What is the relationship between the width of a signal in time and the width of its Fourier transform?

What is the relationship between the width of a signal in time and the width of its Fourier transform?

• A. They are directly proportional.
• B. They are independent.
• C. They are inversely proportional.
• D. They are equal.

Answer: (C)

The key idea is the time–bandwidth trade-off: in Fourier analysis, the duration of a signal in time and the spread of its frequency content are linked so that making the signal last longer in time narrows its spectrum, while shortening it broadens the spectrum. This is captured by an uncertainty-like relation: the product of a time width measure and a frequency-width measure is roughly a constant. So, in general, they are inversely proportional: a longer time width means a smaller frequency width, and a shorter time width means a larger frequency width. For intuition, think of a short pulse like a quick blink in time; it requires many different frequencies to compose it, giving a broad spectrum. A long, slowly varying tone stays confined to a narrow range of frequencies. In the special case of a Gaussian signal, the product of the time and frequency widths reaches the minimum allowed by the uncertainty relation, reinforcing the inverse relationship.

The key idea is the time–bandwidth trade-off: in Fourier analysis, the duration of a signal in time and the spread of its frequency content are linked so that making the signal last longer in time narrows its spectrum, while shortening it broadens the spectrum. This is captured by an uncertainty-like relation: the product of a time width measure and a frequency-width measure is roughly a constant. So, in general, they are inversely proportional: a longer time width means a smaller frequency width, and a shorter time width means a larger frequency width.

For intuition, think of a short pulse like a quick blink in time; it requires many different frequencies to compose it, giving a broad spectrum. A long, slowly varying tone stays confined to a narrow range of frequencies. In the special case of a Gaussian signal, the product of the time and frequency widths reaches the minimum allowed by the uncertainty relation, reinforcing the inverse relationship.