# 0.1 Discrete fourier transform

 Page 1 / 1

## Discrete fourier transform

The Discrete Fourier Transform, from now on DFT, of a finite length sequence $\left({x}_{0},...,{x}_{K-1}\right)$ is defined as

${\stackrel{}{\mathrm{x̂}}}_{k}=\sum _{n=0}^{K-1}{x}_{n}{e}^{-2\pi jk\frac{n}{K}}\phantom{\rule{2.em}{0ex}}\left(k=0,...,K-1\right)$

To motivate this transform think of ${x}_{n}$ as equally spaced samples of a $T$ -periodic signal $x\left(t\right)$ over a period, e.g., ${x}_{n}=x\left(nT/K\right)$ . Then, using the Riemann Sum as an approximation of an integral, i.e.,

$\sum _{n=0}^{K-1}f\left(\frac{nT}{K}\right)\frac{T}{K}\simeq {\int }_{0}^{T}f\left(t\right)dt$

we find

${\stackrel{ˆ}{x}}_{k}=\sum _{n=0}^{K-1}x\left(\frac{nT}{K}\right){e}^{-2\pi j\frac{nT}{K}k/T}\simeq \frac{K}{T}{\int }_{0}^{T}x\left(t\right){e}^{-2\pi jtk/T}dt=K{X}_{k}$

Note that the approximation is better, the larger the sample size $K$ is.

Remark on why the factor $K$ in [link] : recall that ${X}_{k}$ is an average while ${\stackrel{^}{x}}_{k}$ is a sum. Take for instance $k=0$ : ${X}_{0}$ is the average of the signal while ${\stackrel{^}{x}}_{0}$ is the sum of the samples.

From the above we may hope that a development similar to the Fourier series [link] should also exist in the discrete case. To this end, we note first that the DFT is a linear transform and can berepresented by a matrix multiplication (the “exponent” $T$ means transpose):

${\left({\stackrel{ˆ}{x}}_{0},...,{\stackrel{ˆ}{x}}_{K-1}\right)}^{T}=DF{T}_{K}·{\left({x}_{0},...,{x}_{K-1}\right)}^{T}.$

The matrix ${DFT}_{K}$ possesses $K$ lines and $K$ rows; the entry in line $k$ row $n$ is ${e}^{-2\pi jkn/K}$ . A few examples are

${DFT}_{1}=\left(\begin{array}{c}1\end{array}\right)\phantom{\rule{1.em}{0ex}}{DFT}_{2}=\left(\begin{array}{cc}1& 1\\ 1& -1\end{array}\right)\phantom{\rule{1.em}{0ex}}{DFT}_{4}=\left(\begin{array}{cccc}1& 1& 1& 1\\ 1& -j& -1& j\\ 1& -1& 1& -1\\ 1& j& -1& -j\end{array}\right)$

The rows are orthogonal The scalar product for complex vectors $x=\left({x}_{1},{x}_{2},...,{x}_{K}\right)$ and $y=\left({y}_{1},{y}_{2},...,{y}_{K}\right)$ is computed as

$x·y={x}_{1}{y}_{1}^{*}+{x}_{2}{y}_{2}^{*}+...+{x}_{K}{y}_{K}^{*},$
where ${a}^{*}$ is the conjugate complex of $a$ . Orthogonal means $x·y=0$ .
to each other. Also, all rows have length Length is computed as $||x||=\sqrt{x·x}=\sqrt{{x}_{1}{x}_{1}^{*}+{x}_{2}{x}_{2}^{*}+...+{x}_{K}{x}_{K}^{*}}=\sqrt{|{x}_{1}{|}^{2}+|{x}_{2}{|}^{2}+...+{|{x}_{K}|}^{2}}$ . $\sqrt{K}$ . Finally, the matrices are symmetric (exchanging lines for rows does not change the matrix). So, the multiplying DFT with its conjugate complex matrix ${\left(DF{T}_{K}\right)}^{*}$ we get $K$ times the unit matrix (diagonal matrix with all diagonal elements equal to $K$ ).

Inverse DFT

From all this we conclude that the inverse matrix of $DF{T}_{K}$ is $IDF{T}_{K}=\left(1/K\right)·{\left(DF{T}_{K}\right)}^{*}$ . Since ${\left({e}^{-\alpha }\right)}^{*}={e}^{\alpha }$ we find

${x}_{n}=\frac{1}{K}\sum _{k=0}^{K-1}{\stackrel{ˆ}{x}}_{k}{e}^{2\pi jk\frac{n}{K}}\phantom{\rule{2.em}{0ex}}\left(n=0,...,K-1\right)$

Spectral interpretation, symmetries, periodicity

Combining [link] and [link] we may now interpret ${\stackrel{ˆ}{x}}_{k}$ as the coefficient of the complex harmonic with frequency $k/T$ in a decomposition of the discrete signal ${x}_{n}$ ; its absolute value provides the amplitude of the harmonic and its argument the phase difference.

If $x$ is even, ${\stackrel{ˆ}{x}}_{k}$ is real for all $k$ and all harmonics are in phase.

Using the periodicity of ${e}^{2\pi jt}$ we obtain ${x}_{n}={x}_{n+K}$ when evaluating [link] for arbitrary $n$ . Short, we can consider ${x}_{n}$ as equally-spaced samples of the $T$ -periodic signal $x\left(t\right)$ over any interval of length $T$ :

${\stackrel{ˆ}{x}}_{k}=\sum _{n=-K/2}^{K/2-1}{x}_{n}{e}^{-2\pi jk\frac{n}{K}}.$

Similarly, ${\stackrel{ˆ}{x}}_{k}$ is periodic: ${\stackrel{ˆ}{x}}_{k}={\stackrel{ˆ}{x}}_{k+K}$ . Thus, it makes sense to evaluate ${\stackrel{ˆ}{x}}_{k}$ for any $k$ . For instance, we can rewrite [link] as

${x}_{n}=\frac{1}{K}\sum _{n=-K/2}^{K/2-1}{\stackrel{ˆ}{x}}_{k}{e}^{2\pi jk\frac{n}{K}}$

Since ${\stackrel{ˆ}{x}}_{k}$ corresponds to the frequency $k/T$ , the period $K$ of ${\stackrel{ˆ}{x}}_{k}$ corresponds to a period of $K/T$ in actual frequency. This is exactly the sampling frequency (or sampling rate) of the original signal( $K$ samples per $T$ time units). Compare to the spectral repetitions.

However, the period $T$ of the original signal $x$ is nowhere present in the formulas of the DFT (cpre. [link] and [link] ). Thus, if nothing is known about $T$ , it is assumed that the sampling rate is 1 (1 sample per time unit), meaning that $K=T$ .

FFT

The Fast Fourier Transform (FFT) is a clever algorithm which implements the DFT in only $Klog\left(K\right)$ operations. Note that the matrix multiplication would require ${K}^{2}$ operations.

Matlab implements the FFT with the command fft(x) where $x$ is the input vector. Note that in Matlab the indices start always with 1! This means that the first entry ofthe Matlab vector $x$ , i.e. $x\left(1\right)$ is the sample point ${x}_{0}=x\left(0\right)$ . Similar, the last entry of the Matlab vector $x$ is, i.e. $x\left(K\right)$ is the sample point ${x}_{K-1}=x\left(\left(K-1\right)T/K\right)=x\left(T-T/K\right)$ .

what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Got questions? Join the online conversation and get instant answers!