Luciano Egusquiza Castillo: The Discrete Fourier Transform: Some Properties and Applications

Abstract.

The discrete Fourier transformation (DFT) is a map that takes a finite sequence from one domain into another. Typically, from time to frequency, generating the amplitude and phase of the constituent complex-sinusoidal components, which are its basis. The aim of this thesis is to gain an under- standing of how the discrete Fourier transformation (DFT) is constructed in terms of its mathematical foundation. Some of its properties are delved into, amongst a few are linearity, one-to-oneness, periodicity, and its property of simplifying convolution operations.

Once a mathematical foundation and some properties have been explored, the fast Fourier transformation (FFT) is introduced. The FFT is an algo- rithm used in practice in order to reduce the number of complex multiplications needed to perform the DFT. This is done by exploiting some nice properties of the DFT matrix. Such as its symmetries, the fact that it is unitary, and that its inverse has similar elements.

Finally, applicative examples are demonstrated. A case of denoising a signal by applying the FFT is shown. Following the denoising example is an example of differentiation, where the error is compared with the Euler for- ward method. Lastly, an example of integration using the FFT is presented.