CHAPTER 15Fourier Transform
In this chapter, we introduce the implementation of the Fourier Transform, in particular, the discrete Fourier transform (DFT) and illustrate its usage for time series analysis. The DFT allows us to transform a time series from a time domain into a frequency domain and understand the periodicity, if any, of the contributing components. In order to be able to introduce the DFT, we first start with complex numbers. After introducing the DFT, we conclude with a bonus section on quaternions and fractals.
15.1 COMPLEX NUMBERS
Complex numbers are native to some languages, but this is not the case in q, which does not go beyond real numbers. What is a complex number? For a full answer, see for example Agarwal et al. (2011). In a nutshell, a complex number is a number which can be written as 
, where 
and 
are real numbers and 
. It is the last property of 
, which distinguishes complex numbers from reals as there is no real number which would satisfy this condition. ...
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