At first glance, finding its Fourier transform seems impossible. The Fourier transform of a function ( f(t) ) is:
confirming the result. | Function | Fourier Transform | |----------|------------------| | ( u(t) ) (unit step) | ( \pi\delta(\omega) + \frac1i\omega ) | | ( \textsgn(t) ) (sign) | ( \frac2i\omega ) | | Constant ( 1 ) | ( 2\pi\delta(\omega) ) | fourier transform step function
The Fourier transform of ( \textsgn(t) ) is ( 2/(i\omega) ) (without a delta, since its average is zero). Thus: At first glance, finding its Fourier transform seems
[ u(t) = \frac12 + \frac12 \textsgn(t) ] At first glance
[ \boxed\mathcalFu(t) = \pi \delta(\omega) + \frac1i\omega ]