Recall CTFS (renormalize a bit)

Figure 1

Figure 2

As T→∞ T , Equation 1 becomes d n =∫−∞∞f⁢t⁢e(−i)⁢(Δ⁢ω⁢n)⁢td t d n t f t Δ ω n t , where Δ⁢ω→0 Δ ω 0 so Δ⁢ω⁢n Δ ω n can approximate any real number.

Let (ω=Δ⁢ω⁢n)∈R ω Δ ω n and F⁢i⁢ω= d n F ω d n .

Then Equation 2 becomes f⁢t=12⁢π⁢∑n∈Z d n ⁢ei⁢(Δ⁢ω⁢n)⁢t⁢Δ⁢ω f t 1 2 n n d n Δ ω n t Δ ω where ∑n∈ZΔ⁢ω n n Δ ω is a Reimann integral, d n d n is F⁢i⁢ω F ω , and Δ⁢ω⁢n Δ ω n is ωω.

Recall FS f⁢t=∑n∈z c n ⁢ei⁢ ω 0 ⁢n⁢t f t n n z c n ω 0 n t builds up periodic or finite length signal from sum of harmonic sinusoids with frequencies that are multiples of ω 0 =2⁢πT ω 0 2 T .

Figure 3

CTFT f⁢t=12⁢π⁢∫−∞∞F⁢i⁢ω⁢ei⁢ω⁢td ω f t 1 2 ω F ω ω t builds up arbitrary signal from sum of sinusoids of all frequencies ω∈R ω R .

Figure 4

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This work is licensed by Richard Baraniuk under a Creative Commons Attribution License (CC-BY 1.0), and is an Open Educational Resource.

Last edited by Jeffrey Silverman on Jun 29, 2004 1:35 pm -0500.