I've got 129 samples of some signal. I have to discover what signal it is (what components it has) and try to reconstruct it.
so after I have done FFT i wanted to find phase of signal so i try:
phs=unwrap(angle(y));
pha_wek=(0:length(y)-1)'/length(y)*129;
plot(pha_wek,phs)
Result I've got it's not what I expected. Can you point me what is wrong ?
The plot looks exactly right to me, depending upon what the original signal is. The discontinuities in the phase angle are due to zero-crossings in the magnitude. However, the phase jumps are less than pi (which is what you might expect if the input signal were originally real and had been windowed when applying the FFT). So my guess is that this signal is some form of complex modulation with some discrete frequencies.
Generally viewing the phase tells you a lot less about the signal than its magnitude. It is essential for reconstructing the signal, but much harder to interpret (particularly as it is subject to dramatic changes with a simple shift in the time domain).
You might be better plotting the phase without using the unwrap command In this case, the issue is the unwrap command - its aim is to limit jumps in phase to be less than pi, which is why the final phase is around -4pi.
I'm not sure what your challenge is, but it isn't clear what you mean by reconstructing the samples. You've already got these, so what's the issue? (Are you being asked to reconstruct interpolated samples)?
Related
How can I capture big changes in graph like this:
It is clear that there are 3 sudden curves which can be seen by eyes.
Can anyone help me to determine the start of each of them?
Matlab has findpeaks function in signal toolbox (I think) doing exactly this.
Without that, a primitive solution can be done by finding the derivative of the signal and checking the zero crossings for local maximums and minimums. Obviously if your signal is noisy you need to smooth it or handle false positives afterwards.
Ok here is what i need to do:
I want to do some tracking using Kalman filter(possibly adaptive).My measurements(when they are available) are very good with very small error from the real measurements. In some cases though the measurements jump to a value,completely off from the correct position i am looking for, and then after few frames the come back to their correct position.
The problem is that if my filter(not adaptive) has specific values for Measurement Noise Covariance(R) and State Error Covariance(Q) matrices the results are not very accurate,because even for these 1% of cases i have to do a compromise between R and Q.
So i decided to use an adaptive Kalman filter as they do in here: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.367.1747&rep=rep1&type=pdf
They estimate the measurement noise covariance matrix based on the innovation sequences.
Basically, they are using a moving window on previous samples and the calculate the covariance of the error between the previous measurements-prior estimations. For eg 5 past measurements and the 5 prior estimations.When a faulty measurement comes under the window, the covariance increases and thus the R increases also.
But in practice the R increases(but not enough) so in the next step the estimation is still good but just a bit towards the the faulty measurement.In the next step(because now the the previous estimation has moved a bit towards the measurement) the R becomes smaller with result the new estimation to go even closer to the measurements, and so on and so forth.
In the end after a few frames the estimations follow the faulty measurements. Here is a plot to understand better what i mean.
https://www.dropbox.com/s/rkv0tjcm4s54kv3/untitled.tif
Maybe what i am trying to do is completely wrong and can't be done with the adaptive Kalman filter.Maybe someone who has worked extensively with Kalman Filter in the past and he has faced this problem before can help.
Any idea is welcome!
Before the answer, I want to be sure I got the problem you have right.
You have measurements, some of them are good (Low measurement noise) yet others are outliers.
The problem you're having is tuning the measurement noise covariance matrix.
Practically, you tune for the good measurements.
Outliers measurements are rejected by using the Error Covariance.
If the innovation falls outside an ellipse you define using the Error Covariance Matrix the measurement is rejected.
Whenever a measurement is rejected you just apply the prediction step again and wait for another measurement.
Yes the problem is exactly this.
However i manage to solve it without the need to define any ellipse.What i was doing was correct except the fact that was not working if i had a lot of(lets say fifty) consecutive outliers.
This is normal if you think the size of your window.If it is for example only 10 samples and you have 20 outliers obviously it won't work.But for 5 consecutive outliers work perfectly.Generally i haven't used any threshold as you propose("if the innovation falls outside an ellipse") reject the measurements.I keep the measurements but in the same time when i start to have outliers the Error measurement covariance becomes very large.So the estimation is based more in previous estimation than in current measurement.
If i used your method which is indeed more logical(reject the current measurement,if it is an outlier based on a threshold) i have the problem that i have to define this threshold a priori,right?Maybe i am missing something..
Imagine I have a rectangular reference value for the position/displacement x and I need to smooth it.
The math for translatoric movements is quite simple:
speed: v = x'
acceleration: a = v' = x''
jerk. j = a' = v'' = x'''
I need to limit all these values. So I thought about using rate limiters in Simulink:
This approach works perfect for ramp signals, as you can see in the following output:
BUT, my reference signals for x are no ramps, they are rectangles/steps. Hence the rate limiters are not working, because the derivatives they get to limit are already infinite and Simulink throws an error. How can I resolve this problem? Is there actually a more elegant way to implement the high order rate-limiters? I guess this approach could be unstable in some cases.
continue reading: related question
Even though it seems absurd, the following approach is working: integration and instant derivation does the trick:
leading to:
More elegant, faster and simpler solutions for the whole smoothing problem are highly appreciated!
It's generally not a good idea to differentiate signals in Simulink because of numerical issues, I would advise to start with the higher order derivatives (e.g. acceleration) and integrate, much more robust numerically. This is what the doc about the derivative block says:
The Derivative block output might be very sensitive to the dynamics of
the entire model. The accuracy of the output signal depends on the
size of the time steps taken in the simulation. Smaller steps allow a
smoother and more accurate output curve from this block. However,
unlike with blocks that have continuous states, the solver does not
take smaller steps when the input to this block changes rapidly.
Depending on the dynamics of the driving signal and model, the output
signal of this block might contain unexpected fluctuations. These
fluctuations are primarily due to the driving signal output and solver
step size.
Because of these sensitivities, structure your models to use
integrators (such as Integrator blocks) instead of Derivative blocks.
Integrator blocks have states that allow solvers to adjust step size
and improve accuracy of the simulation. See Circuit Model for an
example of choosing the best-form mathematical model to avoid using
Derivative blocks in your models.
See also Best-Form Mathematical Models for more details.
I was trying to do something similar. I was looking for a "Smooth Ramp". Here is what I found:
A simpler approach is to combine ramp with a second order lag. Then the signal approachs s-shape. And your derivatives will exist and be smooth as well. Only thing to remember is that the 2nd or lag must be critically damped.
Y(s) = H(s)*X(s) where H(s) = K*wo^2/(s^2 + 2*zeta*wo*s + wo^2). Here you define zeta = 1.0. Then the s-shape is retained for any K and wo value. Note that X(s) has already been hit by a ramp. In matlab or any other tools, linear ramp and 2nd lag are standard blocks.
Good luck!
I think the 'Transfer Fcn' block is what you're looking for.
If you leave the equation in the default form 1/(s+1) you have a low-pass filter which can be tuned to what you need by changing the numerator and denominator coefficients.
I have a signal in matlab and what to calculate the instantaneous phase for a specific band. I want to filter the signal into this range (using a bandpass filter) and then get the instantaneous phase. I know that there are problems using some filters with non-linear phase responses, is there any way to get around this? I have found some information online about back filtering the signal but it's still a little unclear. I'd like to avoid using wavelets (they're probably overkill here). Thanks.
Unless you resort to noncasual techniques (like the filtfilt suggested in the comment by nibot), you will always have some phase distortion. Linear phase FIRs with a delay D will add a phase of 2*pi*f*D, while nonlinear phase IIRs will add phase that is not linearly dependent on f.
In both cases, it is easy to compute the phase distortion (for example, use freqz(num, den) for IIRs) and account for that distortion when interpreting the resulting measurement. Of course, you'll have meaningless results when the phase changes significantly over the frequency range you are interested in - but that's a different issue.
I have FFT outputs that look like this:
At 523 Hz is the maximum value. However, being a messy FFT, there are lots of little peaks that are right near the large peaks. However, they're irrelevant, whereas the peaks shown aren't. Are the any algorithms I can use to extract the maxima of this FFT that matter; I.E., aren't just random peaks cropping up near "real" peaks? Perhaps there is some sort of filter I can apply to this FFT output?
EDIT: The context of this is that I am trying to take one-hit sound samples (like someone pressing a key on a piano) and extract the loudest partials. In the image below, the peaks above 2000 Hz are important, because they are discrete partials of the given sound (which happens to be a sort of bell). However, the peaks that are scattered about right near 523 seem to be just artifacts, and I want to ignore them.
If the peak is broad, it could indicate that the peak frequency is modulated (AM, FM or both), or is actually a composite of several spectral peaks, themselves each potentially modulated.
For instance, a piano note may be the result of the hammer hitting up to 3 strings that are all tuned just a tiny fraction differently, and they all can modulate as they exchange energy between strings though the piano frame. Guitar strings can change frequency as the pluck shape distortion smooths out and decays. Bells change shape after they are hit, which can modulate their spectrum. Etc.
If the sound itself is "messy" then you need a good definition of what you mean by the "real" peak, before applying any sort of smoothing or side-band rejection filter. e.g. All that "messiness" may be part of what makes a bell sound like a real bell instead of an electronic sinewave generator.
Try convolving your FFT (treating it as a signal) with a rectangular pulse( pulse = ones(1:20)/20; ). This might get rid of some of them. Your maxima will be shifted by 10 frequency bins to teh right, to take that into account. You would basically be integrating your signal. Similar techniques are used in Pan-Tompkins algorithm for heart beat identification.
I worked on a similar problem once, and choosed to use savitsky-golay filters for smoothing the spectrum data. I could get some significant peaks, and it didn't messed too much with the overall spectrum.
But I Had a problem with what hotpaw2 is alerting you, I have lost important characteristics along with the lost of "messiness", so I truly recommend you hear him. But, if you think you won't have a problem with that, I think savitsky-golay can help.
There are non-FFT methods for creating a frequency domain representation of time domain data which are better for noisy data sets, like Max-ent recontruction.
For noisy time-series data, a max-ent reconstruction will be capable of distinguising true peaks from noise very effectively (without adding any artifacts or other modifications to suppress noise).
Max ent works by "guessing" an FFT for a time domain specturm, and then doing an IFT, and comparing the results with the "actual" time-series data, iteratively. The final output of maxent is a frequency domain spectrum (like the one you show above).
There are implementations in java i believe for 1-d spectra, but I have never used one.