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Is there a more optimal peak location estimator specifically for triangle waves in AGWN, compared with peak estimation for generic waveforms?

For a "smooth" waveform in noise, a common peak locator method might involve a linear-phase low-pass filter or an overdetermined polynomial curve fit (by regression, et.al.), followed by a maxima/minima search. But if the signal is known to be a sharp triangle wave (which may imply that any sampling included/aliased some non-filtered frequency content above Fs/2, if that is relevant), or portion thereof, and not a low order polynomial or other smooth waveform, can this information be used to improve the triangle peak location estimation?

If so, how?

Added: Less than 1 period of the triangle wave may be available. But assume at least one extrema is in the data window.

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  • $\begingroup$ hotpaw2, can you show some plots for best/worst case scenarios? $\endgroup$
    – Spacey
    May 20 '12 at 21:01
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    $\begingroup$ @hotpaw2 Can you edit your question to clarify a few things? First, you need to define 'optimal'. Are you looking for the minimum-variance unbiased estimator (MVUE)? Or something that is optimal in some other sense? Second, can you define 'peak location'? Is this the fundamental frequency of your triangle wave? Also, can you provide a mathematical definition of the triangle wave (although it shouldn't matter too much)? Thanks. To answer your question, absolutely. Typically, the more you know about the signal the better estimator you can build. But we need more information about the problem. $\endgroup$
    – Bryan
    May 20 '12 at 21:14
  • $\begingroup$ Not the frequency, but the point at which the 1st derivative discontinously reverses (e.g. for instance dy/dx goes from +1 to -1). $\endgroup$
    – hotpaw2
    May 21 '12 at 0:34
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You may be able to determine this in the frequency domain. The noise signal is broad band and has low spectral density at any one frequency. The triangle, however, has all energy concentrated in just a few frequencies, so these should stick out in a short term Fourier spectrum. You can match against your expected spectrum (odd harmonics, amplitude proportional to 1/n^2).

The harmonics of a triangle are all in phase. Since your triangle wave is arbitrarily located inside your analysis window, the harmonics will appear to have a linear phase. You can probably model your measured phase (after proper unwrapping) as a linear phase using a simple least squares error approach. This should give you the exact position of the peak of the triangular wave with respect to the beginning for your analysis window.

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My first guess would be to fit a sine wave + noise model to the data (using something like ESPRIT or MUSIC) ; use the estimated amplitude as the peak value, and $\frac{2 \pi k + \frac{\pi}{2} - \phi}{\omega}$ as peak locations. My intuition is that approximating a triangle wave by a sine wave isn't such a bad assumption - given how fast the spectrum of a triangle wave decays (only odd harmonics, with a $\frac{1}{n ^ 2}$ decay).

Of course, works only if the amplitude and frequency of the triangle wave you want to monitor is stationary over a sufficiently large amount of samples.

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  • $\begingroup$ "My first guess would be to fit a sine wave + noise model to the data (using something like ESPRIT or MUSIC)", Pichenettes, I am familiar with both MUSIC and ESPRIT but how do you mean to use them to fit a triangle waveform to a sinusoid? Do you mean just just use MUSIC/ESPRIT to do a frequency estimation of the triangular waveform, or is there something more to it? $\endgroup$
    – Spacey
    May 21 '12 at 0:07
  • $\begingroup$ The MUSIC algorithm requires you to estimate or know the number of discrete sinusoids a priori. Therefore, it is not necessary and in fact detrimental to model as a single sinusoid, as you will average in eigenvectors belonging to the signal subspace. If one were to use the MUSIC algorithm, it would be wise to look at the worst case for the number of harmonics and be conservative in using this number. You can always leave out some noise subspace vectors in the average with less consequences. $\endgroup$
    – Bryan
    May 21 '12 at 0:17
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    $\begingroup$ @Mohammad: Just estimate 1 sinusoid from the signal - I don't think it'll lock to anything else than the fundamental of the triangle wave. $\endgroup$ May 21 '12 at 1:13

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