Periodicity recognition

Periodic signals lurk everywhere. A well understandable example is our weather. Since our climate is very complex and changes obviously, no two years are the same.?Nevertheless, natural phenomena can be recognized.

As you may know, our planet moves in an elliptical orbit around our sun. A whole period of time in which the same point in this orbit is passed again, we call a year.?Furthermore, the earth rotates around its own axis 365 times in one year. The following figure shows the outside air temperature over an entire year with 8760 hours at the same place in Cologne, Germany in 2022.

A proven method for the recognition of periodic signals is the discrete Fourier transform. This mathematical tool converts a signal in the time domain to the frequency domain, where you can see the amplitude of all oscillations in the entire spectrum.?Instead of discrete values over n time samples, we will get discrete amplitudes over k frequency samples.

In principle, any signal can be examined with the Fourier transform. However, only regular parts of the signal are highlighted. Since the Fourier transform is a symmetric operation, it is sufficient to represent only half of all 8760 frequency components, as shown in the following figure.

As you have already guessed, you are looking at the same picture of the outside air temperature as before, but in the frequency domain. If we now zoom in, we can see three essential pieces of information.

1. Mean temperature, 12.2°C, represented by k = 0
2. Annual periodic temperature, 4.3 K (peak), represented by k = 1
3. Daily periodic temperature, 1.1 K (peak), represented by k = 365

We know from experience that the outside air temperature varies from day to night in many places around the world. But here is the proof. What else do we see? The annual periodic temperature shows the four seasons spring, summer, fall and winter. The peak values do not represent the overall peak outside air temperature, since the time domain signal is the sum of all frequency components.

Since this conclusion is difficult to follow in the frequency domain, we filter the signals and do an inverse Fourier transform. The following figure shows the outdoor air temperature in the time domain, with the individual harmonics superimposed in each case.

You can see that the annual periodic temperature reaches its maximum in mid-July, while the minimum falls in mid-January. To check if this is not a coincidence, let's take a look at other cities in Germany, which you can see on the map below.?

Seems that cologne has the warmest outside air temperature in average. If you would like to meet there for a K?lsch or need support with development tasks, feel free to get in touch with me.