Confused about DFT symmetric behavior for real-valued signals?

Question

I have a simple and might be a preliminary question about symmetric property of DFT of real valued signals. From definition we know, DFT coefficients of a real-valued signal is conjugate symmetric, so $X_{N-k}=X_{k}^*$ . However, I cant see this in calculated DFT, for example consider the below:

N=1000;
fs=10^6;
t=(0:N-1)./fs;
x=2*sin(2*pi*t*10000);
plot(x);
figure;
stem((abs((fft(x)))));

The non-zero bins are 11 and 991, where I expected them to be 11 and 989, even considering that indexes starts at 1 in MATLAB,and subtracting bin numbers by 1, the non-zero bins must be 10th and 990th bins.

Would any one please point out, where is my misunderstanding?

Answer 1

$N=1000$ and your signal frequency is $10000$. So there should be a peak at bin $k=10$. And another one at $N-k=1000-10=990$. However since MATLAB starts indexing at $1$ instead of zero, you see peaks at $11$ and $991$.