I thought that the magnitude spectra of real signals was symmetric. When I take the absolute value of the fft of a sine wave the result is not symmetric. There seems to be a peak at the first coefficient causing non symmetry.
When I plot the absolute value of the fft of a cosine wave I see symmetry.
The reason I am asking is because I was working on a project that involved taking the fft of audio data that was known to be real. When we did that, we only used half the spectrum for our analysis because the magnitude spectra of real signals is symmetric.
Specifically, the case I am talking about is when plotting $$x=\sin( 2 \pi n)$$ Then finding the magnitude spectra,
The result does not have perfect symmetry.
Is this correct? Can someone clear up my confusion on why I am seeing non-symmetric properties when I view the magnitude spectra of a sine wave?
Here I used matlab,
n=0:100; x=sin(2*pi*n); X=abs(fft(x)); plot(X)
As you can see, there is some symmetry but the fft is not completely symmetric. Why is this? sin(2*pi*n) is definitely a real signal.
edit: It seems that matlab does approximations with this sine signal and gives small values when it should be 0 for all n. This may have solved my problem then and be the reason why the fft magnitude did not seem periodic. So question, is there any cases that symmetry does not hold for real signals? Can the 2*pi periodicity of the fft be related to nyquist sampling frequency at all?
What I had thought was going on was something similar to this:
x= [1 2 3 4] X= fft(x);
which gives the result
X= [10.0000 2.8284 2.0000 2.8284]
which is still not periodic. Which is strange to me because x= [1 2 3 4] is a real signal correct?