EECS20N: Signals and Systems

Sinusoidal Signals

Consider a continous-time signal

x(t) = cos(ω0 t)

for some real constant ω 0. Use Eulers relation to write this as a sum of two complex exponentials, and then use linearity of the CTFT to find

X(ω) = π δ (ω - ω 0 ) + π δ (ω + ω 0 )

where δ is the Dirac delta function. What this says is that a cosine in the time domain is concentrated at two frequencies in the frequency domain, one the negative of the other (which should not be surprising).