sin = e^jwt - e^-jwt
cos = e^jwt + e^-jwt
This is why the FFT of a sine wave generates significant content at Fc and (Fs-Fc). A complex input will have different magnitudes for at least some of the content in the frequencies between Fs/2 and Fs. (these correspond to -Fs/2 to 0).
eg:
Code:
f = sin(2*pi*100/1000 * [0:1023]); % 1024 samples of 100hz sampled at 1khz
fd = f .* exp(-i*50/1000 * [0:1023]));
fu = f .* exp(+i*50/1000 * [0:1023]));
figure(); plot(20*log(abs(fft(f))));
figure(); plot(20*log(abs(fft(fd))));
figure(); plot(20*log(abs(fft(fu))));
in the above, you will see a peak near 100hz and near 900hz (aka -100Hz). The second plot, the peaks move to 50hz and 850Hz (aka -150Hz). the third plot moves the peaks to 150Hz and 950Hz (aka -50Hz). Also, in all three plots there will be more than just a single line -- 100 hz @ 1khz sampling rate doesn't end at a full cycle for 1024 samples. Thus it is not a basis vector (a frequency that exactly corresponds to a frequency bin).