# What is the formula for the frequency spectrum?

A signal $$f[n]$$ is given, the corresponding DTFT as $$F(e^{j\omega})$$ and a plot of the frequency spectrum $$f(t)$$. Unfortunately I can't find a formula for the frequency spectrum in my documents.

When I plot $$|F(e^{jt})|$$ it looks exactly like the given plot of the frequency spectrum f(t) but I don't feel comfortable to "guess" a formula by comparing plots. Searching for this in the internet confuses me more than it helps.

What is the formula for the frequency spectrum?

• I did. No frequency spectrum is mentioned there. – Martin Mar 26 '20 at 13:13
• That can't be true because the continuous time Fourier transform is complex whereas the frequency spectrum plot is real. – Martin Mar 26 '20 at 13:33
• Yes, a Fourier transform has a magnitude and phase response. If you just want to show "where the signal energy is", then you are looking for the magnitude response. – Engineer Mar 26 '20 at 13:42

You are likely looking for the "Power Spectral Density" which is determined by taking the complex conjugate multiplication of the Fourier Transform. The complex conjugate process will give the expected real result and represents a power quantity (which has no phase). For the DFT this is often presented in normalized fashion such as dBFS (dB relative to full scale) or dBc (dB relative to the total power in the signal). If the time domain signal is in volts, then scale the DFT by $$1/N$$ and the complex conjugate multiplication will represent the power in each bin in terms of Watts assuming a 1 ohm load (or adjust further for true power if the resistance is known). Since it is now a power quantity to convert to dB use $$10Log_{10}(F(\omega)F^*(\omega))$$. Where $$F(\omega)$$ represents the DFT or DTFT result with proper scaling, depending on which units are used.

Further as a "density" the result will be power per unit of frequency (such as dBc/Hz). The DFT and DTFT without any further windowing has an equivalent noise bandwidth that is the inverse of the time duration $$T$$ of the time domain signal, so therefore is equal to $$1/T$$ (in whatever units are used for $$T$$). Windowing complicates this further, but see this post for more details on that:

How to calculate resolution of DFT with Hamming/Hann window?