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Continuous-Time Fourier Transform

Module by: Phil Schniter

Summary: Review of the continuous-time fourier transform.

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Laplace Transform

X c s=∫-∞∞ x c tⅇ-ⅈωtdt X c s t x c t ω t (1)

s∈ℂ s

Continuous-Time Fourier Transform

X c ⅈΩ=∫-∞∞ x c tⅇ-ⅈΩtdt X c Ω t x c t Ω t (2)

t∈ℝ t Ω∈ℝ Ω

X c t=12π∫-∞∞ x c Ωⅇ-ⅈΩtdΩ X c t 1 2 Ω x c Ω Ω t (3)
Figure 1: Before CTFT
Figure 1 (signal.png)
Figure 2: After CTFT
Figure 2 (ctft.png)

Notes:

  1. X c Ω X c Ω is the Laplace Transform evaluated at s=ⅈΩ s Ω .
  2. When the X c t X c t is "smooth", X c Ω X c Ω is concentrated around Ω=0 Ω 0 and dies off as |Ω|→∞ Ω .
  3. When X c t X c t is real valued, X c Ω X c Ω is conjugate symmetric.
  4. When X c t X c t is "bandlimited", X c Ω=0 X c Ω 0 for all |Ω|> Ω 0 Ω Ω 0 , where Ω 0 Ω 0 is the cutoff frequency.

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