I want to design a score or signature function based on a time series signal. Usually, the signal has ups and downs.
For a given time window, I desire to design the score function based on the number of times it fluctuates, the duration of the fluctuations, and the magnitude of the fluctuations. I am wondering what kind of math I can use to design the function. I am not sure if the statistical features (mean, median, and so on) would be enough to design unique function such that two time windows would be distinguishable.
Thanks!
Summary statistics will not give you what you want... but it can still be useful.
Things you can try:
Zero crossings on the signal will give you number of fluctuations. You'll have to use some central tendency value to move the signal about the 0 line in order to do this. Alternatively you can use FFT on the original to find the harmonic frequency as part of the score.
Could define the duration of fluctuations as the difference between zero crossings divided by two (since one fluctuation will reach the 0-line twice).
Magnitude can be done by finding the local minima and maxima - check out some packages with peak finding functions. You might want to use the mean or median to rule out local minima and maxima that fall on the wrong side of the line. Alternatively, finding the zero crossings on the derivative signal and then mapping them back to the original will give you all the local minima and maxima as well.
Related
I have a requirement where I have to verify the transmit power out of a device as measured at its connector is within 2 dB of its expected value over 95% of test measurements.
I am using a signal analyzer to analyze the transmitted power. I only get the average power value, min, max and stdDev of the measurements and not the individual power measurements.
Now, the question is how would I verify the "95% thing" using average power, min, max and stdDev. It seems that I can use normal distribution to find the 95% confidence level.
I would appreciate if someone can help me on this.
Thanks in anticipation
The way I'm reading this, it seems you are a statistical beginner, so if I'm wrong there, the rest of this answer will probably be insultingly basic, and I'm sorry.
Anyway, the idea is that if a dataset is normally distributed, and all the observations are independent of one another, then 95% of the data points will fall within 1.96 standard deviations of the mean.
Do you get identical estimates of average power every time you measure, or are there some slight random differences from reading to reading? My guess is that it's the second. If you were to measure the power a whole bunch of times, and each time you plotted your average power value on a histogram, then that histogram of sample means would have the shape of a bell curve. This bell curve of sample means would have its own mean and standard deviation, and if you have thousands or millions of data points going into the calculation of each average power reading, it's not horrible to assume that it is a normal distribution. The explanation for this phenomenon is known as the 'central limit theorem', and I recommend both the Khan academy's presentation of it as well as the wikipedia page on it.
On the other hand, if your average power is the mean of some small number of data points, like for instance n= 5, or n= 30, then assumption of a normal distribution of sample means can be pretty bad. In this case, your 95% confidence interval around the average power goes from qt(0.975,n-1)*SD/sqrt(n) below the average to qt(0.975,n-1)*SD/sqrt(N) above the average, where qt(0.975,n-1) is the 97.5th percentile of the t distribution with n-1 degrees of freedom, and SD is your measured standard deviation.
Im fairly new to onset detection. I read some papers about it and know that when working only with the time-domain, it is possible that there will be a large number of false-positives/negatives, and that it is generally advisable to work with either both the time-domain and frequency-domain or the frequency domain.
Regarding this, I am a bit confused because, I am having trouble on how the spectral energy or the results from the FFT bin can be used to determine note onsets. Because, aren't note onsets represented by sharp peaks in amplitude?
Can someone enlighten me on this? Thank you!
This is the easiest way to think about note onset:
think of a music signal as a flat constant signal. When and onset occurs you look at it as a large rapid CHANGE in signal (a positive or negative peak)
What this means in the frequency domain:
the FT of a constant signal is, well, CONSTANT! and flat
When the onset event occurs there is a rapid increase in spectrial content.
While you may think "Well you're actually talking about the peak of the onset right?" not at all. We are not actually interested in the peak of the onset, but rather the rising edge of the signal. When there is a sharp increase in the signal, the high frequency content increases.
one way to do this is using the spectrial difference function:
take your time domain signal and cut it up into overlaping strips (typically 50% overlap)
apply a hamming/hann window (this is to reduce spectrial smudging) (remember cutting up the signal into windows is like multiplying it by a pulse, in the frequency domain its like convolving the signal with a sinc function)
Apply the FFT algorithm on two sucessive windows
For each DFT bin, calculate the difference between the Xn and Xn-1 bins if it is negative set it to zero
square the results and sum all th bins together
repeat till end of signal.
look for peaks in signal using median thresholding and there are your onset times!
Source:
https://adamhess.github.io/Onset_Detection_Nov302011.pdf
and
http://www.elec.qmul.ac.uk/people/juan/Documents/Bello-TSAP-2005.pdf
You can look at sharp differences in amplitude at a specific frequency as suspected sound onsets. For instance if a flute switches from playing a G5 to playing a C, there will be a sharp drop in amplitude of the spectrum at around 784 Hz.
If you don't know what frequency to examine, the magnitude of an FFT vector will give you the amplitude of every frequency over some window in time (with a resolution dependent on the length of the time window). Pick your frequency, or a bunch of frequencies, and diff two FFTs of two different time windows. That might give you something that can be used as part of a likelihood estimate for a sound onset or change somewhere between the two time windows. Sliding the windows or successive approximation of their location in time might help narrow down the time of a suspected note onset or other significant change in the sound.
"Because, aren't note onsets represented by sharp peaks in amplitude?"
A: Not always. On percussive instruments (including piano) this is true, but for violin, flute, etc. notes often "slide" into each other as frequency changes without sharp amplitude increases.
If you stick to a single instrument like the piano onset detection is do-able. Generalized onset detection is a much more difficult problem. There are about a dozen primitive features that have been used for onset detection. Once you code them, you still have to decide how best to use them.
I have a question regarding how to determine the Duration of notes given their Onset Locations.
So for example, I have an array of amplitude values (containing short) and another array of the same size, that contains a 1 if a note onset is detected, and a 0 if not. So basically, the distance between each 1 will be used to determine the duration.
How can I do this? I know that I have to use the Sample Rate and other attributes of the audio data, but is there a particular formula that I can use?
Thank you!
So you are starting with a list of ONSETS, what you are really looking for is a list of OFFSETS.
There are many methods for onset detection (here is a paper on it) https://adamhess.github.io/Onset_Detection_Nov302011.pdf
many of the same methods can be applied to Offset Detection:
Since the onset is marked by an INCREASE in spectral content you can measure a decrease in Spectral content.
take a reasonable time window before and after your onset. (.25-.5s)
Chop up the window into smaller segments and take 50% overlapping Fourier transforms.
compute the difference between the fourier co-efficient between two successive windows decreases and only allow negative changes in SD.
multiple your results by -1.
pick the peaks off of the results
Voila, offsets.
(look at page 7 of the paper listed above for more detail about spectrial difference function, you can apply a modified (as above) version of it_
Well, if your samplerate in Hz is fs, then the time between two nodes is equal to
1/fs * <number of zeros between the two node-ones>
Very simple :-)
Regards
I'm implementing a 'filter sweep' effect (I don't know if it's called like that). What I do is basically create a low-pass filter and make it 'move' along a certain frequency range.
To calculate the filter cut-off frequency at a given moment I use a user-provided linear function, which yields values between 0 and 1.
My first attempt was to directly map the values returned by the linear function to the range of frequencies, as in cf = freqRange * lf(x). Although it worked ok it looked as if the sweep ran much faster when moving through low frequencies and then slowed down during its way to the high frequency zone. I'm not sure why is this but I guess it's something to do with human hearing perceiving changes in frequency in a non-linear manner.
My next attempt was to move the filter's cut-off frequency in a logarithmic way. It works much better now but I still feel that the filter doesn't move at a constant perceived speed through the range of frequencies.
How should I divide the frequency space to obtain a constant perceived sweep speed?
Thanks in advance.
The frequency sweep effect you're referring to is likely a wah-wah filter, named for the ubiquitous wah-wah pedal.
We hear frequency in terms of octaves, and sweeping through octaves with a logarithmic scale is the way to linearize it. Not to sound dismissive, but it sounds like what you're doing is physically and mathematically correct. (You should spent as much time between 200 and 400 Hz as you do between 2000 and 4000 Hz, etc.) You just don't like how it sounds. And that's quite okay on both counts -- audio is highly subjective.
To mix things up a bit, one option would be to try the Bark scale, which is based on psychoacoustics and the structure of the ear. As I understand it, this is designed to spend equal amounts of time in each of your ear's internal "bandpass filters".
You could always try a quadratic or cubic function between 0 and 1. Audio potentiometers often use a few piecewise quadratic or cubic sections to get their mapping.
Winging it, but try this:
http://en.wikipedia.org/wiki/Physics_of_music#Scales "The following table shows the ratios between the frequencies of all the notes of the just major scale and the fixed frequency of the first note of the scale."
There is then a chart showing fractional values between 1 and 2, and if you tweak your timing to match, you may get what you wish. While the overall progression is still logarithmic, the stepping between each one should divide up into equal stepped 8ths (a bit jumpy).
Put another way, every half second adjust one note up. Each octave (I think) will cover twice the frequency range of the prior octave.
EDIT: Also, you'll find the frequencies here: http://en.wikipedia.org/wiki/Middle_C#Designation_by_octave (doesn't the programmer in you wish that C0 was exactly 16hz?)
Without any user interaction, how would a program identify what type of waveform is present in a recording from an ADC?
For the sake of this question: triangle, square, sine, half-sine, or sawtooth waves of constant frequency. Level and frequency are arbitrary, and they will have noise, small amounts of distortion, and other imperfections.
I'll propose a few (naive) ideas, too, and you can vote them up or down.
You definitely want to start by taking an autocorrelation to find the fundamental.
With that, take one period (approximately) of the waveform.
Now take a DFT of that signal, and immediately compensate for the phase shift of the first bin (the first bin being the fundamental, your task will be simpler if all phases are relative).
Now normalise all the bins so that the fundamental has unity gain.
Now compare and contrast the rest of the bins (representing the harmonics) against a set of pre-stored waveshapes that you're interested in testing for. Accept the closest, and reject overall if it fails to meet some threshold for accuracy determined by measurements of the noisefloor.
Do an FFT, find the odd and even harmonic peaks, and compare the rate at which they decrease to a library of common waveform.. peak... ratios.
Perform an autocorrelation to find the fundamental frequency, measure the RMS level, find the first zero-crossing, and then try subtracting common waveforms at that frequency, phase, and level. Whichever cancels out the best (and more than some threshold) wins.
This answer presumes no noise and that this is a simple academic exercise.
In the time domain, take the sample by sample difference of the waveform. Histogram the results. If the distribution has a sharply defined peak (mode) at zero, it is a square wave. If the distribution has a sharply defined peak at a positive value, it is a sawtooth. If the distribution has two sharply defined peaks, one negative and one positive,it is a triangle. If the distribution is broad and is peaked at either side, it is a sine wave.
arm yourself with more information...
I am assuming that you already know that a theoretically perfect sine wave has no harmonic partials (ie only a fundamental)... but since you are going through an ADC you can throw the idea of a theoretically perfect sine wave out the window... you have to fight against aliasing and determining what are "real" partials and what are artifacts... good luck.
the following information comes from this link about csound.
(*) A sawtooth wave contains (theoretically) an infinite number of harmonic partials, each in the ratio of the reciprocal of the partial number. Thus, the fundamental (1) has an amplitude of 1, the second partial 1/2, the third 1/3, and the nth 1/n.
(**) A square wave contains (theoretically) an infinite number of harmonic partials, but only odd-numbered harmonics (1,3,5,7,...) The amplitudes are in the ratio of the reciprocal of the partial number, just as sawtooth waves. Thus, the fundamental (1) has an amplitude of 1, the third partial 1/3, the fifth 1/5, and the nth 1/n.
I think that all of these answers so far are quite bad (including my own previous...)
after having thought the problem through a bit more I would suggest the following:
1) take a 1 second sample of the input signal (doesn't need to be so big, but it simplifies a few things)
2) over the entire second, count the zero-crossings. at this point you have the cps (cycles per second) and know the frequency of the oscillator. (in case that's something you wanted to know)
3) now take a smaller segment of the sample to work with: take precisely 7 zero-crossings worth. (so your work buffer should now, if visualized, look like one of the graphical representations you posted with the original question.) use this small work buffer to perform the following tests. (normalizing the work buffer at this point could make life easier)
4) test for square-wave: zero crossings for a square wave are always very large differences, look for a large signal delta followed by little to no movement until the next zero crossing.
5) test for saw-wave: similar to square-wave, but a large signal delta will be followed by a linear constant signal delta.
6) test for triangle-wave: linear constant (small) signal deltas. find the peaks, divide by the distance between them and calculate what the triangle wave should look like (ideally) now test the actual signal for deviance. set a deviance tolerance threshold and you can determine whether you are looking at a triangle or a sine (or something parabolic).
First find the base frequency and the phase. You can do that with FFT. Normalize the sample. Then subtract each sample with the sample of the waveform you want to test (same frequency and same phase). Square the result add it all up and divide it by the number of samples. The smallest number is the waveform you seek.