# Does it make sense to use a wavelet that is equal to a sine of one period?

Would I get same results with STFT method (short-time Fourier transform) if I try this?

With all due caution, no in both cases (title and body question). I'll start with the second one.

• Continuous wavelets use all dilations of the mother wavelet, which are not accessed with the STFT
• The STFT is complex in general, and the windowed sine is not.

For the first one: I never tried it, and do not remember having seen it in use, and one should check first whether it satisfies the admissibility condition (first thought: seems OK on 0 in Fourier, and decays at infinity, I'd expect families of synthesis wavelets). You can compute the effects in Fourier using SE.Math: Fourier transform of 1 cycle of sine wave.

However, the sine falls abruptly at its edges, without differentiability, in other words it is not smooth enough. So I would not expect a more interesting interpretation of the scalogram than with other continuous wavelet.

But it could be used as an exploratory tool (analysis) for some specific signals. You can think about hybrid methods as well: try to fit (in more or less robust ways) a parametric sine model, at different scales.

• Thanks for your answer. Regarding the first question : I was thinking that by doing this i would get FFT information ( good frequency information because sine has only one frequency component) but also time information. I hope that my question is not that stupid. – John Jun 10 at 11:49
• Unfortunately, a single period of a sine is not " only one frequency component", since it is windowed by a box, hence has cardinal sine type ripples in the frequency domain – Laurent Duval Jun 10 at 11:57
• @John it's certainly not stupid! But it forgets that wavelets are designed to be a trade-off between frequency and spatial resolution. Sadly, due to the math, there's no free lunch: you can't have high frequency resolution and high spatial resolution at the same time; the bandwidth-time product is always limited (that's Heisenberg's uncertainty in signal terms). – Marcus Müller Jun 10 at 11:57
• As Marcus says, it's certainly not stupid, and reveals intuition. Even answers are negative, this should not prevent you from trying, and you'll learn a lot (and possibly prove us wrong). I'd expect difficulties in the time discretization, though – Laurent Duval Jun 10 at 12:08
• I fully agree with Laurent, @John! By the way, often questions that explain what you want to achieve, describe what you've considered so far (STFT and wavelets) and how that failed, asking for advice how to solve the overall problem, are very well-received and get a lot of good answers here. So maybe go and ask a new question along the lines of "I want to achieve…"! – Marcus Müller Jun 10 at 13:56

No, since the STFT gives you complex values by not only correlating to sines of different periods, but also cosines.

Depends on what kind of “sense” you want.

A 1 cycle sine wavelet might give you good time resolution for locating a mostly positive to negative zero-crossing waveform slice, but almost no frequency resolution, as the nearest orthogonal sinewave is a whole octave higher.

Add more cycles and you get more frequency resolution, but less time resolution.

A longer windowed DFT provides more information, as most of the basis vectors have more than one cycle, and the complex, or sine & cosine format also adds some phase information to each slice.