Fourier analysis

The figure below shows the monthly average exchange rate between the Australian dollar and Danish kroner starting from January 1, 1991, and ending on March 2010. The y axis is the price in kroners for 100 dollars and the x axis is the month index after Janary 1, 1991. It is clearly seen that there exists an underlying oscillatory behavior. This could motivate us to buy dollars paying with kroners whenever the exchange rate is low, and then make an exchange back to kroners whenever the rate is high, thereby making a net earning. The exchange rate data are a bit hard to read because of the superimposed noise and by the fact that the periodicity is not strictly a constant. It would therefore be desirable to have a way to extract the main periods or frequencies before we make large investments. A Fourier (or spectral) analysis can help you with this.

Before we analyse the exchange rate data, however, we should try to get an understanding of how the Fourier transform works in...