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So, I'm reading Steven Brunton's book, "Data Driven Science & Engineering", and I'm trying to understand what he means by mode in this following excerpt:

Most natural signals, such as images and audio, are highly compressible. This compressibility means that when the signal is written in an appropriate basis only a few modes are active, thus reducing the number of values that must be stored for an accurate representation. Said another way, a compressible signal $\mathbf{x} \in \mathbb{R}^n$ may be written as a sparse vector $\mathbf{s} \in \mathbb{R}^n$ (containing mostly zeros) in a transform basis $\Psi \in \mathbb{R}^{n \times n}$ : $$ \mathrm{x}=\Psi \mathrm{s} $$

What I'm thinking is that he means the operations (or operators? not sure which one is correct) that the dictionary or the transform basis contains, such as $e^x$ or $\sin(x)$ etc. Those are the examples that I'm thinking due to the DWT and FFT being examples of traditional transform bases. (that is, if I haven't got anything wrong).

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  • $\begingroup$ Welcome to Cross Validated! This might be on-topic here, but you are likely to get better responses on Signal Processing Stack Exchange. $\endgroup$
    – Dave
    Feb 24, 2023 at 15:50
  • $\begingroup$ Thanks Dave! I'll check that out. $\endgroup$
    – Nyquist-er
    Feb 24, 2023 at 15:58

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This is actually not "Mode" in the statistical sense but rather "Mode" in the Wave Theory sense (i.e. a standing wave of a given frequency) https://en.wikipedia.org/wiki/Normal_mode#Mode.

So you're right, the book is just referencing how you'll get few coefficients that aren't zero or close to zero from a Fourier or Laplace transform of a natural signal.

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