FRAT-OFDM vs . FFT-OFDM Systems in Fading AWGN Channels

UDC 621.396.946.2 Al-Shuraifi Mushtaq, Al-Anssari Ali Ihsan, Mikhail Reznikov, Ph.D. 1 Taras Shevchenko National University of Kyiv, Radio Physics Faculty, Department of Radio Engineering and Radio electronic systems, 64/13, Volodymyrska Street, City of Kyiv, Ukraine, 01601, e-mail: Mushtaq_talib2005@yahoo.com, e-mail: mir@univ.kiev.ua. 2 O.S. Popov Odesa National Academy of Telecommunications, Faculty of Telecommunication Systems and Network, 1, Kuznechnaya Street, City of Odesa, Ukraine, 65000, e-mail: ali_eng87@yahoo.com.


Introduction
In this paper, a block diagram of FRAT based on OFDM was designed to decrease Signal to Noise ratio (SNR) and to make comparison between fading channels.
The Study of Finite Radon Transform in Face Recognition was studied by Zhang Yuhua and Wang Xin [1]; proposed implementation of modified FRAT Based OFDM Transceivers was studied by Ali A. Ali and Laith A. Abdul-Rahaim [2], and Design and Simulation of a Proposed Radon Based OFDM Transceiver was studied by Omar M.M. Faraj [3] to proposed FRAT for the first time as a new technique in the implementation of OFDM.
In this paper the 4QAM and 16 QAM modulations were used in small-scale fading channels to improve the increasing gain (dB) in the transmission rate.

Small-Scale Fading Channels
Small scale fading occurs due to multipath components in a channel. It consists of two independent mechanisms: the time spreading of the signal, and the time-varying behaviour of the channel. The time-varying behaviour of the channel causes Doppler shift. The Doppler shift is a measure of the spectral broadening of the signal, caused by the relative motion of the receiver with respect to the transmitter. The Doppler shift ( ) f d is a function of the speed of the receiver. The Doppler shift in the frequency can be written [4]: Where ( ) f d is the change in the frequency of the source seen at the receiver (Doppler frequency), f is the frequency of the source, v is the speed difference between the source and transmitter, and c is the speed of light. This Doppler shift occurs whether the channel is described as slow fading or fast fading. Thus, in a fast fading channel, the coherence time, c T , of the channel is smaller than the symbol duration of the transmitted signal (i.e. >> s c T T ). In a slow fading channel, the channel impulse response changes at a rate much slower than the transmitted baseband signal, ( ) s t , and the symbol period of the signal is much smaller than the coherence time of the channel << s c T T [5].

FFT-based OFDM System
The block diagram of a given system for OFDM is depicted in Fig. 1. First of all, the input serial data stream is formatted into the word size required for transmission, e.g. 2 bit/word for QPSK and 4 bit/word for 16-QAM, and then the format will be shifted into a parallel format. The data is then transmitted in parallel by assigning each word to one sub-carrier in the transmission. After that, the data to be transmitted on each sub-carrier is mapped into one of M -ary PSK or M -ary QAM constellation format, as determined (In addition, differential-PSK (DPSK) and differential-QAM (DQAM) are available). This process will convert data to corresponding value of M -ary constellation which is a complex word, i.e. consists of real and imaginary parts. The training frame (pilot subcarriers frame) will be inserted and sent prior to information frame. This pilot frame will be used to make channel estimation that is used to compensate the channel effects on the signal. After that, the complex words frame and pilots frame will pass to IFFT to generate an OFDM symbol. Zeros will be inserted in some bins of the IFFT in order to make the transmitted spectrum compact and to reduce the adjacent carrier's interference.

Modified FRAT-Based OFDM System
The method has been suggested to apply the Optimum Ordering algorithm to a square matrix of any dimension, especially powers of two, i.e. typical sizes of FFT windows. Thus, in a similar manner to several previous theses, we were able to propose an OFDM system that uses this Modified FRAT as a 2D modulator, other than a mapper. The size of its input window will be 8 8 x , and resulting in an 8 12 x output matrix, i.e. 96 symbols, thus providing a reduction in the required symbol number for each OFDM packet, compared to 128 symbols for conventional OFDM scheme. This system is illustrated in Fig. 2; it differs from the FFT-based OFDM system in several points: The modulator that replaces FFT is twodimensional, i.e. modulates a matrix of data rather than a sequence or a vector.
The method by which the pilot carriers are arranged with the data symbols is essentially a merging process, that converts the pilot sequence into a matrix equal in size to the data matrix, then constructs a bigger matrix in which the rows pilot matrix become the odd rows, and of the data matrix the even ones. The resultant matrix becomes the input to the FRAT mapper.
The zero padding happens inside the modulator, in the optimum ordering step, rather than outside it. This will be discussed thoroughly below.
If we return to the Optimum Ordering pattern, given by the equation 2, (2)

Fig. 2. Block diagram of Modified FRAT-based OFDM
We will notice that the recurring pattern of zeros is in the same locations for several columns (in this case, half of the columns), which is due to two reasons: The first set of columns contains most of the elements in the given directions, so that when the remaining columns sweep the input matrix in other directions, they will find several used elements in the way, i.e. they intersect with the previous  Al-Shuraifi Mushtaq, Al-Anssari Ali Ihsan, Reznikov Mikhail, 2014 directions. As the proposed algorithm states, so that the used elements are marked by zero.
As the procedure aims to generate an independent set of projections, each column in the output matrix has the same pattern that exists in the result vector of 1D FFT that is the complex elements in the upper half are copied as conjugates in the lower half. This note is also applied to the columns including patterns of zeros, as the remaining elements from the sweep are reordered based on the same principle.
This system was applied for two cases: normalized (the input matrix was divided by its highest value), and non-normalized. The nonnormalized case showed better performance than the normalized one. It was found that the power level of the signal can be increased by at least 10 dB. Fig. 3 illustrates the BER performance of the Modified FRAT-OFDM system versus that of the FFT-OFDM, and Fig. 4 depicts the frequency spectrum of a modulated packet in each case.

Simulation and Results
The design of FRAT-based OFDM systems were illustrated and explained in this work. The work gave the simulation results of the proposed systems using MATLAB. The results of both studied systems were taken in three types of channels, and were examined and compared with others. The effects of several parameters of wireless channels on the two systems were investigated. Table 1 shows the parameters of the systems that were used in the simulation.
The performance of the FRAT-OFDM system was illustrated and compared with the FFT-OFDM for the possible two types of channel as well as for two types of modulation, in two cases: normalized and non-normalized. The results obtained for AWGN channel in turn were divided into two cases and demonstrated in details in different cases as presented in table 1.

Conclusions
Fig . 5 illustrates the performance of FRAT-OFDM system for 4-QAM modulation in flat-fading AWGN channel. It can be seen that FRAT-OFDM is better than FFT-OFDM, and the gain is about 15 dB for a BER level of − ⋅ 4 2 10 . Fig. 6 illustrates the performance of normalized FRAT-OFDM (denoted nFRAT-OFDM) system for 4-QAM modulation in flat-fading AWGN channel. It can be seen that nFRAT-OFDM is better than FFT-OFDM, and the gain is about 6 dB for a BER level of −4 10 . Fig. 7 illustrates the performance of FRAT-OFDM system for 4-QAM modulation in selectivefading AWGN channel. It can be seen that FRAT-OFDM is better than FFT-OFDM with 12 dB of SNR, for a BER level of −3 10 . Fig. 8 illustrates the performance of nFRAT-OFDM system for 4-QAM modulation in flat-fading AWGN channel. It can be seen that nFRAT-OFDM is better than FFT-OFDM, and the gain is about 2 dB for a BER level of −3 10 . Table 2 summarizes all the obtained results, with respect to the SNR gain and the corresponding BER levels.