Why is $e^{i2\pi Nf_snT}=1$ in snippet from book?

enter image description here


2 Answers 2


Because $Tf{_s}$=1 and $N$ and $n$ are integers. So the exponent becomes 2$\pi$ times an integer, say $K$. But $e{^{i\theta}} = \cos(\theta)+i\sin(\theta)$. Therefore $e{^{i2K\pi}} = 1$.


A try without maths: $x(t)$ gives you your location in the complex plane at time $t$.

The answer is: because when you walk in circles ($2\pi$) an integer number of times, whatever your rotational speed, or the length of your steps, you finally up end at the same place as in the beginning, like in the latin palindromic sentence:

In girum imus nocte ecce et consumimur igni

which means: "At night we wander in circles and are consumed by fire."

  • 1
    $\begingroup$ It's the first time I see the ecce in that palindrome. $\endgroup$
    – Gilles
    Apr 9, 2016 at 14:27
  • $\begingroup$ From what I still understand from latin, its merely add 4 letters :) $\endgroup$ Apr 9, 2016 at 17:33

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.