# What frequencies are present in the Fourier transform of the Dirac impulse?

When I do the Fourier transform of the Dirac impulse I get a pure sinusoid (or complex exponential, however you wanna call it) but I read in several places that all frequencies are present in the dirac impulse and all of them with the same amplitude. How is this possible? Am I wrong when I perform the transform?

• Note that the complex exponential's independent variable is frequency, so you get a non-zero contribution for all frequencies. – Matt L. Aug 7 '18 at 19:39
• (Very) related: this answer – Matt L. Aug 7 '18 at 19:47

A Dirac impulse $x(t)=\delta(t-d)$ has the continuous-time Fourier transform $X(\Omega)$ of $$\mathcal{F}\{\delta(t-d) \} = 1 e^{-j\Omega d}$$
whose magnitude is $$|X(\Omega)| = 1 ~~~, \text{ for all } \Omega$$ and a phase of $$<X(\Omega) = -\Omega \cdot d$$
So in this case the magnitude is $1$ and hence it's said to contain all frequencies of magnitude $1$. Note that these are differential amplitude components of continuum frequency range as opposed to a finite amplitude of discrete set of frequency components, aka line components.