20PRS - Population = 10PRS - Population = 20PRS - Population = 30

Figure 3: Ackeley PSO Benchmark Function

The sudden drop seen at around 4th iteration demonstrate the change in the rate of convergence of the

PSO algorithm, while the PRS algorithm maintains its apparently exponential decay along the iterations.

4 Numerical Results

To better understand the behavior of the PAPR regarding the variation of the fft length and the increase

of modulation order, computational simulations were developed considering an OFDM system operating under

6

two transmission scenarios:

a) qpsk modulation and three subcarrier setups: 128, 256 and 2048;

b) 512 subcarriers with three modulation order setups: bpsk, qpsk and 16-psk.

The Monte-Carlo simulations (MCS) were run considering 104 OFDM symbols without cyclic prefix, as it

would not affect the results. In the following the numerical results were depicted in terms of probability density

function (pdf) format aiming to better identify the statistical behaviour of the PAPR.

PAPR [dB]4 5 6 7 8 9 10 11 12 13

pdf

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7PAPR Distribution Regarding the N

FFT

OFDM 4-PSK 128-FFTOFDM 4-PSK 512-FFTOFDM 4-PSK 2048-FFT

Figure 4: Probability density function of the PAPR for 104 symbols showing the variation of the number of

subcarriers.

Subcarriers Nfft = 128 Nfft = 512 Nfft = 2048

E[PAPR] 7.3122 8.2762 9.0976

min[PAPR] 4.5857 6.1497 7.3876

Var[PAPR] 0.8240 0.5584 0.4113

Table 1: Simulation results from the main statistical values of PAPR.

In Fig. 4, one is able to identify the increasing PAPR as the number of subcarriers Nfft increases, while in

Table 1, the first and second PAPR statistical moments are depicted for different number of subcarriers Nfft.

From the results, it can be shown that while the mean PAPR increases with the number of subcarriers used,

its variance decreases, narrowing the overall format of the pdf.

By taking the expected value of the PAPR (E[PAPR]) for seven scenarios, from 32-Nfft to 2048-Nfft,

using qpsk modulation, the simulated PAPR values of Fig. 5 could be well approximated by a quadratic

function with a root-mean-square error (rmse) of 0.0135:

E[PAPR] ≈ −0.2843 · log210(Nfft) + 3.041 · log10(Nfft) + 2.13 [dB] (9)

7

log10

(NFFT

)1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4

Mea

n P

AP

R [d

B]

6

6.5

7

7.5

8

8.5

9

Mean PAPR

Simulated PAPRPolynomial Fit

Figure 5: Mean PAPR of qpsk OFDM symbols with logarithmic scale

With the fitting curve of Eq. (9), it is possible to expeditiously calculate the expected PAPR of an OFDM

symbol of qpsk modulation with a given number of subcarriers, without having to analyse numerically such a

system.

Next we investigate the PAPR probability distribution dependency on the modulation order, it can be seen

from Fig. 6 and Table 2 that the mean PAPR as well as its minimum value and variance do not increase

significantly when modulation order increases.

PAPR [dB]5 6 7 8 9 10 11 12 13

pdf

0

0.1

0.2

0.3

0.4

0.5

0.6PAPR Distribution Regarding the Modulation Order

OFDM 2-PSK 512-FFTOFDM 4-PSK 512-FFTOFDM 16-PSK 512-FFTOFDM 32-PSK 512-FFT

Figure 6: Probability Density Function of the PAPR for 104 symbols showing the variation from the modulation

order M.

Modulation Order BPSK QPSK 16-PSK 32-PSK

E[PAPR] 7.8421 8.2785 8.2745 8.2814

min[PAPR] 5.6317 6.2586 6.1899 6.3399

Var[PAPR] 0.7127 0.5495 0.5602 0.5779

Table 2: Simulation results from the variation of the modulation order.

8

4.1 PSO-PTS, PRS-PTS and C-PTS PAPR Reduction Strategies

In order to evaluate the effectiveness of proposed methods, Monte Carlo simulations were made comparing

the conventional partial transmit sequence algorithm (C-PTS) with the PSO-PTS and PRS-PTS algorithm.

Three different scenarios were considered for the simulations, using OFDM symbols with qpsk modulation

and 128 subcarriers. The proposed PSO-PTS, PRS-PTS and the known C-PTS PAPR reduction algorithms

were compared in terms of performance and the approximate number of floating-point operations; the latter

was able to be approximated with the use of the Lightspeed toolbox for MATLAB, available at [17].

The PAPR threshold level taken into account for the three simulation scenarios was 6 dB, empirically

selected as a proper choice for the PSO and PRS algorithms. This threshold resulted in satisfactory trade-offs

between execution time and performance; notice that setting it too low would force the algorithm to run almost

every iteration for each symbol in order to achieve the desired result, whereas a high threshold would make the

algorithm finish too soon, providing poor results. For practical scenarios, the threshold level can be tweaked

to provide better performance or lower execution times, depending on the application. The OFDM system and

PSO parameters considered in the simulations are summarized in Table 3.

Parameter Scenario 1 (Fig.8) Scenario 2 (Fig. 9) Scenario 3 (Fig.10)

Population Size 30 30 30

c1 2 to 0.1 2 to 0.1 2 to 0.1

c2 0.1 to 2 0.1 to 2 0.1 to 2

M 4 4 8

W 4 8 4

Nfft 128 128 128

Modulation Order 4-PSK 4-PSK 4-PSK

Threshold 6 dB 6 dB 6 dB

OFDM Symbols 104 104 103

Table 3: Simulation parameters used throughout the section

It is worth noting that the individual cognitive factor (c1) was linearly decreased with the number of iterations,

while the social cognitive factor (c2) was linearly increased. This assures a more conservative behavior for the

algorithm as the number of iterations grow.

When comparing the cost function curve for the PAPR problem with PSO and PRS, they both show an

apparent exponential decay, being very close to one another for all scenarios investigated in Table 3. Notice

that there is a slightly improvement in fitness values reached by PSO-PTS algorithm for the second scenario.

The main reason why the cost function for the PRS algorithm decreases at a higher rate, therefore achieving

the desired results in less time, is the behavior of the search space for the PAPR problem in the PTS method,

depicted in Fig.12. By not having a clear global minimum nor an apparent optimization tendency, the PSO

heuristic method might struggle to minimize the function, being surpassed in execution speed by the simpler

PRS algorithm in many cases.

9

Iteration1 2 3 4 5 6 7 8 9 10

Mea

n P

AP

R [d

B]

5.74

5.76

5.78

5.8

5.82

5.84

5.86

5.88Cost function decrease

PSO-PTSPRS-PTS

(a) Scenario 1 - M4-W4

Iteration1 2 3 4 5 6 7 8 9 10

Mea

n P

AP

R [d

B]

5.68

5.7

5.72

5.74

5.76

5.78

5.8

5.82

5.84Cost function decrease

PSO-PTSPRS-PTS

(b) Scenario 2 - M4-W8

Iteration1 2 3 4 5 6 7 8 9 10

Mea

n P

AP

R [d

B]

5.62

5.64

5.66

5.68

5.7

5.72

5.74

5.76

5.78Cost function decrease

PSO-PTSPRS-PTS

(c) Scenario 3 - M8-W4

Figure 7: Cost function decrease over iterations for scenarios 1, 2, and 3 of Table 3.

To compare the performance of the three studied algorithms in terms of PAPR reduction, the comple-

mentary cumulative distribution function (CCDF) of the PAPR for the considered OFDM symbols under the

three optimization techniques, as well as the original OFDM signal.

On the first scenario on Fig.8 the three algorithms converged to a PAPR of approximately 7.1 dB in 10−4

probability, resulting in an almost 4 dB decrease when compared to the original OFDM signal. Although the

three methods achieved the same performance in terms of the maximum PAPR for the 104 OFDM symbols,

by looking at the mean PAPR per symbol, the PSO-PTS algorithm managed to achieve 93.7% of the reduction

accomplished by C-PTS, whereas the PRS-PTS algorithm exceeded the PSO-PTS results at 100.3%.

In terms of speed, C-PTS took approximately 435.47 KFLOPS per OFDM symbol, PSO-PTS managed

to do the job in 176.38 KFLOPS and PRS-PTS in 100.74 KFLOPS.

10

PAPR [dB]0 1 2 3 4 5 6 7 8 9 10 11 12

Pr(

PA

PR

>P

AP

R0)

[%]

10-4

10-3

10-2

10-1

100

101

102 CCDF Comparison M4-W4

C-PTS M4-W4PSO-PTS M4-W4PRS-PTS M4-W4Original OFDM Signal

Figure 8: Comparison between PAPR reduction techniques in an OFDM signal with QPSK modulation and

128 subcarriers, using 4 branches and 4 phase factors

On the next scenario, depicted in Fig.9, the three algorithms were compared with 4 branches and 8 phase

factors. The CCDF shows slightly better results than Fig.8, with PSO-PTS and C-PTS achieving around 6.7

dB at 10−4 probability and PRS-PTS achieving around 6.9 dB. Regarding the mean PAPR per OFDM symbol,

PSO-PTS obtained 80.94% of the PAPR reduction accomplished by C-PTS; on the same subject, PRS-PTS

managed to attain 99.84% of the PSO-PTS performance.

When comparing the number of required floating-point operations per technique, PRS-PTS is again the

best alternative at 73.1 KFLOPS per OFDM symbol, in contrast to 125.54 KFLOPS of the PSO-PTS and

the enormous 6967.3 KFLOPS in the C-PTS algorithm.

PAPR [dB]0 1 2 3 4 5 6 7 8 9 10 11 12

Pr(

PA

PR

>P

AP

R0)

[%]

10-4

10-3

10-2

10-1

100

101

102CCDF Comparison M4-W8

C-PTS M4-W8PSO-PTS M4-W8PRS-PTS M4-W8Original OFDM Signal

Figure 9: Comparison between PAPR reduction techniques in an OFDM signal with QPSK modulation and

128 subcarriers, using 4 branches and 8 phase factors

11

On the last simulated scenario, the number of branches was increased to 8 while the phase factors were

kept in 2. The CCDF at Fig.10 displays better results than the previous scenarios, as well as a larger difference

between the C-PTS and the heuristic algorithms. The mean PAPR per symbol on the PSO-PTS technique

was 64.6% of the C-PTS; comparing PRS-PTS to PSO-PTS, the former surpassed the latter with 101.13% of

the achieved mean PAPR.

In terms of floating-point operations, C-PTS required around 211615.7 KFLOPS per OFDM symbol,

PSO-PTS required 153.86 KFLOPS and PRS-PTS 94.64 KFLOPS.

PAPR [dB]0 1 2 3 4 5 6 7 8 9 10 11 12

Pr(

PA

PR

>P

AP

R0)

[%]

10-4

10-3

10-2

10-1

100

101

102CCDF Comparison M8-W4

C-PTS M8-W4PSO-PTS M8-W4PRS-PTS M8-W4Original OFDM Signal

Figure 10: Comparison between PAPR reduction techniques in an OFDM signal with QPSK modulation and

128 subcarriers, using 8 branches and 4 phase factors

The Table 4 summarizes the simulation results commented above, with the addition of the PAPR reduction

obtained at 10−4 probability.

Scenario 1 (M4-W4) Scenario 2 (M4-W8) Scenario 3 (M8-W4)

Method C-PTS PSO PRS C-PTS PSO PRS C-PTS PSO PRS

E[PAPR] dB 5.64 5.75 5.74 5.30 5.68 5.68 4.75 5.65 5.63

PAPR Reduction dB 3.74 3.74 3.74 4.86 4.85 4.68 6.38 5.58 5.57

KFLOPS/symbol 435.37 176.38 100.74 6967.3 125.54 73.1 211615.6 153.86 94.64

Table 4: Simulation results

The results in Fig.11 demonstrate the computational advantages of the heuristic algorithms developed herein,

both requiring a lot less operations than C-PTS to achieve similar PAPR reduction. On both cases studied, the

PRS-PTS algorithm has the best computational performance. The results are shown in KFLOPS (103 FLOPS)

per OFDM symbol.

On Fig.11a the number of branches in which the OFDM symbol is partitioned was increased from 2 to 64

in powers of 2, while the number of phase factors was kept at 2. As expected, the C-PTS algorithm requires

a rapidly increasing amount of floating point operations. On the other hand, the heuristic algorithms displayed

a much smaller grow in complexity; the PRS-PTS algorithm achieved a minimum of 116 KFLOPS/symbol,

12

whereas the PSO-PTS showed a minimum of 201 KFLOPS/symbol, both on M = 8 branches, in constrast to

the 1020 KFLOPS required for C-PTS at M = 64 branches.

Fig 11b shows the increase in complexity of the C-PTS algorithm as the number of available phase factors

grows while the number of branches was kept at 4. On the other hand, both heuristic algorithms shows a

decrease in complexity, saturating at around 100 KFLOPS per symbol.

Number of Branches (M)2 4 8 16 32 64

KF

LOP

S/S

ymbo

l

100

102

104

106

108

1010

1012FLOPS Comparison

PSO-PTSPRS-PTSMeasured C-PTSEstimated C-PTS

(a) Variation of the number of branches (M)

Phase Factors (W)2 4 8 16 32 64

KF

LOP

S/S

ymbo

l101

102

103

104

105

106

107

108FLOPS Comparison

PSO-PTSPRS-PTSMeasured C-PTSEstimated C-PTS

(b) Variation of the number of phase factors (W)

Figure 11: Computational complexity comparison among the three studied algorithms.

It is worth noting that for 16, 32 and 64 branches and phase factors, the FLOPS calculation of the C-PTS

algorithm was estimated based on the number of phase factors combinations, WM , and the available measured

results. Also, the above results do not include the floating-point operations required by the M parallel ifft, each

one growing at O(n log n) rate. Although according to [18], thanks to the complexity of modern computers

there is no longer any clear connection between operation counts and fft speed, one should keep in mind that

by increasing the number of branches, the computational time required will increase linearly, as the M parallel

ifft operations will have the same number of points, N.

5 Conclusions

Reducing the PAPR of OFDM signals is of paramount importance to modern communication systems. A

low PAPR is a prerequisite of power efficient RF electronics in an OFDM transmission system. Although some

methods of PAPR reduction show suitable performance results, some of them achieve it at the expense of

prohibitive computational time, such as the conventional partial transmit sequence technique when the dimension

order of optimization increases (M and W increases), which makes them impractical for realistic scenarios.

With the results from Table 4 it is possible to say that for every scenario simulated on this paper, both

heuristic optimization methods required much less floating-point operations to minimize the PAPR, while

maintaining at least 80% of the performance of the conventional PTS method. With that in mind, the pseudo-

random search algorithm outperforms particle swarm optimization for the PTS problem in terms of floating-point

operations per OFDM symbol, yet achieving very close or even better results than the former in terms of PAPR

reduction. This result can be explained by the complexity of the minimization cost function, displayed at Fig.12.

13

The lack of an optimization tendency or a clear global minimum makes the PRS algorithm more suited to the

problem.

Appendices

A PTS Search Space

In order to investigate the PTS search space through which the proposed algorithms must find the global

minimum, three setups were considered for one random OFDM symbol, providing WM phase factor combi-

nations, respectively 164, 84 and 44. Thus, the PAPR level for each phase factor combination was calculated

and displayed in the compositions in Fig.12. For each sector formed by a ω3 and ω4 pair, all the combinations

of ω1 and ω2 were calculated.

2π

3π/2

ω4

π

π/2π/2

π

ω3

3π/2

2π4

6

8

10

PA

PR

[dB

]

2π7π/4

3π/25π/4

ω4

π

3π/4π/2

π/4π/4π/2

3π/4

ω3

π

5π/43π/2

7π/42π

5

7

9

10

8

6

PA

PR

[dB

]

a) M = 4, W = 4 b) M = 4, W = 8

c) M = 4, W = 16

Figure 12: Search space for a 128 subcarriers OFDM symbol using M = 4 branches and different W phase

factors

14

B FLOPS Characterization

The table below has the FLOP count for each operation considered on the complexity estimation of the

PTS algorithms.

Function Name FLOPS Estimative

a ∈ R + b ∈ R 1

a ∈ C + b ∈ C 2

a ∈ R · b ∈ R 1

a ∈ C · b ∈ C 6

abs(x) 4

a/b 8

ex 40

log(x) 20√

x 8

FFT, IFFT 4 · n · log2(n)

A ∈ Cn×m × B ∈ Ca×b n · b(2m − 1)∑m

i=1 An×i ∈ Cn×m m(n − 1)∑n

i=1 Ai×m ∈ Cn×m n(m − 1)

R{xn×m} 11(m + n)

Table 5: FLOPS estimates table

References

[1] A. Goldsmith, Wireless Communications. Cambridge University Press, 2005.

[2] M. Gouda and M. Hussien, “Partial transmit sequence PAPR reduction method for LTE OFDM systems,”

4th International Conference on Intelligent Systems Modelling and Simulations (ISMS), January 2013.

[3] P. Varahram, W. F. Al-Azzo, and B. M. Ali, “A low complexity partial transmit sequence scheme by use

of dummy signals for PAPR reduction in OFDM systems,” IEEE Transactions on Consumer Electronics,

November 2010.

[4] T. Wattanasuwakull and W. Benjapolakul, “PAPR reduction for OFDM transmission by using a method

of tone reservation and tone injection,” 5th International Conference on Information, Communications and

Signal Processing, 2005.

[5] Y. Z. Jiao, X. J. Liu, and X. A. Wang, “A novel tone reservation scheme with fast convergence for PAPR

reduction in OFDM systems,” 5th IEEE Consumer Communications and Networking Conference, January

2008.

[6] P. Phoomchusak and C. Pirak, “Adaptive tone-reservation PAPR technique with optimal subcarriers allo-

cation awareness for multi-user OFDMA systems,” 13th International Conference on Advanced Communi-

cation Technology (ICACT), February 2011.

15

[7] Y. Inoue, H. Tsutsui, and Y. Miyanaga, “Study of PAPR reduction using coded PTS in 8x8 MIMO-

OFDM systems,” International Symposium on Intelligent Signal Processing and Communications Systems,

November 2013.

[8] D. Phetsomphou, S. Yoshizawa, and Y. Miyanaga, “A partial transmit sequence technique for PAPR

reduction in MIMO-OFDM systems,” International Symposium on Communications and Information Tech-

nologies (ISCIT), October 2010.

[9] V. Sudha, S. Balan, and S. Kumar, “Performance analysis of PAPR reduction in OFDM system with

distortion and distorion less methods,” 2014 International Conference on Computer Communication and

Informatics, January 2014.

[10] H.-L. Hung, Y.-F. Huang, C.-M. Yeh, and T.-H. Tan, “Performance of particle swarm optimization tech-

niques on PAPR reduction for OFDM systems,” IEEE International Conference on Systems, Man and

Cybernetics, October 2008.

[11] T. Abrão, Ed., Search Algorithms for Engineering Optimization. InTech, February 2013.

[12] W. Baocheng, W. Tiane, and W. Zenghui, “The implementation of parallel ant colony optimization algo-

rithm based on MATLAB,” 3rd Global Congress on Intelligent Systems (GCIS), November 2012.

[13] H. Liang, Y.-R. Chen, Y.-F. Huang, and C.-H. Cheng, “A modified genetic algorithm PTS technique for

PAPR reduction in OFDM systems,” 15th Asia-Pacific Conference on Communications, October 2009.

[14] T. Araújo and R. Dinis, “On the accuracy of the gaussian approximation for the evaluation of nonlinear

effects in OFDM signals,” IEEE Transactions on Communications, November 2011.

[15] S. Muller and J. Huber, “A novel peak power reduction scheme for OFDM,” in Personal, Indoor and

Mobile Radio Communications, 1997. Waves of the Year 2000. PIMRC ’97., The 8th IEEE International

Symposium on, vol. 3, Sep 1997, pp. 1090–1094 vol.3.

[16] C. Tellambura, “Use of m-sequences for OFDM peak-to-average power ratio reduction,” Electronics Letters,

vol. 33, no. 15, pp. 1300–1301, Jul 1997.

[17] T. Minka, “Lightspeed MATLAB toolbox,” http://research.microsoft.com/en-us/um/people/minka/

software/lightspeed, accessed: 2015-12-30.

[18] M. Frigo and S. G. Johnson, “The design and implementation of FFTW3,” Proceedings of the IEEE,

February 2005.

16

Anexo BOtimização do Nível de IBO em

Sistemas OFDM

1

Input Back-Off Optimization in OFDM SystemsCaio Henrique Azolini Tavares, José Carlos Marinello Filho, Cristiano Magalhães Panazio, Taufik Abrão

Abstract—A key aspect of orthogonal frequency divisionmultiplexing (OFDM) that comes along with the highspectral efficiency is the high peak-to-average power ratio(PAPR), which impacts on the power efficiency of high-power amplifiers (HPA) at the transmitter by requiring abroad dynamic range, as well as high resolution analog-to-digital converters. If a portion of the OFDM signals fallsbeyond the saturation point of the HPA, the output signalwill be clipped, resulting in intermodulation at the symbolbandwidth, as well as out-of-band emissions due to thedistortion. In this letter we have encountered the optimuminput back-off (IBO) level at the HPA on AWGN channelfor a given noise power, concerning the signal-to-noise-plus-distortion ratio (SNDR). Our formulation and numericalresults show that the optimum IBO level depends only onthe channel noise power and HPA saturation power.

Keywords – Orthogonal frequency division multiplexing(OFDM); peak-to-average power ratio (PAPR); input back-off (IBO); solid state high-power amplifier (HPA); opti-mization

I. INTRODUCTION

Many of the recent wireless communication technolo-gies employ orthogonal frequency division multiplexing(OFDM) to achieve a reliable and spectral-efficient highdata-rate link with robustness against frequency selectivefading. These advantages are obtainable through the useof a large number of orthogonal narrowband subcarrierswith a certain level of spectral overlaping in conjunctionwith the use of the so-called cyclic prefix.

However, one of the main disadvantages of OFDMsignals is the high peak-to-average power ratio (PAPR)that occurs after the inverse fast fourier transform (IFFT)in the transmitter. This effect is due to the phase com-bination of adjacent subcarriers at time domain. A high-PAPR signal can increase the distortion noise on theoutput signal at the high-power amplifier (HPA) trans-mitter device, due to the limited linear operation rangeof the HPA [1].

There are many ways to deal with this problem,such as distortion and distortionless PAPR reductiontechniques, introduction of pre-distortion in the OFDMsignal aiming to minimize the effects of the non-linear

C. Tavares, T. Abrão and J. Marinello are with Department ofElectrical Engineering, State University of Londrina, Parana, Brazil.C. Panazio is with Laboratory of Communications and Signals,Polytechnic School of the University of São Paulo, São Paulo,Brazil. E-mail: caio.tavares11@gmail.com; taufik@uel.br; zecar-los.ee@gmail.com; cpanazio@lcs.poli.usp.br

region by lowering the operation point of the HPA viaa suitable input back-off (IBO) level [2].

Distortionless PAPR reduction techniques can achievea certain level of PAPR reduction usually at the expenseof increased complexity at the transmitter and/or receiversides. Moreover, some techniques require the transmis-sion of side information and this overhead reduces thespectral efficiency of the system. On the other hand,signal clipping above a certain threshold is one of thesimplest and most effective techniques among that whichcauses distortion. Therefore, this technique has beenadopted in this letter.

When clipping is employed and the HPA operates ina low IBO region, it can introduce in-band and out-of-band interference [3] [4] while the bit error rate (BER)performance of the system may be degraded. On theother hand, by deliberately increasing the IBO level,the amplifier is set into a less energy efficient operatingpoint. According to [1], in a typical OFDM device,the transmit power accounts for only 8% of the totalpower consumption at the transmitter, whereas 41% ofpower is wasted by the HPA, and the power consumptionof all other circuit devices is about 51%. Therefore,the choice of the IBO level highly impacts on theenergy efficiency and the signal-to-noise-plus-distortionratio (SNDR). Hence, in this work, the power efficiencyversus SNDR trade-off is analytically optimized.

The rest of this letter is organized as follows. Thesystem model is described in Section II, while theproposed IBO optimization methodology is discussed inSection III. Illustrative numerical results are explored insection IV. Final remarks and conclusions are offered inSection V.

II. SYSTEM MODEL

An OFDM symbol can be written as the IFFT ofN phase-shift keying (PSK) or quadrature amplitudemodulation (QAM) modulated symbols, according to:

s(n) =1√N

N−1∑k=0

mkej2πnk/N , 0 ≤ n ≤ N − 1,

(1)where mk is the data symbol at the k-th subcarrier, Nis the FFT length, and s = [s(0), s(1) . . . s(N − 1)]T isthe OFDM symbol vector, while {·}T is the transposeoperator. With no loss of generality, a QPSK modulationhas been adopted in this work.

2

For the OFDM symbol in (1), we can define the PAPRas:

PAPR =

max0≤n≤N−1

|s(n)|2

E {|s(n)|2} , (2)

where E {·} is the expectation operator.Considering the Rapp model for a solid state power

amplifier (SSPA) [5], the HPA AM/AM transfer functioncan be written as:

so(n) = fA(s(n)) =G · s(n)

[1 +

(G · |s(n)|Asat

)2p] 1

2p

(3)

where Asat is the amplifier saturation voltage, p is thesmoothness factor and G is the amplifier voltage gain,considered as unitary in this letter. If we consider a pre-distorter before the HPA stage with a transfer functionfP (s(n)) = f−1A (s(n)), the combined transfer functioncan be treated as a soft envelope limiter: [6]

so(n) =

{G · s(n), for G · |s(n)| < Asat,

Asat · ejarg{s(n)}, for G · |s(n)| ≥ Asat.(4)

The IBO level is defined as the quotient of theamplifier saturation power and the input signal meanpower. From [7], it can be written as:

γ2 =A2

sat

Pin=

A2sat

E {|s(n)|2} . (5)

For large N , the real and imaginary parts of s(n),i.e. x(n) and y(n), respectively, can be considered asGaussian distributed random variables (r.v.), which leadsto Rayleigh distributed amplitudes. From [6], [8], theoutput of a memoryless nonlinear process of Gaussiandistributed r.v. x(n) and y(n) leads to an attenuatedsignal component plus a distortion noise:

xo(n) = αx(n) + v(n) (6)

yo(n) = αy(n) + w(n)

with v(n) and w(n) Gaussian distributed r.v. and uncor-related with x(n) and y(n).

By writing the signal plus distortion noise as so(n) =xo(n) + jyo(n), the attenuation factor α can be writtenas [1]:

α =E [so(n)s

∗(n)]E [s(n)s∗(n)]

, (7)

where {·}∗ is the complex conjugate operator.Assuming the input as Gaussian distributed, the atten-

uation factor α can be rewritten in terms of the IBO levelγ2 as [8]:

α = 1− e−γ2

+

√π

2γ · erfc(γ) (8)

where erfc(γ) =2√π

∫∞γ e−t

2

dt.

Considering a communication system operating underAWGN channel model with a white noise power η2t , wecan define the asymptotic signal-to-noise ratio (SNR) as

SNRsat =A2

sat

η2t, (9)

attained under an ideal distortionless amplifier in whichE{|so(n)|2

}= Psat, namely the maximum output power

of the HPA defined in (4). Thus, the system describedhereby do not operate with SNRsat due to the distortionintroduced by the amplifier, but the definition of SNRsat

allows us to analyze the performance of the describedsystem in a simplified way.

III. IBO OPTIMIZATION PROCEDURE

By considering a realistic approach, the HPA param-eters, such as the saturation voltage, are fixed; hence,in order to increase the IBO level, one must decreasethe average power of the input signal. Furthermore, ifthe noise power of the AWGN channel is also fixed(as it really is for a fixed temperature), the signal-to-noise-plus-distortion ratio (SNDR) could be optimizedas described in the following. Note that the approachadopted herein is different from that of [1], [6], in whichfor assuring a given SNR and IBO level, either thechannel noise power or the HPA saturation power isscaled. Notice that if the IBO level is considerably low,the system performance will be limited by distortion;on the other hand, if it is too high, which is obtainedby reducing the average input signal power, the systemperformance will be limited by thermal noise.

From Eq. (6), the power of the distortion noise atthe output of the soft envelope limiter is given byη2d = E

{v2(n) + w2(n)

}. Based on the assumption that

the input signal is complex Gaussian distributed, the totaloutput power of the HPA is related with the cumulativedistribution function (CDF) of the input power, which isexponentially distributed:

Pout = E{|so(n)|2

}= (1− e−γ2

) · Pin. (10)

Thus, by defining the output signal power as Psig =α2 · Pin, the output distortion noise power will be η2d =Pout − Psig. Finally, the SNDR can be defined as:

SNDR =Psig

η2d + η2t. (11)

Substituting (5), (9), and (10) into (11) we have:

SNDR =

α2A2sat

γ2

(1− e−γ2

)A2sat − α2A2

sat

γ2+

A2sat

SNRsat

. (12)

3

By multiplying numerator and denominator by γ2

A2sat

:

SNDR =α2

1− e−γ2 − α2 + γ2

SNRsat

. (13)

Notice the SNDR depends on the asymptotic signal-to-noise ratio SNRsat, the IBO level γ2, and the attenuationfactor α. In order to find the maximum value of theSNDR, we differentiate (13) with respect to the IBOlevel and equal it to zero, yielding:

erfc(γopt)γopt

=1√

π4 · SNRsat

. (14)

For details, see the Appendix.Unfortunately, we cannot arrive at a closed-form solu-

tion for the optimal IBO level γ2opt. However, it can befound using a numeric method such as Newton-Raphson,yielding an iterative square root IBO level procedure withupdating equation:

γk+1 =SNRsat

[√π2 erfc(γk) + γk · e−γ

2k

]

1 + SNRsat · e−γ2k

, (15)

which converges to the optimal value after few itera-tions; indeed, in the simulations discussed in followingSection, four iterations have been sufficient to achieveconvergence, setting γ0 = 1.

IV. NUMERICAL RESULTS

Based on (13), we have plotted the SNDR curvesaccording the IBO variation in Fig. 1, taking into accountseven SNRsat scenarios ranging from 0 to 30 dB andQPSK modulation. Then, with the aid of (14), (15), theoptimum IBO levels were calculated for each SNRsat

scenario, as well as the respective optimal SNDR’s. Theyare also indicated in Fig. 1. As expected, the resultsdemonstrate that by increasing SNRsat, the system devel-ops a distortion-limited performance, in which to achievethe desired optimum SNDR, higher IBO levels must beemployed.

Next, to verify the effect of the input signal PAPRon the optimum IBO level from Eq. (14), Monte Carlosimulations were carried out with three OFDM setups,each with a different number of subcarriers. Hence, theirSNDR’s were evaluated from (7), (13).

IBO dB-4 -2 0 2 4 6 8

SN

DR

dB

-5

0

5

10

15

20

25

30

SNRsat

0dB

SNRsat

5dB

SNRsat

10dB

SNRsat

15dB

SNRsat

20dB

SNRsat

25dB

SNRsat

30dB

SNDRopt

Figure 1. SNDR for different SNRsat scenarios, along with thepredicted optimum SNDR and IBO levels.

The results on Fig. 2 show that despite the decrease inthe output signal SNDR with the number of subcarriersincrease, the input PAPR does not influence the predictedoptimum IBO level. Interestingly, this is in sharp contrastto the common belief that higher PAPR signals requireHPA operating under higher IBO levels. Indeed, afurther elaborated analysis would take into account thesystem spectral mask constraints, limiting the spectralregrowth introduced by the clipping process, and is thecontinuity of this work.

IBO dB0 0.5 1 1.5 2 2.5 3 3.5 4

SN

DR

dB

8.4

8.45

8.5

8.55

8.6

8.65

8.7

8.75

8.8

8.85

8.9

128 subcarriers, E[PAPR] = 7.28dB512 subcarriers, E[PAPR] = 8.27dB2048 subcarriers, E[PAPR] = 9.09dB

Figure 2. Simulated SNDR for different number of subcarriers,considering SNRsat = 10 dB

In addition to the SNDR results, it is possible topredict the bit error rate for QPSK modulation un-der AWGN channel with Pb = Q

{√SNDR

}, where

Q {x} = 12erfc

(x√2

). Thus, we have also investigated

the BER performance of the OFDM system for differentSNRsat levels, as depicted in Fig. 3, by means of MonteCarlo simulations. Besides, the optimal IBO levels cal-culated from (14)-(15) for each SNRsat scenario are alsoindicated, in conjunction with the predicted BER.

4

IBO dB-4 -2 0 2 4 6 8

BE

R

10-6

10-5

10-4

10-3

10-2

10-1

100

SNRsat

0dB

SNRsat

5dB

SNRsat

10dB

SNRsat

15dB

BERopt

Figure 3. Simulated BER results, in addition to the predictedoptimum BER at IBOopt.

V. DISCUSSION AND FINAL REMARKS

For OFDM systems the IBO level is an essentialmeasure of energy efficiency of the HPA. A high IBOdelivers a distortion-free output signal, but at the expenseof energy efficiency reduction, while a low IBO resultsin distortion and interference. In this letter, we investigatethe problem from a realistic perspective, in which bothHPA saturation power and noise power are fixed. Thus,IBO level is adjusted by scaling the average power of theinput signal. Based on the SNDR analytical expression,we derived the optimum IBO level of an OFDM systemoperating in AWGN channels, maximizing the SNDRof the output signal, given that the HPA and channelcharacteristics remain unchanged. With this result, it ispossible to set a realistic and more appealing trade-offbetween spectral efficiency and energy efficiency for theOFDM system. Additionally, Monte Carlo simulationswere conducted in order to validate our analytical find-ings.

It is worth noting that the input signal’s PAPR doesnot influence the optimum IBO level. Indeed, analyticalexpression (14) was derived based on the assumptionthat the input signal is Gaussian distributed. By reducingthe input PAPR, one would limit the input probabilitydensity function to that specific level and increase theattenuation factor α defined by (7), but the approximationto Gaussian distribution remains valid.

As the continuity of this work, we will investigate theeffects of fading and dispersive channels, as well as theimpact of the out-of-band interference introduced by (4),in the optimum IBO level encountered.

APPENDIX

DERIVATION OF THE OPTIMUM IBO LEVEL

By differentiating (13) with respect to γ2 and equallingto zero:

d(SNDR)

dγ2=2α · dαdγ2

[1− e−γ2

opt − α2 +γ2opt

SNRsat

]

[1− e−γ2

opt − α2 +γ2opt

SNRsat

]2 −

(16)

−α2[e−γ

2opt − 2α · dαdγ2 + 1

SNRsat

]

[1− e−γ2

opt − α2 +γ2opt

SNRsat

]2 = 0

From (16) one can equal the numerator to zero andrearrange the terms, yielding:

1− e−γ2opt +

γ2optSNRsat

e−γ2opt +

1

SNRsat

=

α

2dα

dγ2

(17)

By expanding the terms α and dαdγ2 , the right side of

equation above simplifies to:

α

2dα

dγ2

=2− 2e−γ

2opt + γopt

√π · erfc(γopt)√

π · erfc(γopt)γopt

+ 2e−γ2opt

(18)

Finally, by manipulating the terms of (18) into (17), onecan get (14).

REFERENCES

[1] T. Jiang, C. Li, and C. Ni, “Effect of PAPR reduction onspectrum and energy efficiencies in OFDM systems with class-aHPA over AWGN channel,” IEEE Transactions on Broadcasting,September 2013.

[2] Y. Rahmatallah and S. Mohan, “Peak-to-average power ratioreduction in OFDM systems: A survey and taxonomy,” IEEECommunications Surveys Tutorials, March 2013.

[3] H.-G. Ryu, T. P. Hoa, N. T. Hieu, and J. Jianxue, “BER analysisof clipping process in the forward link of the OFDM-FDMAcommunication system,” IEEE Transactions on Consumer Elec-tronics, November 2004.

[4] N. Taspinar, D. Karaboga, M. Yildirim, and B. Akay, “PAPRreduction using artificial bee colony algorithm in OFDM sys-tems,” Turkish Journal of Electrical Engineering and ComputerSciences, 2011.

[5] C. Rapp, “Effects of HPA-nonlinearity on a DPSK/OFDMsignal for a digital sound broadcasting system,” Proceedings ofthe Second European Conference on Satellite Communications,October 1991.

[6] H. Ochiai and H. Imai, “Performance of the deliberate clippingwith adaptive symbol selection for strictly band-limited OFDMsystems,” IEEE Journal on Selected Areas in Communications,November 2000.

[7] G. Li and G. Stuber, Orthogonal Frequency Division Multiplex-ing for Wireless Communications. Springer, 2006.

[8] H. Rowe, “Memoryless nonlinearities with gaussian inputs: Ele-mentary results,” The Bell System Technical Journal, September1982.

41

Referências

ABRãO, T. (Ed.). Search Algorithms for Engineering Optimization. [S.l.]: InTech, 2013.12

ARAúJO, T.; DINIS, R. On the accuracy of the gaussian approximation for theevaluation of nonlinear effects in OFDM signals. IEEE Transactions on Communications,November 2011. 18

BAOCHENG, W.; TIANE, W.; ZENGHUI, W. The implementation of parallel antcolony optimization algorithm based on MATLAB. 3rd Global Congress on IntelligentSystems (GCIS), November 2012. 12

GOLDSMITH, A. Wireless Communications. [S.l.]: Cambridge University Press, 2005.11

GOUDA, M.; HUSSIEN, M. Partial transmit sequence PAPR reduction method forLTE OFDM systems. 4th International Conference on Intelligent Systems Modelling andSimulations (ISMS), January 2013. 11

HUNG, H.-L. et al. Performance of particle swarm optimization techniques on PAPRreduction for OFDM systems. IEEE International Conference on Systems, Man andCybernetics, October 2008. 12

INOUE, Y.; TSUTSUI, H.; MIYANAGA, Y. Study of PAPR reduction using coded PTSin 8x8 MIMO-OFDM systems. International Symposium on Intelligent Signal Processingand Communications Systems, November 2013. 12

JIAO, Y. Z.; LIU, X. J.; WANG, X. A. A novel tone reservation scheme withfast convergence for PAPR reduction in OFDM systems. 5th IEEE ConsumerCommunications and Networking Conference, January 2008. 12

LASORTE, N.; BARNES, W. J.; REFAI, H. H. The history of orthogonal frequencydivision multiplexing. IEEE Global Telecommunications Conference, December 2008. 11

LIANG, H. et al. A modified genetic algorithm PTS technique for PAPR reduction inOFDM systems. 15th Asia-Pacific Conference on Communications, October 2009. 12

MAHAFENO, I. M.; LOUëT, Y.; HéLARD, J.-F. PAPR reduction method for OFDMsystems using dedicated subcarriers: a proposal for the future DVB-T standard. IEEEInternational Symposium on Broadband Multimedia Systems and Broadcasting, April2008. 11

MOLISCH, A. Wireless Communications. Wiley, 2010. (Wiley - IEEE). ISBN9780470666692. Disponível em: <https://books.google.com.br/books?id=vASyH5-jfMYC>. 12

OCHIAI, H.; IMAI, H. Performance of the deliberate clipping with adaptive symbolselection for strictly band-limited OFDM systems. IEEE Journal on Selected Areas inCommunications, November 2000. 16

Referências 42

OCHIAI, H.; IMAI, H. Performance analysis of deliberately clipped OFDM signals.IEEE Transactions on Communications, January 2002. 16

PHETSOMPHOU, D.; YOSHIZAWA, S.; MIYANAGA, Y. A partial transmit sequencetechnique for PAPR reduction in MIMO-OFDM systems. International Symposium onCommunications and Information Technologies (ISCIT), October 2010. 12

PHOOMCHUSAK, P.; PIRAK, C. Adaptive tone-reservation PAPR techniquewith optimal subcarriers allocation awareness for multi-user OFDMA systems. 13thInternational Conference on Advanced Communication Technology (ICACT), February2011. 12

ROWE, H. Memoryless nonlinearities with gaussian inputs: Elementary results. TheBell System Technical Journal, September 1982. 16

RYU, H.-G. et al. BER analysis of clipping process in the forward link of theOFDM-FDMA communication system. IEEE Transactions on Consumer Electronics,November 2004. 12

SUDHA, V.; BALAN, S.; KUMAR, S. Performance analysis of PAPR reduction inOFDM system with distortion and distorionless methods. 2014 International Conferenceon Computer Communication and Informatics, January 2014. 12

WATTANASUWAKULL, T.; BENJAPOLAKUL, W. PAPR reduction for OFDMtransmission by using a method of tone reservation and tone injection. 5th InternationalConference on Information, Communications and Signal Processing, 2005. 12

WEN, J.-H. et al. Coding schemes applied to peak-to-average power ratio (PAPR)reduction in OFDM systems. International Wireless Communications and MobileComputing Conference, August 2008. 12