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Useful Fourier Transforms

Module by: Richard Baraniuk. E-mail the author

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Square Pulse / Box Car / Unit Gate / Square Window

Figure 1.

Figure 1
Figure 1 (f1.png)
Rⅈω=∫-∞∞rtⅇ-ⅈωtdt=∫-TT1ⅇ-ⅈωtdt=ⅇ-ⅈωt-ⅈω|-TT=-1ⅈωⅇ-ⅈωT−ⅇⅈωT R ω t r t ω t t T T 1 ω t T T ω t ω 1 ω ω T ω T (1)
since ⅇ-ⅈωT−ⅇⅈωT=-2ⅈsinωT ω T ω T 2 ω T we get:
Rⅈω=2sinωTω=2TsinωTωT R ω 2 ω T ω 2 T ω T ω T (2)
which is a sinc!

recall:

Moving average impulse responce, p. B560.
Figure 2.
Figure 2: Zeros are at ωT=nπ ω T n , which means ω=nπT ω n T and Rⅈω R ω is real.
Figure 2 (f2.png)

Triangle

Figure 3.

Figure 3
Figure 3 (f3.png)
Bⅈω=∫-∞∞btⅇ-ⅈωtdt B ω t b t ω t (3)

Delta function

Figure 4.

Figure 4
Figure 4 (f4.png)
Δⅈω=∫-∞∞δtⅇ-ⅈωtdt=1 Δ ω t δ t ω t 1 (4)
Figure 5.
Figure 5
Figure 5 (f5.png)
  • Delta function contains an equal mix of all frequencies.
  • It is in some sense "white" (like white light is made up of colors of all frequencies).

Shifted Impulse

Figure 6.

Figure 6: ft=δt− t 0 f t δ t t 0 .
Figure 6 (f6.png)
Fⅈω=∫-∞∞δt− t 0 ⅇ-ⅈωtdt=ⅇ-ⅈω t 0 F ω t δ t t 0 ω t ω t 0 (5)
which is a complex sinusoid in frequency (Figure 7).
Figure 7
Figure 7 (f7.png)

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