chapter_3.3.lyx

#LyX 2.0 created this file. For more info see http://www.lyx.org/
\lyxformat 413
\begin_document
\begin_header
\textclass scrbook
\begin_preamble
\setcounter{chapter}{3}
\usepackage{graphicx}
\usepackage{pict2e}
\usepackage{graphpap}
\usepackage{color}
\usepackage{bm}
\end_preamble
\use_default_options false
\maintain_unincluded_children false
\language english
\language_package default
\inputencoding auto
\fontencoding global
\font_roman default
\font_sans default
\font_typewriter default
\font_default_family default
\use_non_tex_fonts false
\font_sc false
\font_osf false
\font_sf_scale 100
\font_tt_scale 100

\graphics default
\default_output_format default
\output_sync 0
\bibtex_command default
\index_command default
\paperfontsize default
\spacing single
\use_hyperref false
\papersize default
\use_geometry false
\use_amsmath 1
\use_esint 1
\use_mhchem 1
\use_mathdots 1
\cite_engine basic
\use_bibtopic false
\use_indices false
\paperorientation portrait
\suppress_date false
\use_refstyle 0
\index Index
\shortcut idx
\color #008000
\end_index
\secnumdepth 3
\tocdepth 3
\paragraph_separation indent
\paragraph_indentation default
\quotes_language english
\papercolumns 1
\papersides 1
\paperpagestyle default
\tracking_changes false
\output_changes false
\html_math_output 0
\html_css_as_file 0
\html_be_strict false
\end_header

\begin_body

\begin_layout Section
The Fourier Transform
\end_layout

\begin_layout Standard
Up to now, we looked at functions on a fixed interval (for convenience,
\begin_inset Formula $[-\pi,\pi]$
\end_inset

), and their representation as a Fourier Series, i.e.
 as a linear combinations of sines and cosines, where the period of each
 sine is an integer multiple of the base frequency 
\begin_inset Formula $\frac{1}{2\pi},$
\end_inset

 i.e.
 the frequencies are 
\begin_inset Formula $\frac{1}{2\pi},$
\end_inset


\begin_inset Formula $\frac{2}{2\pi},$
\end_inset


\begin_inset Formula $\frac{3}{2\pi},$
\end_inset

 etc...
 If we want to look at multiple functions at the same times, this is a somewhat
 annoying restriction.
 For example, one can not find an interval on which both 
\begin_inset Formula $f(x)=$
\end_inset


\begin_inset Formula $\sin(\sqrt{2}x)$
\end_inset

 and 
\begin_inset Formula $g(x)=\sin(x)$
\end_inset

 have 
\begin_inset Quotes eld
\end_inset

simple
\begin_inset Quotes erd
\end_inset

 Fourier Series.
 If we choose the interval 
\begin_inset Formula $[-\pi,\pi]$
\end_inset

, then the Fourier Series of 
\begin_inset Formula $f(x)$
\end_inset

is quite complicated, and has non-zero coefficients at all frequencies,
 as 
\begin_inset Formula $\sqrt{2}$
\end_inset

is not an integer.
\end_layout

\begin_layout Standard
One way around this problem is to use the 
\emph on
Fourier Transform
\emph default
, rather than the
\emph on
 Fourier Series
\emph default
, to represent a function in the frequency domain.
 We get rid of the fixed interval 
\begin_inset Formula $[-T,T]$
\end_inset

 by letting 
\begin_inset Formula $T\rightarrow\infty,$
\end_inset

 and by that, the sum 
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\noun off
\color none

\begin_inset Formula $\sum_{k=-\infty}^{\infty}c_{k}e^{ikx}$
\end_inset

 over discrete coefficients 
\begin_inset Formula $c_{k}$
\end_inset

 turns into an integral over the 
\family default
\series default
\shape default
\size default
\emph on
\bar default
\noun default
\color inherit
spectral densities 
\begin_inset Formula $c(\omega),$
\end_inset

 
\emph default
for any real-valued frequency 
\begin_inset Formula $\omega\in\mathbb{R}.$
\end_inset


\end_layout

\begin_layout Standard
Thus, we can write 
\begin_inset Formula 
\[
f(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}c(w)e^{ixw}dx,
\]

\end_inset

 where 
\begin_inset Formula $c(w)$
\end_inset

 is the 
\emph on
Fourier Transform
\emph default
 of 
\begin_inset Formula $f(x),$
\end_inset

 which is often written as 
\begin_inset Formula $\hat{f}(w)$
\end_inset

.
 Conversely, 
\begin_inset Formula $f(x)$
\end_inset

is the 
\emph on
Inverse Fourier Transform
\emph default
 of 
\begin_inset Formula $c(w).$
\end_inset

 
\begin_inset Formula $f(x)$
\end_inset

and 
\begin_inset Formula $c(w)$
\end_inset

 are sometimes referred to as a 
\emph on
Fourier Transform Pair
\emph default
.
 
\end_layout

\begin_layout Standard
We can calculate the Fourier Transform of any function 
\begin_inset Formula $f(x)$
\end_inset

by 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
\hat{f}(w)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(x)e^{-iwx}dx,
\]

\end_inset


\end_layout

\begin_layout Standard
provided that this integral exists.
 The notations and conventions can be ambiguous and vary between different
 communities : The FT is also sometimes defined by 
\begin_inset Formula $\hat{f}(w)=\int_{-\infty}^{\infty}f(x)e^{-i\omega x}dx$
\end_inset

 or by 
\begin_inset Formula $\hat{f}(w)=\int_{-\infty}^{\infty}f(x)e^{-2\pi i\omega x}dx.$
\end_inset

 These different conventions don't, in principle, matter, but one has to
 be careful not to be confused or get results which are off by, say, a factor
 of 
\begin_inset Formula $2\pi.$
\end_inset


\end_layout

\begin_layout Standard
In practice, we can rarely use 
\begin_inset Quotes eld
\end_inset

actual
\begin_inset Quotes erd
\end_inset

 functions 
\begin_inset Formula $f(x),$
\end_inset

 but rather have to work with discrete approximations, i.e.
 
\begin_inset Formula $f(x)$
\end_inset

 is only given by a finite set of functions values 
\begin_inset Formula $f_{1},$
\end_inset


\begin_inset Formula $f_{2},$
\end_inset


\begin_inset Formula $\ldots,f_{n}$
\end_inset

 In this case, one should use the 
\emph on
Discrete Fourier Transform
\emph default
, which is defined to be the vector with 
\begin_inset Formula $m-th$
\end_inset

 entry 
\begin_inset Formula 
\[
\hat{f}_{m}=\sum_{k=1}^{n-1}f_{k}e^{-\frac{2\pi i}{n}km}.
\]

\end_inset


\end_layout

\begin_layout Standard
Sometimes, one can read sentences like 
\begin_inset Quotes eld
\end_inset

Figure X shows the Fast Fourier Transform of data Y
\begin_inset Quotes erd
\end_inset

 in papers.
 This is slight nonsense, as the Fast Fourier Transform is an 
\emph on
algorithm
\emph default
 for computing the DFT, and not something one could plot.
 Of course, they mean that the DFT of the data is shown.
\end_layout

\begin_layout Standard

\end_layout

\end_body
\end_document