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/|/Tx(t)e -jwtdt

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-tu(t)
y(t) = e-tu(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