A chirp signal, in which the angular frequency rises uniformly from ω₀€“Δω/2 to ω₀+ Δω/2 over a time T, can be written as exp(iφ(t)), where the phase φ(t) is given by

for |t| ‰¤ T/2. This signal is then passed through a delay-line, whose effect on a component of angular frequency w can be represented as multiplication by the factor 

Show that the signal that emerges from the delay-line is a carrier of angular frequency ω₀, modulated by an envelope of width 2Ï€/Δω.

()- + 27 exp( - io(T/) (00 + o/2)