Prove fourier coefficients of an odd discrete signal is $a_n = -a_{-n}$

Question

I'm self studying signal and system. I've come across this property: fourier coefficients of an odd discrete signal is $a_n = -a_{-n}$, how can this be proved?

Answer 1

First the signal must be periodic. Second if the signal is $x[n]$ we have $$x[n]=\sum_{k=0}^{N-1}a_ne^{j2\pi kn}$$and $$a_k={1\over N}\sum_{n=0}^{N-1}x[n]e^{-i2\pi kn}$$if $x[n] $ is odd we have$$a_{-k}{={1\over N}\sum_{n=0}^{N-1}x[n]e^{i2\pi kn}\\={1\over N}\sum_{n=0}^{N-1}-x[-n]e^{i2\pi kn}\\=-{1\over N}\sum_{n=1-N}^{0}x[n]e^{-i2\pi kn}\\=-{1\over N}\sum_{n=0}^{N-1}x[n]e^{-i2\pi kn}\\=-a_{k}}$$hence the proof is complete $\blacksquare$