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.
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Notes
- 1.
Programming languages such as, for example Turbo-Pascal, C, Fortran, ... feature random generators that can be called as functions. Their efficiency varies considerably.
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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):
-
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?
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d.
What do we get for \(|F_j|^2\)?
4.6
Pure Noise
Given the random input series containing numbers between \(-0.5\) and \(0.5\).
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a.
What does the Fourier transform of a random series look like (see Fig. 4.27)?
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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):
where RND is a random numberFootnote 1 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 \(\omega _1:\omega _2:\omega _3=5:7:9\)?
4.8
Go on the Ramp (for Gourmets only)
Given the input series:
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!):
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 \(\sum _{k=1}^{N/2} 1/\sin ^2(\pi (2k-1)/N)\).
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Butz, T. (2015). Discrete Fourier Transformation. In: Fourier Transformation for Pedestrians. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-16985-9_4
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DOI: https://doi.org/10.1007/978-3-319-16985-9_4
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