# amplitude and phase from the Fourier transform equation

This is a snapshot from some text I'm currently reading which aims to describe how the amplitude and phase of a seasonal cycle can be determined:

I think I understand the text but I'm unsure when they refer to 'using the Fourier transform'. Does the equation that follows that comment qualify as a Fourier transform? On first reading I thought that this meant that X was the complex array returned by fft, but according to the description, this doesn't seem to be the case.

I have written this up in matlab as:

datev = datevec(dates);
uyear = unique(datev(:,1)); % this is the unique years of data
t = 0.5:1:11.5; t = t';
for j = 1:length(uyear);
x = yt(datev(:,1) == uyear(j));
yx(j,:) = exp(2.*pi.*1i*t/12).*x;
end
Yx = 2/12.*sum(yx,2);
Amp = abs(Yx);
phase = angle(Yx);


According to the text does this make sense? Also, how can this be described as a Fourier transform?

Your computation of yx, Yx, Amp, phase look okay to me. It seems to me that this is from the analysis equation of a discrete-time periodic signal (with different notations of course) where the coefficients $\{c_k\}$ are calculated as follows:
$$c_k = \frac 1 N \sum_{n = 0}^{N - 1}x(n)\exp\left(\frac{-j2\pi kn}{N}\right)$$
It is not a Fourier transform as such but rather the amplitude of the (Fourier) harmonics at each year. In your equation you have a positive exponent (and not negative, a matter of choice I'm guessing in the definition of the synthesis-analysis equations). The 2 is for both positive and negative contributions. The multiplicative $2\pi$ you have is probably a typo.