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# Continuous-Time Fourier Transform

Module by: Richard Baraniuk. E-mail the author

Summary: Introduction to the continuous-time Fourier Transform.

## Theme:

Fourier representation for infinite-length "aperiodic" signals

Alternative notations for CTFT: Fω F ω and Fiω F ω

Recall that Fourier Series builds up periodic or finite length signal from sum of harmonic sinusoids with frequencies that are multiples of ω o =2πT ω o 2 T

CTFT builds up arbitrary signal from sum of sinusoids of all frequencies ωR ω

Fiω=fte(iωt)dt F ω t f t ω t
(1)
ft=12πFiωeiωtdω f t 1 2 ω F ω ω t
(2)

## Note:

CTFT is totally symmetrical except for the 12π 1 2 goes with δω δω
Alternate normalizations include :
• 12π 1 2 in both the above equations.
• Do not use rad/sec frequency ω ω but rather Hz frequency γ γ. Then all the 12π 1 2 's go away.

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