I have to compute the cross-correlation between two transient signals with non-zero mean. I read that the function xcorr in MATLAB works properly only with zero-mean inputs.
Since these signals represent transient phenomena, it doesn't make sense to me to subtract the mean value.
My objective is to compute the time delay between the maximum values of the two signals. The signals are not exactly correlated-similar, but I guess this is always the case.
If i try to compute the time delay using xcorr, I get close results to what I expect (i.e the time delay checked visually by checking where the maximum of the two signals are) only using the UNBIASED options.
Why is that? Does the unbiased routine subtract the mean values from both my signals?
Now I am working on sound localization algorithm. I use TDOA method. I use xcorr function in matlab. Suppose the two signals are sine waves. I found if phase difference of the two waves is pi, I can get two peaks in cross correlation result, which values are equal. So I don't konw whose phase is lead and which one is lag. If phase difference is more than pi,for example one wave is lead another several periods, the result is wrong. I used voice signal to test the algorithm, I found if the phase difference was more than pi, the results sometimes were wrong.
I have read several papers, most of which use cross correlation algorithm to identify sound location. But I don't know how to solve my problem.
I have two (or more) time series that I would like to correlate with one another to look for common changes e.g. both rising or both falling etc.
The problem is that the time series are all fairly noisy with relatively high standard deviations meaning it is difficult to see common features. The signals are sampled at a fairly low frequency (one point every 30s) but cover reasonable time periods 2hours +. It is often the case that the two signs are not the same length, for example 1x1hour & 1x1.5 hours.
Can anyone suggest some good correlation techniques, ideally using built in or bespoke matlab routines? I've tried auto correlation just to compare lags within a single signal but all I got back is a triangular shape with the max at 0 lag (I assume this means there is no obvious correlation except with itself?) . Cross correlation isn't much better.
Any thoughts would be greatly appreciated.
Start with a cross-covariance (xcov) instead of the cross-correlation. xcov removes the DC component (subtracts off the mean) of each data set and then does the cross-correlation. When you cross-correlate two square waves, you get a triangle wave. If you have small signals riding on a large offset, you get a triangle wave with small variations in it.
If you think there is a delay between the two signals, then I would use xcorr to calculate the delay. Since xcorr is doing an FFT of the signal, you should remove the means before calling xcorr, you may also want to consider adding a window (e.g. hanning) to reduce leakage if the data is not self-windowing.
If there is no delay between the signals or you have found and removed the delay, you could just average the two (or more) signals. The random noise should tend to average to zero and the common features will approach the true value.
I have two datasets at the time (in the form of vectors) and I plot them on the same axis to see how they relate with each other, and I specifically note and look for places where both graphs have a similar shape (i.e places where both have seemingly positive/negative gradient at approximately the same intervals). Example:
So far I have been working through the data graphically but realize that since the amount of the data is so large plotting each time I want to check how two sets correlate graphically it will take far too much time.
Are there any ideas, scripts or functions that might be useful in order to automize this process somewhat?
The first thing you have to think about is the nature of the criteria you want to apply to establish the similarity. There is a wide variety of ways to measure similarity and the more precisely you can describe what you want for "similar" to mean in your problem the easiest it will be to implement it regardless of the programming language.
Having said that, here is some of the thing you could look at :
correlation of the two datasets
difference of the derivative of the datasets (but I don't think it would be robust enough)
spectral analysis as mentionned by #thron of three
etc. ...
Knowing the origin of the datasets and their variability can also help a lot in formulating robust enough algorithms.
Sure. Call your two vectors A and B.
1) (Optional) Smooth your data either with a simple averaging filter (Matlab 'smooth'), or the 'filter' command. This will get rid of local changes in velocity ("gradient") that appear to be essentially noise (as in the ascending component of the red trace.
2) Differentiate both A and B. Now you are directly representing the velocity of each vector (Matlab 'diff').
3) Add the two differentiated vectors together (element-wise). Call this C.
4) Look for all points in C whose absolute value is above a certain threshold (you'll have to eyeball the data to get a good idea of what this should be). Points above this threshold indicate highly similar velocity.
5) Now look for where a high positive value in C is followed by a high negative value, or vice versa. In between these two points you will have similar curves in A and B.
Note: a) You could do the smoothing after step 3 rather than after step 1. b) Re 5), you could have a situation in which a 'hill' in your data is at the edge of the vector and so is 'cut in half', and the vectors descend to baseline before ascending in the next hill. Then 5) would misidentify the hill as coming between the initial descent and subsequent ascent. To avoid this, you could also require that the points in A and B in between the two points of velocity similarity have high absolute values.
I am a beginner in MATLAB and I should perform a spectral analysis of an EEG signal drawing the graphs of power spectral density and spectrogram. My signal is 10 seconds long and a sampling frequency of 160 Hz, a total of 1600 samples and have some questions on how to find the parameters of the functions in MATLAB, including:
pwelch (x, window, noverlap, nfft, fs);
spectrogram (x, window, noverlap, F, fs);
My question then is where to find values for the parameters window and noverlap I do not know what they are for.
To understand window functions & their use, let's first look at what happens when you take the DFT of finite length samples. Implicit in the definition of the discrete Fourier transform, is the assumption that the finite length of signal that you're considering, is periodic.
Consider a sine wave, sampled such that a full period is captured. When the signal is replicated, you can see that it continues periodically as an uninterrupted signal. The resulting DFT has only one non-zero component and that is at the frequency of the sinusoid.
Now consider a cosine wave with a different period, sampled such that only a partial period is captured. Now if you replicate the signal, you see discontinuities in the signal, marked in red. There is no longer a smooth transition and so you'll have leakage coming in at other frequencies, as seen below
This spectral leakage occurs through the side-lobes. To understand more about this, you should also read up on the sinc function and its Fourier transform, the rectangle function. The finite sampled sequence can be viewed as an infinite sequence multiplied by the rectangular function. The leakage that occurs is related to the side lobes of the sinc function (sinc & rectangular belong to self-dual space and are F.Ts of each other). This is explained in more detail in the spectral leakage article I linked to above.
Window functions
Window functions are used in signal processing to minimize the effect of spectral leakages. Basically, what a window function does is that it tapers the finite length sequence at the ends, so that when tiled, it has a periodic structure without discontinuities, and hence less spectral leakage.
Some of the common windows are Hanning, Hamming, Blackman, Blackman-Harris, Kaiser-Bessel, etc. You can read up more on them from the wiki link and the corresponding MATLAB commands are hann, hamming,blackman, blackmanharris and kaiser. Here's a small sample of the different windows:
You might wonder why there are so many different window functions. The reason is because each of these have very different spectral properties and have different main lobe widths and side lobe amplitudes. There is no such thing as a free lunch: if you want good frequency resolution (main lobe is thin), your sidelobes become larger and vice versa. You can't have both. Often, the choice of window function is dependent on the specific needs and always boils down to making a compromise. This is a very good article that talks about using window functions, and you should definitely read through it.
Now, when you use a window function, you have less information at the tapered ends. So, one way to fix that, is to use sliding windows with an overlap as shown below. The idea is that when put together, they approximate the original sequence as best as possible (i.e., the bottom row should be as close to a flat value of 1 as possible). Typical values vary between 33% to 50%, depending on the application.
Using MATLAB's spectrogram
The syntax is spectrogram(x,window,overlap,NFFT,fs)
where
x is your entire data vector
window is your window function. If you enter just a number, say W (must be integer), then MATLAB chops up your data into chunks of W samples each and forms the spectrogram from it. This is equivalent to using a rectangular window of length W samples. If you want to use a different window, provide hann(W) or whatever window you choose.
overlap is the number of samples that you need to overlap. So, if you need 50% overlap, this value should be W/2. Use floor(W/2) or ceil(W/2) if W can take odd values. This is just an integer.
NFFT is the FFT length
fs is the sampling frequency of your data vector. You can leave this empty, and MATLAB plots the figure in terms of normalized frequencies and the time axis as simply the data chunk index. If you enter it, MATLAB scales the axis accordingly.
You can also get optional outputs such as the time vector and frequency vector and the power spectrum computed, for use in other computations or if you need to style your plot differently. Refer to the documentation for more info.
Here's an example with 1 second of a linear chirp signal from 20 Hz to 400 Hz, sampled at 1000 Hz. Two window functions are used, Hanning and Blackman-Harris, with and without overlaps. The window lengths were 50 samples, and overlap of 50%, when used. The plots are scaled to the same 80dB range in each plot.
You can notice the difference in the figures (top-bottom) due to the overlap. You get a cleaner estimate if you use overlap. You can also observe the trade-off between main lobe width and side lobe amplitude that I mentioned earlier. Hanning has a thinner main lobe (prominent line along the skew diagonal), resulting in better frequency resolution, but has leaky sidelobes, seen by the bright colors outside. Blackwell-Harris, on the other hand, has a fatter main lobe (thicker diagonal line), but less spectral leakage, evidenced by the uniformly low (blue) outer region.
Both these methods above are short-time methods of operating on signals. The non-stationarity of the signal (where statistics are a function of time, Say mean, among other statistics, is a function of time) implies that you can only assume that the statistics of the signal are constant over short periods of time. There is no way of arriving at such a period of time (for which the statistics of the signal are constant) exactly and hence it is mostly guess work and fine-tuning.
Say that the signal you mentioned above is non-stationary (which EEG signals are). Also assume that it is stationary only for about 10ms or so. To reliably measure statistics like PSD or energy, you need to measure these statistics 10ms at a time. The window-ing function is what you multiply the signal with to isolate that 10ms of a signal, on which you will be computing PSD etc.. So now you need to traverse the length of the signal. You need a shifting window (to window the entire signal 10ms at a time). Overlapping the windows gives you a more reliable estimate of the statistics.
You can imagine it like this:
1. Take the first 10ms of the signal.
2. Window it with the windowing function.
3. Compute statistic only on this 10ms portion.
4. Move the window by 5ms (assume length of overlap).
5. Window the signal again.
6. Compute statistic again.
7. Move over entire length of signal.
There are many different types of window functions - Blackman, Hanning, Hamming, Rectangular. That and the length of the window and overlap really depend on the application that you have and the frequency characteristics of the signal itself.
As an example, in speech processing (where the signals are non-stationary and windowing gets used a lot), the most popular choices for windowing functions are Hamming/Hanning of length 10ms (320 samples at 16 kHz sampling) with an overlap of 80 samples (25% of window length). This works reasonably well. You can use this as a starting point for your application and then work on fine-tuning it a little more with different values.
You may also want to take a look at the following functions in MATLAB:
1. hamming
2. hanning
I hope you know that you can call up a ton of help in MATLAB using the help command on the command line. MATLAB is one of the best documented softwares out there. Using the help command for pwelch also pulls up definitions for window size and overlap. That should help you out too.
I don't know if all this info. helped you out or not, but looking at the question, I felt you might have needed a little help with understanding what windowing and overlapping was all about.
HTH,
Sriram.
For the last parameter fs, that is the frequency rate of the raw signal, in your case X, when you extract X from audio data using function
[X,fs]=audioread('song.mp3')
You may get fs from it.
Investigate how the following parameters change the performance of the Sinc function:
The Length of the coefficients
The Following window functions:
Blackman Harris
Hanning
Bartlett