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ABBREVIATIONS
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<h1>2 Basics</h1>
<ol>
<li>
<h2>What is the FT/ IFT?</h2>
The Fourier transform (FT) is an integral transform, which attaches a suitable original function
f(t) to its image function or Fourier transformed F(ν).<br />
The original function f(t) is piecewise continuous and absolute integrable. Typical its domain is time.
<br />
The image function F(ν) is a function of frequency ν.
<br />
<br />
The inverse Fourier transform (IFT) is the inverse function of the Fourier transform. <br />
It transforms an image function F(ν) into its original function f(t). By using the IFT a function
of time can be generated from a frequency spectrum.
<br />
<br />
</li>
<li>
<h2>What does the FT supply?</h2>
The FT analyzes, whether a function is periodic. In order to do this the analyzed function f(t) is
compared with Euler′s formula
<img src="Formeln/KomplDrehzeiger-Dateien/image001.gif" class="Textfluss"/>, which states:
<p align="center" >
<img src="Formeln/KomplDrehzeigerFormel-Dateien/image001.gif" />
</p>
The product 2πν describes the angular frequency ω. j is the imaginary unit.<br />
Hence the continuous Fourier transform is defined for every real numebr ν as:<p align="center" >
<img src="Formeln/KFT-Dateien/image001.gif" />
</p>
By using Euler′s formula a function can be tested on the similarity to both a sine (imaginary part)
and a cosine (real part).<br />
<br />
</li>
<li>
<h2>How is the FT used in signal processing?<br />
DFT/ IDFT and FFT/ IFFT
</h2>
To analyze discrete signals as used in signal processing, the discrete Fourier transfrom is used.<br />
It attaches a finite sequence of numbers f<sub>n</sub> with n = 0, 1, 2, ... to an image sequence F<sub>k</sub>
with k = 0, 1, 2, ...:
<p align="center" >
<img src="Formeln/DFT-Dateien/image001.gif" />
</p>
The complex factor <img src="Formeln/KomplDrehzeigerFolge-Dateien/image001.gif" class="Textfluss"/>
is Euler′s formula attuned to a sequence.<br />
<br />
The inverse discrete Fourier transfrom (IDFT) is the inverse function of the DFT. <br />
The original sequence f<sub>n</sub> can be derived from the image sequence F<sub>k</sub> by:
<p align="center" >
<img src="Formeln/IDFT-Dateien/image001.gif" />
</p>
In practice audio signals are scanned with a "scan comb" (delta pulse sequence) in equidistant
time intervals t<sub>n</sub> = nΔt with n = 0, 1, 2, ..., m and the scan time Δt > 0, so that
a discrete signal is produced. So the function exists as measuring points and is described as the sequence
f<sub>n</sub>. <br />
Here you have to notice, that the scan times Δt need to be chosen small enough. Then the scanned
sequence f<sub>n</sub> reflects the signal adequate, so that the signal can be reproduced out of the
measuring points. <br />
<br />
In the modern computer science a fast, very effective algorithm, the fast Fourier transform (FFT), is adopted
in order to calculate discrete Fourier transforms with a high m. The number of realized computations using
the FFT is proportional to N·log<sub>2</sub>(N). This is much less than using the DFT with its
∝N² operations. <br />
The inverse function of the FFT is the inverse fast Fourier transfrom (IFFT).<br />
<br />
<b>Principle of the FFT algorithm</b><br />
The algorithm of the FFT decomposes the DFT into both, one transformed function with even
"c<sub>2k</sub>" and one with odd "c<sub>2k+1</sub>" Fourier coefficients: <br />
<p align="center" > DFT </p>
<p align="center" >
<img src="Formeln/FFTAlgorithmusDFT-Dateien/image001.gif" />
</p>
<p align="center" >
<img src="Bilder-HTML/PfeilFFTSkaliert.jpg" />
</p>
<p align="center" > FFT </p>
<table border="0" align="center" >
<tbody>
<tr>
<td> <img src="Formeln/FFTAlgorithmusgerade-Dateien/image001.gif" /> </td>
<td><div id="Abstand" > </div></td>
<td><img src="Formeln/FFTAlgorithmusungerade-Dateien/image001.gif" /> </td>
</tr>
</tbody>
</table>
with <img src="Formeln/FFTAlgorithmusW-Dateien/image001.gif" class="Textfluss"/>,
until there is left only one even and one odd Fourier coefficient "c". <br />
<br />
</li>
<li>
<h2>Specialties in application of the FFT in signal processing
</h2>
In signal processing audio signals are scanned with a "scan comb" (Delta pulse seqeunce) to
produce a discrete signal. So generally 44100 measured values are scanned per second. When scanning
there is always a refelection to 0.5 · sample rate, <br />
In other words 0.5 · 44100 Hz = 22050 Hz, what conforms to the maximal hearable frequency
domain of the human ear. <br />
<br />
The reason for the reflection is made clear in the following image:
<p align="center" >
<img src="Bilder-HTML/BegruendungSpiegelungSkaliert.png" />
</p>
The upper graphic shows two different waves, one with 0.3 · sample rate and one with
0.7 · sample rate. The red lines describe those measured values, that are saved during
the scanning. You can see that both waves result in the same scanned vaules. Hence, the scanned
signals as well as their frequency spectra after FFT are always symmteric to 0.5 · sample rate.
For this reason only the first half of the frequency spectrum of the transformed function is relevant.
<br />
<br />
The following images show frequency spectra with and without reflection. <br />
<table border="0" >
<tbody>
<tr>
<td> <img src="Bilder-HTML/DarstellungFFTmitSpiegelungSkaliert.png" /> </td>
<td><img src="Bilder-HTML/ReellwertigeDarstellungFFTSkaliert.png" /> </td>
</tr>
<tr>
<td align="center" >Spectrum with reflection</td>
<td align="center" >Spectrum without reflection</td>
</tr>
</tbody>
</table>
<br />
<br />
Furthermore only the different frequencies are important, which are included in a signal.
The phase shift, that is to say whether you have a sine or a cosine, is ignored, because the human ear
can not notice it.
<br />
For better clarity in plotting the frequency spectrum the absolute value of the transformed function
("abs(FFT)") is calculated and imaged. At this a real-valued diagram results out of the complex one.
The x-axis states the frequency and the y-axis states the absolute value of the transformed function.
The complex diagram is not suitable for processing the frequency spectrum, because identifying the contained
frequencies is not that easy.
<br />
<br />
The images below make this circumstance understandable once more:
<table border="0" >
<tbody>
<tr>
<td> <img src="Bilder-HTML/KomplexeDarstellungFFTSkaliert.png" /> </td>
<td><img src="Bilder-HTML/ReellwertigeDarstellungFFTSkaliert.png" /> </td>
</tr>
<tr>
<td align="center" >complex diagram</td>
<td align="center" >real-valued diagram</td>
</tr>
</tbody>
</table>
</li>
</ol>
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