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Continuous-Time Fourier Transform

Module by: Richard Baraniuk

Summary: Introduction to the continuous-time Fourier Transform.

Theme:

Fourier representation for infinite-length "aperiodic" signals

Figure 1
Fourier Series
Fourier Series (f1.png)
Figure 2
Fourier Transform
Fourier Transform (f2.png)

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 ω∈ℝ ω

Fⅈω=∫-∞∞ftⅇ-ⅈωtdt F ω t f t ω t (1)
ft=12π∫-∞∞Fⅈωⅇⅈωtdω 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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