# Is a time domain spectrum obtainable from a frequency domain spectrum? [closed]

I know for a fact that a frequency domain spectrum can be obtained from a time domain spectrum using a Fourier transform - but can you do the reverse?

Also what are the advantages and disadvantages of the frequency and time domain spectra?

## 1 Answer

Can you do the reverse?

Yes, mathematically the Fourier transform can be reversed. If we define the spectrum $$S(\omega)$$ as the Fourier transform of a time-domain signal $$f(t)$$,

$$S(\omega) = \mathcal{F}[f(t)] = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(t)\mathrm{e}^{-\mathrm{i}\omega t}\,\mathrm{d}t$$

then the inverse Fourier transform is simply given by

$$f(t) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}S(\omega)\mathrm{e}^{\mathrm{i}\omega t}\,\mathrm{d}\omega$$

Wikipedia has a page on this which is rather technical, but any decent book on Fourier transforms should cover this, so a mathematics book targeted at physicists/chemists may be easier to digest.

What are the advantages and disadvantages of the frequency and time domain spectra?

The time domain signal (in NMR, "free induction decay") is for the most part, not particularly useful. It is the Fourier transform which makes it valuable for chemists, because the frequencies that appear in the spectrum are related to transitions between different energy states:

$$\Delta E = \hbar\omega$$

In general, probing these energy levels is the main point of spectroscopy, regardless of what range of energies you are using.

• I just like to add to @orthocresol's excellent answer that there might be situations in which you can extract time-domain information from a frequency spectrum. The lifetime of excited states is for instance inversely proportional to the linewidth of a spectral line (this follows from the FT). This principle can, for instance, be used in studies on chemical kinetics to probe very fast processes. – Paul Nov 12 '18 at 16:43