# Fourier transform of unit step

I was reading pdf by caltech and in one of its section, Fourier transform of Unit step signal is calculated but I am confused, how this can be possible if region of convergence for Laplace transform ($$1/s$$) of unit step signal does not contain imaginary axis?

And if above case is possible then if it given that impulse response of a system is Unit step then frequency response of it should also exist and equal to $$H(ω)= πδ(ω) + 1/jω$$ and then can we calculate Fourier transform of output by computing $$H(ω)X(ω)$$ where $$X(ω)$$ is Fourier transform of input?

• @user215805: That's a pretty complex topic, but the Fourier transform is defined for tempered distributions, including functions in $L^1$ and $L^2$ (absolutely and square integrable functions), and distributions of compact support. For some math background start on this wikipedia page. – Matt L. Jun 1 at 18:49