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As far as I know a $\sin(x)$ which is one cycle long can be made into a wavelet and then we can use its dilated and translated versions to represent another function, effectively a wavelet transform.

We have wavelets like Haar and others, why not have a single cycle sinusoid as a wavelet?

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So you want to take the function $w(t)$:

  • $w(t) = \sin(t)$, if $-\pi < t < \pi$,
  • $w(t) = 0$, elsewhere,

and construct a family of functions $w\left(\frac{t-b}{a}\right)$ for wavelet-like analysis.

  • First, from a continuous wavelet transform (CWT) point of view, it satisfies a necessary condition related to the Admissibility condition for wavelets. But it is not sufficient, and exhibits derivative discontinuities that may hamper analysis (to be described later).

  • Second, I am not aware of a sound orthogonal discrete scheme for such a function. The reason could be that it does not satisfy MRA axioms.

  • Third: to focus on meaning: a windowed portion of a sine might lack regularity, or vanishing moments, to make it a really useful wavelet (compared to others).

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  • $\begingroup$ for the second pont, what if we define s(t) such that it becomes a wave between 0 and 1 rather than -pi and +pi? $\endgroup$ – quantum231 Feb 3 at 17:25
  • $\begingroup$ Between $0$ and $1$, the integral of a sine does not vanish, so you are pretty sure this is not a wavelet (which requires that the integral is zero) $\endgroup$ – Laurent Duval Feb 3 at 18:01
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A 1-cycle sinewave wavelet corresponds to imaginary component basis vector for bin 1 of a rectangularly windowed DFT result, and thus has the same problems of the lowest bins near DC of a rectangular windowed FFT.

The frequency response of a rectangular windowed FFT is a Sinc function with an expanse of large ripples in the response. The main lobe of the frequency response would be very wide relative to the frequency of that wavelet's basis. And the complex conjugate image, or negative frequency response, is nearby to the lowest bin(s); and thus their side lobes can potentially cause interference with the positive frequency response, depending on phase.

e.g. the wavelet would be highly subject to interference from noise and any other spectral content that wasn’t purely sinusoidal and exactly integer periodic relative to the wavelets length.

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