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Let $$x(t) = A \text{rect}_T({t-\tau})$$I calculated its fourier transform through the direct way: $$X(\omega) = Ae^{-i\omega \tau} \int_{-T/2}^{T/2} e^{-i \omega t} dt = Ae^{-i\omega \tau} \frac{\sin(T \omega/2)}{\omega /2}$$ but I want to see if there are other clever methods, infinite thanks to everyone!

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  • $\begingroup$ Are you saying the Fourier Transform isn't clever enough?? ^^ $\endgroup$
    – Jdip
    Nov 16, 2023 at 2:22
  • $\begingroup$ exactly, the stupid questions said give multiple answers ! @Jdip $\endgroup$ Nov 16, 2023 at 2:23
  • $\begingroup$ hmm. give multiple answers to what exactly? I'm assuming this is homework, what is the homework question exactly? $\endgroup$
    – Jdip
    Nov 16, 2023 at 2:25
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    $\begingroup$ no it isnt a homework otherwise i would already find solution from my other classmates, this is an open question in class, saying find other methods to find the fourier transform of that function @Jdip $\endgroup$ Nov 16, 2023 at 2:28
  • $\begingroup$ Well, you’re already using a clever method since you’re using the frequency shift property! $\endgroup$
    – Jdip
    Nov 16, 2023 at 2:49

2 Answers 2

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There are two main strategies to simplify the calculation of the Fourier Transform.

  1. Use Fourier Transform properties
  2. "Deconstruct" the time domain functions into other functions with easier or already known Fourier Transforms.

Mix and match as needed.

You already have used strategy #1 by applying the time shift property

$$x(t-\tau) \leftrightarrow X(\omega)\cdot e^{-j\omega \tau}$$

Matt L. answer shows another example using the integration/differentiation properties.

An example for using strategy #2 would be to express the rectangle function as the difference of two unit steps

$$x(t) = \text{rect}_T(t-\tau) = u(-T/2-\tau)-u(T/2-\tau)$$

However, that FT of the unit step is (IMO) more complicated than that of a rectangle, so this may not help much. Then again, you could try to leverage the differentiation property again using $u(t) = \frac{\partial}{\partial t}\delta(t)$ and you end up with Matt's answer again.

A better example would be the FT of a triangle wave. A triangle can be expressed as the convolution of two rectangles. Convolution in the time domain is equal to multiplication in the frequency domain. The FT of rectangle is a $\text{sinc}()$ so the FT of a triangle is $\text{sinc}^2()$

Conclusion: you can try any combination of properties and deconstructions in any order. Sometimes it helps, but sometimes it doesn't and you still have to do the original integral in all its glory.

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  • $\begingroup$ Using the differentiation property to derive the Fourier transform of $u(t)$ is problematic: $\mathscr{F}\{\delta(t)\}=1$ could make one think that $\mathscr{F}\{u(t)\}=1/j\omega$, which is incorrect. $\endgroup$
    – Matt L.
    Nov 16, 2023 at 13:19
  • $\begingroup$ Agreed. Hence my assessment of "complicated". $\endgroup$
    – Hilmar
    Nov 16, 2023 at 13:45
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With such a simple function it is probably easiest to directly solve the Fourier integral. It's also wise to commit such simple Fourier identities to memory.

Another relatively simple method I can think of for this specific function is computing the Fourier transform of the function's derivative. Since the derivative is just two Dirac impulses, you can write down the Fourier transform without any calculation. Then you use the integration property of the Fourier transform to derive the transform of the original function, which in this case just reduces to dividing by $j\omega$:

\begin{align*} G(j\omega) &= \mathscr{F}\{f'(t)\} \\ F(j\omega) &= \frac{G(j\omega)}{j\omega} \end{align*}

I trust that you can fill in the necessary details.

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