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Fourier TransformDISCUSSION 13
CLASS X
OutlineWhy Fourier Transform?
Relationship with Laplace transform
Cases of convergence
Properties
Practices
Q&A
Why Fourier Transform? Periodic signals ⇔ Fourier series
Aperiodic signals? Treat aperiodic signals as special cases of periodic signals with an infinite period.
Fourier series ⟹ Fourier Transform
How it works:
Relationship with Laplace transform Fourier transform can be treat as a special case of the bilateral Laplace transform.
𝑠 = 𝜎 + 𝑗𝜔 → jω
Using Laplace Transforms to find Fourier Transforms𝑓(𝑡) is positive-time ⟹𝐹 𝑓 𝑡 = 𝐿 𝑓 𝑡 𝑠=𝑗𝜔
𝑓(𝑡) is negative-time ⟹𝐹 𝑓 𝑡 = 𝐿 𝑓 −𝑡 𝑠=−𝑗𝜔
Question after class: What’s the condition can we use Laplace Transforms to find Fourier Transforms?
Cases of convergenceUsual targets we need to transform: Limit Case
How to deal with a signal 𝑓 𝑡 without convergence?• Treat 𝑓 𝑡 as the Limit Case
• Example:
• Check the duality property: ℱ 𝑓 𝑡 = 𝐹(𝜔) → ℱ 𝐹 𝑡 = 2𝜋𝑓(−𝜔)
What do you find?
Properties
Real part, 𝐴 𝜔 = 𝐴 −𝜔
Imaginary part, 𝐵 𝜔 = −𝐵 −𝜔
If 𝑓(𝑡) is an even function, 𝐹(𝜔) is:• Real, 𝐵 𝜔 = 0
• Even & 𝐴 𝜔 = 2 0∞𝑓 𝑡 cos 𝜔𝑡 𝑑𝑡
If 𝑓(𝑡) is an odd function, 𝐹(𝜔) is:• Imaginary, 𝐴 𝜔 = 0
• Odd & 𝐵 𝜔 = −2 0∞𝑓 𝑡 sin 𝜔𝑡 𝑑𝑡
Practice 1 Find the Fourier transform of the “sine-wave pulse”.
Practice 1-Solution
Practice 2 Determine the signal f(t) whose Fourier transform is shown below. (Hint: Use the duality property.)
Practice 2-Solution
Practice 2-Solution
Practice 3 Use the Fourier transform to find 𝑖(𝑡) in the circuit if 𝑣𝑠 𝑡 = 10𝑒−2𝑡𝑢 𝑡 .
Practice 3-SolutionWe may convert the voltage source to a current source as shown below.
Combining the two resistors gives 1Ω. The circuit now becomes that shown below.
Practice 3-Solution
Practice 4 Determine 𝑣𝑜 𝑡 in the transformer circuit below.
Practice 4-Solution
Exercises
Exercises
Q&A
End