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I have a couple of resources I have from my university I had being checking and I found this:

Find Fourier Series coefficients of x(2t+3). x(t) is continuous and periodic by T.

I see this solution: enter image description here

But the 3/2 is seemingly wrong. Am I right? It supposed to be 3, not 3/2. Or am I missing something? They claim that:

enter image description here

Please let me know if this resource is wrong or I am having the mistake and it shall be 3/2 some why.

Vitali Pom
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1 Answers1

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Ok, I see now: They did it straight from the computation. The 3/2 is because they divided T by two in $e^{j3k{\frac {2\pi} T}}$.

It’s important, because it tells us that the first intuition we have that we can apply scaling to Fourier Series or Fourier Transform and then apply transformation to the Fourier Transform is wrong. We might mistakenly think that we can scale time, apply transform, and then apply shift by 3 to the transform to move by 3 units in time. As you can see it does not produce the same result, since the shift is by 3/scaling-factor and not straight 3.

Vitali Pom
  • 174
  • 8