# I have fft data. How can I predict from the fft data whether the data is of sine or cosine wave?

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### Answers (1)

Rick Rosson
on 13 Nov 2015

Edited: Rick Rosson
on 13 Nov 2015

You can tell from the phase. If the phase is 0 degrees, then it's a cosine. If the phase is -90 degrees, then it's a sine. If +/-180 degrees, it's -cosine. And if +90 degrees, it's -sine. If the phase is anything else, it's a mixture of both sine and cosine.

In any event, regardless of the phase, you can always represent it as

a*cos(2*pi*f*t + phi)

where a is the amplitude, f is the frequency (in hertz), t is time (in seconds), and phi is the phase angle (in radians).

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Rick Rosson
on 15 Nov 2015

Edited: Rick Rosson
on 15 Nov 2015

What happens if you have a sine and a cosine where both are at the same frequency?

a*cos(2*pi*f*t) - b*sin(2*pi*f*t)

Then you will see a single peak in the magnitude response at frequency f even though the signal has two components: a sine and a cosine.

Not to worry, it is possible to show that this signal is mathematically equivalent to:

c*cos(2*pi*f*t + phi)

You can find the value of c from the magnitude response, and the value of phi from the phase response. Then, if you are interested, you can compute the values of a and b as:

a = c*cos(phi)

b = c*sin(phi)

As you can see, these parameters represent a right triangle (in the complex plane), where a and b are the legs of the triangle, c is the hypotenuse, and phi is the angle opposite to b.

Jaime López
on 18 Apr 2022

Hi.

I know it is a very old issue but:

Is there any reason for the cosine to be 0 degrees and the sine -90 degrees and not the other way around, sine 0 degrees and cosine 90 degrees?

Also, altough it is maybe too big a question, is this always the case?, i mean, everytime someone performs a fft, the way to plot it is always c*cos(2*pi*f*t + phi) instead of c*sin(2*pi*f*t + phi)?

Thanks in advance

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