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A periodic signal $x(t)$ has a trigonometric Fourier series expansion

$$x( t )= a_{0}+\sum_{n=1}^{ \infty } ( a_{n} \cos n\omega _{0}t+b_{n}\sin n\omega _{0}t )$$

If $x(t)= -x(-t)=-x(t-\frac{\pi }{\omega _{0}})$, we can conclude that

  1. $a_n$ are zero for all $n$ and $b_n$ are zero for $n$ even
  2. $a_n$ are zero for all $n$ and $b_n$ are zero for $n$ odd
  3. $a_n$ are zero for $n$ even and $b_n$ are zero for $n$ odd
  4. $a_n$ are zero for $n$ odd and $b_n$ are zero for $n$ even
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