1
$\begingroup$

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:

enter image description here

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?

$\endgroup$
5
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.