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# CTFT Development 1

Module by: Richard Baraniuk. E-mail the author

Recall CTFS (renormalize a bit)

Define Δω=2πT= ω 0 Δ ω 2 T ω 0 .

d n =T c n =T2T2fte(i)(Δωn)td t d n T c n t T 2 T 2 f t Δ ω n t
(1)
ft=12πnz d n ei(Δωn)tΔω f t 1 2 n n z d n Δ ω n t Δ ω
(2)

As T T , Equation 1 becomes d n =fte(i)(Δωn)td t d n t f t Δ ω n t , where Δω0 Δ ω 0 so Δωn Δ ω n can approximate any real number.

Let (ω=Δωn)R ω Δ ω n and Fiω= d n F ω d n .

## Forward CTFT

Fiω=fte(iωt)d t F ω t f t ω t
(3)

Then Equation 2 becomes ft=12πnZ d n ei(Δωn)tΔω f t 1 2 n n d n Δ ω n t Δ ω where nZΔω n n Δ ω is a Reimann integral, d n d n is Fiω F ω , and Δωn Δ ω n is ωω.

## Inverse CTFT

ft=12πFiωeiωtd ω f t 1 2 ω F ω ω t
(4)

Recall FS ft=nz c n ei ω 0 nt 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 =2πT ω 0 2 T .

CTFT ft=12πFiωeiωtd ω f t 1 2 ω F ω ω t builds up arbitrary signal from sum of sinusoids of all frequencies ωR ω R .

Table 1: CTFT
analysis synthesis
Fiω=fte(iωt)d t F ω t f t ω t ft=12πFiωeiωtd ω f t 1 2 ω F ω ω t

## Note:

Totally symmetrical, except for 12π 1 2 goes with ω ω .
Alternate normalizations:
1. 1 2 2π 1 2 2 in both above equations.
2. Don't use radsec rad sec frequency ω ω but rather Hz Hz frequency γ γ (substitute ω=2πγ ω 2 γ in above). Then all 12π 1 2 's go away.

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