I want to calculate the Fourier series of a signal using FFT on matlab. And I came across the following unexpected issue. Mere example
If I define a grid and then compute the fft as:
M=59;
x= deal(1*(0:M-1)/M);
y=3*cos(2*pi*x);
Yk=fftshift(fft2(y)/(M));
which gives me the exact analytic values expected: Yk(29)=1.5; Yk(31)=1.5; zeros anything else
but if I define the grid as, and repeat the fft calculation:
x=0:1/(M-1):1;
y=3*cos(2*pi*x);
Yk=fftshift(fft2(y)/(M));
got the Yk's values completely screwed up
This is an annoying issue since I have to analyse many signals data that was sampled as in the second method so the Yk's values will be wrong. Is there a way to workaround this? an option to tell something to the fft function about the way the signal was sampled. Have no way to resample the data in the correct way.
The main reason to avoid have spectral leaking, is that I do further operations with these Fourier terms individually Real and Imag parts. And the spectral leaking is messing the final results.
The second form of sampling includes one sample too many in the period of the cosine. This causes some spectral leaking, and adds a small shift to your signal (which leads to non-zero imaginary values). If you drop the last point, you'll cosine will again be sampled correctly, and you'll get rid of both of these effects. Your FFT will have one value less, I don't know if this will affect your analyses in any way.
x = 0:1/(M-1):1;
y = 3*cos(2*pi*x);
Yk = fftshift(fft2(y(1:end-1))/(M-1));
>> max(abs(imag(Yk)))
ans =
1.837610523517500e-16
Related
I am trying to use the ifft function in MATLAB on some experimental data, but I don't get the expected results.
I have frequency data of a logarithmic sine sweep excitation, therefore I know the amplitude [g's], the frequency [Hz] and the phase (which is 0 since the point is a piloting point).
I tried to feed it directly to the ifft function, but I get a complex number as a result (and I expected a real result since it is a time signal). I thought the problem could be that the signal is not symmetric, therefore I computed the symmetric part in this way (in a 'for' loop)
x(i) = conj(x(mod(N-i+1,N)+1))
and I added it at the end of the amplitude vector.
new_amp = [amplitude x];
In this way the new amplitude vector is symmetric, but now I also doubled the dimension of that vector and this means I have to double the dimension of the frequency vector also.
Anyway, I fed the new amplitude vector to the ifft but still I don't get the logarithmic sine sweep, although this time the output is real as expected.
To compute the time [s] for the plot I used the following formula:
t = 60*3.33*log10(f/f(1))/(sweep rate)
What am I doing wrong?
Thank you in advance
If you want to create identical time domain signal from specified frequency values you should take into account lots of details. It seems to me very complicated problem and I think it need very strength background on the mathematics behind it.
But I think you may work on some details to get more acceptable result:
1- Time vector should be equally spaced based on sampling from frequency steps and maximum.
t = 0:1/fs:N/fs;
where: *N* is the length of signal in frequency domain, and *fs* is twice the
highest frequency in frequency domain.
2- You should have some sort of logarithmic phases on the frequency bins I think.
3- Your signal in frequency domain must be even to have real signal in time domain.
I hope this could help, even for someone to improve it.
I have two sensors seperated by some distance which receive a signal from a source. The signal in its pure form is a sine wave at a frequency of 17kHz. I want to estimate the TDOA between the two sensors. I am using crosscorrelation and below is my code
x1; % signal as recieved by sensor1
x2; % signal as recieved by sensor2
len = length(x1);
nfft = 2^nextpow2(2*len-1);
X1 = fft(x1);
X2 = fft(x2);
X = X1.*conj(X2);
m = ifft(X);
r = [m(end-len+1) m(1:len)];
[a,i] = max(r);
td = i - length(r)/2;
I am filtering my signals x1 and x2 by removing all frequencies below 17kHz.
I am having two problems with the above code:
1. With the sensors and source at the same place, I am getting different values of 'td' at each time. I am not sure what is wrong. Is it because of the noise? If so can anyone please provide a solution? I have read many papers and went through other questions on stackoverflow so please answer with code along with theory instead of just stating the theory.
2. The value of 'td' is sometimes not matching with the delay as calculated using xcorr. What am i doing wrong? Below is my code for td using xcorr
[xc,lags] = xcorr(x1,x2);
[m,i] = max(xc);
td = lags(i);
One problem you might have is the fact that you only use a single frequency. At f = 17 kHz, and an estimated speed-of-sound v = 340 m/s (I assume you use ultra-sound), the wavelength is lambda = v / f = 2 cm. This means that your length measurement has an unambiguity range of 2 cm (sorry, cannot find a good link, google yourself). This means that you already need to know your distance to better than 2 cm, before you can use the result of your measurement to refine the distance.
Think of it in another way: when taking the cross-correlation between two perfect sines, the result should be a 'comb' of peaks with spacing equal to the wavelength. If they overlap perfectly, and you displace one signal by one wavelength, they still overlap perfectly. This means that you first have to know which of these peaks is the right one, otherwise a different peak can be the highest every time purely by random noise. Did you make a plot of the calculated cross-correlation before trying to blindly find the maximum?
This problem is the same as in interferometry, where it is easy to measure small distance variations with a resolution smaller than a wavelength by measuring phase differences, but you have no idea about the absolute distance, since you do not know the absolute phase.
The solution to this is actually easy: let your source generate more frequencies. Even using (band-limited) white-noise should work without problems when calculating cross-correlations, and it removes the ambiguity problem. You should see the white noise as a collection of sines. The cross-correlation of each of them will generate a comb, but with different spacing. When adding all those combs together, they will add up significantly only in a single point, at the delay you are looking for!
White Noise, Maximum Length Sequency or other non-periodic signals should be used as the test signal for time delay measurement using cross correleation. This is because non-periodic signals have only one cross correlation peak and there will be no ambiguity to determine the time delay. It is possible to use the burst type of periodic signals to do the job, but with degraded SNR. If you have to use a continuous periodic signal as the test signal, then you can only measure a time delay within one period of the periodic test signal. This should explain why, in your case, using lower frequency sine wave as the test signal works while using higher frequency sine wave does not. This is demonstrated in these videos: https://youtu.be/L6YJqhbsuFY, https://youtu.be/7u1nSD0RlwY .
I am having a samll problem while converting a spectrum to a time series. I have read many article sand I htink I am applying the right procedure but I do not get the right results. Could you help to find the error?
I have a time series like:
When I compute the spectrum I do:
%number of points
nPoints=length(timeSeries);
%time interval
dt=time(2)-time(1);
%Fast Fourier transform
p=abs(fft(timeSeries))./(nPoints/2);
%power of positive frequencies
spectrum=p(1:(nPoints/2)).^2;
%frequency
dfFFT=1/tDur;
frequency=(1:nPoints)*dfFFT;
frequency=frequency(1:(nPoints)/2);
%plot spectrum
semilogy(frequency,spectrum); grid on;
xlabel('Frequency [Hz]');
ylabel('Power Spectrum [N*m]^2/[Hz]');
title('SPD load signal');
And I obtain:
I think the spectrum is well computed. However now I need to go back and obtain a time series from this spectrum and I do:
df=frequency(2)-frequency(1);
ap = sqrt(2.*spectrum*df)';
%random number form -pi to pi
epsilon=-pi + 2*pi*rand(1,length(ap));
%transform to time series
randomSeries=length(time).*real(ifft(pad(ap.*exp(epsilon.*i.*2.*pi),length(time))));
%Add the mean value
randomSeries=randomSeries+mean(timeSeries);
However, the plot looks like:
Where it is one order of magnitude lower than the original serie.
Any recommendation?
There are (at least) two things going on here. The first is that you are throwing away information, and then substituting random numbers for that information.
The FFT of a real sequence is a sequence of complex numbers consisting of a real and imaginary part. Converting those numbers to polar form gives you magnitude and phase angle. You are capturing the magnitude part with p=aps(fft(...)), but you are not capturing the phase angle (which would involve atan2(...)). You are then making up random numbers (epsilon=...) and using those to replace the original numbers when you reconstruct your time-series. Also, as the FFT of a real sequence has a particular symmetry, substituting random numbers for the phase angle destroys that symmetry, which means that the IFFT will in general no longer be a real sequence, but a sequence of complex numbers - and again, you're only looking at the real portion of the IFFT, so you're throwing away information again. If this is an audio signal, the results may sound somewhat like the original (or they may be completely different), but the waveform definitely won't match...
The second issue is that in many implementations, ifft(fft(...)) will scale the result by the number of points in the signal. There are several different ways to avoid that, with differing results, but sometimes more attractive in different scenarios, depending on what you are trying to do. You can either scale the fft() result before you do the ifft(), or scale the ifft() result at the end, or in some cases, I've even seen both being scaled by a factor of sqrt(N) - doing it twice has the end result of scaling the final result by N, but it is a bit less efficient since you do the scaling twice...
I am using the FFT function in Matlab in an attempt to analyze the output of a Travelling Wave Laser Model.
The of the model is in the time domain in the form (real, imaginary), with the idea being to apply the FFT to the complex output, to obtain phase and amplitude information in the frequency domain:
%load time_domain field data
data = load('fft_data.asc');
% Calc total energy in the time domain
N = size(data,1);
dt = data(2,1) - data (1,1);
field_td = complex (data(:,4), data(:,5));
wavelength = 1550e-9;
df = 1/N/dt;
frequency = (1:N)*df;
dl = wavelength^2/3e8/N/dt;
lambda = -(1:N)*dl +wavelength + N*dl/2;
%Calc FFT
FT = fft(field_td);
FT = fftshift(FT);
counter=1;
phase=angle(FT);
amptry=abs(FT);
unwraptry=unwrap(phase);
Following the unwrapping, a best fit was applied to the phase in the region of interest, and then subtracted from the phase itself in an attempt to remove wavelength dependence of phase in the region of interest.
for i=1:N % correct phase and produce new IFFT input
bestfit(i)=1.679*(10^10)*lambda(i)-26160;
correctedphase(i)=unwraptry(i)-bestfit(i);
ReverseFFTinput(i)= complex(amptry(i)*cos(correctedphase(i)),amptry(i)*sin(correctedphase(i)));
end
Having performed the best fit manually, I now have the Inverse FFT input as shown above.
pleasework=ifft(ReverseFFTinput);
from which I can now extract the phase and amplitude information in the time domain:
newphasetime=angle(pleasework);
newamplitude=abs(pleasework);
However, although the output for the phase is greatly different compared to the input in the time domain
the amplitude of the corrected data seems to have varied little (if at all!),
despite the scaling of the phase. Physically speaking this does not seem correct, as my understanding is that removing wavelength dependence of phase should 'compress' the pulsed input i.e shorten pulse width but heighten peak.
My main question is whether I have failed to use the inverse FFT correctly, or the forward FFT or both, or is this something like a windowing or normalization issue?
Sorry for the long winded question! And thanks in advance.
You're actually seeing two effects.
First the expected one goes. You're talking about "removing wavelength dependence of phase". If you did exactly that - zeroed out the phase completely - you would actually get a slightly compressed peak.
What you actually do is that you add a linear function to the phase. This does not compress anything; it is a well-known transformation that is equivalent to shifting the peaks in time domain. Just a textbook property of the Fourier transform.
Then goes the unintended one. You convert the spectrum obtained with fft with fftshift for better display. Thus before using ifft to convert it back you need to apply ifftshift first. As you don't, the spectrum is effectively shifted in frequency domain. This results in your time domain phase being added a linear function of time, so the difference between the adjacent points which used to be near zero is now about pi.
I am trying to compare two data sets in MATLAB. To do this I need to filter the data sets by Fourier transforming the data, filtering it and then inverse Fourier transforming it.
When I inverse Fourier transform the data however I get a spike at either end of the red data set (picture shows the first spike), it should be close to zero at the start, like the blue line. I am comparing many data sets and this only happens occasionally.
I have three questions about this phenomenon. First, what may be causing it, secondly, how can I remedy it, and third, will it affect the data further along the time series or just at the beginning and end of the time series as it appears to from the picture.
Any help would be great thanks.
When using DFT you must remember the DFT assumes a Periodic Signal (As a Superposition of Harmonic Functions).
As you can see, the start point is exact continuation of the last point in harmonic function manner.
Did you perform any Zero Padding in the Spectrum Domain?
Anyhow, Windowing might reduce the Overshooting.
Knowing more about the filter and the Original data might be helpful.
If you say spike near zero frequencies, I answer check the DC component.
You seem interested by the shape, so doing
x = x - mean(x)
or
x -= mean(x)
or
x -= x.mean()
(I love numpy!)
will just constrain the dataset to begin with null amplitude at zero-frequency and to go ahead with comapring the spectra's amplitude.
(as a side-note: did you check that you approprately use fftshift and ifftshift? this has always been the source of trouble for me)
Could be the numerical equivalent of Gibbs' phenomenon. If that's correct, there's no way to remedy it except for filtering.