# What are the modes of a transform basis?

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).

• What I think he means when he says mode is that when we are transforming a signal lets say a sum of 20 and 40 KHz tone signals when we do a transform (fourier) in this case then we will only see spikes at 20 and 40 KHz (assuming ideal signal and no leakage just to get the idea) hence only couple of modes (20 and 40 KHz) are active in this case. Hope that makes sense.. Feb 25, 2023 at 16:49