Magnitude and Phase Spectra, Amplitude Scaling

Effect of Amplitude Scaling:

The Fourier transform of y(t) = ax(t) is Y(w) = aX(w) . Therefore the magnitude spectrum of y(t) is given by

|Y(w)| = |a| |X(w)|

The phase spectrum of y(t) is given by

<Y(w) = <X(w), if a > 0
<Y(w) = p + <X(w), if a <0

Compare the two exponential signals

x(t) = e^-tu(t)
y(t) = 3 x(t) = 3 e^-tu(t)

with the corresponding magnitude spectra

and phase spectra

and also the two exponential signals

x(t) = e^-tu(t)
y(t) = -3 x(t) = -3 e^-tu(t)

with the corresponding magnitude spectra

and phase spectra