Question

The continuous-time Fourier transform transforms a continuous-time function x(H) into a function X(ω) of a real-variable ω as follows:

X(ω)=∫−[infinity][infinity]x(t)e⁻ʲωᵗdt
Using only the continuous-time Fourier transform definition above, find the Fourier transform of the continuous-time impulse δ(t).

Answer 1

The Fourier transform of the continuous-time impulse function δ(t) is 1, as integrating the product of δ(t) with any other function over all time yields the value of that function at the point of the impulse.

Step-by-step explanation:

The question asks to find the Fourier transform of the continuous-time impulse function δ(t) using the continuous-time Fourier transform. The Fourier transform is given by:

X(ω)=∫∞-∞x(t)e⁻ᵃωᵗdt

When x(t) is the impulse function δ(t), the transform simplifies to:

X(ω)=∫∞-∞δ(t)e⁻ᵃωᵗdt = e⁻ᵃω(0) = 1

Thus, the Fourier transform of the continuous-time impulse function δ(t) is 1, independent of ω.

Relevant Information

An impulse function, also known as a Dirac delta function, is used to represent a signal that occurs at a single point in time. In the context of continuous-time signals and systems, the impulse function has a crucial property: when integrated over time, it yields a value of 1 if the integration interval includes the impulse, and 0 otherwise.