# Ofdm System Simulation And Papr Reduction Computer Science Essay

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Communication technology has been widely applied in the world. With the development of communication technology, people can now communicate to anyone in anywhere. [1] Digital communications is the most important technology in the world. It has been applied in many areas of the communications such as the TV, radio and the internet. [2]

OFDM (Orthogonal Frequency Division Multiplexing) technology becomes the most popular technology in wireless communication. It can be seen as a special method of multicarrier modulation (MCM). [3] In OFDM system, the series bit information is converted to the parallel form to be transmitted after specific modulation method such as PSK or QAM so that the parallel data is modulated separately. With the development of IFFT in hardware, IFFT is applied in OFDM system instead of filters. Comparing with the traditional multicarrier modulation method like single carrier modulation and FDM, OFDM technology can significantly reduce the ISI (Inter Symbol Interference) and ICI (Inter Channel Interference) during the transmission so that there is much less distortion than that in the traditional modulation methods.

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There are also some disadvantages of OFDM technology which are the PAPR (Peak to Average Power Ratio) problem and the frequency offset problem. PAPR problem could bring the power amplifier to the nonlinear range, this results in great distortion. [4] PAPR problem can be solved by several methods such as the clipping and filtering, [5] SLM (Selective Mapping) [6] [7], PTS (Partial Transmission Sequence) [8] and PS (Pulse Shaping) [9] and other methods which will be discussed in detail in this report. Frequency offset problem is not include in this report.

## 1 Introduction

The significant technology, wireless communication, with the character of communicating between two sides in the movable environment becomes more and more popular in the modern world. Orthogonal Frequency Dividing Multiplexing (OFDM) technologies play a very important role in wireless communication as it allows transmitting the high data rate information by using many subcarriers. Also, the implementation of FFT and IFFT makes OFDM technology more realistic and Cyclic Prefix solves the problem of Inter Symbol Interference in communication system. [10]

However, there is still one major drawback which has bad effect on the communication system. High peak to average power ratio (PAPR) problem is discussed in this report. As a consequence of high PAPR, the signal is distorted in the nonlinear region of high power amplifier. Distortion gives a result of high bit error rate (BER) which is not expected in the system. Eight methods have been introduced to solve the problem: Clipping and Filtering, Windowing, [11] Coding [12], SLM (Selective Mapping), PS (Pulse Shaping), PTS (Partial Transmission Sequence) Tone Injection (TI) and Tone Rejection (TR). [13 14] There is no best method in PAPR reduction, so method should be chosen according to the actual situation and request.

In this report there are three parts which are introduction, OFDM system, PAPR reduction and the simulation result. The introduction gives the history, the advantages and disadvantages of OFDM. OFDM system description is talked in the second part. Next part discusses the four PAPR reduction methods and gives the result and the analysis. Finally, conclusion and reference are shown.

## 1.1 Back ground

The first communication form is telegram which came up in 1837. After telegram, communication mode has been changed from the telegram to telephones, internet and mobile communication. Also it has been changed from wired to wireless, optical fiber. OFDM is the most important technology in the communications for its cost and good performances.

In 1960s, OFDM idea came up but was not applied in the practice because there was no hardware support. [15]

1966, <<Synthesis of band-limited orthogonal signals for multi channel data transmission>> was written by R.W.Chang. It described the principle of transmission of no ISI and ICI( inter channel interference).

1982, Wernstei and Ebert made great progress in OFDM.DFT was applied to the demodulation of OFDM. For solving the problem of ISI and ICI they inserted symbol in the time domain. [16] Peled and Ruiz brought the idea of (cyclic prefix, CP) to make OFDM better on the Orthogonality.

As the development of DSP, the advantages of OFDM have become obvious; the technologies of Fourier Transform and demodulations have been applied very well in OFDM. OFDM has a very good future in communication system.

## 1.2 Disadvantages and advantages of OFDM

## 1.21 Advantages of OFDM

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Three are five main advantages of OFDM:

OFDM can efficiently reduce inter symbol interference (ISI) which is caused by interferes with subsequent symbols. After the high bit rate signal is transmitted through the series-to-parallel part, the length of each subcarrier is longer than before. Also there is another method to remove ISI which is mention in the last part-cyclic prefix. [17]

In the past, a lot of filters are used to separate the subcarriers. This method is simple but the efficiency is low, also more and more filters are needed. [18] [19] [20] In OFDM system, the overlap can be allowed because of the Orthogonality, so comparing with the other method, OFDM is more efficient especially in the area of frequency spectrum.

The modulation of OFDM is achieved by FFT or DFT. As the development of DSP, FFT and IFFT will be simpler to achieve.

OFDM can be easily combined with other system, like OFDMA system. Also MC-CDMA and OFDM-TDMA are the result of combination of OFDM. [21]

As the wireless has the characteristic of selectivity, it is impossible for all the subcarriers to have the deep fading. So in the OFDM system, it is more efficient to use the high SNR subcarrier in order to have the better performance. [22]

## 1.22 Disadvantages of OFDM

Although OFDM is the most popular technology in mobile communication, it has two main disadvantages.

OFDM system is easily affected by frequency; this is because there is overlap on each subcarrier. The orthogonality must be exactly right. As the transmission is changeable in wireless channel, there will always be frequency shift in the transmission like Doppler shift. The frequency shift will always affect the orthogonality of the subcarriers which could result ICI. This is the main disadvantage of OFDM. [23]

OFDM has high peak-to-average power ratio(PAPR), comparing with single-carrier, the output of modulation in OFDM is the addition of many subcarrier signals. The power of the addition is much bigger than the average power which result PAPR. [24]

## 2 OFDM System

## 2.1 OFDM definition

## 2.11 Single carrier modulation

Before discussing OFDM, single carrier modulation should be learnt. In single carrier modulation, the input information is directly modulated by the one and the only one carrier. There is one major drawback in this method: if the carrier is broken during the transmission, the system will be destroyed. Figure one shows the single carrier modulation. [25]

C:\Users\xiaojiba\AppData\Local\Temp\D1W18_W{}$01641DK77[6YH.jpg

Figure Single Carrier Spectrum

## 2.12 FDM Technology

FDM refers to Frequency Dividing Multiplexing which is a form of multiple carrier modulation (MCM). Assuming high data rate information is going to be transmitted, this situation is very common in the communication because the data rate of most information like speech, image and video is very high. Instead of transmitting the signal directly into the channel (Single carrier modulation), FDM converts the information into several sub-information and then passes them to corresponding subcarriers which will be modulated. [26]

Comparing to single carrier modulation, FDM can still transmit the signal if one of the subcarrier is broken. However, this method also has one main disadvantage: it waste too much bandwidth to prevent there is no interference between two subcarriers. Figure 2 shows the FDM principle.

Figure Signal spectrum in FDM

## 2.13 OFDM technology

OFDM refers to Orthogonal Frequency Dividing which is the development of FDM. It is very similar to FDM and the only difference is that in OFDM system every two adjacent subcarriers are orthogonal, this means that there can be overlap between the subcarriers, so it can save almost 50% of the bandwidth. [27]

In OFDM, the subcarrier contains two sine or cosine functions which are orthogonal. The orthogonality means in the (

## =

In the sine function it is the same as above.

As mentioned above, for the frequency is from satisfy the signal of one subcarrier is:

It can be seen from the function, the subcarriers are orthogonal. The signal would be orthogonal after demodulation as long as the synchronization is exactly right. The formula of OFDM is:

S(t)=

This formula is same as FDM, but the spectrum is different from the FDM's. The subcarriers in OFDM are narrower than that in FDM. Figure 1 shows the OFDM principle.

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Figure Signal Spectrum In OFDM

## 2.2 OFDM System Description

## 2.21 OFDM System structure

Figure OFDM system structure

As shown in the figure above, there are three main parts which are Transmitter, Channel and Receiver in OFDM system. The Transmitter and Receiver are discussed together as the latter is just the opposite process of the former in order to recover signals to the original ones.

## 2.22 Transmitter and Receiver

## 2.22.1 Encoding and Decoding:

The encoding technology converts the binary information to its input and generates a codeword including not only the original information but also the redundant bits. The Decoder is used to recover the message to the original information. In the project, convolutional code is used. [28]

Convolutional code is an error correcting code. The structure is shown below:

Figure Convolutional Coding Structure

As shown above, m(t) is the input binary bits. There are two outputs in the structure defined and . is the generator polynomial. In OFDM system, polynomial is applied and its binary form is 001011011, 001111001. Therefore the generator polynomial is (1+ and (1+. The structure of is shown below:

Figure Sonvolutional Encoding Structure

## 2.22.2 Interleaving and Deinterleaving:

Interleaving re-arranges a sequence of bits to new positions so as to reduce error. The deinterleaving performs the inverse process. [29]

There are two classes of interleaving which are block interleaving and Pseudo-random interleaving. In this project, block interleaving is applied. The process of block interleaving is shown below:

Figure Principle Of Interleaving

The original message is 1011100100011010. After the interleaving the message has become 1101000010011110. Therefore the burst error in the message is converted to a single bit error and it is not serious problem.

## 2.22.3 Modulation and Demodulation:

Analog signal is modulated to digital signal before the transmission and also the digital signal is modulated to analog signal before receiving. This process is called modulation. There are 4 kinds of modulations in digital communication which are Amplitude Shift Keying (ASK), Frequency Shift Keying(FSK), Phase Shift Keying (PSK) and finally Quadrature Amplitude Modulation(QAM). In the OFDM system, PSK and QAM are most useful in the practice. [30]

## 2.22.31 ASK and FSK

There are three characteristics in an analog signal which are amplitude, frequency and phase. Analog signal would change to digital signal after changing any of the three characteristics or changing two of three (QAM).

In ASK, the amplitude of the carrier signal is defined to represent the digital level of the signal. For example, the high amplitude could be defined as 1 and low amplitude is defined to be 0, or vice versa. However, the designers never use ASK because it is affected by noise a lot.

In FSK, the variation of frequency is defined to be 0 or 1. It is better than ASK because it avoids the problems form the noise. However, the physical capability of the carriers is a big problem in the practice.

## 2.22.32 PSK

The variation of phase is defined to be binary 0 or one in PSK. It simply avoids the noise which is caused by the high amplitude. PSK could be extended to M-PSK in order to represent more bits. In this case, the phase would be shift in a smaller value. For example, in 8-PSK, is shift instead of but it can represent 3 bits.

There is a main disadvantage of PSK that is need of the high quality of the equipment so as to distinguish the difference of the phase.

## 2.22.33 QAM

QAM is the combination of PSK and ASK. Both phase and amplitude are changing when the modulation is processing. Also noise is a problem using the variation of amplitude in QAM but the variation is smaller than that in ASK, so QAM becomes the most popular method in OFDM system design. [31]

A 16-QAM diagram is given below:

C:\Users\xiaojiba\AppData\Local\Temp\6K_338}1950S(_DBDD62@ZH.jpg

Figure Modulation Using 16-QAM [8]

## 2.22.34 Modulation in OFDM

One OFDM symbol contains many subcarriers signal which are modulated and in each subcarrier is modulated by PSK and QAM. One OFDM symbol can be written as:

S(t)=,

A basic modulation model of OFDM is given below:

C:\Users\xiaojiba\AppData\Local\Temp\GVA5MFDU}D2R51988{Y__20.jpg

Figure Basic Modulation Model

## 2.22.4 S2P and P2S:

Series to parallel and parallel to series perform the task of conversion of the signals. The principle of OFDM is to transmit the series data in parallel form in order to reduce the ISI and ICI. In the receiver P2S performs the inverse process.

## 2.22.5 IFFT and FFT:

FFT is the most important part in the OFDM system. First, the formula is given :

πik/N)

It can be seen that is the inverse discrete Fourier transform (IDFT). Similarly, at the receiving part, is the discrete Fourier transform of. So after the DFT the in time domain has been changed to in frequency domain. [32]

In practice, fast Fourier Transform (FFT) is applied instead of DFT. This is because DFT is more complex than FFT. For N-point the number of multiplications is in DFT, but in FFT only (N/2)lo(N) multiplications are needed. As the number of subcarriers is increasing, more and more multiplications are needed.

There are other algorithms of FFT which can be simpler than the form mentioned above.

## 2.22.6 Cyclic Prefix and Cyclic Prefix removal:

Guard interval should be added after the IFFT step in order to reduce the inter symbol interference. In this project Cyclic prefix is used. The principle of cyclic prefix is to move the last few bits of the IFFT output to the front of it so that the symbol period is longer than the channel spread. [33] The processing of adding CP is shown below:

C:\Users\xiaojiba\AppData\Local\Temp\S(IF[Z[@I1AR1V325Q]FQSQ.jpg

Figure Principle of Cyclic Prefix

## 2.3 Channel

Channel in communication system refers to the physical media which is applied to transmit the signal from the transmitter to the receiver. There are plenty kinds of media such as the wired line, optical fiber lines. [34]

In OFDM system, the channel is wireless that is to say the air is the physical media. Wireless channel brings a lot of advantages. For example, there will be no cable and area problem. Also it is more robust against the disaster. However, wireless channel also gives much more noise than wired channel such as the random noise- additive white Gaussian noise (AWGN) and man-made noise.

## 3. PAPR reduction:

## 3.1 PAPR Description

Peak to average power ratio (PAPR) is equal to the average power of the signal divided by maximum power of the signal. [35] The equation in term of dB is shown below:

PAPR (dB) =10

## =

is OFDM symbol which is the sum of all the signals had been modulated respectively. It is easy to calculate the maximum equal to N and the average power is equal to 1and the PAPR function is equal to N. This means as the subcarriers increasing which should be as many as possible the PAPR is increasing too. The PAPR is so high that the signal will in huge distortion which is not expected. So it is necessary to reduce the PAPR to improve the system performance. A PAPR figure is shown below:

4.jpg

Figure Peak to Average Power Ratio

## 3.2 CCDF of PAPR

In order to display the performance of PAPR technology, complementary cumulative distribution function (CCDF) which is commonly used in practice is introduced. [36] CCDF of PAPR means the possibility that PAPR of the message exceeds the threshold. [37]

The derivation of CCDF of PAPR is given below:

First, the CDF of the amplitude is:

F(k)=1-exp(k);

Then, CCDF is derived as

P(PAPR>k)=1-P(PAPR≤k)

## =

=1-

In this derivation one assumption is made: the signals samples are uncorrelated.

An example is given below to show how CCDF can display the performance of PAPR:

Figure Example of CCDF of PAPR In Selective Mapping

The CCDF is commonly used as the above figure, the blue one is the original CCDF of PAPR while the purple one is the one after using SLM method. The horizontal axe shows the threshold while the vertical axe shows the possibility. For example, the original signal has a PAPR that is larger than 11.1 dB with a possibility of 0.1 percent. On the other hand, after using SLM method the signal has a PAPR which is larger than 6.4 dB with a possibility of 0.1 percent. It also can be seen from the figure that by applying the SLM method gives a reduction from 12.1 dB to 6.6. Therefore this method gives 5.5 dB reductions. Generally speaking, the closer the CCDF line to the vertical axis, the greater its PAPR reduction performance.

## 3.3 PAPR reduction methods

There are a lot of methods to reduce the PAPR. In this project 8 methods are used which are clipping and filtering, windowing, coding, selective mapping, partial transmit sequence, pulse shaping, tone injection and tone rejection. These are discussed respectively.

3.31 Clipping and Filtering

In OFDM system, as the possibility of appearing maximum power is very small, clipping technology is the most direct way to reduce the PAPR. However, clipping is a nonlinear process which would bring the in band distortion and out band interference. This could reduce the BER performance and the spectrum efficiency. Although the out band interference is reduced by add the filter after clipping, PAPR comes up again. If the digital signal is clipping directly, all the clipping noise will fall in to the in band, furthermore, this noise cannot be reduced by the filter. [38]

To prevent this, several 0s is added after the input signals and these signals are sampled using longer IFFT. Note that, if the threshold is achieved, the magnitude is filtered.

## 3.31.1 Analysis in mathematical

Assuming A={} is the original input sequence with the number of carriers N, is the information of carrier K. The OFDM symbol with the central frequency 0 then can be represented as:

S (t) =, 0≤t<

is the period of OFDM symbol.

The OFDM symbols are sampled at the âˆ†t=The signal then can be represented as:

=s(nâˆ†t)=

N=0,1,…,JN-1,

is the phase shift with the definition:

## =-

This phase sift is used to help analyses. The equation can also be written in other form which is:

As can be seen above, continuous OFDM signal can be estimated by JN points IDFT.

Filtering:

The function of low pass filter is to filter the out band interference which is generated by clipping. The signal after clipping and phase shift is processed by IDFT, the JN points sequence generating distortion is given as:

## ={,,…,}

Here, =DFT(JN,{}) is the original sequence after distortion. So the original sequence is:

## ={,,…,}

This sequence then is modulated by N points IDFT, in other word, J=1. Finally, the clipping and filtering is design successfully.

## 3.31.2 Clipping and filtering simulation result and analysis

Simulation is based on Matlab 2009b in this project. The number of carriers is 256 and the length of the cyclic prefix is 25% of the data length. Block interleaving and convolutional coding are also applied and QAM is the modulation mode. Assuming the channel is AWGN channel, calculate 100 times and four times clipping and filtering in every loop. The result is shown below:

Figure CCDF of PAPR Using Repeated Clipping and Filtering

It can be seen from figure, the original PAPR is reduced from 11 dB to 8.1 dB for first clipping and filtering. After four times clipping and filtering, original PAPR has been reduced to 6.2 dB. Therefore repeated clipping and filtering can greatly reduce PAPR of the signal. The out band interference is not increased as the new filter is used. In practice, this method is limited only by in bad noise. On the other hand, as the clipping noise is introduced at the transmitter, it would fad by the channel effect so that the BER performance is a little better.

The following figure shows the relationship between BER and SNR:

Figure BER Performance of OFDM System Using Clipping and Filtering

## 3.32 Windowing:

Although clipping and filtering is the most simple and direct way to reduce PAPR, the process of clipping and filtering is a nonlinear operation. This nonlinear operation will bring the in band noise and out band reflection. Actually, clipping is the multiplication of OFDM signal and rectangle function. When the signal magnitude is small than the threshold, the value of the rectangle function is 1. When the magnitude is beyond the threshold, the rectangle function value is less than 1. The OFDM signal spectrum after clipping is the convolution of the spectrum before clipping and the spectrum of windowing. [39]

In order to solve the problem of out band interference of clipping, non-rectangle window function is multiplied by the larger peak value. Actually if the better spectrum can be received, any window function is allowed to use. In order to reduce the out band interference, the ideal window should have the narrowest bandwidth. On the other hand, in the time domain, the window function should not be too wide since the wide window means more sampled signals are affected so that the BER performance is bad. The better window functions are cosine, Kaiser and Hamming.

## 3.33 Selective Mapping method

The principle of selective mapping is that the output of the modulation is multiplied by D different phase sequences and transmitter chooses the result with the lowest PAPR to transmit. For example, the output of the modulation is A=[1 1 1 1 1 1 1 1 ] and there are four phase sequences which are=[ 1 1 1 1 1 -1 -1 1], =[ 1 1 1 1 -1 -1 -1 -1], =[1 -1 1 -1 1 -1 1 -1] and =[-1 -1 -1 -1 1 1 1 1]. Therefore there are four results after A*, A*, A* and A*. Among these four results, A* has the lowest PAPR, so is chosen to be the phase sequence and transmitted.

The receiver must perform the exactly opposite operation like the transmitter to recover the original message. So the receiver need to know which phase sequence has been transmitted. The simplest way to do this is that transmitter transmits another package which includes the information of the phase sequence but it must make sure there is no error in that package. [40] [41]

So the phase sequence will be encoded too. Generally for a D-carrier transmitter, it will need bits to transmit.

In practice, it is very convenient to choose the phase sequence from { ±1, ±j } as it does not need to do the multiplications so that the hardware complexity is greatly reduced. Also it is suitable for any kind of modulation methods. However, the disadvantage of SLM is that it needs D times IFFT operations. So the cost of SLM is higher than other methods.

There is another method in SLM which is the implementation of the previous one. The phase sequences are not random anymore. They are also different but generated in an m loop. As the spectrum of m is often flat, the spectrum of the result is also flat so as to reduce the peak value.

## 3.33.1 Selective mapping simulation result and analysis

Simulation is based on Matlab 2009b in this project. The number of carriers is 256 and the length of the cyclic prefix is 25% of the data length. Block interleaving and convolutional coding are also applied and QAM is the modulation mode. Assuming the channel is AWGN channel, calculate 10000 times. The result is shown below:

Figure CCDF of PAPR in OFDM System Using Selective Mapping

Notice that in this OFDM system, there are 32 phase sequences which have been divided into five groups: M=1, M=2, M=4, M=8, M=16 and M=32. M=1 means there is only one phase sequence which is multiplied with the output of the modulation. Actually, M=1 makes no changes of the PAPR, in other word; it can be seen as the original signal. M=2 means there are two phase sequences which are multiplied with the modulated signal. After the multiplications, the one between the two phase sequences with lower PAPR is chosen and transmitted. The principle of M=2, M=4, M=8, M=16 and M=32 are all the same.

It can be seen in Figure 15, with M=1 PAPR exceeded 11 dB with a probability of 0.1 percent. With M=2 PAPR exceeded 9.6 dB with a probability of 0.1 percent. When the M is equal to 32, PAPR exceeded 7.2 with a probability of 0.1 percent. This result gives the best number of phase sequence which is 32. Generally speaking, the larger number of the phase sequences, the better the PAPR reduction performance.

Figure BER Performance in OFDM System Using Selective Mapping

## 3.34 Partial Transmit Sequence (PTS)

The principle of PTS is similar to SLM while the structure of phase sequence is different. First, the output of the modulation is converted to V sub-vectors which must not overlap. Therefore the result is:

D=

Each of the subcarrier is multiplied by the same phase sequence, so

## )

## =

## =

The principle of the deviation is based on the linear property of IFFT, this is better because d can be constructed after IFFT, therefore it does not need additional processes of IFFT. The structure is shown below:

Figure Transmitter Structure of OFDM System Using Partial Transmit Sequence

Similarly to SLM, the phase sequences can be chosen from { ±1, ±j } in order to reduce the number of multiplications. [42]

As mentioned above, the output of the modulation is converted to V sub-vectors. There are three ways which are adjacent, interleaving and random form. Whatever form is chosen, it must make sure there is no overlap of the sequence. The structures of these three forms are shown below:

Notice that all the sequences are divided by two in these three figures and assume the result of the modulation is [1 1 1 1 1 1 1 1 1 1]

1): interleaving form

Figure Interleaving Form of Data Division

By interleaving, the sequence is converted to two subsequences which are [0 1 0 1 0 1 0 1 0 1] and [1 0 1 0 1 0 1 0 1 0]

2): random form

Figure Random Form of Data Division

Using the random form, the sequence is converted to [ 0 1 1 1 0 0 0 1 0 0] and [1 0 0 0 1 1 1 0 1 0]

3): adjacent form

Figure Adjacent Form of Data Division

In the adjacent form, the sequence is converted to [ 0 0 0 0 0 1 1 1 1 1] and [1 1 1 1 1 0 0 0 0 0].

Among these three forms, random form has the best performance to reduce PAPR.

At the transmitter, the signal with the lowest PAPR is transmitted. Also the transmitter should transmit the information of the phase sequences so that the receiver can know which phase sequence is used. If each the subsequence is modulated by different mode, then no phase sequence information is needed to transmit. Therefore the receiver does not need the inverse process and the structure of receiver can be simplified.

## 3.34.1 Implementation of PTS

There is a new method to choose the phase sequence. The phases of the samples can be seen as continuous phases and choose the suitable phase of phase sequence in order to make the amplitude of the samples small. The best phase sequence can be found by repeating the above process.

The output signal of the PTS is given by:

Y= *

Where J is the oversampling factor and N is the number of the carriers. Θ is the phase sequence. The principle of PTS is to choose the best θ to reduce the PAPR. Choose the best θ to make the magnitude of y smallest. In other word, this can be done by make the max|y(θ)| smallest.

For a given θ,

And

Choose

Θ=

Therefore the lowest magnitude of the sample signal is:

## -+-…

From the above equation, it is easy to find phase sequence which can always reduce the signal magnitude. The next step is to repeat this process to find the best phase sequence which can best reduce PAPR.

## 3.34.2 Partial Transmit Sequence simulation result and analysis

Simulation is based on Matlab 2009b in this project. The number of carriers is 256 and the length of the cyclic prefix is 25% of the data length. Block interleaving and convolutional coding are also applied and QAM is the modulation mode. Assuming the channel is AWGN channel, calculate 10000 times. Interleaving form is used when dividing the vectors .The result is shown below:

Figure CCDF of PAPR in OFDM System Using Partial Transmit Sequence

It can be seen in the above figure, the PAPR of the original exceeded 11 dB with a probability of 0.1 percent while using PTS method the PAPR exceeded 11 dB with a probability of 0.1 percent. It has shown PTS method can reduced PAPR. However, in this simulation the PAPR reduction performance is not as good as the one using Selective Mapping method. This situation is caused by two reasons: one, there are 32 phase sequences in Selective Mapping part while there are only 4 phase sequences here. Another reason is that the signal here is divided using interleaving form rather than random form, random form is the best one to have good performance. Note that random form will increase the complexity and the system speed. By these two reasons the PAPR performance is worse than that using Selective Mapping method.

Figure BER Performance in OFDM System Using Partial Transmit Sequence

## 3.35 Pulse shaping

Pulse shaping is another method to reduce PAPR. It chooses the suitable time domain impulse of each subcarrier in OFDM modulation so that the transmitter does not need additional IFFT process. [43]

The structure of PS method at the transmitter is shown below:

Figure Transmitter Structure of OFDM System Using Pulse Shaping

N is the number of the carriers. There are 3 steps in the above structure.

First, the message is modulated and the symbol rate is 1/.

Second, the modulated data is converted from series to parallel form.

Third, each subcarrier is multiplied by the impulse and transmitted separately.

From the above structure, the OFDM symbol can be written as:

x(t)=

is the modulated data on subcarrier k. T is length of OFDM symbol. is the impulse on k subcarrier.

The definition of PAPR is :

PAPR (dB) =10

For a OFDM system with MPSK mode, assume the denominator is 1, the maximum PAPR is therefore written as:

So is the function of N subcarriers and the impulse on each subcarrier. The PS method to reduce PAPR is then derived: if there is a suitable group of impulse , PAPR can be reduced.

For a given OFDM system with N subcarriers, if there is a group of different impulses for the subcarriers and

## ;

the maximum PAPR of the signal is ≤N. is equal to N if and only if the pulse is rectangular and this situation is the normal OFDM system. This result suggests that every shape of the pulse can reduce except for the rectangular pulse.

## 3.36 Encoding Techniques.

The principle of Encoding techniques is to use the different codes which are form different encoding techniques to choose the smaller PAPR code for transmission. [44]

For a BPSK OFDM system with N subcarriers, all the possible PAPR values are calculated below:

K1

K2

K3

K4

PAPR(dB)

0

0

0

1

2.32

0

0

1

0

2.32

0

1

0

0

2.32

0

1

1

1

2.32

1

0

0

0

2.32

1

0

1

1

2.32

1

1

0

1

2.32

1

1

1

0

2.32

0

0

1

1

3.73

0

1

1

0

3.73

1

0

0

1

3.73

1

1

0

0

3.73

0

1

0

1

6.02

1

0

1

0

6.02

0

0

0

0

6.02

1

1

1

1

6.02

Table PAPR Values of Different Combinations of Data

It can be seen from the figure, the above eight code words which are {0001 0010 0100 0111 1000 1011 1101 1110} have smaller PAPR. If there is a coding method which can encode the data to the above eight code words the PAPR can be reduced.

Using Golay Complementary Sequences (GCS) is a good way to encode the data for two reasons: first, this technology can reduce the PAPR to less than 3 dB. Furthermore this method has no relationship with the input data and the number of subcarriers. Second, GSC has the ability of error correcting, so the performance of the system is better.

## 3.37 Tone Injection and Tone Rejection

Correction vector is applied to reduce PAPR in Tone Injection (TI). Using this method, QAM constellation is expanded so that one data is represented by many points. PAPR is reduced by choosing the suitable constellation points. [45]

The figure is shown below to explain the principle of TI:

Figure Expanded Constellation in Tone injection

The solid points are the original constellation and TI expand this constellation into 9 large ones, so one data is represented by many expanded constellation. Receiver must know the information of this mapping method in order to recover the message.

The principle of Tone Rejection (TR) is to change the frequency spectrum of the noise so that the noise is distributed in the high frequency area while SNR is low.

## 3.4 The standard of choosing method

This project is to implement a new method to reduce peak-to-average power ratio (PAPR) in order to have better performance in the practice. [46]

First, think about how to know the method to reduce PAPR is good or bad.

There are four factors which should be considered:

(1): The complexity of the system. Basically, the complexity of OFDM system should be low. If the system is complexity, it is very difficult to achieve in the practice.

(2): the number of the subcarriers. The characteristic of OFDM is that it can change the series subcarriers in to parallel forms which are orthogonal through the channel, so the number of subcarriers must not be limited. A good method to reduce PAPR should accept any number of subcarriers.

(3): method of modulation: it should not be limited and chosen by the customer according the situation of need.

(4): the performance of OFDM system: the comparison should be done when choosing the method to reduce the PAPR. For example, in the same environment, (bit error rate) BER is a good way to compare. Choose the better performance for OFDM system.

## 3.5 Comparison of PAPR reduction methods

PAPR is the main problem of OFDM technology because low PAPR reduces the consumption of the energy and increase the efficiency of the power amplifier.

PAPR reduction can be divided into three large areas: Clipping technology, Coding technology and Possibility technology. Each of the method has its advantages and disadvantages.

## 3.51 Clipping technology:

Clipping and filtering, window and window cancelling methods are parts of this technology. The main advantage of clipping is simple and direct. However it is nonlinear operation so that it will bring the in-band noise and out-band interference. Filtering after the clipping will reduce the out-band interference but the peak value of PAPR could appear again.

In order to reduce the out-band interference, window method is introduced. It does not use the rectangular window but using the non-rectangular window like Hamming window to do the multiplication with the power that is exceeded the threshold. However, the performance is worse than clipping and filtering because it brings more distortion of the signal.

## 3.52 Coding technology

Coding technology is linear operation and completely avoids the peak value of the signal. It will not bring the distortion comparing with clipping technology. However, the complexity is much higher than clipping technology. The main disadvantage is that it reduces the data rate so this method is only suitable in the OFDM system with less subcarriers.

## 3.53 Possibility technology

There are five main methods in this technology which are selective mapping (SLM), partial transmit sequence (PTS), pulse shaping (PS), tone injection (TI) and tone rejection.

Clipping and coding technology is based on reducing the peak values of the signal. Possibility technology is base on reducing the probability of appearing peak value of the signal. This technology is based on the linear operation so that there is less distortion than clipping. The disadvantage is that the calculation is very complex.

A figure is shown for comparison of these methods:

## ã€€

Distortion

Power

Data Rate Loss

Clipping and Filtering

Yes

Decrease

No

Windowing

Yes

Decrease

No

Window Cancelling

Yes

Decrease

No

Coding

No

No

Yes

Selective Mapping

No

No

Yes

Partial Transmit Sequence

No

No

Yes

Pulse Shaping

No

No

Yes

Tone Injection

No

Increase

No

Tone Rejection

No

Increase

Yes

Table Comparison of Methods of PAPR Reduction

It can be seen in the above figure, each method has its advantage and disadvantage, so there is no best method. Method must be chosen according the practical situation. For example, if the system does not care the processing speed and only using MPSK to modulate the signal, coding is a good one to choose. Finally, although there is no best method, the probability technology has a better future than other methods.

## 4 Conclusion

OFDM technology is the most popular communication technology in the modern world. However, PAPR problem is the major disadvantage of OFDM system. In this project, the principle of OFDM technology has been shown and the disadvantages and advantages have been talked. Also PAPR problems is main part of this project and eight methods have been discussed to reduced PAPR which are Clipping and Filtering, Windowing, Coding, SLM (Selective Mapping), PS (Pulse Shaping), PTS (Partial Transmission Sequence) Tone Injection (TI) and Tone Rejection (TR). These eight methods have been further divided into three large areas which are clipping technology, encoding technology and probability technology. Clipping and encoding method is base on reducing the peak value of the signal while probability is base on reducing the possibility which the peak values could appear. Probability technology has a good future in PAPR reduction. Now, there is no best method to reduce PAPR. Methods should be chosen according to the practical situation. Finally CCDF is a good measure of PAPR reduction performance. The simulation result is given to shows the relationship between PAPR and CCDF.