Fourier Transformation

Abstract

Fourier transformation is a very important tool for signal analysis but also helpful to simplify the solution of differential equations or the calculation of convolution integrals. In this chapter we discuss the discrete Fourier transformation as a numerical approximation to the continuous Fourier integral. It can be realized efficiently by Goertzel’s algorithm or the family of fast Fourier transformation methods. Computer experiments demonstrate trigonometric interpolation and nonlinear filtering as applications.

Problems

Problem 7.1 Discrete Fourier Transformation

In this computer experiment for a given set of input samples

$$\begin{aligned} f_{n}=f(n\frac{T}{N})\quad n=0\ldots N-1 \end{aligned}$$

(7.69)

• the Fourier coefficients

  $$\begin{aligned} \widetilde{f}_{\omega _{j}}=\sum _{n=0}^{N-1}f_{n}e^{-i\omega _{j}t_{n}}\quad \omega _{j}=\frac{2\pi }{T}j,\quad j=0\ldots N-1 \end{aligned}$$

  (7.70)

  are calculated with Görtzel’s method 7.3.1.

• The results from the inverse transformation

  $$\begin{aligned} f_{n}=\frac{1}{N}\sum _{j=0}^{N-1}\widetilde{f}_{\omega _{j}}e^{i\frac{2\pi }{N}nj} \end{aligned}$$

  (7.71)

  are compared with the original function values $f(t_{n})$.

• The Fourier sum is used for trigonometric interpolation with only positive frequencies

  $$\begin{aligned} f(t)=\frac{1}{N}\sum _{j=0}^{N-1}\widetilde{f}_{\omega _{j}}\left( e^{i\frac{2\pi }{T}t}\right) ^{j}. \end{aligned}$$

  (7.72)

• Finally the unphysical high frequencies are replaced by negative frequencies (7.24). The results can be studied for several kinds of input data.

Fig. 7.3

(Screenshot from computer exercise 7.2) Top The input signal is rectangular with Gaussian noise. Middle The Components of the Fourier spectrum (red) below the threshold (green line) are dropped. Bottom the filtered signal is reconstructed (blue)

Problem 7.2 Noise Filter

This computer experiment demonstrates a nonlinear filter.

First a noisy input signal is generated.

The signal can be chosen as

• monochromatic $\sin (\omega t)$

• the sum of two monochromatic signals $a_{1}\sin \omega _{1}t+a_{2}\sin \omega _{2}t$

• a rectangular signal with many harmonic frequencies $\text {sign}(\sin \omega t)$

Different kinds of white noise can be added

• dichotomous $\pm 1$

• constant probability density in the range $[-1,1]$

• Gaussian probability density

The amplitudes of signal and noise can be varied. All Fourier components are removed which are below a threshold value and the filtered signal is calculated by inverse Fourier transformation. Figure 7.3 shows a screenshot from the program.