# Inverse Fourier Transforms

* InverseDTFT*:

*ContPeriodic*→

_{2π}*DiscSignals*, such that if

*x*=

*InverseDTFT*(

*X*), then ∀ ν ∈

*Reals*,

x(n) = (1/2π) ∫_{ (− π to π )}X(ω)e^{iω n}dω

(This is like a Fourier series expansion, in the that it expresses a signal as a sum (integral) of weighted complex exponentials.)

* InverseCTFT*:

*ContSignals*→

*ContSignals*, such that if

*x*=

*InverseCTFT*(

*X*), then ∀

*ω*∈

*Reals*,

x(t) = (1/2π) ∫_{ (− ∞ to ∞ )}X(ω)e^{iω t}dω

(This too is like a Fourier series expansion, in the that it expresses a signal as a sum (integral) of weighted complex exponentials.)

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