Adding Harmonics of a sine wave together

I have extracted harmonics of a sine wave through 'fft' in matlab. I considered the phase to define the signs of each harmonics - a two sided psd is shown in fig below. However when I add the first 8 harmonics, I do not get the exact sine wave as shown in fig below. Can some one help?

[Syed AWM]

from psd plot, beyond the 5th harmonic amplitude becomes negligible, though I added 8 harmonics

[Bjorn Gustavsson]

To get help from others you cannot remove the plot with the original function and your sum-of-sines approximations.

Answer 1

[Bjorn Gustavsson]

You have to take into account that the fft returns complex Fourier-coefficient - that is both real and imaginary components. So you have to use both sin and cos - terms in your sum, also the average of the signal is not zero so you should have a negative (?) DC-component giving you a constant off-set, that is absent from your plot.

HTH

[Syed AWM]

by 0H in that plot I mean the 0th Harmonic or DC component which has an amplitude as shown in the PSD plot. While adding them as sine or cos the frequency of 0th harmonic would be zero - which that is why it has zero contribution.

[Bjorn Gustavsson]

No.

The Fourier-series approximation of your periodic signal is:

For your second "original curve" (orange with '.-') it is not the case that the average (DC-component) is zero, therefore a_0 is smaller than zero, and cos(0) is equal to 1. Since it is non-zero you have to use that term in your sum-of-Fourier-terms. It seems rather plausible that some of the other cosine-coefficients also are non-zero and they also have to be included.