## Effect of Time Truncating:

The Fourier transform of y(t) = x(t)
for |t| < T, and y(t) = 0
for |t| > T is

Y(w) = _{-T/}|^{/T}x(t)e^{
-jwt}dt
We can also write y(t) as the product of
x(t) and p(t),
where p(t) is the unit-amplitude rectangular pulse
of width 2T centered at t = 0.
Then the Fourier transform relationship is a convolution:

Y(w) =
(1/2p)
X(w) * P(w)
Via either expression, the magnitude and phase spectra of y(t)
are related to the magnitude and phase spectra of x(t)
in a rather complicated fashion.

Compare the two exponential signals

x(t) = e^{-t}u(t)

y(t) = e^{-t}u(t)
for t <1, and y(t)= 0 for t> 1

with the corresponding magnitude spectra

and phase spectra

return to Magnitude and Phase Spectra page