I am new to this site, and new to calculus, clearly. I apologize if this is a silly question, but I cannot find the answer anywhere explained in a way that I can understand.
This is the original waveform, f(t):
g(t) = {1, 3, 0, -2, 1, 3, 0, -2}
Using this original waveform, and applying the original Fourier transform, I ended up with the graph of the frequency spectrum of g(t). I took the complex numbers that resulted, and calculated the magnitude, and multiplied the resulting by $\frac{2}{N\tau}$ to normalize them. This resulted in points on the spectrum graph. I cannot post images here, so the points are:
G(f) = {1, 0, 2.5, 0, 0, 0, 2.5, 0}
I know that this graph, aside from the 0Hz aspect, is symmetrical, so there are only actually two frequencies used.
I have been tasked with using the inverse Fourier transform to calculate g(1), g(2), and g(4). However I am not quite clear how to do this.
This is what we're given for the inverse Fourier series. It is the original inverse Fourier transform integration function rewritten as a summation:
$$\ g(t) = \sum_{n=0}^{N-1} \Biggl[G\Biggl(\frac{n}{N\tau}\Biggr)\Biggr] e^{i2\pi(\frac{n}{N\tau})}\frac{1}{N\tau}$$ The frequencies that are graphed on the spectrum (so $\frac{n}{N\tau}$) are: $\frac{0}{8}, \frac{1}{8}, \frac{2}{8}, \frac{3}{8}, \frac{4}{8}, \frac{5}{8}, \frac{6}{8}$, and $\frac{7}{8}$.
Of course, N=8, $\tau$ (sampling interval) = 1 in this case, so $N\tau$ = 8 in all cases. I understand that $\frac{1}{N\tau}$ is there to define the width of the "strips" used to integrate. The frequency variable represents an amplitude, and the Euler's section represents the angle.
From my understanding, I am trying to get the points of the original waveform by using the inverse FT on the points of the spectrum. So, for example, g(2) would equal 3, since the waveform is at 3 at 2sec in g(t). Calculating g(0) is very easy, since you are summing the values of G(f). But how do I get from point A to point B using this formula above? Do I sum up the values of the spectrum waveform (which ends up being 6) and multiply this by the second half of the equation, or am I multiplying the value at that specific $G\frac{n}{N\tau}$ point?
I am supposed to show my work, so computing this on the computer is not an option. My professor has never actually taught us this part, only the original Fourier transform, but expects us to be able to do it.
Thank you in advance for any assistance