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Coherence Calculation

Discussion in 'Electronic Design' started by David L. Jones, Nov 30, 2005.

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  1. A rather unusual question...
    I am looking for a way to calculate the coherence value of two signals
    which are several cycles of a fixed frequency sinusoidal like waveform.
    i.e. I need a single value in the range 0-1 for the coherence of the
    two waveforms.

    I have tried calculating the coherence using the standard formula:
    |Sxy|^2 / (Sxx.Syy)
    where Sxy is the Cross Power Spectrum and Sxx and Syy are the AutoPower
    Spectrums
    and then extracting the value of the single frequency I am interesed in
    from the frequency domain response.
    But the coherence specturm calcuation using this technique is only
    valid with averaged data samples, and I only have *one* set of sampled
    data for each waveform, so I always get a result of 1.0 regardless of
    the actual coherence between the two waveforms.

    Does anyone know of a way to calculate coherence, or a "coherence like"
    result for *non-averaged* data that gives a result from 0 to 1 for two
    similar sine waves?

    Any help appreciated.

    Thanks
    Dave :)
     
  2. Real_McCoy

    Real_McCoy Guest

    No - I always average.

    McC
     
  3. Peter K.

    Peter K. Guest

    Using coherence on pure (or close-to-pure) sinusoids is usually a bad
    idea -- because, if the sinusoids are of the same frequency (i.e. are
    phase or frequency locked), they'll be "completely coherent", and if
    they're not they won't be.
    I don't think it's because you only have one set of data, I think it's
    because you're only looking at one frequency --- that of the sinusoids
    of interest.
    Start back at square one: tell us why you're really interested in the
    sine waves! :)

    Do they have a phase-shift between them? -> Use correlation to find it.

    Do they have a frequency-shift between them? -> Mix (multiply) them and
    look at the beat frequency.

    You'll need to supply the background before we can help further, I
    think.

    Ciao,

    Peter K.
     
  4. Yes, I am aware it's not the best of ideas, as coherence is usually
    done and calculated over a frequency domain with broadband data.

    The two waveforms will in practice not be pure sinusoids, but will have
    minor noise and distortion components, so in theory a coherence
    calculation is possible. In fact, the goal is to have them as
    completely coherent as possible (as it is for most systems) i.e. no
    noise or distortion is added to the system, and all of the output
    signal is due entirely to the input signal.
    No, it's the same over the entire frequency domain. The standard
    coherence function as presented relies on averaging. If you have
    calculate coherence with no averaging you get a result of 1.0 in every
    frequency bin. This is why every dynamic signal analyser will not allow
    you to display coherence without turning on averaging.
    For mostly political reasons (don't ask!) I require a "coherence"
    number of 0 to 1 to indicate the "quality" of one waveform compared to
    a reference.
    The standard way to do this is with a coherence function, but in this
    case I do not have the averaged data sets available to do this usign
    the standard technique.

    In the end I may have to compare the signals in the time domain instead
    of using the coherence function in the frequency domain. I have a way
    to do this to give a 0 to 1 "coherence like" number, but I am wondering
    if anyone knows how to do it in the frequency domain using a coherence
    function?

    I know there are many other ways to compare two waveforms, but I need a
    0 to 1 "coherence" indication display.

    Dave :)
     
  5. Alan Peake

    Alan Peake Guest

    Try Average Cross-Correlation
    R12(t) = (Integration sign from -infinity to +infinity)
    v1(t)*v2(t)*(T+t)dt
    This is for non-peiodic waveforms.
    reference: Principles of Communication Systems - Taub+Schilling

    Alan
     
  6. Thanks Alan.
    My waveform will however be periodic, several full cycles of fixed
    frequency data.
    So probably not suitable?

    Dave :)
     
  7. Peter K.

    Peter K. Guest

    So the outcome you'd like is for the "coherence" to be 1.
    OK. That's a little strange, but perhaps you're right about the
    averaging... though I'm not exactly sure what you mean.

    Dumb question: why can't you just block the data you have into,
    admittedly smaller, datasets so you _can_ use the averaging technique?

    That's what matlab does: if I do this:

    x = sin(0.090823*[0:1023]+rand(1)*2*pi) + randn(1,1024);
    y = sin(0.090823*[0:1023]+rand(1)*2*pi) + randn(1,1024);

    and then try

    cohere(x,y,1024)

    everything is 1, but if I go

    cohere(x,y,128)

    I don't get the frequency resolution, but I do get something that's 1
    around the right frequency and not 1 outside of that.

    Bear in mind that this example is very fake: even with random phases,
    the freqyencies are effectively "locked".
    OK. Ain't politics a killer? :)
    Total harmonic distortion (THD)? That'll start at zero and probably never
    get close to 1. If you need "good" to be close to 1, how about: 1 - THD?

    Thanks for taking the time to explain more about the problem.

    Let us know if any of this is a help.

    Ciao,

    Peter K.
     
  8. In Matlab

    coh=rand;

    ;)
     
  9. Peter K.

    Peter K. Guest

    Stan Pawlukiewicz wrote:

    :)

    Ciao,

    Peter K.
     
  10. Yep. That is the desired outcome for any measurement system, which is
    why coherence measurement is of such importance, especially in the
    vibration and acoustics systems I am involved with.
    Unfortunately I don't have the necessary math skills to explain it, but
    the coherence function is inherently meaningless if you don't do it
    over averaged data. Your Matlab example below shows that the result is
    1 for all frequency bins, and that is exactly my problem too.
    If you are interested, this might explain it a little better:
    http://www.educatorscorner.com/media/Exp96.pdf
    Yes, I thought about this, but I don't think I have sufficent data to
    do this.
    I'm already getting 10Hz frequency bins on the output of my FFT power
    spectrum functions, and the signal I want to measure is 40Hz. Not
    exactly crash-hot resolution :-(
    Also, I might only have several cylces in total to work with (it is
    user selected in software). In fact, in some cases I might only have 1
    cycle to work with, so my coherence calculation function might have to
    handle that eventuality too.
    The #1 killer of every cool project!
    Gave that a brief thought, but I suspect it may be be a bit "touchy"
    and not robust enough. I would actually need a THD "envelope" pass/fail
    band as my two signals may (or may not) have significant THD, but have
    still perfect coherence.
    Not sure how you would scale that to a 0-1 result either...

    In any case, I've already got a time domain analysis method in mind
    that will work, based on scaled means and error difference between the
    two waveforms. I just have to go through the motions to eliminate the
    possibility of any other frequency domain coherence method.
    Thanks for your help Peter. It is good to know that Matlab gives the
    same result I am getting in my LabWindows/CVI software.

    Regards
    Dave :)
     
  11. In Labwindows/CVI:
    coherence=Random(0.8,1.0);

    Tempting, very tempting! ;-)

    The stupid thing is, I could most likely get away with it, having just
    shifted our managerial focus to being "results driven"!

    Dave :)
     
  12. Peter K.

    Peter K. Guest

    You poor b*st*rd! :)

    Ciao,

    Peter K.
     
  13. It's ok, it'll change again next month, regular as clockwork!

    Dave :)
     
  14. Hi David,

    It seems that it is unecessary to go to the frequency domain for this
    problem, since all you want to do compare time-domain images of short
    duration. As you probably discovered, you get impulses in the
    frequency domain for signals of low spectral variance.

    One way to calculate the "coherence" in the time domain for
    short-duration signals is:

    1. take the time-domain samples of two signals and treat them as
    vectors in N-space.
    2. Normalize each vector so as to kill any power advantage that one
    might have over the other. They will retain their positive and
    negative components.
    3. Take the scalar product between X & Y to yield a value from -1 to
    +1.
    4. Shift the scalar product +1 to get value between 0 and 2.
    5. Divide result by 2 to get value between 0 and 1.

    If you prefer, you can avoid steps 4 and 5, so you could end up with
    confidences of
    +1 = strongly correlated
    0 neutral
    -1 strongly uncorrelated

    Naturally, the hills and valeys will dominate the output of the
    correlation if you use this, so if you are really interested in
    phase/magnitude correlation, you can do something similar:

    1. Normalize both signals (before or after DFT - after relieves any
    cofactor fuzzies in FT)
    2. Take DFT
    3. Treating phase and magnitude separately, take scalar products to
    yield -1 < x > +1.

    But again, for signals that are both in phase with same spectral
    content, you should not be surprised to get something very close to 1
    on both accounts, as the frequency domain will contain weak noise
    dominated by impulses convolved with sin(x)/x.

    -Le Chaud Lapin-
     
  15. Have you looked at one of the Bendat and Piersol books, like Random
    Data? Most of the stuff they do is vibration analysis and I do recall
    they cover coherence in some detail.
     
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