Common Fourier Transform Pairs

Common Fourier Transform Pairs#

	Name	\(f(t)\)	\(F(\omega)\)	Remarks
1.	Dirac delta	\(\delta(t)\)	\(1\)	Constant energy at all frequencies.
2.	Time sample	\(\delta(t-t_0)\)	\(e^{-j\omega t_0}\)
3.	Phase shift	\(e^{j\omega_0 t}\)	\(2\pi\delta(\omega - \omega_0)\)
4.	Signum	\(\operatorname{sgn} t\)	\(\displaystyle{\frac{2}{j\omega}}\)	also known as sign function
5.	Unit step	\(u_0(t)\)	\(\displaystyle{\frac{1}{j\omega}+\pi\delta(\omega)}\)
6.	Cosine	\(\cos \omega_0 t\)	\(\pi\left[\delta(\omega-\omega_0)+\delta(\omega+\omega_0)\right]\)
7.	Sine	\(\sin \omega_0 t\)	\(-j\pi\left[\delta(\omega-\omega_0)-\delta(\omega+\omega_0)\right]\)
8.	Single pole	\(e^{-at}u_0(t)\)	\(\displaystyle{\frac{1}{j\omega + a}}\)	\(a \gt 0\)
9.	Double pole	\(te^{-at}u_0(t)\)	\(\displaystyle{\frac{1}{(j\omega + a)^2}}\)	\(a \gt 0\)
10.	Complex pole (cosine component)	\(e^{-at}\cos \omega_0 t\;u_0(t)\)	\(\displaystyle{\frac{j\omega + a}{(j\omega + a)^2+\omega_0^2}}\)	\(a\gt 0\)
11.	Complex pole (sine component)	\(e^{-a t}\sin \omega_0 t\;u_0(t)\)	\(\displaystyle{\frac{\omega}{(j\omega + a)^2+\omega_0^2}}\)	\(a\gt 0\)