# What is the physical interpretation of the absolute value of a fourier transformed signal, $\left| F(t)\right|$?

If one has some oscillating voltage signal, for example: $$V(t) = V_{max}\cos(2 \pi \nu_{0}t) e^{-\gamma t}$$ and you take the Fourier transform of this in the usual way to get: $$\hat{V}(\nu) = V_{max}\int^{+\infty}_{0} \cos(2 \pi \nu_{0}t) e^{-\gamma t} \ dt$$ Now I often see in literature the absolute value squared taken of the result, so $$\left|\hat{V}(\nu) \right|^{2}$$ the physical interpretation of which it is the energy per unit frequency.

My question is, if $$\left|\hat{V}(\nu) \right|^{2}$$ is energy per unit frequency, then what is the interpretation of $$\left|\hat{V}(\nu) \right|$$?

The $$\cos$$ function in your example is a so called power signal, not an energy signal, as it is infinitly long it has infinite energy, so to assess its energy in a practical way, one refers to its power (energy per time).
• if $x(t)$ has units of Volts, then the unit of $X(f)$ is $Volts \cdot s$ (Volts-second). – Fat32 Apr 11 '19 at 14:07