Similarly, what happens if a signal is very broad in the time domain regarding its Fourier transform?

• A. It becomes even broader in frequency.
• B. It becomes very narrow in frequency.
• C. It becomes zero for all frequencies.
• D. It becomes periodic.

Answer: (D)

When a signal lasts for a long or very broad interval in time, its Fourier transform becomes concentrated in a narrow range of frequencies. In other words, there is an inverse relationship between how long a signal persists and how wide its spectrum is: long duration tends to shrink the spectral bandwidth. For an infinitely long constant signal, the spectrum collapses to a single frequency component (a delta at DC). The spectrum does not become periodic just because the time-domain signal is broad in time; periodicity in the spectrum arises when the time-domain signal itself is periodic, which produces a line spectrum at harmonic frequencies. So the correct interpretation is that the Fourier transform becomes very narrow in frequency.