# Discrete Fourier series of an odd signal

Assuming the signal shown below :

I have found an expression for fourier series coeffecients as the following: $$a_{k} = \frac{1}{5}+\frac{j}{5}\sin{\frac{2\pi}{5}k}$$

Which matches with what the books suggests as an answer. My confusion is this : why are they not purely imaginary ?

I tried to prove this property the following way: Signal is real and even: $$\implies x[n] = x^{*}[n] = -x[-n]$$
with * denoting the conjugate. And using properties of conjugation, time-reversal and linearity we get : $$a_{k} = a^{*}_{-k} = -a_{-k}$$

Then I concluded : $$a_{k} = -a_{-k} \implies a_{k}\ are\ odd\ in\ k$$ $$a^{*}_{-k} = -a_{-k} \implies a_{k}\ are\ purely\ imaginary$$

What is wrong with all the above? Is any step in my reasoning wrong or is the solution to the problem incorrect ?

• What value should $x[0]$ have if the signal was odd? Dec 28, 2022 at 10:53
• @MattL. It should be zero, which is not the case here. Dec 28, 2022 at 11:51
• On a second look now, I see that there was a mistake in the assumption of "the signal is odd". I don't know how that escaped me. But is the proof sound? Dec 28, 2022 at 11:53