# Discrete Fourier transform - Norms of complex input signals and their transformation

Given a signal $$\mathbf{z} \in \mathbb{C}^n$$ and its Discrete Fourier transform $$\hat{\mathbf{z} }$$, does $$||\mathbf{z}|| = ||\hat{\mathbf{z} }||$$ hold?

The question is given to me like this with no additional details. Information about what kind of norm is also not given. Does anyone have an idea what the question might be looking for?

## 1 Answer

I may consider them as L^2 then following theorem holds.

https://en.wikipedia.org/wiki/Plancherel_theorem

https://de.wikipedia.org/wiki/Satz_von_Plancherel

https://ms.mcmaster.ca/craig/craig-pde-chap5.pdf

Also keen to learn if there is any other theorems.

• So the relation does not hold, correct? – Karla Feb 2 at 11:40
• It holds but L^2 norm. – jomegaA Feb 2 at 11:41