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  • Generally speaking, what analyses are necessary and sufficient for the detection of periodicities in an n-dimensional signal amounting to a discretely sampled density distribution over n-dimensional space?

Assume that the signal-to-noise ratio and sampling frequency in all dimensions are adequate for detection of any periodicities that are present.

  • Would an n-dimensional Fourier transform and an n-dimensional autocorrelogram suffice?
  • Would these be limited to detecting periodicities that are aligned with one of the basis vectors of the n-D space?

Suggested references for further reading would be appreciated.

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Would these be limited to detecting periodicities that are aligned with one of the basis vectors of the n-D space?

Well, if you say "sampling is sufficient", then it follows that the whole space is the span of the base vectors (in fact, you usually have them orthogonal).

From that follows that a non-base-vector aligned oscillation can be represented by a linear combination of oscillations that are aligned with base vectors.

From that follows that yes, with your requirement that sampling frequency is sufficient to begin with, the usual methods of 1D frequency analysis work.

Note that you can't generally detect all periodicities with just a sufficiently fast sampling – if something is periodic with a period longer than you observation window, you have no information about that. So, observation length must be sufficient, too, unless your signal model restricts both highest and lowest frequency.

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