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Can I define frequency for an non periodic signal?

  1. Oct 10, 2012 #1
    Do non periodic signals have frequency? Because my pretty general rule

    f = 1/T​

    says that they have zero frequency.

    But suppose i analyze a voice signal. We generally associate a term frequency with them. If you ever had used audacity you might have noticed that the graph is quite random and its nearly impossible to find any periodicity in them. Then why don't they have zero frequency?

    I hope that you understand my post because my English is not so good as i am not a native English speaker.
     
  2. jcsd
  3. Oct 10, 2012 #2
    Yes, non periodic signal can be defined in frequency. Look up Fourier analysis and Fourier Transforms.
     
  4. Oct 10, 2012 #3
    The voice signal can be thought of as lets say 15Khz signal modulated by some slow frequency signal at 'around' 5Hz. (The former being periodic but the later non-periodic)
    Although the overall signal isn't periodic but we can always talk about the the high frequency component present on the non periodic signal.
     
  5. Oct 10, 2012 #4

    sophiecentaur

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    Transforming time domain information into the frequency domain is 'just' a theoretical idea which happens to be useful. One needs to be careful about what it actually means, in the real world, in each particular situation.
    For instance, you have to remember that any analysis you do will be of a finite number of samples (not even a continuum of information) and that it then assumes that the sequence repeats itself. Under many circumstances, when you do a simple FT on the data, you will generate artifacts which may lead you to wrong conclusions. You need to take this into account.
    We glibly assume that a spectrum analyser display of a passage of sound or a portion of Radio Spectrum is showing the 'true' spectrum of what's there. It may be masking the very thing that we happen to be looking for and this goes for how we analyse a set of time domain data.

    This is often dealt with by suitable 'windowing' your data. There is no end to the complications involved here and you need to decide just how deeply you want to go into it but just bear in mind that you need to treat the whole business carefully. :smile:
     
    Last edited: Oct 10, 2012
  6. Oct 10, 2012 #5
    If the signal is approximately periodic you can define an approximate period or frequency...

    This is done commonly with musical sounds, which are not periodic and shall not be (periodic sounds are very artificial to us and don't resemble real instruments). We hear a pitch, and electronic tuners can (often...) measure a pitch.

    If a signal is far from periodic, you can take its spectrum and tell for instance "the maximum power density is around this frequency" or "90% of the power is within this band". Only periodic signals have clear frequency lines in their spectrum.
     
  7. Oct 10, 2012 #6
    Analysis of aperiodic signal = analysis of series of periodic segments

    Sophiecentaur pretty much summed it up, I'll just say it in other words.

    Audio signals theoretially are not periodic. What happens in processing of such signals is that we take a section (called a frame, a window, a segment, etc.) of a certain length (25ms, 4096 samples, ...) and proclaim that this segment repeats indefinitely in the original signal (which is obviously not true). This creates a periodic signal which we can further analyse with Fourier transform for instance. This is what you see in Audacity or on any real-time spectral analyzer.

    Since this segment (and the resulting periodic signal assembled from this segment at all) is not the exact representation on the original signal (especially around the ends where we "glued" it together) the resulting spectrum suffers from artifacts (spurious spectral lines) that were not in the original signal. To minimize these artifacts we weigh the segment with "window functions" that suppress these incorrect spectral lines.

    This is a well-known problem with Fourier transform of non-periodic signals and there have been entire books written about it.

    Check this article about spectral leakage.
     
  8. Oct 29, 2012 #7
    Picture worth many words.

    Here is a spectrogram of me saying "Eating Pigs Makes Men Fat.":
    EatingPigs_Gauss2048_4xOverlap.png

    Several things to note:
    1) Predetermined, frequency-band rules for representing the voice are meaningless.

    2) This is windowed analysis. I told RX to use 2048 frequency bands and muliple window widths, since higher frequencies resolve in less time. In addition, the windows overlap in time, to improve time detail.

    3) THe waveform is displayd in blue.

    4) The weights that make up each (tiny) window, at the moment of transformation are coefficients of infinitely repeating periodic functions. In the spectrogram, all other periods are discarded. We pump out enough windows to "image" the whole recording. Each window is a fragment of the overall picture. You must determine how to bin the coefficients to satisfy the exact question you are asking about the data.

    For example, in this recording, I am speaking with a gravely throat voice. You can see in the waveform that the last vowel is stuttered with peaks: the motor-like rattle of my throat. But the spectrogram window is too long to capture it. The frequencies are correct, and better represent the vowel. Reduce the # of bands, and the stutter is visible, but the frequency representation is junk.

    The function is unknown. It does not become known by transformation. Hopefully you can see, however, how it could be useful to analyze signals in this representation.
    There are an infinite ways to break each recording up into individual weights to fill the 2D image. You can hang yourself letting the transform lead you around by the nose. Instead, define those parameters which make what you want to do possible.

    5) In this case, the function is finite. I opened my mouth, and for a time there was sound. The words did not repeat forever, after the recording stopped. But I may have to record many many versions of each sound, from different speakers, if I want to have enough information to identify one of these words reliably.

    6) Sustained horizontal lines are harmonic content. In general, vertical columns of (what look like TV static) are noise.

    7) We (and not he spectrogram) know what the function is. For example, the letter "t" is almost all noise. However, as far as we are concerned, it is "signal." To identify it, I might look for noise of a particular duration, distribution across the spectrum, relationship to surrounding signal, etc.

    8) To call the voice a function, I must construct both the rule sets for identifying each element of speech, and collect the core data (e.g. samples of real voices) which those rules will use to compare new data. If my model is not both descriptive and predictive, it is not a scientific model.
    Can the voice really be represented by one frequency modulating another?


    9) Although in general audio software lags behind image processing, from a mathematical and physical point of view the way forward is plain:
    Between the waveform (amplitude/time) and a (series of) spectral representation(s) of my choice, the question of properly modeling the voice is a topological one. I hope it is clear from the image just how useful a spectral image can be for defining "similar" signals. The spectrogram can be processed as an image. One giant question is how then to rebin my data as I work, so that image processing entails acceptable audio in the linear waveform.

    10) The comment about periodic signals not being like real instruments is misleading. Pure sine waves sound nothing like instruments because they're not anything like natural harmonic (periodic) functions. Vibrating strings and columns of air are the same thing as a series of harmonics. Even if the materials were ideal, these would not sound the way a synthesizer produces them. Then, of course, the signal is in continuous convolution with the amplification apparatus, etc. It's not because the light falling on your desk is poorly modeled by both a straight-line vector to the sun and by a static, single radius around the sun of one wavelength, that "real light" does not behave like a particle and a wave.
    The point is to model the phenomenon in sufficient detail to distinguish description from storytelling.


    I am not attempting to sound authoritative. These are all open questions for me. It's hard to find discussions about the subject where the questions are left open. Everyone seems eager to shut the door on slender assumptions. I'd like to keep them open until we can describe the problems, and not tell little tales about them which shut their mouths prematurely.

    Of course, if I have made faulty assumptions myself, I want to know. It's tough going to see straight, and I do it alone, so don't be shy. Corrections are precious.

    Cheers.
     
    Last edited: Oct 30, 2012
  9. Oct 29, 2012 #8
    Oh. I chose the sentence because the sequence of vowels:
    EAting PIgs mAkes mEn fAt
    IPA {i} {I} {e} {ε} {æ}

    ... illustrates some characteristics of formants. Below the blue waveform, there are bright clumps of frequency bands, separated by a region with less energy. These curves can be used to identify vowels. From speaker to speaker, certain of the anatomical mechanics are the same.

    If you take defining "the same" seriously for signals, this is a heck of a puzzle. I think this is the kind of thing SophieCentaur is alluding to with "no end" to the complications. Formants: consistently iidentifiable elements of human speech.

    And yet.
    ...
     
  10. Oct 30, 2012 #9

    Bobbywhy

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    Slumberland, Welcome to Physics Forums!

    Your spectrogram of "Eating pigs makes men fat" is an excellent visual representation of our voiced formants, harmonics, and noise. Your written explanation is superb! You've already made a fine contribution to our forum by helping others to learn.

    Cheers,
    Bobbywhy
     
  11. Oct 30, 2012 #10
    um. thanks!
    *blush*
    *awkward feet*

    I forgot the OP's specific question! It's buried in 6) the vertical columns of noise.

    I would trade "zero frequency" for "zero period":
    f(t) = f(t +0).
    Which is every function. :)

    Uncorrelated noise spits out noise. AllFrequenciesAtOnce, T=0. Percussive strikes, noise, friction, etc.
     
  12. Oct 30, 2012 #11

    sophiecentaur

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    If you are analysing speech, specifically, then the time window needs to be sub-syllabic - or the analysis will 'blur' the required information. If you look up 'voiceprint' then you can find the parameters that have been arrived at for universally accepted measurement systems but there won't be a 'correct' set of parameters.
     
  13. Oct 30, 2012 #12
    Real instruments produce signals that are NOT periodic and shall not be. We do perceive this and need it to recognize an instrument. Synthesis from harmonics, which creates a periodic signal, fails to imitate an instrument whatever precise the spectrum is.

    This is current knowledge for nearly 20 years, though still not widespread enough.

    After 100 years of failing to imitate music instruments by harmonic spectrum, which anyone can test with a simple software, and after researchers succeeded just by introducing non-periodic elements to the sound, I don't see why some people go on spreading these false ideas.
     
  14. Oct 30, 2012 #13
    *Sigh*
    Please talk physics and math. Are you giving up the initial assumption that musical tones are harmonic? Into what global context are you introducing these non-periodic elements?

    If you have a different groundwork, please substantiate your position carefully and at length. Start at the beginning. You have a disturbance propagating in space at a constant speed in all directions from a source. Or are you giving all this up? That would be novel.

    Appeals to experience and authority are not data.

    Clarify. Your comments continue to be misleading. A person who does not know that you're taking for granted the opposite assumption that you put in bold, will be misled. Wrong sections of books. Wrong physical assumptions to frame a problem.

    I know. I can't more-know the problem you're driving at. The state of sound design tools makes me so angry I changed the course of my life to tackle it. But it's not my fault or my problem if other people don't reason their way through problems, and don't listen objectively. The "current knowledge" you describe can be reconstructed by touching a string, employing vibrato, pedaling, using real materials. By making a few obvious observations about how instruments are played, constructed. Or simply by trying to construct an ideal material. :)

    But I am interested in the knowledge which cannot be so easily demonstrated by schoolchildren.

    Articulate your frustration in a way that is instructive or useful. You have fabricated a dichotomy. It does not describe or alleviate the problem before us. Stop cramming multiple, complex problems into blanket statements. Take them apart. These are not difficult problems. But they ARE problems. They are both numerous and specific.

    Let me give you an example. One of yours, back to you.

    My claim is that there are simple, identifiable problems with the way electronic synthesizers construct and distribute harmonics. I can put it into a question:

    What are the differences between electrically generated harmonics and those produced by a cello?

    Be specific.

    The failure of (most) synthesizer (manufacturers) to model physical phenomena -- the relative mathematical and logical poverty of the sound design field-- are not evidence against the vibrating string problem.
    Or the problem of convolution.
    Or the problem of transfer functions.
    Or the problem of energy loss.
    Or the problem of defining noise, and distinguishing the propagation of waves and noise in space and time.
    Or...
    Or...

    we will probably have to start a new thread. And from here on out I will expect to talk math and physics.
     
    Last edited: Oct 30, 2012
  15. Oct 31, 2012 #14

    sophiecentaur

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    @enthalpy and @ slumberland

    You are arguing at cross purposes. Here is some Maths and some Physics:
    If you analyse the sound from any instrument you will not get a periodic function because the word 'periodic' implies that it carries on forever. What comes from musical instruments consists of Overtones, which are never exact harmonics and can be many % different from the corresponding harmonics (so that puts the mockers on the idea of 'periodic'). Furthermore, it is the attack phase of most instruments that determines the characteristic sound and the waveform at that time is, by definition, very much not periodic.
    Synthesising (electronically) a musical instrument is a pretty thankless task. After all, it's hard enough to produce a 'copy' of the sound from one (physical) instrument using another instrument that's be made in the same sort of way and then the Physics is working in your favour. Sampling and tone shifting is also not much good because the mechanics just don't 'scale' up or down.
     
  16. Oct 31, 2012 #15

    AlephZero

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    If this is anything more than a rant, you will have to start by explaing what you think those "problems" actually are.

    If you want to discuss this subject using "science and math", it would help if you used some science and math in your explanations. At the very least, we might then be able to estimate how much science and math you already know, and respond accordingly.

    From your posts so far, I can't tell if (1) you are saying that over-simplistic models about "musical sounds" are taught in elementary physics courses, apparently without any realization that they are over-simplistic if not just plain wrong, or (2) you trying to dismiss everything that is currently known about acoustics as "wrong" and somehow start again.

    FWIW I entirely agree with you about (1), but I don't think (2) will lead anywhere useful.

    IMO your distinction between "noise" and "waves" is irrelevant. The definition of what is a "musical sound" (or a "wave") has nothing to do with physics. If it is a scientific question at all (which I doubt) the relevant science is anthropology, not physics.
     
  17. Nov 1, 2012 #16
    I am trying to. There is no point to introducing the vibrating string if we are not in agreement that there is such a thing as harmonic motion, and that this motion describes the boundary conditions imposed on an elastic cord stopped at both ends, and then displaced.

    For signal analysis this is equivalent to the assumption of periodicity.

    Okay. Here is what I will do.

    We have to agree to the same definitions and use them consistently. Let's back up.

    I will start a new thread shortly, so we can compare definitions and come to a common agreement.


    Then I have a favor then, to ask.
    I would prefer that you respond, always, in whatever detail is necessary to answer the question at hand. It is my responsibility to acquaint myself with the necessary material to understand that answer.

    Neither.

    To 1) It is my experience that authors are very careful to list their assumptions and offer places I can read further to understand them. I think when we read through the first time, we often have no picture of what those ramifications could be, and later blame the author.
    I find it hard to reconcile how a text is often spoken about, with the words on the page.
    Which is not to say that there are no such omissions and errors. I ran across the vibrating string in DiffEQ, I have no experience with the texts you describe. Sorry. I can be unhappy about such things in principle if that helps.

    To 2) I accept the entirety of classical mechanics. I suggest we use it.

    The distinction between noise and signal is both mathematical and physical. THere is a lot about this I do not know. I will stick with a basic example: white noise is uncorrelated. I think everyone will agree this is "noise." Pass it through an (the?) autocorrelation function, and there is no frequency band in which the function spends more time than any other. The new thread is for definitions, so I will post more there. The terminology is consistent for both audio and image processing. Of course there is no physics in a value judgment.

    I don't mind being checked for sociological assumptions, but I do not find them serviceable for physical description. I would prefer you assumed I was willing to take full responsibility for the mathematical and physical ramifications of my assertions. That will be entertaining for everyone, including me, and I will learn much faster.

    A wave satisfies the wave equation. I use H.F. Weinberger's A First Course In Partial Differential Equations.
     
  18. Nov 1, 2012 #17

    K41

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    Why does period have to involve noise and signals? Why don't we look at three dimensional body dynamical systems :P
     
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