Computational Performances of OFDM Using Different FFT Algorithms

In this paper, an intuitive comparison of the computational performance of orthogonal frequency division multiplexing (OFDM) system has been made in terms of complex calculations required using different Fourier transform techniques. The different transform techniques are introduced such as discrete Fourier transform (DFT) and various types of fast Fourier transform (FFT) as 2-radix FFT, 4-radix FFT etc. and the very recent very fast Fourier transform (VFFT). With intuitive mathematical analysis, it has been shown that with the reduced complexity that VFFT can offer, OFDM performance can be greatly improved in terms of calculations needed.


Introduction
Orthogonal Frequency Divisional Multiplexing (OFD-M) is a modulation scheme that allows digital data to be efficiently and reliably transmitted over a radio channel, even in multi-path environments [1].OFDM transmits data by using a large number of narrow bandwidth carriers.These carriers are regularly spaced in frequency, forming a block of spectrum.The frequency spacing and time synchronization of the carriers is chosen in such a way that the carriers are orthogonal, meaning that they do not cause interference to each other.In OFDM system, Discrete Fourier Transforms (DFT)/Fast Fourier Transforms (FFT) are used instead of modulators.The computational complexity of implementing DFT/FFT/Very Fast Fourier Transform (VFFT) has been calculated in an OFDM system and compared their performance.At the current state of wireless communication techniques facing ever increasing demand of high data rates, single carrier systems are offering limited solutions due to frequency selectivity of wideband channel resulting in severe complexities in equalizer design at the receiver end [2].OFDM, as a multicarrier system, has become an effective modulation technique for next generation wireless communication methods.Using FFT algorithms provides speed enhancements for data processing for OFDM systems.This technique is being used for Digital Audio Broadcasting (DAB), Digital Video Broadcasting (DVB), Wireless Local Area Network (WLAN), Wireless Metropolitan Area Network (WMAN), Multi Band-OFDM Ultra Wide Band (MB-OFDM-UWB) etc.Also it is used in wired communication systems such as Asymmetric Digital Subscriber Line (ADSL) and Power Line Communication (PLC) [3].In the next section, the basic OFDM system model using FFT has been described.

OFDM System Model
OFDM is a kind of frequency division multiplexing (FDM) technique in which we divide a data stream into a number of bit streams which are transmitted through sub-channels [4].The characteristics of these sub-channels are that they are orthogonal to each other.As the data that are transmitted through a sub-channel at a particular time are only a portion of the data transmitted through a channel so bit rate in a sub-channel can be kept much low.After splitting the data in N parallel data streams each stream is then mapped to a tone at a unique frequency and combined together using the Inverse Fast Fourier Transform (IFFT) to yield the time domain waveform to be transmitted [5].After IFFT is done, the time domain signals are then converted to serial data and cyclic extension is added to the signal.Then the signal is transmitted.At the receiving side we do the reverse process to get original data from the received one [5,6].
In case of deep fade, several symbols in single carrier is damaged seriously, but in parallel transmission each of N symbol is slightly affected.So even though the channel is frequency selective, the sub-channel is flat or slightly frequency selective.This is why OFDM provide good protection against fading [7].In an OFDM system there are N number of sub-channels.If N is high then it will be very complex to design a system with N modulators and demodulators.Fortunately, it can be implemented alternatively using DFT/FFT to reduce the high complexity.A detailed system model for OFDM system is shown in Figure 1 [6,7].

Discrete Fourier Transform (DFT) Algorithm
For a sequence of data x(n) of length N, the DFT may be expressed as [6]     For each value of k direct computation of x(k) involves N complex multiplications and N − 1 complex additions.So, to compute N values of the DFT, complex multiplications and N 2 − N complex additions.Direct computation of DFT is inefficient because of inability to use symmetrical and periodical properties of phase factor W N for computation.These properties are symmetry property, W   W ; periodicity property:

Radix-2 FFT Algorithm
For speed demanding applications which requires im- . So the expression can be written as, where F 1 (k) and F 2 (k) are the 2 N -point DFTs of the sequences f 1 (m) and f 2 (m) respectively.Since F 1 (k) and F 2 (k) are periodic, with period 2 N , we can evaluate: For Equations ( 5) and ( 6) After simplification, we get In case of N-data points expressed employ radix-4 algorithm instead of radix-2 algorithm for more efficient estimation.If we consider a column-wise mapping for x(n) as n = Ml + m and row-wise mapping as k = Mp + q and N be factored as N= LM, then we can write [6], Let us consider L = 4, M = N/4, so it results in, , where,

Very Fast Fourier Transforms (VFFT) puting
The VFFT is a new approach in the field of com DFT and attributed to mainly Simon Shepherd [8].It can be implemented either by various level of approximation or replacing the FFT exactly with floating point accuracy.G-Matrix is a suitable tool in this regard with different quantization level concept.
Fourier matrix F N is normalized by 1 N in the G-matrix.Important properties of G-matrix are outlined as follows [8,9]: * Non-unitary matrix , where A of the th i row of N G is defined as Row-gain is unity for the Fourier matrix, whereas N G is not necessarily unity.
the the ities the This causes non-uniform power spectral density in transmitted signal and non-uniform noise power in received demodulated signal.This is the main disadvantage in this G-matrix design.To recover this n-level quantization is applied, where n = 2 x ; x is integer.

Computational Performance Analyses
In order to compare the computational complex among the different Fourier transforms on OFDM, calculations based on the OFDM block sizes have been performed which are given in Table 1.In case of multiplication computation for VFFT, the speed improvement factor is as high as 204.8 for large block size 1024.That indicates the multiplication complexity is decreased by 99.5% and 20% from DFT to FFT to VFFT.This is due to the computational savings done by VFFT with complexity reduced to only of the order of N.This rate of complexity reduction in terms multiplication would be a huge factor for OFDM systems to meet the demand of multi-user high data traffic environment.It can also be seen that, the advantage of implementing VFFT only becomes significant as the size of OFDM block (N) is increased.
In Table 2, the comparison of the different FFT algorithms evolved has been shown.We can see that higher radix FFT algorithms reduces the number of both multiplications and additions and as such can give less complexity for OFDM calculations for larger FFT blocks [10,11].For example, for OFDM block size of 64, multiplication complexity reduces by almost 23% and addition complexity reduces by almost 6%.This less complexity leads to more robust and effective multicarrier system design for OFDM.It also assures optimum performance in terms of high speed data processing for communication systems using OFDM technique.
Figure 2 summarizes the results shown in Tables 1  and 2 for DFT, FFT & VFFT algorithms.VFFT algorithm surely can save a lot of computational power but it has been often effected by a slight degradation of 1dB bit error rate (BER) performance [8,12].But still it will provide the robust & reliable performance needed in terms of computation for OFDM systems with ever increasing demand multi user support using high data rate applications.Overall, FFT algorithms provide the different  ired Quality of Service (QoS) and n ork specifications.
an OFDM system depends on OFDM system DFT/FFT works as a omplexity of DFT/FFT decreases, then d Dr. ll their helpful advice and

Conclusion
The performance of DFT/FFT as in an modulator.If the c the speed of OFDM system increases.VFFT will certainly increase the speed of OFDM system with the drawbacks of non-uniformity of Power Spectral Density (PSD) and non uniformity of Signal to Noise Ratio (SNR) among the sub-carriers.Even though VFFT's implementation is in primary stages, but it is emerging as the compact solution for OFDM systems used in next generation communication networks.
[1] B. E. E. P. Lawrey, "Adaptive Techniques for Multi-User OFDM," Ph.D niveristy, Townsville, 2001, pp Second Asian Himalayas Internatechniques to deliver wireless communication system designs with requ et-w plementation of FFT algorithm, FFT algorithms are decimated in terms of time or frequency.If we consider N points of data points where it can be factored as