[attachment=3418]

---

A Technique for Orthogonal Frequency Division Multiplexing
Frequency Offset Correction

Abstract:
The Orthogonal Frequency Division Multiplexing (OFDM) is a Multi Carrier Modulation technique employing Frequency Division Multiplexing of orthogonal subcarriers. It is also sometimes called as discrete multitone modulation (DMT). It is better than CDMA in high bit rate and Broadband Communications .It is a multi carrier technique, which is very sensitive to Frequency Offset between transmitter and receiver. This paper discusses the effects of frequency offset on the performance of OFDM digital communications. The main problem with frequency offset is that it introduces interference among multiplicity of carriers in the OFDM signal. It is shown that to maintain signal to interference ratios of 20 dB or greater for the OFDM carriers, offset is limited to 4% or less of the intercarier spacing. Also the paper describes a method to estimate frequency offset using a repeated data symbol. A maximum likelihood estimation (MLE) algorithm is used. Since the intercarrier interference energy and signal energy both contribute coherently to the estimate, the algorithm generates extremely accurate estimates even when the offset is far too great to demodulate the data values. Also, the estimation error depends only on total symbol energy so it is insensitive to channel spreading and frequency selective fading.

---

Presented By:
Mr.S.KATHIRAVAN Mr. P.PONMANI
III-YR ECE III-YR CSE
MEPCO SCHLENK ENGG.COLLEGE.

---

Introduction:
Orthogonal frequency-division multiplexing (OFDM), also sometimes called discrete multitone modulation (DMT), is a transmission technique based upon the idea of frequency division multiplexing (FDM). In FDM, multiple signals are sent out at the same time, but on different frequencies. Most people are familiar with FDM from the use of radio and television: normally, each station is designated to broadcast at a particular frequency or channel. OFDM takes this concept further: In OFDM, a single transmitter transmits on many different orthogonal (independent) frequencies (typically dozens to thousands). (Because the frequencies are so closely spaced, each one only has room for a Narrowband signal). This, coupled with the use of advanced modulation techniques on each component, results in a signal with high resistance to interference.
Application:
OFDM is used in many communications systems such us: ADSL, Wireless LAN, DAB, DVB, UWB and PLC.
Characteristics:
An OFDM baseband signal is the sum of a number of orthogonal sub-carriers, with data on each sub-carrier being independently modulated commonly using some type of quadrature amplitude modulation (QAM) or phase-shift keying (PSK). This composite baseband signal is typically used to modulate a main RF carrier.
The benefits of using OFDM are many, including high spectrum efficiency, resistance against multipath interference (particularly in wireless communications), and ease of filtering out noise (if a particular range of frequencies suffers from interference, the carriers within that range can be disabled or made to run slower)..
The above figure represents theOFDM signals.Here there are 5 carriers.when any one of the carrier is in positive peak , all the others are at zero.thus only one signal is received at a particular frequency.Thus the ICI is reduced.
Frequency offset:
The frequency offset in the OFDM symbols represents the frequency shift between the transmitted and the received symbols.This case arises as a result of frequency change in the receivers during transmission.As a result of this frequency offset problem, there arises a condition where the orthogonality of the individual carriers is affected. Thus the inter symbol interference (ICI) arises.

In the figure ,the frequency offset Af is present.as a result , at the output sampling is affected.This leads to the reduction in the performance of the OFDM system.
Method:
The method used in this paper has been developed to correct frequency offset errors in digital communication systems employing OFDM as the method of modulation. There are two deleterious effects caused by frequency offset; one is the reduction of signal amplitude in the output of signal amplitude in the output of the filters matched to each of the carriers and the second is introduction of ICI(intercarrier interference) from the other carriers which are no longer orthogonal to the filter. Because, in OFDM, the carriers are inherently closely spaced in frequency compared to the channel bandwidth, the tolerable frequency offset becomes a very small fraction of the channel bandwidth. Maintaining sufficient open loop frequency accuracy can become difficult in links, such as satellite links with multiple frequency translations.
We have presented the algorithm to estimate frequency offset from the demodulated signals in the receiver. The algorithm extends to OFDM, with important differences, a method for single carrier MPSK. The technique involves the repetition of a data symbol and comparison of the phases of each of the carriers between the successive symbols. Since the modulation phase values are not changed, the phase shift of each of the carriers between successive repeated symbols is due to the frequency offset. The offset is estimated using a maximum likelihood estimate (MLE) algorithm.
The basic block diagram of OFDM is in the Figure:
Generalized OFDM System Model:
I/p data
Symbol Mapping
Serial to
Parallel
IFFT
Parallel to
Serial
Fig3:Baseband Model of OFDM system
Channel
Parallel to
Serial
FFT
Serial to
Parallel
Frequency offset estimation using MLE algorithm)
In an OFDM transmission symbol is repeated, one receives, in the absence of noise , the
2 N point sequence
K
rn=(1/N)[ZXkHke2njn(k+e)/N] k=-K
n=0,1,...,2N-1.
The kth element of the first N points of the above equation is
N-1
Rik=[Zrne-2njnk/N]
n=0
k=0,1,2,..N-1
and the kth element of the second half of the sequence is
2N-1
R2k=[Zrne-2njnk/N]
n=0 2N-1
=[Zrn+Ne-2njnk/N]
n=0 k=0,1,..,N-1.
From the first equation
r =r e2nJe >R =R e2nje rn+N=rne ->R2k=R1ke
including the AWGN one obtains
Y1k=R1k+W 1k;
Y2k=R1ke2nje+W2k; K=0,1,2,...,N-1.
It is found that between the first and second DFT's both the ICI and the signal are altered in exactly the same way, by a phase shift proportional to frequency offset. Therefore, if offset e is estimated using observations, it is possible to obtain accurate estimates even when the offset is too large for satisfactory data demodulation.
The maximum likelihood estimate (MLE) of e is given by K
e1=(1/2n)tan1[ZIm(Y2kY*1k)/
k=-K
K
[ZRe(Y2kY*1k)]
k=-K
in the absence of noise the angle of Y2kY*1k is 2IIe. Statistical properties of the estimate:
The conditional mean and variance of e given e and {Rk} can be approximated as follows. Consider the complex products Y2kY 1k from which we estimate e. For a given e, subtract the corresponding phase, 2IIe, from each product to obtain the tangent of the phase error.
K tan[2n(e1-e)]= [ZIm(Y2kY*1ke-2nje)/
k=-K
K
[ZRe(Y2kY*ike-2nje)] k=-K
For Ie1-eI<<1/2n, the tangent can be approximated by its argument so that
e1-e~
K[ZIm(R1k+W2ke-2nje)(R*1k+W*1k)]/
k=-K
K
ZRe(R1k+W2ke- 2nje)(R*1k+W*1k)]
k=-K
At high signal to noise ratios, a condition compatible with successful communications signaling, may be approximated by
K
e1-e~ [ZIm(W2k R* 1ke-2nje+R1kW* 1k)]/
k=-K
K
[ZRe(R1k2)] k=-K
From which we find that E[e-eIe,{Rk}]=0.
Therefore, for small errors the estimate is conditionally unbiased. The conditional variance of the estimate is easily determined for above equation. Var[eIe,{Rk}]=1/{(2n)2(Es/No)} Where
N-1
Es=(T/N)ZIrJ2
n=0
is the total symbol energy. Since the total energy is the sum of the energies of the 2K+1 carriers, the error variance of the offset estimate will in practice be very low.
Acquisition:
In the event that the frequency offset is greater than + or -1/2 of the carrier spacing, a strategy for initial acquisition to bring the offset within the limits of the algorithm must be developed. A continuous symbol stream occurs in the applications such as digital audio broadcast. A second possibility is that OFDM communications such as digital radio. Here we envision that the session interval will include one or more repeated symbols.
The basic strategy for initial frequency offset acquisition, in either case, is to shorten the dft's and use larger carrier spacings such that the phase shift doesnot exceed +or - n. The frequency offset in Hz is 8=e/T=eAf where Af is the inter carrier spacing and T is the symbol interval. Let us assume that the initial frequency offset is not greater than + or-5max. Then
Af initial>2Smax
determines the minimum initial carrier spacing, and the corresponding dft lengths. If the average power of the shortened symbols is kept the same, the variance of the estimate of einitial will be larger than for the longer data symbols,since it estimates the fraction of carrier spacing, corresponds to a proportionately larger fractional offset for the longer data symbols. However, the MLE estimate is so accurate that in practice the initial estimate still may be adequate. If not, it is refined by following the shortened repeated symbols by a repetition of the first full length data symbol.
Codes:
1.calculation of frequency offset:
clc; clear all; close all;
x1= randint(1,1020);%generating a random number
for i=1:1020
if (x1(i)>0.5)
x1(i)=1;
else
x1(i)=0;
end end
disp(x1);
x2=reshape(x1,4,255);%to convert 1 by 1020 to 255 by 4 for n=1:4
x3(:,n)=ifft(x2(:,n),255);%ifft of the random signal end
y=awgn(x3,10);%adding the awgn noise for p=1:4
xx(:,p)=fft(x3(:,p),255);%taking fft of transmitted signal end
x7=reshape(xx,1,1020); x8=abs(x7);
for v=1:1020
if (abs(x8(v))-abs(x1(v)))<=0.5%to convert into a 1 or 0
x8(v)=0;
end end
q=0;
for c=1:1020 if(x1©==x8©)
q=q+1;%number of error free signals end end
disp(q);%displays the number of signals without error xy=xx.*exp(i*2*pi*.005);%taking fft after introducing a phase shift
figure(2); subplot(2,2,1);
plot(xx);%fft value without phase shift
title('normal fft'); subplot(2,2,2);
plot(xy);%fft value with phase shift title('fft after a phase shift');
xx=reshape(xx,1,1020); xy=reshape(xy,1,1020);
z1=[zeros(1,1020)]; y1=[zeros(1,1020)]; for k=1:1:1020 if k==1
y1(1)=imag(xy(k)*conj(xx(k)));%calculating the %estimate phase shift according to the algorithm
z1(1)=real(xy(k)*conj(xx(k)));
else
y1(k)=imag(y1(k-1))+imag(xy(k)*conj(xx(k)));
z1(k)=real(z1(k-1))+real(xy(k)*conj(xx(k))); end end
e2=(1/(2*pi))*atan(imag(y1)/real(z1));%estimated phase
%shift
z2=[zeros(1,1020)]; y2=[zeros(1,1020)];
for b=1:1:1020
if b==1
y2(1)=imag(xx(b)+(conj(xx(b))*exp(-2*pi*i*.005))); %calculating the difference in phase shifts betweeen %the estimated and the actual phase shifts z2(1)=(abs(xx(b)).A2);
else
y2(b)=imag(y2(b-1))+imag(xx(b)+(conj(xx(b))*exp(-2*pi*i*.005))); z2(b)=z2(b-1)+(abs(xx(b)).A2);
end end
e1=(1/(2*pi))*(y2/z2);%difference in phase shiftsof estimated and the actual disp(e1)
e=e2-e1;%the phase shift disp(abs(e));
freqoff=.2*1.5;%frequency offset for a phase shift of %0.2 hz and inter carrier spacing of 1.5 hz
disp(freqoff);
2.generation of OFDM symbols:
clc;
x1= randint(1,150);%generating a random number
figure(1);
subplot(4,1,1);
plot(x1);%plots the random signal
title('random signal'); x2=x1';%converting 1 by 150 to 150 by 1 x3=ifft(x2,150);%taking ifft y=awgn(x3,30);%adding the awgn noise
disp(y);
x4=fft(y,150);%taking fft of the received signal
x5=abs(x4');
disp(x5);
for p=1:1:150
if x5(p)>0.5
x5(p)=1;
elseif x5(p)<0.5
x5(p)=0;%converting complex to 1 and 0 end end
q=0;
for i=1:1:150 if (x1(i)-x5(i))==0
q=q+1;%counting the number of error free signals end end
disp(q); subplot(4,1,2);
plot(x5);%plots the output of the receiver title('received signal');
Conclusions:
An algorithm for maximum likelihood estimate (MLE) of frequency offset using the dft values of a repeated data symbol has been presented. It has been shown that for small error in the estimate is conditionally unbiased and is the sense that the variance is inversely proportional to the number of the carriers in the OFDM signal. Furthermore, both the signal values and the ICI, contribute coherently to the estimate so that it is possible to obtain very accurate estimates even when the offset is too great, that is there is too much ICI to demodulate the data values. Since the estimate error depends only on total symbol energy, the algorithm works well in the multipath channels.
However, it is required that the frequency offset as well as the channel impulse response be constant for a period of two consequtive symbols.
References:
1. .R.W.Chang," synthesis of band limited orthogonal signals for multi
channel data transmission", Bell syst.Tech. J, vol 45, pp. 1775-1796, Dec
1966.
S.Darling, "On digital -sideband modulators, IEEE Trans. Circuit
2. Theory, vol CT-17, pp. 409-414, Aug. 1970.
3. S.B.Weinstein and P.M.Ebert, "Data transmission by frequency division multiplexing using the discrete Fourier transform", IEEE Trans. Commn. Technol., vol. COM-19, p.p.628-634, Oct.1971.
4. J.A.Cbingham,"Multicarrier modulation for data transmission: An idea whose time has come", IEEE Commn. Mag., vol. 28, PP. 17-25, Mar.1990.

---