Fourier Transform - ShanghaiTech

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