Discrete Fourier Transformation

Abstract

This chapter deals with the discrete Fourier transformation. Here, a periodic series in the time domain is mapped onto a periodic series in the frequency domain. Definitions of the discrete Fourier transformation and its inverse are given. Linearity, convolution, cross-correlation, and autocorrelation are treated as well as Parseval’s theorem. The sampling theorem is illustrated with a simple example. Data mirroring, cosine- and sine-transformations, as well as zero-padding are discussed. It concludes with the Fast Fourier Transformation algorithm by Cooley and Tukey.

Playground

4.1

Correlated

What is the cross correlation of a series $\{f_k\}$ with a constant series $\{g_k\}$? Sketch the procedure with Fourier transforms!

4.2

No Common Ground

Given is the series $\{f_k\}=\{1,0,-1,0\}$ and the series $\{g_k\}=\{1,-1,1,-1\}$.

Calculate the cross correlation of the two series.

4.3

Brotherly

Calculate the cross correlation of $\{f_k\}=\{1,0,1,0\}$ and $\{g_k\}=\{1,-1,1,-1\}$, use the Convolution Theorem .

4.4

Autocorrelated

Given is the series $\{f_k\}=\{0, 1, 2, 3, 2, 1\}$, $N=6$.

Calculate its autocorrelation function . Check your results by calculating the Fourier transform of $f_k$ and of $f_k\otimes f_k$.

4.5

Shifting around

Given the following input series (see Fig. 4.25):

$$\begin{aligned} f_0 =1, \qquad f_k=0\qquad \text {for }k=1, \ldots , N-1. \end{aligned}$$

• a. Is the series even , odd , or mixed ?

• b. What is the Fourier transform of this series?

• c. The discrete “$\delta$-function” now gets shifted to $f_1$ (Fig. 4.26). Is the series even, odd, or mixed?

• d. What do we get for $|F_j|^2$?

Fig. 4.25

Input-signal with a $\delta$-shaped pulse at $k=0$

Fig. 4.26

Input-signal with a $\delta$-shaped pulse at $k=1$

Fig. 4.27

Random series

4.6

Pure Noise

Given the random input series containing numbers between $-0.5$ and $0.5$.

• a. What does the Fourier transform of a random series look like (see Fig. 4.27)?

• b. How big is the noise power of the random series, defined as:

  $$\begin{aligned} \sum _{j=0}^{N-1} |F_j|^2 ? \end{aligned}$$

  (4.57)

  Compare the result in the limiting case of $N\rightarrow \infty$ to the signal power of the input $0.5\cos \omega t$.

4.7

Pattern Recognition

Given a sum of cosine functions as input, with plenty of superimposed noise (Fig. 4.28):

$$\begin{aligned} f_{k}= & {} \cos \frac{5\pi k}{32}+\cos \frac{7\pi k}{32}+\cos \frac{9\pi k}{32}+15(\mathrm {RND}-0.5) \\ \nonumber&\text {for } k = 0, \ldots , 255, \end{aligned}$$

(4.58)

where RND is a random number¹ between $0$ and $1$.

How do you look for the pattern Fig. 4.29 that’s buried in the noise, if it represents the three cosine functions with the frequency ratios $\omega _1:\omega _2:\omega _3=5:7:9$?

Fig. 4.28

Input function according to (4.58)

Fig. 4.29

Theoretical pattern (“toothbrush”) that’s to be located in the data set

4.8

Go on the Ramp (for Gourmets only)

Given the input series:

$$\begin{aligned} f_k = k \ \mathrm {for\ } k=0,1,\ldots ,N-1. \end{aligned}$$

Is this series even, odd, or mixed? Calculate the real and imaginary part of its Fourier transform. Check your results using Parseval’s theorem . Derive the results for $\sum _{j=1}^{N-1}1/\sin ^2(\pi j/N)$ and $\sum _{j=1}^{N-1}\cot ^2(\pi j/N)$.

4.9

Transcendental (for Gourmets only)

Given the input series (with $N$ even!):

$$\begin{aligned} f_{k} = \left\{ \begin{array}{ll} k &{} \mathrm {for\ } k = 0, 1, \ldots , \frac{N}{2}-1 \\ N-k &{} \mathrm {for\ } k = \frac{N}{2}, \frac{N}{2}+1, \ldots , N-1 \end{array} \right. . \end{aligned}$$

(4.59)

Is the series even, odd, or mixed? Calculate its Fourier transform. The double-sided ramp is a high-pass filter (cf. Sect. 5.2), which immediately becomes clear considering the periodic continuation. Use Parseval’s theorem to derive the result for $\sum _{k=1}^{N/2} 1/\sin ^4(\pi (2k-1)/N)$. Use the fact that a high-pass does not transfer a constant in order to derive the result for $\sum _{k=1}^{N/2} 1/\sin ^2(\pi (2k-1)/N)$.