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CTFT Development 3

Module by: Richard Baraniuk. E-mail the author

The forward and inverse Laplace transforms are defined as:

F⁢s=∫−∞∞f⁢t⁢e−(s⁢t)d t F s t f t s t
(1)
f⁢t=12⁢π⁢i⁢∫a−i⁢∞a+i⁢∞F⁢s⁢es⁢td s f t 1 2 s a a F s s t
(2)
substitute in s=i⁢ω s ω and let a=0 a 0 (integrate along i⁢ω ω axis in complex plane) and we have:
F⁢i⁢ω=∫−∞∞f⁢t⁢e−(i⁢ω⁢t)d t F ω t f t ω t
(3)
f⁢t=12⁢π⁢∫−∞∞F⁢i⁢ω⁢ei⁢ω⁢td ω f t 1 2 ω F ω ω t
(4)
In other words
F⁢i⁢ω=F⁢s| s=i⁢ω F ω s ω F s
(5)
which explains the " i⁢ω ω " notation.

Example 1

f∈L2R f L 2 but not L′R L .

Figure 1
Figure 1 (f1.png)
∫|f⁢t|d t =∞ t f t and ∫|f⁢t|2d t <∞ t f t 2 .

Example 2

f∈L′R f L but not L2R L 2 .

Figure 2
Figure 2 (f2.png)
∫|δ⁢t|d t =1 t δ t 1 and ∫|δ⁢t|2d t =∞ t δ t 2 .

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