- Research Article
- Open Access

# Amplitude PDF Analysis of OFDM Signal Using Probabilistic PAPR Reduction Method

- Hyunseuk Yoo
^{1}Email author, - Frédéric Guilloud
^{1}and - Ramesh Pyndiah
^{1}

**2011**:983915

https://doi.org/10.1155/2011/983915

© Hyunseuk Yoo et al. 2011

**Received:**24 June 2010**Accepted:**19 January 2011**Published:**15 February 2011

## Abstract

To reduce the peak-to-average power ratio (PAPR) of an orthogonal frequency division multiplexing (OFDM) modulation scheme, one class of methods is to generate several OFDM symbols (candidates) carrying the same information and to select for transmission the one having the lowest PAPR. We derive a theoretical amplitude probability density function (PDF) of the selected OFDM symbol using order statistics. This amplitude PDF enables one to derive the signal-to-noise-plus-distortion ratio (SNDR) as a function of the number of candidates. Based on the SNDR derivation, theoretical error performance and statistical channel capacity are provided for this class of methods. The results match the simulations and make the system design easier.

## Keywords

- Probability Density Function
- Orthogonal Frequency Division Multiplex
- Orthogonal Frequency Division Multiplex System
- Orthogonal Frequency Division Multiplex Symbol
- Orthogonal Frequency Division Multiplex Signal

## 1. Introduction

Orthogonal frequency division multiplexing (OFDM) is a multicarrier multiplexing technique, where data is transmitted through several parallel frequency subchannels at a lower rate. It has been popularly standardized in many wireless applications such as Digital Video Broadcasting (DVB), Digital Audio Broadcasting (DAB), High Performance Wireless Local Area Network (HIPERLAN), IEEE 802.11 (WiFi), and IEEE 802.16 (WiMAX). It has also been employed for wired applications as in the Asynchronous Digital Subscriber Line (ADSL) and power-line communications.

A significant drawback of the OFDM-based system is its high Peak-to-Average Power Ratio (PAPR) at the transmitter, requiring the use of a highly linear amplifier which leads to low power efficiency [1]. Moreover, when an OFDM signal level works on the nonlinear area of amplifier, the OFDM signals go through nonlinear distortions and degrade the error performance.

The various approaches to alleviate this problem in OFDM-based systems can be classified into five categories: clip effect transformation [2], coding [3], frame superposition using reserved tones [4], expansible constellation point: tone injection [4] and active constellation extension [5], and probabilistic solutions [6–13].

The principle of probabilistic methods is to reduce the probability of high PAPR by generating several OFDM symbols (multiple candidates) carrying the same information and by selecting the one having the lowest PAPR. The probabilistic method can also be classified into two strategies: subblock partitioning strategy and entire block strategy. The subblock partitioning strategy, such as partial transmit sequence (PTS) [6–8], divides frequency domain signals into several subblocks. On the other hand, the entire block strategy, such as selected mapping (SLM) [8–10] and interleaving [11–13], considers the entire block for generating multiple candidates.

In this paper, we consider the entire block strategy of the probabilistic methods to generate multiple candidates. First, the probability density function (PDF) for the multiple candidate system is analyzed. When the candidate having the lowest PAPR is selected, the PDF of the amplitude of a selected OFDM symbol becomes the function of the number of candidates . We apply the analyzed PDF (as a function of ) to Ochiai's method [13] for obtaining the signal-to-noise-plus-distortion ratio (SNDR) as a function of . Then, the SNDR (as a function of ) can be used for analytical error performance. Note that in [13], the authors used the Rayleigh PDF (single candidate) for obtaining the error performance of multiple candidate cases. However, we suggest using our PDF (multiple candidates) to obtain the theoretical error performance and also the statistical channel capacity for the multiple candidate system.

The paper is organized as follows: in Section 2, we describe the multiple candidate OFDM system, and analyze the PDF for the system. In Section 3, we derive the theoretical performance, such as the SNDR (as a function of ), and error rate, and also statistical channel capacity. In Section 4, an extension of the results to an oversampled SLM model, implementing the "clipping and filtering" technique [14], is tackled. Finally, we conclude this paper in Section 5.

## 2. Multiple Candidate System

### 2.1. Description

The clipped th candidate is transmitted to the receiver with its side information, where the side information contains the information of and it is used for recovering the original data. The side information protection depends on the various protection strategies, such as no side information method [9, 10] or coded side information method [12].

However, in this paper, for analyzing the pure effect of increasing for the multiple candidate system, we assume that the side information is sent without errors.

Throughout this paper, the following are also assumed: according to the central limit theorem, the complex OFDM signal, which consists of a number of independent orthogonal subcarriers, is modeled as a complex Gaussian process with Rayleigh envelope distribution. In addition, since the OFDM modulation is strictly band limited, we consider only in-band distortion.

### 2.2. PDF Analysis

where and .

where .

## 3. Theoretical Performance

### 3.1. for Multiple Candidate System

Now, we apply (8) to obtaining the signal-to-noise-plus-distortion ratio (SNDR) as a function of , by using Ochiai's method [13]. The authors in [13] used the Rayleigh PDF, , to obtain the SNDR of a multiple candidate system. However, as shown in Figure 2, the PDF of amplitude of the selected candidate is not Rayleigh PDF anymore, being the function of . Therefore, we use the PDF of (8), , to obtain the SNDR of multiple candidate system, and hereafter we will use as a function of , instead of SNDR.

For that, the PAPR threshold for clipping is defined as , where the input power and is the maximum permissible amplitude for the multiple candidate system.

### 3.2. Error Rate

Since we assume that the side information is transmitted without errors, the BER of QPSK-modulated signal over the AWGN channel is given by . Furthermore, QPSK symbol error rates (SER) are as follows: .

where is the channel attenuation which is Rayleigh distributed with .

### 3.3. Channel Capacity

where is the two-dimensional PDF of received symbol with the Gaussian noise variance in each dimension.

## 4. Application: Oversampling and Filtering

We present an extension of the multiple candidate system: combination with an oversampling and filtering technique [14]. For the single-candidate system, an OFDM symbol with a large is usually assumed to have a Gaussian PDF in the real and imaginary parts. However, for the multiple candidate system, this Gaussian assumption no longer holds. In this section, we show mathematical non-Gaussian PDF for the multiple candidate system.

### 4.1. Presentation of Extended Model

Then, the candidate with the minimum PAPR is selected, and clipped by the soft limiter, where , , as in (8). The clipped signal goes through a band pass filter (BPS) which removes out-of-band frequency components, yielding a filtered signal which will be converted into an analog signal .

where SNR denotes the signal-to-noise ratio for the channel, and denotes the signal-to-distortion ratio of the th subcarrier for candidate system.

where is given in (9) and is the autocorrelation function of the clipped signal.

where denotes the expectation operation.

### 4.2. Inaccuracy of Gaussian Assumption

For the single candidate case, since are assumed to be Gaussian distributed, is expressed as a joint Gaussian PDF [14, 18]. However, for the multiple candidate case , since the amplitude of the selected candidate is not Rayleigh distributed, such as (8), this Gaussian assumption no longer holds. In the rest of this paper, we consider the PDF of for the multiple candidate case.

where is given by (8).

which denotes the mathematical non-Gaussian PDF of or .

## 5. Conclusion

We study the probability density function (PDF) analysis and the signal-to-noise-plus-distortion ratio (SNDR) of a multiple candidate system for reducing the PAPR in OFDM modulation system. Since the selected OFDM symbol (candidate) has an amplitude PDF which is function of the number of candidates , the derived is also the function of , and it can be used for estimation of theoretical error performance and statistical channel capacity.

In this paper, the side information is assumed not to be erroneous for analyzing the pure effect of multiple candidates. The analytical estimation matches well the simulation results, and with this study, we conclude that the more the candidates, not only the better PAPR reduction performance, but also the better error performance and the more gain of channel capacity, under the assumption of side information transmission without error, and at the expense of computational complexity for IFFT circuits.

Furthermore, the amplitude PDF analysis enables one to apply to a probabilistic PAPR reduction system jointly with "oversampling and filtering" technique. In this application, since the selected candidate is not complex Gaussian distributed, more investigation for SNDR is required.

Our analytical approach to obtaining the implies that the estimation of error rate is achievable without time-consuming simulation, making system level design easier. Note that the error floor level is usually decreased by implementing channel coding techniques. In our future work, we will take channel coding into account for error performance analysis.

## Authors’ Affiliations

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This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.