Fourier Transforms

DTFT: DiscSignals Ž ContPeriodic_2p , such that if X = DTFT (x), then " w Î Reals,

X(w) = S_{(m
= - ¥
to ¥
)} x(m)e^-imw

To verify that this is periodic with period 2p, just evaluate X(w + 2p) to show that it is equal to X(w) for all w.

CTFT: ContSignals Ž ContSignals, such that if X = CTFT (x), then " w Î Reals,

X(w) = ň_{(- ¥
to ¥
)} x(t)e^{-iw
t} dt

Each of these transforms a time-domain signal into a frequency-domain representation.