EECS20N: Signals and Systems

Fourier Transforms

DTFT: DiscSignals → ContPeriodic_2π , such that if X = DTFT (x), then ∀ ω ∈ Reals,

X(ω) = ∑_{(m
= − ∞
to ∞
)} x(m)e^−imω

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

CTFT: ContSignals → ContSignals, such that if X = CTFT (x), then ∀ ω ∈ Reals,

X(ω) = ∫_{(− ∞
to ∞
)} x(t)e^{−iω
t} dt

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

Week 12