What does a Fourier transform do to a time-domain signal?

• A. Integrates the signal to obtain energy.
• B. The Fourier transform converts time-domain signals into a frequency-domain representation of amplitude and phase versus frequency.
• C. Differentiates the signal with respect to time.
• D. It yields only the magnitude spectrum, not the phase.

Answer: (B)

The Fourier transform reveals the frequency content of a signal by expressing a time-domain signal as a function of frequency. It converts x(t) into a complex spectrum X(f) that tells you how much of each frequency is present and what phase that frequency component has. In other words, you get a frequency-domain representation where the magnitude |X(f)| shows the strength of each frequency and the phase ∠X(f) shows the relative timing of those components. This is why the transform provides both amplitude and phase information across frequencies, not just a single number or just magnitude.

It’s not simply an energy integral, and it’s not a time-domain differentiation. Although energy considerations relate to the spectrum via results like Parseval’s theorem, the transform itself is about decomposing the signal into its sinusoidal components across frequencies. Also, the transform yields a complete spectrum (complex values), so you can recover the original time signal with the inverse transform; focusing only on magnitude loses phase information, which matters for accurately reconstructing the waveform.