How to recover $f(t)$ from Fourier Transform of its absolute value $\mathcal{F}|f(t)|$?

1. What is the absolute value of Fourier transform?
2. What is Fourier transform of T?
3. Why do we use absolute value of FFT?

What is the absolute value of Fourier transform?

For each frequency, the magnitude (absolute value) of the complex value represents the amplitude of a constituent complex sinusoid with that frequency, and the argument of the complex value represents that complex sinusoid's phase offset. If a frequency is not present, the transform has a value of 0 for that frequency.

What is Fourier transform of T?

The Fourier Transform is a mathematical technique that transforms a function of time, x(t), to a function of frequency, X(ω). It is closely related to the Fourier Series. If you are familiar with the Fourier Series, the following derivation may be helpful.

Why do we use absolute value of FFT?

Absolute value is useful if you want to know exact value(say 5V) of a certain sinusoid. In case you want to just compare spectrum components you might display the output in dB. 2) I suppose you talk about real FFT. If you compute FFT for a complex set of input values there is no symmetry.