# Peak location estimation for a triangle wave in AGWN?

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.

• hotpaw2, can you show some plots for best/worst case scenarios? May 20, 2012 at 21:01
• @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. May 20, 2012 at 21:14
• Not the frequency, but the point at which the 1st derivative discontinously reverses (e.g. for instance dy/dx goes from +1 to -1). May 21, 2012 at 0:34

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).