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Useful Fourier Transforms

Module by: Richard Baraniuk. E-mail the author

Square Pulse / Box Car / Unit Gate / Square Window

Figure 1.

Figure 1
Figure 1 (f1.png)
R⁢i⁢ω=∫−∞∞r⁢t⁢e−(i⁢ω⁢t)d t =∫−TT1⁢e−(i⁢ω⁢t)d t =e−(i⁢ω⁢t)−(i⁢ω)|−TT=(−1i⁢ω)⁢(e−(i⁢ω⁢T)−ei⁢ω⁢T) 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−(i⁢ω⁢T)−ei⁢ω⁢T=−(2⁢i⁢sinω⁢T) ω T ω T 2 ω T we get:
R⁢i⁢ω=2⁢sinω⁢Tω=2⁢T⁢sinω⁢Tω⁢T 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 ω⁢T=n⁢π ω T n , which means ω=n⁢πT ω n T and R⁢i⁢ω R ω is real.
Figure 2 (f2.png)

Triangle

Figure 3.

Figure 3
Figure 3 (f3.png)
B⁢i⁢ω=∫−∞∞b⁢t⁢e−(i⁢ω⁢t)d t B ω t b t ω t
(3)

Delta function

Figure 4.

Figure 4
Figure 4 (f4.png)
Δ⁢i⁢ω=∫−∞∞δ⁢t⁢e−(i⁢ω⁢t)d t =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: f⁢t=δ⁢t− t 0 f t δ t t 0 .
Figure 6 (f6.png)
F⁢i⁢ω=∫−∞∞δ⁢t− t 0 ⁢e−(i⁢ω⁢t)d t =e−(i⁢ω⁢ 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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