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 .
Related
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
I have asked this question on DSP.SE before, but my question has got no attention. Maybe it was not so related to signal processing.
I needed to divide a discrete audio signal into segments to have some statistical processing and analysis on them. Therefore, segments with fixed local mean would be very helpful for my case. Length of segments are predefined, e.g. 512 samples.
I have tried several things. I do use reshape() function to divide audio signal into segments, and then calculate means of every segment as:
L = 512; % Length of segment
N = floor(length(audio(:,1))/L); % Number of segments
seg = reshape(audio(1:N*L,1), L, N); % Reshape into LxN sized matrix
x = mean(seg); % Calculate mean of each column
Subtracting x(k) from each seg(:,k) would make each local mean zero, yet it would distort audio signal a lot when segments are joined back.
So, since mean of hanning window is almost 0.5, substracting 2*x(k)*hann(L) from each seg(:,k) was the first thing I tried. But this time multiplying by 2 (to make the mean of hanning window be almost equal to 1) distorted the neighborhood of midpoints in each segments itself.
Then, I have used convolution by a smaller hanning window instead of multiplying directly, and subtracting these (as shown in figure below) from each seg(:,k).
This last step gives better results, yet it is still not very useful when segments are smaller. I have seen many amazing approaches here on this site for different problems. So I just wonder if there is any clever ways or existing methods to obtain zero local means which distorts an audio signal less. I read that, this property is useful in some decompositions such as EMD. So maybe I need such decompositions?
You can try to use a moving average filter:
x = cumsum(rand(15*512, 1)-0.5); % generate a random input signal
mean_filter = 1/512 * ones(1, 512); % generate a mean filter
mean = filtfilt(mean_filter, 1, x); % filtfilt is used instead of filter to obtain a symmetric moving average.
% plot the result
figure
subplot(2,1,1)
plot(x);
hold on
plot(mean);
subplot(2,1,2)
plot(x - mean);
You can tune the filter by changing the interval of the mean filter. Using a smaller interval, results in lower means inside each interval, but filters also more low frequencies out of your signal.
I try to find the strongest frequency component with Matlab. It works, but if the datapoints and periods are not nicely aligned, I need to zero-pad my data to increase the FFT resolution. So far so good.
The problem is that, when I zero-pad too much, the frequency with the maximal power changes, even if everything is aligned nicely and I would expect a clear result.
This is my MWE:
Tmax = 1024;
resolution = 1024;
period = 512;
X = linspace(0,Tmax,resolution);
Y = sin(2*pi*X/period);
% N_fft = 2^12; % still fine, max_period is 512
N_fft = 2^13; % now max_period is 546.1333
F = fft(Y,N_fft);
power = abs(F(1:N_fft/2)).^2;
dt = Tmax/resolution;
freq = (0:N_fft/2-1)/N_fft/dt;
[~, ind] = max(power);
max_period = 1/freq(ind)
With zero-padding up to 2^12 everything works fine, but when I zero-pad to 2^13, I get a wrong result. It seems like too much zero-padding shifts the spectrum, but I doubt it. I rather expect a bug in my code, but I cannot find it. What am I doing wrong?
EDIT: It seems like the spectrum is skewed towards the low frequencies. Zero-padding just makes this visible:
Why is my spectrum skewed? Shouldn't it be symmetric?
Here is a graphic explanation of what you're doing wrong (which is mostly a resolution problem).
EDIT: this shows the power for each fft data point, mapped to the indices of the 2^14 dataset. That is, the indices for the 2^13 data numbered 1,2,3 map to 1,3,5 on this graph; the indices for 2^12 data numbered 1,2,3 map to 1,5,9; and so on.
.
You can see that the "true" value should in fact not be 512 -- your indexing is off by 1 or a fraction of 1.
Its not a bug in your code. It has to do with the properties of the DFT (and thus the FFT, which is merely a fast version of the DFT).
When you zero-pad, you add frequency resolution, particularly on the lower end.
Here you use a sine wave as test, so you are basically convolving a finite length sine with finite sines and cosines (see here https://en.wikipedia.org/wiki/Fast_Fourier_transform details), which have almost the same or lower frequency.
If you were doing a "proper" fft, i.e. doing integrals from -inf to +inf, even those low frequency components would give you zero coefficients for the FFT, but since you are doing finite sums, the result of those convolutions is not zero and hence the actual computed fourier transform is inaccurate.
TL;DR: Use a better window function!
The long version:
After searching further, I finally found the explanation. Neither is indexing the problem, nor the additional low frequency components added by the zero-padding. The frequency response of the rectangular window, combined with the negative frequency components is the culprit. I found out on this website explaining window functions.
I made more plots to explain:
Top: The frequency response without windowing: two delta peaks, one at the positive and one at the negative frequency. I always plotted the positive part, since I didn't expect to need the negative frequency components. Middle: The frequency response of the rectangular window function. It is relatively broad, but I didn't care, because I thought I'd have only a single peak. Bottom: The frequency response of the zero-padded signal. In time domain, this is the multiplication of window function and sine-wave. In frequency domain, this amounts to the convolution of the frequency response of the window function with the frequency response of the perfect sine. Since there are two peaks, the relatively broad frequency responses of the window overlap significantly, leading to a skewed spectrum and therefore a shifted peak.
The solution: A way to circumvent this is to use a proper window function, like a Hamming window, to have a much smaller frequency response of the window, leading to less overlap.
From my understanding, when using the cpsd function as such:
[Pxy,f] = cpsd(x,y,window,Ns,NFFT,Fs);
matlab chops the time series data into smaller windows with size specified by you. And the windows are shifted by Ns data point. The final [Pxy, f] are an average of results obtained from each individual window. Please correct me if I am wrong about this process.
My question is, if I use angle(Pxy) at a specific frequency, say 34Hz. Does that give me the phase difference between signal x and y at the frequency 34Hz?
I am having doubt about this because if Pxy was an average between each individual window, and because each individual was offset by a window shift, doesn't that mean the averaged Pxy's phase is affected by the window shift?
I've tried to correct this by ensuring that the window shift corresponds to an integer of full phase difference corresponding to 34Hz. Is this correct?
And just a little background about what I am doing:
I basically have numerous time-series pressure measurement over 60 seconds at 1000Hz sampling rate.
Power spectrum analysis indicates that there is a peak frequency at 34 Hz for each signal. (averaged over all windows)
I want to compare each signal's phase difference from each other corresponding to the 34Hz peak.
FFT analysis of individual window reveals that this peak frequency moves around. So I am not sure if cpsd is the correct way to be going about this.
I am currently considering trying to use xcorr to calculate the overall time lag between the signals and then calculate the phase difference from that. I have also heard of hilbert transform, but I got no idea how that works yet.
Yes, cpsd works.
You can test your result by set two input signals, such as:
t=[0:0.001:5];
omega=25;
x1=sin(2*pi*omega*t);
x2=sin(2*pi*omega*t+pi/3);
you can check whether the phase shift calculated by cpsd is pi/3.
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.