Frequency response of unit impulse function

Question

Could someone throw some light on how to get the frequency response of unit impulse function.

I am not from EE, but I need it for my wavelet study.

Answer 1

Just apply the definition of the Fourier transform

$$X(j\omega)=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt$$

and use the property of the Dirac delta impulse

$$\int_{-\infty}^{\infty}\delta(t)f(t)dt=f(0)$$

Then you get for the Fourier transform of $x(t)=\delta(t)$

$$X(j\omega)=\int_{-\infty}^{\infty}\delta(t)e^{-j\omega t}dt=e^{-j\omega\cdot 0}=1$$