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Inverse Fourier transform of δ(ω)

Short answer

Inverse FT of δ(ω): x(t)= (1/2π) ∫ δ(ω) e^{jωt} dω = 1/(2π) by sifting. Hence δ(ω) ↔ 1/(2π), a constant DC signal, meaning all spectral content concentrated at zero frequency.

Find inverse Fourier Transform of CT unit impulse signal δ(ω)

Use the inverse Fourier transform with X(ω)=δ(ω)X(\omega) = \delta(\omega):

x(t)=12π∫−∞∞δ(ω)ejωtdωx(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty}\delta(\omega)e^{j\omega t}d\omega

By the sifting property, ∫δ(ω)f(ω)dω=f(0)\int\delta(\omega)f(\omega)d\omega = f(0), and here f(0)=ej0⋅t=1f(0) = e^{j0\cdot t} = 1:

x(t)=12π⋅1=12πx(t) = \frac{1}{2\pi}\cdot 1 = \frac{1}{2\pi}

Answer: δ(ω)↔F−112π\delta(\omega) \xleftrightarrow{F^{-1}} \frac{1}{2\pi}, a constant (pure DC) signal. Equivalently, 1↔2πδ(ω)1 \leftrightarrow 2\pi\delta(\omega): a constant signal has all its content (its power) at zero frequency.

✓ Answer checked by Claude Opus 5.5 (high effort)

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