# Inter Carrier Interference (ICI) in OFDM due to frequency offset In this post, let us evaluate the impact of frequency offset resulting in Inter Carrier Interference (ICI) while receiving an OFDM modulated symbol. We will first discuss the OFDM transmission and reception, the effect of frequency offset and later we will define the loss of orthogonality and resulting signal to noise ratio (SNR) loss due to the presence of frequency offset. The analysis is accompanied by Matlab/Octave simulation scripts.

## OFDM transmission

As discussed in the post on Understanding an OFDM transmission,  for sending an OFDM modulated symbol, we use multiple sinusoidals with frequency separation is used, where is the symbol period. The information to be send on each subcarrier is multiplied by the corresponding carrier and the sum of such modulated sinusoidals form the transmit signal. Mathematically, the transmit signal is,

The interpretation of the above equation is as follows:
(a) Each information signal multiplies the sinusoidal having frequency of .
(b) Sum of all such modulated sinusoidals are added and the resultant signal is sent out as .

## OFDM reception

In an OFDM receiver, we will multiply the received signal with a bank of correlators and integrate over the period . The correlator to extract information send on subcarrier  is.

The integral,
,

where
takes values from till .

## Frequency offset

In a typical wireless communication system, the signal to be transmitted is upconverted to a carrier frequency prior to transmission. The receiver is expected to tune to the same carrier frequency for down-converting the signal to baseband, prior to demodulation. Figure: Up/down conversion However, due to device impairments the carrier frequency of the receiver need not be same as the carrier frequency of the transmitter. When this happens, the received baseband signal, instead of being centered at DC (0MHz), will be centered at a frequency , where
.

The baseband representation is (ignoring noise),

, where

is the transmitted signal and

is the frequency offset.

## Effect of frequency offset in OFDM receiver

Let us assume that the frequency offset is a fraction of subcarrier spacing i.e.
.

Also, for simplifying the equations, lets us assume that the transmitted symbols on all subcarriers,

.

The output of the correlator for sub-carrier is,

.

For  ,

The integral reduces to the OFDM receiver with no impairments case.

However for non zero values of , we can see that the amplitude of the correlation with subcarrier includes

• distortion due to frequency offset between actual frequency  and the desired frequency .
• distortion due to interference with other subcarriers with with desired frequency . This term is also known as Inter Carrier Interference (ICI).

## Simulation Model

1. Generates an OFDM symbol with all subcarriers modulated with .

2. Introduce frequency offset and add noise to result in =30dB.

3. Perform demodulation at the receiver.

4. Find the difference between the desired and actual constellation.

5. Compute the rms value of error across all subcarriers.

6. Repeat this for different values of frequency offset. Figure: Error Magnitude vs frequency offset for OFDM

## Observations

The theoretical results are computed by .

Quite likely the simulated results are slightly better than theoretical results because the simulated results are computed using average error for all subcarriers (and the subcarriers at the edge undergo lower distortion).

## 65 thoughts on “Inter Carrier Interference (ICI) in OFDM due to frequency offset”

1. mahender says:

hidden morkov model and selected applications in speech recognition in mat lab implementation . in which i didn,t get an idea that can restructure the model by using matlab code.please can you help me to proceed to write a code in mat lab.and can u help me to modification in the existing hidden model in speech recognition so that we will get benefit .

1. Krishna Sankar says:

@mahender: sorry, i have not studied much on the topic of speech recognition

2. turki khadija says:

3. tejashri says:

hello…
i am doing my PG project in ofdm communication…..can u plz give me the matlab code for ICI SELF CANCELLATION SCHEME IN OFDM SYSTEM according to IEEE 802.11a standard….or suggest some reference material…which will help me…..

thanks…..

1. Krishna Sankar says:

@tejashri: I ‘ve not not studied that topic. For OFDM related posts, please checkout
https://dsplog.com/category/ofdm/

4. kareem hamed says:

please Krishna i work the 4Pam OFDM AWGN code and the code not run i want ask you what wrong in the program my email:engkareem.aast@hotmail.com:
clc
clear all
nFFT = 64; % fft size
nDSC = 52; % number of data subcarriers
nBitPerSym = 52; % number of bits per OFDM symbol (same as the number of subcarriers for PAM)
nSym = 10^4; % number of symbols
N = 10^5; % number of symbols
alpha4pam = [-3 -1 1 3]; % 4-PAM alphabets
EbN0dB = [0:10]; % bit to noise ratio
EsN0dB = EbN0dB + 10*log10(nDSC/nFFT) + 10*log10(64/80); % converting to symbol to noise ratio

for ii = 1:length(EbN0dB)

% Transmitter
ipMod = randsrc(1,nBitPerSym*nSym,alpha4pam);

ipMod = reshape(ipMod,nBitPerSym,nSym).’

% Assigning modulated symbols to subcarriers from [-26 to -1, +1 to +26]
xF = [zeros(nSym,6) ipMod(:,[1:nBitPerSym/2]) zeros(nSym,1) ipMod(:,[nBitPerSym/2+1:nBitPerSym]) zeros(nSym,5)] ;

% Taking FFT, the term (nFFT/sqrt(nDSC)) is for normalizing the power of transmit symbol to 1
xt = (nFFT/sqrt(nDSC))*ifft(fftshift(xF.’)).’;

% Appending cylic prefix
xt = [xt(:,[49:64]) xt];

% Concatenating multiple symbols to form a long vector
xt = reshape(xt.’,1,nSym*80);

% Gaussian noise of unit variance, 0 mean
nt = 1/sqrt(2)*[randn(1,nSym*80) + j*randn(1,nSym*80)];

% Adding noise, the term sqrt(80/64) is to account for the wasted energy due to cyclic prefix
yt = sqrt(80/64)*xt + 10^(-EsN0dB(ii)/20)*nt;

yt = reshape(yt.’,80,nSym).’; % formatting the received vector into symbols
yt = yt(:,[17:80]); % removing cyclic prefix

% converting to frequency domain
yF = (sqrt(nDSC)/nFFT)*fftshift(fft(yt.’)).’;
yMod = yF(:,[6+[1:nBitPerSym/2] 7+[nBitPerSym/2+1:nBitPerSym] ]);

% demodulation
r = real(yt); % taking only the real part

ipHat(find(r= 2/sqrt(5))) = 3;
ipHat(find(r>=-2/sqrt(5) & r=0 & r<2/sqrt(5))) = 1;

% converting modulated values into bits
ipBitHat = (ipHat+3)/2;
ipBitHat = reshape(ipBitHat.',nBitPerSym,nSym).';

% counting the errors
nErr(ii) = size(find(ipBitHat – ipMod),2);

end

simBer = nErr/N;
theoryBer = 0.75*erfc(sqrt(0.2*(10.^(Es_N0_dB/10))));
close all
figure
semilogy(Es_N0_dB,theoryBer,'b.-');
hold on
semilogy(Es_N0_dB,simBer,'mx-');
axis([-3 20 10^-5 1])
grid on
legend('theory', 'simulation');
xlabel('Es/No, dB')
ylabel('bit Error Rate of 4 pam OFDM')
title('bit error probability curve for 4-PAM modulation')

1. Krishna Sankar says:

@kareem: Are you getting zero BER in no noise case?
How are the bits getting converted to modulation symbols?

5. Ahmed says:

Hello,

Can you told me, how to simplify

sum(a^2)/sum(a)

where a is a vector of N complex variables.

1. Krishna Sankar says:

@Ahmed: How about sum(a.^2)./sum(a) in matlab ? Note the dot (.)

6. nanu nadoda says:

hi krishna
find here with program for standard OFDM simulator in MATLAB
%%OFDM simulator
ep = [0 .15 .3 ];
% Es/No
EbNo=0:30;
% NS = number of OFDM symbols to transmit
NS = 100;
% Modulation type
modulation = ‘psk’;
M = 4
% N = number of carriers in OFDM symbol
N = 52;
% BPS = number of bits per OFDM symbol
BPS = N*log2(M);
% index for carriers in OFDM symbol
carriers = (1:52) + 2;
% IFFT size
ifftsize = 64;
% Input Bit Stream is normally Distributed
input_bit_stream = sign(randn(1,BPS*NS));
input_bit_stream(input_bit_stream == -1) = 0;
% PERFORM SERIAL TO PARALLEL CONVERSION
parallel_data = reshape(input_bit_stream, log2(M));
% PERFORM MODULATION
modulated_data = dmodce(parallel_data, 1, 1, modulation, M);
% CREATE OFDM SYMBOLS
disp(‘Transmitting OFDM symbols’)
%Normalize
%modulated_data = (modulated_data/max(abs(modulated_data)));
for ll= 1:length(ep)
for l=1:length(EsNo)
k = 1;
for n = 1:NS
ofdm_symbol = zeros(1,ifftsize);
% Map modulated data to FFT bins in OFDM symbol
ofdm_symbol(carriers) = modulated_data(k:k+51);
% Time Signal to transmit
tx_signal = ifft(ofdm_symbol,ifftsize);
% DOPPLER SHIFT
rx_signal = tx_signal.*exp((j*pi*ep(ll)/ifftsize)*(0:ifftsize-1));
noise = sqrt(1/(2*log2(M)*10^(EsNo(l)/10)))*(randn(1,64)+j*randn(1,64));
rx_signal = rx_signal + noise;
% FFT
% Extract data from carriers in OFDM symbol
k = k + 52;
end
% PERFROM DEMODULATION
%PERFORM PARALLEL TO SERIAL CONVERSION
% CALCULATE BER
BER(ll,l)= sum(xor(input_bit_stream, output_bit_stream))/length(input_bit_stream);
end
end
===================================================
i am experiencing problem in reshape. if you can rectify then i will be highly oblige.

1. Krishna Sankar says:

@nanu: Am sure you will be able to identify the problem in reshape() yourself 🙂

7. Thavamaran says:

I am surprised as well Krishna, its the non-linear dynamics of the laser. I am still analyzing now. thanks anyway

1. Krishna Sankar says:

@Thavamaran: Ok, good luck

8. Thavamaran says:

Krishna, thanks for the quick reply. I dont get your question on linearly varying phase? But the constellation always has 45 or 135 or 225 or 315 degree of phase shift, so it has about 90 degree difference between them.

Anyway symbol timing synch has a lot of method, which method can I look into, do you have any code referring to timing synch?

For receiver portion, im using correlator receiver, not PLL cause I feel its pointless cause correlator works extremely well for my back-to-back.

1. Krishna Sankar says:

@Thavamaran: Not sure why you are seeing a fixed phase shift. Kinda surprising.

9. eng_dina says:

hello please I try to estimate frequency offset in mimo ofdm using pn sequences have you an exaples for matlab codes could help me i make acquistion and tracking please advise it’s important .thanks

1. Krishna Sankar says:

@eng_dina: Sorry, I do not have the matlab codes.

10. Thavamaran says:

Hi Krishna, this is my first post in your website. This is a very straight forward example. I have built my OFDM system and my back-to-back works perfectly fine. But when I perform optical OFDM, which is part of my work, at the receiver, the constellation point shifted by 45 degree, then I just compensate the phase, but end up I realize that its nonlinear phase shift.

But I dont think I have a problem with CFO, cause its ideal. Do you think I need timing symbol synchronization? Please advise. Thanks.

1. Krishna Sankar says:

@Thavamaran: Are you having a linearly varying phase across subcarriers? If yes, it might be related to the symbol timing synchronization.

11. Obaid says:

Hi krishna

Kindly tell me why are we doing this: theoryErr = (theoryErr-1);

Regards
Obaid

12. Shaimaa says:

Thank you for replying, I will see the post then I will back to discuss you.

1. Krishna Sankar says:

@Shaimaa: Ok

13. Shaimaa says:

The simulated curve was around 50% and increased slightly as EbNo increases.
Is it necessary to compensate for frequency offset before demodulating the bpsk symbols as in Rayleigh fading channel??????????

14. Shaimaa says:

xt = xt.*exp(j*2*pi*freqOffsetkHz_v*(1e3/20e6)*[0:length(xt)-1]);
with freqOffsetkHz_v values=10 or 20 kHz
in your script_ber_bpsk_ofdm then run the program but the simulated curve was so bad. Is that expected

1. Krishna Sankar says:

@Shaimaa: If there is frequency error, then the intercarrier interference can result in poor BER. However, I have not done the simulations to see how much bad it can get.

15. PRIYA says:

hi…
what are the problems are occur in high PAPR?

1. Krishna Sankar says:

@PRIYA: Power amplifier efficiency is comprimised

16. eng_dina says:

thanks

17. invizible says:

Hi krishna,
I hope you are doing fine and great. I have seen your code and its great. here you have supposed that all the input symbols are modulated like ak=1, if I am right for this reason you are generating all ones as the input i.e.
ipBit = ones(1,nBitPerSym*nSym) > 0.5; (kindly correct me if I am wrong)

My point is that when we generate random bits i.e. ones and zeros, then the simmulated error is higher than the theoretical error …
kindly help in this regard …

1. Krishna Sankar says:

@invizible: Oh, is it. I have not tried that.

18. eng_dina says:

Idon’t try to simulte cfo for mimo system only but for mimo ofdm I propose a new scheme that targets MMIO OFDM systems which have unsynchronized oscillators such that CFO of individual paths have to be estimated separately. This scheme may also apply to OFDM systems with multi-user access. The new method, which is similar to Moose’s method, estimates the CFO by measuring the carrier phase difference between 2 identical successive training sequences embedded in the preambles. In order to make CFO estimates be more time efficient,I allow 2 transmitter antennas transmit their training sequence concurrently in every time period, except the first and the last period. I use Frank-Zadoff code with different phase shifts in the training sequences in different antennas. Due to the good correlation property of Frank-Zadoff code, it helps reduce the interference caused by the concurrent transmissions from other antennas.
please reply me it’s urgent Ineed the code it’s really important and thanks alot

1. Krishna Sankar says:

@eng_dina: Good luck with you are algorithm investigation. Hope you have made the simulations and obtained satisfactory results.

19. Ahmed says:

Could you help me to find the theoretical BER of OFDM with Walsh-Hadamard over the multipath transmission

1. Krishna Sankar says:

@Ahmed: Sorry, I do not have the required information

20. eng_dina says:

Idon’t try to simulte cfo for mimo system only but for mimo ofdm I propose a new scheme that targets MMIO OFDM systems which have unsynchronized oscillators such that CFO of individual paths have to be estimated separately. This scheme may also apply to OFDM systems with multi-user access. The new method, which is similar to Moose’s method, estimates the CFO by measuring the carrier phase difference between 2 identical successive training sequences embedded in the preambles. In order to make CFO estimates be more time efficient,I allow 2 transmitter antennas transmit their training sequence concurrently in every time period, except the first and the last period. I use Frank-Zadoff code with different phase shifts in the training sequences in different antennas. Due to the good correlation property of Frank-Zadoff code, it helps reduce the interference caused by the concurrent transmissions from other antennas.
cfo for mimo only but for mimo ofdm system

21. eng_dina says:

Idon’t try to simulte cfo for mimo system only but for mimo ofdm I propose a new scheme that targets MMIO OFDM systems which have unsynchronized oscillators such that CFO of individual paths have to be estimated separately. This scheme may also apply to OFDM systems with multi-user access. The new method, which is similar to Moose’s method, estimates the CFO by measuring the carrier phase difference between 2 identical successive training sequences embedded in the preambles. In order to make CFO estimates be more time efficient,I allow 2 transmitter antennas transmit their training sequence concurrently in every time period, except the first and the last period. I use Frank-Zadoff code with different phase shifts in the training sequences in different antennas. Due to the good correlation property of Frank-Zadoff code, it helps reduce the interference caused by the concurrent transmissions from other antennas.
cfo for mimo only but for mimo ofdm system

22. eng_dina says:

Idon’t try to simulte After examining some synchronization I propose a new scheme that targets MMIO OFDM systems which have unsynchronized oscillators such that CFO of individual paths have to be estimated separately. This scheme may also apply to OFDM systems with multi-user access. The new method, which is similar to Moose’s method, estimates the CFO by measuring the carrier phase difference between 2 identical successive training sequences embedded in the preambles. In order to make CFO estimates be more time efficient,I allow 2 transmitter antennas transmit their training sequence concurrently in every time period, except the first and the last period. I use Frank-Zadoff code with different phase shifts in the training sequences in different antennas. Due to the good correlation property of Frank-Zadoff code, it helps reduce the interference caused by the concurrent transmissions from other antennas.
cfo for mimo only but for mimo ofdm system

23. eng_dina says:

please I study for my master in frequency synchronization in mimo ofdm system but i have problem with the matlab code to simulate to find out if estimation of the CFO on one path is affected by the CFO values of the adjacent paths and examine the estimator accuracy in term of its mean and variance

f_max = f_Tofdm/(M_sub*T_s); % f_max =maximal Doppler frequency (Hz),M_sub*T_s = OFDM Symbol period (sec),
%f_ndopp = f_max*M_sub*T_s % Normalized Maximum Doppler Spread Freq.

% Area parameter
% ra Rural Area
% tu Typical Urban
% ht Hilly Terrain
AREA = ‘no’ ;

% start simulation loops
for ee = 1 : length(EcNo)
ee
[theo_fo_var, rfo_var, rfo] = FOE_mimo_fading(EcNo(ee), tfo_array, NTX, NRX, M_sub, N, f_max, AREA, T_s, model, L);
plot_theo_fo_var(:,:,ee)=theo_fo_var;
plot_rfo_var(:,:,ee)=rfo_var;
plot_theo_rfo(:,:,ee)=tfo_array;
plot_rfo(:,:,ee)=rfo;
end

% plot results
for ii_tx = 1 : NTX
for ii_rx = 1 : NRX

% rearrange array order for easier plotting
vect_plot_theo_rfo = permute(plot_theo_rfo(ii_tx,ii_rx, :), [2 3 1]);
vect_plot_rfo= permute(plot_rfo(ii_tx,ii_rx, :), [2 3 1] ) ;
vect_plot_theo_fo_var= permute(plot_theo_fo_var(ii_tx,ii_rx,:),[2 3 1]);
vect_plot_rfo_var= permute(plot_rfo_var(ii_tx,ii_rx, :), [2 3 1]);

%plot mean
figure;
plot(EcNo, vect_plot_theo_rfo, ‘:’ , ‘Linewidth’,2) ;
hold;
plot(EcNo, vect_plot_rfo,’o-‘,’MarkerSize’ , 6,’Linewidth’, 1) ;
grid on;
t_str=sprintf(‘New_ Theo and Est Freq Offset %s M=%d N=%d %dx%d path:tx%d-rx%d fo=%1.3f’,AREA,M_sub,N,NTX,NRX,ii_tx,ii_rx,tfo_array(ii_tx,ii_rx));
title(t_str);
xlabel(‘EcNo SNR range and step (in dB)’);
ylabel(‘Norm Freq Offset’) ;
legend(‘Theo’,’Est’,’Location’,’Northeast’);
% save graph
fstr=sprintf(‘new_m%d_%dx%dp%d%d_%s’,M_sub,NTX,NRX,ii_tx,ii_rx,AREA);
hgsave(fstr);

% plot variance
figure;
semilogy(EcNo, vect_plot_theo_fo_var,’:’,’Linewidth’,2);
hold;
semilogy(EcNo, vect_plot_rfo_var,’o-‘,’MarkerSize’,6,’Linewidth’,1);
grid on;
grid minor;
t_str=sprintf(‘New_ Est Var and CRLB %s M=%d N=%d %dx%d path:tx%d-rx%d fo=%1.3f’,AREA,M_sub,N,NTX,NRX,ii_tx,ii_rx,tfo_array(ii_tx,ii_rx));
title(t_str);
xlabel(‘EcNo (in dB)’);
ylabel(‘Variance’);
legend(‘CRLB’,’Est’,’Location’,’Northeast’);
% save graph
fstr=sprintf(‘new_y%d_%dx%dp%d%d_%s’,M_sub,NTX,NRX,ii_tx,ii_rx, AREA) ;
hgsave(fstr);
end
end
% rename variables for export
new_plot_theo_fo_var=plot_theo_fo_var;
new_plot_rfo_var=plot_rfo_var ;
new_plot_theo_rfo=plot_theo_rfo;
new_plot_rfo=plot_rfo;

% save all variables
fstr=sprintf(‘new_f%d_%dx%d_%s.mat’,M_sub,NTX,NRX,AREA);
save(fstr);
% save variables for plotting
fstr=sprintf(‘new_p%d_%dx%d_%s.mat’,M_sub,NTX,NRX,AREA);
save(fstr,’new_plot_theo_fo_var’,’new_plot_rfo_var’,’new_plot_theo_rfo’,’new_plot_theo_rfo’);

% end of file
and this is the core of the code

1. Krishna Sankar says:

@eng_dina: How are you modeling the CFO for MIMO systems?

If all the chains have a common RF clock, then all the chains will have similar CFO and the estimate from all the chains can be combined to improve the accuracy of the CFO estimation.
If the chains have independent RF clock, then we need to estimate CFO on each chain independently.

24. eng_dina says:

thanks alot for your helpful work please I work in my master in frequency synchronization in mimo ofdm and I have problem in the matlab code try to examine the estimator accuracy in term of its mean and variance and find out if estimation of the CFO on one path is affected by the CFO values of the adjacent paths

25. joel says:

Hello,

can You explain why is sample frequency (20e6) in the term freqOffsetkHz_v(ii)*(1e3/20e6) necessary ? We need delta_f so it shall be actually -200:200kHz and not delta_f*sample_time …

best regards

26. Obaid says:

Hi krishna

Can you please provide any references (books or web links) for SNR loss calculations (analytical) for AWGN and dispersive channels.

Regards.
Obaid

27. Ayesa says:

Sir,
I want to find BER for 4QAM in OFDM using AWGN. If i take the difference between reciver’s demodulated symbols and actual symbols(at transmitter) and devide the difference by the length of actual symbols, can i get the BER?
Thanks.

28. shindeujwala says:

Hello Sir!
i gone through above code.it is useful to understand ici. but i have queery that how to cancel ici due to phase offset due to local oscillator frequency offset.
thank you.

29. r.kalidoss says:

Can i think, delay spread is equal to total number of multiple paths in the system

1. Krishna Sankar says:

@r.kalidoss: Delay spread is computed using the amplitude of the taps and their delay values.

30. cyrus says:

Hi there…for one more question..

ipBit = ones(1,nBitPerSym*nSym) > 0.5; % random 1’s and 0’s

if i change “ones” to “rand”, the result seems to have differences. Can you explain it?

Thanks very much 🙂

1. Krishna Sankar says:

@cyrus:
With ak’s randomly distributed, the interference term slightly changes. How much difference did you see?

31. cyrus says:

1.I would like to know that why we need to have 1e3 in the following expression.
xt = xt.*exp(j*2*pi*freqOffsetkHz_v(ii)*(1e3/20e6)*[0:length(xt)-1]);

2.As you mentioned about,choosing -5 to 5 as subcarriers outside that range has negligible interference.I still have a bit of confuse about it.Can i change to others values?

1. Krishna Sankar says:

@cyrus:
1/ To convert kHz to Hz
2/ Sure, you can.

32. Treasa says:

Hi there,
I’m a bit confused about the theorical results calculation..
theoryErr =sum(1./(j*2*pi*([-5:5]+delta)).*(exp(j*2*pi*([5:5]+delta))-1));

could someone explain to me please how ([-5:5]+delta) relates to (k-m+delta)? Well I of course get the delta part just not sure how -5:5 was chosen?

T

1. Krishna Sankar says:

@Treasa: I just chose -5 to 5 as subcarriers outside that range has negligible interference.

33. Obaid says:

referring to last line of this post:
why the subcarriers at the edge undergo lower distortion ???

1. Krishna Sankar says:

@Obaid: Well, at the edge there will be subcarriers only at the left or right and not both. Hence there will be lower interference.

34. Khaja Rasool says:

Hi krishna,
thank you so much for ur help and posts in this website,

i am working on Next generation WLAN 802.11n Simulation.
I need ur help
i would like get the simulation for MIMO OFDM with MMSE equalizer .and also with MIMO OFDM STBC codes.

pls help me .i would greatful if you help me.
thank you once again.

Sk.khaja rasool.
Pls: do reply to this mail ID khajarasool_17@yahoo.com

35. r.kalidoss says:

Can i think, due to change of frequency i.e. m/T to m+q/T (Frequency offset) orthogonality of the signal change?

1. Krishna Sankar says:

@kalidoss: Yes, you are right.

36. r.kalidoss says:

Krishna, Can we think frequency offset is due to doppler shift…..apart from carrier mismatch in the receiver….

1. Krishna Sankar says:

@kalidoss: I have not discussed Doppler in this post. I guess the effect of Doppler might not be as simple to understand as is the case with frequency offset. I will write about Doppler soon.

37. r.kalidoss says:

First time i saw the frequency offset problem in OFDM with simple example.Its fine working. Then to how to mitigate this problem.

Any method available to solve this issue.