# what is the initial phase of the signal y=sin(2*pi*x)? zero or -π？

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summer on 29 Sep 2015
Commented: summer on 30 Sep 2015
According to the explanation on the internet, the phase of the y=A*sin(2*pi*x+Phi) is 2*pi*x+Phi, and the initial phase is the result when x is zero.so the initial phase of the signal y=sin(2*pi*x) is zero. But when i use fft to calculate the initial phase , i got the result is -90. The matlab code is as follows.
I want to know why the two result is different.thank you very much.
t=linspace(0,2-0.0625,32);%
x=sin(2*pi*t);
figure,plot(t,x);
X=fft(x);
figure,stem(abs(X));
figure;
stem(angle(X)/pi*180);

Chris Turnes on 29 Sep 2015
When taking the FFT of an input array, the first value in the resulting output is the DC bias of the input . Since your input signal x is a pure tone sinusoid that aligns with one of the frequency bins of the DFT, the DC component should be 0.
However, because of round-off error, you don't get exactly 0; instead, you get -8.6529e-16. When you take the phase of this element, it's the same as taking the phase of -1, and you get -pi radians.
However, if you plot the magnitude of your FFT result
plot(abs(X))
you'll see that the frequency of your pure tone sinusoid corresponds to the third DFT bin. So, if you look at the angle of this component, you'll find
>> angle(X(3))/pi*180
ans =
-90.0000
So, now your answer is -90 degrees. Why? The phase of sin(0) is 0 if you're defining the phase relative to a zero-phase sine -- but the phase of the DFT coefficients is relative to a zero-phase cosine. This is why you get a phase of -pi/2 in radians (-90 degrees).
If you need to convince yourself this is the case, try repeating the process, but replace x with
x=cos(2*pi*x-pi/2)
You'll see you get the same FFT results (minus some roundoff error).
Incidentally, if you wanted the phase of the FFT results to be relative to a zero-phase sine, you don't have to do much different; you just have to multiply your input by 1j:
>> X = fft(1j*x);
>> angle(X(3))*180/pi
ans =
-1.5876e-14
summer on 30 Sep 2015
This question confused me several days, your answer is clear and explicit.Thank you very much.And if you're good at the signal processing,I have another question about the phase spectrum of square wave.could you help to answer that question?The address is :https://cn.mathworks.com/matlabcentral/answers/245724-how-to-calculate-the-phase-spectrum-of-the-square-wave-with-fft Thank you.