When you perform the Haar wavelet transform, you take the sums and differences, then at each stage, you multiply the entire signal by $\small\sqrt2$.
When taking the inverse transform, you multiply the signal by $\frac{1}{\sqrt2}$ for each iteration.
What does this "normalization" really represent?
Answer
As I understand it, the normalization is because the Haar wavelet conserves energy of the signal. In that, when you take signal from one domain to another, you aren't supposed to add energy to it, (although conceivably you might lose energy).
The normalization is just a way to ensure that the energy of your Haar-transformed signal in the Haar-domain has the exact same energy as your signal in the original domain.
Intuitively speaking, Haar, Fourier, etc, are all just basis transformations, which intuitively just mean that you are looking at the signal in a different way, (technically, through a different set of bases). Therefore, if all you are doing is looking at a signal differently, its energy cannot/should not change.
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