# OFDM Frequency Data

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OFDM is a particular kind of multicarrier transmission scheme in OFDM .In OFDM single data stream of high data rate is transmitted over low data rate sub carriers. OFDM can be declared as modulation technique. The main advantage of OFDM is it's increased strength against frequency selective fading. In a conventional FDM system, even a single corrupted carrier can cause other carriers to corrupt, but in an OFDM system, only a little percentage of the sub carriers will be affected. The concept of OFDM became practice able after the introduction of FFT in 1960.

2.1.1 Orthogonality

In OFDM the total available bandwidth is divided in to N non overlapping sub channels In OFDM these large number of sub channels are transmitted. The frequency separation is chosen in such manner that all the sub carriers should be mathematically orthogonal to each other. Thus the avoidance of spectral overlapping of sub-carriers minimizes Inter Channel Interference. This allows the proper demodulation of the symbol streams without the necessity of no overlapping spectra. Carriers are divided with an interval of 1/T, where T represents the period of one symbol. Another way of specifying the sub-carrier orthogonality condition is to require that each sub-carrier have exactly integer number of cycles in the interval the frequency spectrum of an OFDM transmission is given below. several familiar multiplexing schemes are intrinsically orthogonal. Multiple information signals can be transmitted with TDM on single channel by assigning unique time slot to each information signal. In this way information signal from single source is transmitted in unique time slot preventing interference due to orthogonal nature of TDM. . In the frequency domain FDM is orthogonal in nature, as different information signals are transmitted on mutually independent carriers. The term OFDM has been reserved for a exceptional form of FDM.

. Condition of orthogonality between different functions can be proved as given below

If any two different functions within the set are multiplied, and integrated over a symbol period, the result is zero, for orthogonal functions.

Multiple sub carriers

0 f1 f2 f3 f4 f5 f6 f7 f8

Fig 2.1 Frequency domain representation

Where N = 1, 2, 3…

Orthogonality in sub carriers saves almost 50% of the bandwidth as compared to the conventional frequency modulation.

Fig 2.2 Traditional FDM vs OFDM

2.1.2 Guard Time

The major advantage of OFDM is its resistance against multipath fading so as to cause inters symbol interference (ISI). Inter symbol period ‘ISI' due to the multipath delay spread is reduced by increasing the symbol period (converting high data rate information signal into ‘N' parallel low rate information signals). The guard time interval is always taken larger than the channel delay spread. Data transmission is more vulnerable to ISI (inter symbol interference) and ICI (inter channel interference) without insertion of guard time. So to reduce both the ISI plus the ICI, part of the OFDM symbol is copied in to the back of called ‘cyclic prefix (CP)' is cyclically extended into the guard period. This cyclic prefix help retain the orthogonality of the sub-carriers by ensuring that the delayed replicas of the OFDM symbol always have an integer number of samples within the FFT interval. This guard interval contains no information and is removed at the receiver before demodulation. The cyclic prefix should be as long as the significant part of the channel impulse response experienced by the transmitted signal. Cyclic prefix benefit us in twin ways. The first benefit is the removal of Inter symbol interference, The second benefit is transformation of the linear convolution with channel impulse response into a cyclic convolution. As a cyclic convolution in the time domain translates to multiplication in the frequency domain, the sub carrier remains orthogonal and there is no ICI.

Cyclic Prefix [3]

Source: IEEE 802.16

Fig 2.4 *Cyclic prefix insertion[4]*

so we can do away with inter symbol interference and inter channel interference by cyclically extending the OFDM symbol into the guard period and confirming it that the delay spread is less than the guard time. If multipath delay spread exceeds than guard time only by three percent (3%), then the sub carriers will lose orthgonality between each other.

- Symbol Duration

Signal to noise ratio SNR plays an important role and must be taken in to consideration in every transmission technique. SNR always places limitations on some parameters of the signals. In OFDM symbol transmission we can minimize the SNR loss by taking the symbol duration larger than guard time, because guard time interval contains no information bits. On the other hand large symbol duration means more number of sub carriers and thus adding complexities. A realistic design option for the symbol time is to be at least five times the guard time, which causes an acceptable SNR loss.

- Number of sub _carriers

The spacing between the sub carriers depends solely on the symbol intervals. When the symbol interval is fixed, the spacing between sub carriers is obtained by taking the inverse of symbol interval(less than guard time interval).The number of sub carriers can be obtained as the total existing bandwidth divided by sub carrier spacing.

2.2 OFDM symbol Generation

First of all a stream of ones and zeros is generated randomly in serial fashion before picking actual speech signal. The size of array is restricted to the size of FFT and number of symbols.

2.2.1PSK Symbol Mapping

The data rate can be defined as the quantity of bits transmitted per second. The bandwidth can be used efficiently in the form of high data rate

by mapping these bits in to symbol. Symbols are produced by combination of M bits each M set of bits representing 2^{M }symbols^{. }At the same time care must be taken to compromise between receiver complexity and quantity of bits used in symbol generation. Noise limit is abridged by increasing number of bits at the cost of increased Receiver's complexity in level detection.

2.2.2 Binary PSK Symbols

From programming point of view in binary psk symbol mapping, the input array of bits is cracked into n×m matrix (n is number of rows and m is number of columns).Care must be exercised that the two adjacent bits are stored in two dissimilar columns. The number of columns is to be 2 in this scenario. With this reason first split the array into a two row matrix with n columns and then goes for its transpose. We can get a

Column matrix of 1×n of n symbols by multiplying the above transposed

matrix by 2×1 with elements 2 and 1 in every row correspondingly in turn

to accomplish binary to decimal conversion.

2.2.3 M-ary PSK Symbols

The input information bits are assembled in to N×M

Matrix for M -aray PSK symbol mapping by using RESHAPE command

following the same method regarding row operation and transpose like

that incase of binary symbol mapping. In this scenario multiplication is

Performed with 1×M column matrix.

We can get a column matrix of 1×n of 2M M-ary symbols by knowing the value by 2×C (C= 0,1,2,3…) from second matrix .

2.2.4 Phase assignment

Phase assignment is a major step in OFDM systems. After generating symbols different phases uniformly spaced are assigned to different Symbols from look up table in the range of 1-2M from 0-2M-1. Each assigned phase should be unique to its respective symbol by keeping in mind grey code sequence. The phases are assigned to the symbols from the lookup table in such a way that each phase index corresponds to symbol value. There is only one bit difference between two adjacent symbols in gray coding. It is quite sure that is an error of one bit incase of erroneous detection.

.

2.2.5 Serial to Parallel

At this stage Serial information stream of bits is converted in to parallel stream before taking IFFT. As IFFT is taken on parallel information stream to transmit data on different sub-carriers. This serial to parallel conversion can be achieved with MATLAB command RESHAPE.

2.2.6 IFFT

IFFT is taken on parallel data stream for the creation of sub_ carriers.

IFFT is favored more than IDFT due to its less calculation and computational

Effectiveness. The time discrete equivalent is the inverse discrete Fourier transform (IDFT), which is given by the following equation, where the time t is replaced by a sample number n.

As IFFT always require less number of calculations. The complexity of IDFT increases linearly with number of sub carriers increasing. When 16 point transform is taken, IDFT takes 256 multiplications against 32 multiplications for IFFT. After taking IFFT on parallel stream, each of the N channels bits appear at a different Sub_ carrier frequency. Carriers on the out side which are left unmodulated are set to zero amplitude are known as “virtual carriers”.

IFFT Block diagram

2.3 GUARD BAND INSERTION

Guard band insertion between OFDM symbols is

Pre requisite to minimize ISI interference.OFDM signal exhibit lesser symbol rate

than single carrier transmission system. As the number of sub-carriers is increased to a number N in OFDM, The symbol rate is condensed by N times in OFDM than single carrier transmission system. This low symbol rate helps trim down ISI due to multipath delay spread. Multi path propagation occurs due to signals are bounced off from different objects. The level of symbol must be same during FFT period to avoid ISI. The

Signal can suffer from ISI due to mix up of different symbols during the FFT interval.

The incorporation of guard band acts as shield to minimize ISI. The length of guard band is picked to be 20% of the length of symbol interval. Guard band is a repeated copy of an OFDM symbol which is attached at the beginning of each symbol. This cyclic prefix help retain the orthogonality of the sub-carriers by ensuring that the delayed replicas of the OFDM symbol always have an integer number of samples within the FFT interval. The length of symbol interval increases after incorporation of guard band and it becomes as given below.

T_{s} = T_{g} + T_{fft}

T_{s} = total length of symbol in samples.

T_{g} = length of guard period in sample.

T_{fft} = size of IFFT used to generate OFDM symbol.

OFDM system Model

Fig 2.5 : OFDM System Model [5]

2.3.1 Guard period is added for two main purposes.

1) Protection against time offset

2) Protection against ISI

2.3.2 Protection against time offset

Guard band protects OFDM signal against time offset due to multi path time spread delay. As FFT is employed on received signal on the receiver side to get phase and amplitude of sub-carriers .The length of FFT on receiver side must be equal to the length of IFFT on transmitter side in order to maintain orthogonolity. After the insertion of guard period, signal size becomes of T_{g} + T_{fft} .Only T_{fft }samples are required to decode the signal. The receiver can recover signal accurately for time offset upto the size of guar period. Time offset introduces phase rotation in all sub_carriers, which can be removed later by channel equalization.

2.3.3) Protection against ISI.

The preservation of orthogonality necessitates that Amplitude and phase of sub_ carriers should be stable through out symbol period. Symbols always come across a change in amplitude and phase at the edge of every symbol. The time delay dispersal of symbols owing to multi path propagation produces transient change in amplitude and phase. Gaurd period insertion absorbs this very transient change and helps to make it die down. So ISI eliminates to some extent by taking guard period greater than multipath time delay.

2.3.4 Draw Backs of Guard Period Insertion

There are several disadvantages to the insertion of guard period between OFDM symbols.

1) The symbols time period changes as “T_{g}+T_{fft}^{” }.The symbols lose orthgonality

because there is no further integer number of cycles over the fresh time period.

2) Being a duplicate of piece of symbol, guard band hold worthless data which is

detached on receiver side prior to taking FFT. For this reason the SNR corrupts and

decreases eventually.

2.3.5 PARALLEL TO SERIAL CONVERSION

After the addition of guard band, the data is converted from parallel to serial stream at the end of transmitter for transmission on the radio channel. Parallel streams of data can not be sent on single channel as it requires parallel transmission medium. We use a single a single carrier as in FDM for transmission of channel. We modulate this serial data with high frequency. some time from Matlab simulation point of view transmission channel is considered as low pass filter, then

The data is not modulated with high frequency.

References

[4] From: Gaining Spectral Efficiency with OFDM

Publish Date: Sep 6, 2006

[4]source:http://www.google.com.pk/imgres?imgurl=http://zone.ni.com/cms/images/devzone/tut/a/a276be30736.jpg&imgrefurl=http://zone.ni.com/devzone/cda/tut/p/id/3370&h=84&w=118&sz=10&tbnid=OI7iJ_6cliwJ:&tbnh=84&tbnw=118&prev=/images%3Fq%3D%2B%2Bdiagram%2Bof%2BDividing%2Bthe%2BBandwidth%2Bin%2BOFDM&hl=en&sa=X&oi=image_result&resnum=1&ct=image&cd=1

[3] Source: IEEE 802.16

[5]Source:http://www.wirelesscommunication.nl/reference/chaptr05/ofdm/ofdmmath.htm