International Journal of Scientific & Engineering Research, Volume 6, Issue 5, May-2015 777

ISSN 2229-5518

Use of PAPR Control and Sub-sliding Windowing Techniques in the improvement of the Bit Error Rate in IFFT-FFT and IDWT-DWT Based OFDM Systems

Arnab Mohapatra, Prabira Kumar Sethy

Abstract— the paper aims at providing an efficient MATLAB Code in obtaining significant reduction in the Bit Error Rate (BER) Curve for two different Orthogonal Frequency Division Multiplexing (OFDM) Systems: FFT-IIFT OFDM System and an IDW T-DWT OFDM System, based on the knowledge obtained from Peak to Average Power Ratio (PAPR) Control and Sub-Sliding W indowing techniques. W hile the IFFT System has been BPSK Modulated, the IDW T system has been QAM Modulated. These modulation techniques are used in keeping with the state-of-the-art wireless communication systems and services.

Index Terms— Fast Fourier Transformations (FFT), Inverse Fast Fourier Transformations (IFFT), Peak to Average Power Ratio (PAPR), Sub-Sliding W indowing techniques, Discrete Wavelet Transformations (DWT), Inverse Discrete Wavelet Transformations (IDWT), BPSK, QAM, MATLAB 2012(B), ISI, ICI.

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ith an increase in the demand for faster and more reliable communication systems, efficient Multiple Access Tech- niques like Orthogonal Frequency Division Multiple

Access(OFDMA) techniques have become the need of the hour. OFDMA is preferred to other Multiple Access techniques like CDMA, TDMA or FDMA for the reason that it is free from the adverse effects of Inter-symbol Interference (ISI) and Inter Car- rier Interference(ICI) and hence is best suited for wireless broadband services like WiMax and WiBro. [6] OFDM allows the high rate data stream to be broken into parallel low rate data streams which are then transmitted over the channel using sub- carriers that are “orthogonal” or statistically independent to each other. This allows the overlapping of the subcarrier fre- quencies in the transmitter section of the OFDM while they can be extracted in the receiver section without any interference, thereby increasing the spectral efficiency of the system.

Due to the use of AWGN channel as a communication channel, the subcarriers are exposed to flat fading which can be corrected by using a single tap delay line filter. Because of the existence of orthogonality among the carriers with respect to one another, the system is free from any Inter Carrier Interference (ICI) as seen in CDMA and FDMA systems even without the use a guard-band. But studies have concluded in the fact that all kinds of OFDM systems are susceptible to signal degradation due to Doppler shifts between transmitter and receiver, phase noise due to channel impairment, the inability of the system to sample its bits due to signal amplitude reduction and other power related constraints.

Hence the central theme of the paper is to re-multiply the deg- radation factors to a complex baseband signal to gain improve- ment in the convergence of the BER Plot on a semi-log graph.

An FFT-IFFT Type OFDM System is the most conventional forms of OFDM system where The data symbols (dn,k) are first assembled into a group of block size N and then modu- lated with complex exponential waveform {φk(t)} After modu- lation they are transmitted simultaneously as transmitter data stream.

Mathematically, an IFFT-FFT Based OFDM System can be ex- pressed as:

Where:

• Sn (t) = symbols mapped to chosen constellation

(BPSK/QPSK/QAM etc..,)

• f n = orthogonal frequency

Owing to the fact that the OFDM signal (c(t) in the equation above ) is in time domain, IFFT convenient to be used in the transmitter section where the frequency domain samples are converted to time domain samples. IFFT/FFT equation comes handy in implementing the conversion process from time to frequency to time domain where elimination of the individual sinusoidal multipliers are required in the transmitter/receiver side. In the diagrammatic block representation given below; the critical importance of the removal of sinusoidal frequen- cies can be understood.

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Fig 1: A Standard OFDM Transmitter transformed into an FFT- IFFT Based OFDM Transmitter

In the system, the bits to be transmitted across the channel are first encoded in the coder section and then fed to the serial-to parallel converter where ‘N’ parallel bits are obtained at the output of the SIPO.

The output obtained at the SIPO is then fed to the N-IFFT; at the output of which is fed the cyclic prefix. The cyclic prefix helps in the process of circular convolution. Circular convolution, also known as cyclic convolution of two aperiodic functions (i.e. Schwartz functions) occurs when one of them is convolved in the normal way with a periodic summation of the other function. That situation arises in the context of the Circular convolution theorem. The output of the IFFT is then fed to the PISO, DAC and RF amplifier and after power control is transmitted through the transmitting antenna.

Discrete Wavelet Transformations are preferred to Fast Fourier Transformations for the simple reason that they not only orthogo- nalise the bit stream but also normalise it to reduce redundancy. Other than making the signal statistically independent, it also makes use of the Nyquist frequency to achieve normalization. A Two dimensional Discreet Wavelet Transformer can be pictorially explained as:

DWT

After Quadrature Modulation and addition of the vector, the

decomposition Initialization - CA is characterised by an inver- sion in the signs of the bits and a change from a serial to a par- allel matrix. The signal is then up-sampled and filtered by the LPF coefficients or namely as approximated coefficients.

The modulated signals perform circular convolution with LPF

filter whereas the HPF filter also performs the convolution with zeroes padding the output signal CD. HPF filter contains detailed coefficients or wavelet coefficients. Different wavelet families have different filter length and values of approximat- ed and detailed coefficients. Both of these filters have to satisfy orthonormal bases in order to operate as a wavelet transform pair. In the transmitter part, this signal is simulated using MATLAB command [Xk] = idwt(CA, CD, wv) where wv is the type of wavelet family. On the other hand, the reverse process is simulated using [ca, cd] = dwt(Uk, wv) in the receiver.

The CA signal will be processed to the QAM demodulator for data recovery. However, the CD signal is discarded because it does not contain any useful information.

The block diagram of a DWT based OFDM system could be

represented as:

Fig 3: An IDWT-DWT based OFDM system

The sub-sliding windowing technique suggests that a single acknowledgement can be used to indicate the reception of multiple packets between two host computers in a TCP-IP Layer protocol.

The OFDM system being more sensitive to frequency error than other sub-carrier systems, a frequency offset is likely to be pro- duced due to the following reasons:

• Doppler Shift arising out of the relative motion between

Transmitter and Receiver

• Phase noise due to channel impairments of neighbouring carriers

Fig 2: Decomposition and Down-sampling process in a 2D-

Thus the degradation function in term of the frequency offset

(Δf ) can be mathematically expressed as:

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Hence with the re-multiplication of the factor of 10/3 and *π to the *baseband signal, the frequency offset can be removed and a better convergence curve for BER can be created.

Where

• Ak is the complex data of the kth sub-carrier

• Ts is the OFDM symbol period

For a large value of N number of subcarriers, the mean for the Gaussian distributed values is taken as taken as 0 and the var- iance is taken as ½.

• Define FFT size

• Define the number of data subcarriers

• Define the number of bits per OFDM symbol (same as

the number of subcarriers for BPSK)

• Define the number of symbols

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• *Prabira Kumar Sethy is currently posted as Lecturer in Electronics at Sam- balpur University, Odisha,India, PH-+91-9439489214 E-mail:*

psethy@suiit.ac.in

• Define a for loop from 1 to the Length of Bit to Noise ra- tio

• Set the Transmitter by:

• Creating a random function for creating random 1’s and

0’s

• Write the modulation function formula for BPSK modu-

lation

• Group the modulation function into streams multiple symbol

• Assign the modulated symbols to subcarriers from [-26 to

-1, +1 to +26]

• Normalize the power of symbol about to be transmitted

to 1

• Append cyclic prefix onto the symbol to be transmitted

• Concatenate the multiple symbols to form a long vector

• Define a function to create Gaussian noise of unit vari-

ance, 0 mean

• Add noise to the term to keep an account of the energy wasted due to cyclic prefix

• Set the Receiver by:

• formatting the received vector into symbols

• removing cyclic prefix

• convert the symbols in time domain to frequency domain

• Carry out the process of BPSK demodulation

• Reconvert modulated values into bits

• Define a function to count the errors

• End the FOR Loop

• Define a function involving the use of Sub-sliding Win- dowing technique used to improve the BER

• multiplying variance with Simulated Bit Error Rate to

view a Peak to Average Power Ratio Controlled BER Curve

• Define a function for a Theoretical BER curve to serve as a reference.

• Define semilogies to plot the graph and also set up the

parameters for the grid

A part of the MATLAB code used in creating the system is:

• Set the bit to noise ratio parameters

• Convert to symbol to noise ratio

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• Define the Size of signal constellation

• Define the Number of bits per symbol

• Number of symbols used /bits to process

• Set the Oversampling rate at 1

• Define the FFT size

• Define the number of data subcarriers to be used

• Set QAM symbols per OFDM symbol with a value same as that of the number of subcarriers used

• Set the Bit to Noise value parameters

• Define a for loop from 1 to the Length of Bit to Noise

ratio

• Define a Signal Source by creating a binary data stream as a column vector.

• Perform the function of Bit-to-Symbol Mapping

• Convert the bits in x into k-bit symbols.

• Perform Modulation using 16-QAM.

• Group the bits into multiple symbols

• After grouping, use the function of Inverse Discreet

Wavelet Transformation

• Transmit the signal over the AWGN Channel

• At the receiver end, use the function of Discreet

Wavelet Transformation

• Perform the function of Demodulation by Demodulat-

ing the signal using 16-QAM.

• Perform Symbol-to-Bit Mapping by undoing the bit-

to-symbol mapping performed earlier.

• Convert the integers to bits.

• Convert z from a matrix to a vector.

• Go on for BER Computation by compare x and z to obtain the number of errors and the bit error rate.

• Defining the baseband signal by adding x and z

• Multiply Variance with EbNo Values to achieve PAPR control and feed it to the base band signal us- ing the erfc function

• Make the use of pi (Π) as a to create a function

shift/sliding factor

• End the FOR Loop

• Define a function involving the use of Sub-sliding

Windowing technique used to improve the BER

• multiplying variance with Simulated Bit Error Rate to

view a Peak to Average Power Ratio Controlled BER

Curve

• Define a function for a Theoretical BER curve to serve

as a reference.

• Define semilogies to plot the graph and also set up

the parameters for the grid

A part of the MATLAB Code used in coding the system is given as:

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Parameters used are given as:

FFT Size | 64 |

Number of Data Subcarriers | 52 |

Number of Bits per OFDM Symbol | 52 |

Number of Symbols used | 104 |

Bit to Noise Ratio | [1:10] |

The simulation curve obtained for the code mentioned above is given as:

Fig 4: BER Probability Curve for a BPSK Modulated

OFDM System using IFFT and FFT

It can be inferred from the plot that the BPSK system is not affected by the non-linearity of the RF Amplifier as much as it is affected by the Doppler shift or the phase noise aris- ing out of Inter Carrier Interference. Hence sub-sliding technique is a better approach in improving the BER con- vergence curve because on re-multiplication of the variance factor of ½, the PAP ratio for the system increases, causing it to be more prone to error.

The parameters used are given as:

The simulation curve obtained for the curve mentioned above is given as:

Fig 4: BER Probability Curve for a QAM Modulated OFDM System using IDWT and DWT

The plot obtained clearly suggests that both the PAPR Control Technique and the Sub Sliding Windowing technique are help- ful in bringing improvement to the convergence curve of the BER plot for the OFDM system. It also proves that the signal is susceptible to the increase in the complexity of the DAC and the ADC as well as the Inter Carrier Interference and the phase noise. The sub-sliding technique is a more reliable method in reducing the number of erroneous bits in the transmitted sig- nal. Overall, both methods are efficient.

It can be concluded that the IDWT-DWT system of OFDM has better control over the IFFT-FFT system of OFDM. Use of BER controlling techniques like the PAPR Control Technique and the Sub-sliding Windowing technique can further find use in improving the convergence curve for the BER plot; making the OFDM system best suited for Wireless Broadband Communi- cation Systems and can also be used as a Multiple Access Technique (OFDMA) in 4G Communications where these two techniques can find use making the system reliable and effi- cient.

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