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δ δ ( f ) = ∫ ∫ − − ∞ ∞ ∞ ∞ δ δ ( t ) ⋅ ⋅ e − − i 2 π π f t d t = lim a → → 0 1 a ∫ ∫ − − ∞ ∞ ∞ ∞ rect ⁡ ⁡ ( t a ) ⋅ ⋅ e − − i 2 π π f t d t = lim a → → 0 sinc ⁡ ⁡ ( a f ) . {\displaystyle \delta (f)=\int _{-\infty }^{\infty }\delta (t)\cdot e^{-i2\pi ft}\,dt=\lim _{a\to 0}{\frac {1}{a}}\int _{-\infty }^{\infty }\operatorname {rect} \left({\frac {t}{a}}\right)\cdot e^{-i2\pi ft}\,dt=\lim _{a\to 0}\operatorname {sinc} {(af)}.} where the sinc function here is the normalized sinc function. Because the first zero of the sinc function is at f = 1 / a {\displaystyle f=1/a} and a {\displaystyle a} goes to infinity, the Fourier transform of δ δ ( t ) {\displaystyle \delta (t)} is

δ δ ( f ) = 1 , {\displaystyle \delta (f)=1,} means that the frequency spectrum of the Dirac delta function is infinitely broad. As a pulse is shorten in time, it is larger in spectrum.

• Fourier transform

• Square wave

• Step function

• Top-hat filter

• Boxcar function

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