Time and Frequency Domains

Signal definitions in two domains
x(t) = signal amplitude at time t

X(w) = signal amplitude at frequency w, where w = 2*pi*f

Fourier transform
X(w) = INT[-inf to inf] x(t)*e^(-jwt)

X(w) = INT[-inf to inf] x(t) * [cos (wt) - j sin (wt) ]

X(w) = INT[-inf to inf] x(t) cos (wt) - x(t) j sin (wt)

Convolution in one domain corresponds to multiplication in the other domain.
x(t) * y(t) = X(w) (+) Y(w)

x(t) (+) y(t) = X(w) * Y(w)