OFDM : A Mathematical Review

Mathematical review of the Orthogonal Frequency Division Multiplexing is demonstrated in terms of Inter symbol interference, Multi carrier modulated system and cyclic prefix. Modeling of the mathematical equation of the Orthogonal Frequency Division Multiplexing, Inverse fast Fourier transform and fast Fourier transform is explained with the suitable example using MATLAB. Bit error rate performance of OFDM is also presented with the help of statistical computation


INTRODUCTION
Orthogonal frequency division multiplexing (OFDM) is a broadband key technology which is employed in "4G wireless communication system". It is also used in "Long term evolution" (LTE) 4G cellular standard and Worldwide interoperability for microwave access (Wi max) also. These are now day's dominants standards and both are based on OFDM.
LTE -A long term evolution advanced which is latest communication standard in broad band is also based on OFDM. LTE is a wireless communication standard which can support a large bandwidth. In general GSM has a bandwidth of about 200KHz but OFDM can have bandwidth of about 100MHz. Naturally data rate in OFDM will be higher which is used in 3G and 4G to enable data rate of up to 100MBps or more then 500MBps. Several IEEE Wi-Fi wireless local area networks WLAN (802.11, 802.11G, 802.11N, 802.11AC) standards are used for high data rate which are based on the OFDM.

INTER SYMBOL INTERFERANCE
In traditional communication systems let we have a bandwidth and there is a single carrier frequency for it. For better understanding of this concept, let Bandwidth of a channel is 1 MHz, therefore symbol time time is less then delay spread time then we have "inter symbol interference" (ISI). This leads to a problem in communication system which degrades the performance of the wireless communication system. As bandwidth increases our symbol time decreases. This is a significant problem in the broadband channel design.

PRINCIPLE TO AVOID ISI
To avoid the "Inter symbol interference" we split the large bandwidth into smaller bands or sub bands. Therefore in each sub band we will have individual subcarrier. Let N sub bands are there then bandwidth of each sub band will be . . As now delay spread time is more than symbol time d T T >> . Therefore there is no Inter symbol interference which is done with the help of multiple sub bands and sub carriers. Such a system with a multi sub bands and sub carrier is known as a multi carrier modulated system (MCM system) [1]. This is the principle basis of the orthogonal frequency division multiplexing. j kF e π . Let the K th symbol is given by k X then the transmitted signal on K th subcarrier will be

MULTI CARRIER MODULATED TRANSRECEIVER
Net transmitted multi carrier modulated signal will be sum of transmit signals across all N subcarrier and will be defined as: - known as coefficient of the signal y(t). Where is the fundamental time period of the signal. This will give we can extract symbols on lth subcarrier. This is the main principle of the multi carrier modulated system [2].

A. Transmitter schematic
At transmitter we have (0), (1),....... ( 1) X X X N − Symbols. Serial to parallel convertor also known as demultiplexing (DEMUX) is used at transmitter to load to the symbols onto the subcarrier. Then take N pt IFFT of these symbols

B. Receiver Schematic
At receiver we have samples over the wireless channel.
− Therefore, first of all we will remove the cyclic prefix, because it consists of inter block interference. Now we have y y y N − . From these samples we have to detect symbols. Therefore, output of detector module will be because there can be error. Then finally parallel to serial conversion (Multiplexing) is there to convert these back to serial symbol stream [3].

ORTHOGONAL FREQUENCY DIVISION MULTIPLEXING
OFDM is very efficient technique because it employs IFFT/FFT, fast efficient algorithm and no matrix inversion calculation is there. Orthogonal frequency division multiplexing is used in the following technologies: -1. Modern wireless technology 2. 4 G Standards 3. LTE, Wi max, Wi Fi and 802.11 etc ISI channel is described as OFDM overcomes ISI with very low complexity and it is very efficient scheme.
( ) X l is loaded on to the lth subcarrier.

OFDM: A Mathematical Review
N point IFFT is taken at the transmitter is the prefix and known as cyclic prefix CP As we know that This is circular shift of the channel taps or channel filter over sample Therefore output will be y h x v = ⊗ + . Addition of cyclic prefix has converted linear convolution to circular convolution. In frequency domain circular convolution becomes product in FFT domain [4]. N point FFT at receiver Which is known as padding with N-L zero's, N=4 and L=2. Channel coefficient across Put N=4 we will get Malik, P. K. Tripathi, M. P.
These are noise samples across subcarrier.
From equation 10 is it clear that no ISI interface from previous symbol on each subcarrier, ISI has been removed in frequency domain. generating this signal will be difficult because of large number of subcarrier [5]. Because for each subcarrier we need an oscillator i.e. for 100 subcarrier we will be needed 100 oscillators and should be preciously place.
As we know that this principle is known as orthoganility principle. Which means that any two subcarrier k and l in this system is orthogonal.
How to overcome the problem, of large number of subcarrier can be explain as follows. Let a signal be a maximum frequency of B hence h h h L − , which says that L channel taps. From the above equation it can be shown that output does not depend on current input but also on previous input ( ) 1 x n − . Therefore in the time domain this channel has inter symbol interference (ISI ). Therefore, this input output received symbol model consists of ISI. Now let us consider the output of the OFDM system. Output for x(0) will be But if we consider previous symbol. Output for x(0) will be Now there is a problem, we have inter-block interference. We have to remove these inter block interference in OFDM. To remove inter block interference we use cyclic prefix. As we know that Transmitted samples of OFDM system (Sampled version of the MCM channel, which is transmitted over the wireless channel). Prefix of L bar samples from the tail. Now instead of directly transmitting these samples, we are going to add a prefix. We are taking L bar samples.
Now if you look at the first symbol: - Now the interference has been restricted to the same block. Therefore we have avoided inter block interference. There is ISI but there is no inter block inter- Which says that expected value of each W(k) is zero. Since the FFT is a linear combination, the input of noise samples is zero mean. Naturally at the output of the FFT W(k) is also zero mean.
Let us look the variance of each noise samples.
Noise sample w(l) are independent and therefore uncorrelated. Therefore ex- Therefore, noise at the output can be characterize as

RESULT
A MATLAB based program is written to compute the above specified methods and following are the results of the program. For simplicity of the data presentation, following terms are assumed. Number of bits per channel is 32 but can be extended upto 128 Number of subcarrier channel is 4 Total number of bits to be transmitted at the transmitter is 128 Size of each OFDM block to add cyclic prefix 16 Channel is AWGN in nature Random samples are taken for the transmission purpose. These samples are in the binary form. Sample size is kept to be 32. Random generated data to be transmitted Quadrature phase shift keying method is employed here to modulate the binary signal. Following graph shows the output of the QPSK modulator. Following graph shows the output of the receiver at the receiver. i.e received OFDM signal. As the output of the FFT we received the original signal, in this case it seems that it is almost similar to the input. For the proposed statistically computed BER Vs SNR of the OFDM system, it has been observed that it is continuously decreasing as a function of SNR.