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The discrete Fourier transform (DFT) and its inverse (IDFT) are the primary numerical transforms relating time and frequency in digital signal processing. The DFT has a number of important properties relating time and frequency, including shift, circular convolution, multiplication, time-reversal and conjugation properties, as well as Parseval's theorem equating time and frequency energy.

Dft

The discrete Fourier transform (DFT) is the primary transform used for numerical computation in digital signal processing. It is very widely used for spectrum analysis , fast convolution , and many other applications. The DFT transforms N discrete-time samples to the same number of discrete frequency samples, and is defined as

X k n N 1 0 x n 2 n k N
The DFT is widely used in part because it can be computed very efficiently using fast Fourier transform (FFT) algorithms.

Idft

The inverse DFT (IDFT) transforms N discrete-frequency samples to the same number of discrete-time samples. The IDFT has a form very similar to the DFT,

x n 1 N k N 1 0 X k 2 n k N
and can thus also be computed efficiently using FFTs .

Dft and idft properties

Periodicity

Due to the N -sample periodicity of the complex exponential basis functions 2 n k N in the DFT and IDFT, the resulting transforms are also periodic with N samples.

X k N X k x n x n N

Circular shift

A shift in time corresponds to a phase shift that is linear in frequency. Because of the periodicity induced by the DFT and IDFT, the shift is circular , or modulo N samples.

x n m N X k 2 k m N The modulus operator p N means the remainder of p when divided by N . For example, 9 5 4 and -1 5 4

Time reversal

x n N x N n N X N k N X k N Note: time-reversal maps 0 0 , 1 N 1 , 2 N 2 , etc. as illustrated in the figure below.

Original signal
Time-reversed
Illustration of circular time-reversal

Complex conjugate

x n X k N

Circular convolution property

Circular convolution is defined as x n h n m N 1 0 x m x n m N

Circular convolution of two discrete-time signals corresponds to multiplication of their DFTs: x n h n X k H k

Multiplication property

A similar property relates multiplication in time to circular convolution in frequency. x n h n 1 N X k H k

Parseval's theorem

Parseval's theorem relates the energy of a length- N discrete-time signal (or one period) to the energy of its DFT. n N 1 0 x n 2 1 N k N 1 0 X k 2

Symmetry

The continuous-time Fourier transform , the DTFT , and DFT are all defined as transforms of complex-valueddata to complex-valued spectra. However, in practice signals are often real-valued.The DFT of a real-valued discrete-time signal has a special symmetry, in which the real part of the transform values are DFT even symmetric and the imaginary part is DFT odd symmetric , as illustrated in the equation and figure below.

x n real  X k X N k N (This implies X 0 , X N 2 are real-valued.)

Real part of x(k) is even

Even-symmetry in DFT sense

Imaginary part of x(k) is odd

Odd-symmetry in DFT sense
DFT symmetry of real-valued signal

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
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
Google
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
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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 ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, The dft, fft, and practical spectral analysis. OpenStax CNX. Feb 22, 2007 Download for free at http://cnx.org/content/col10281/1.2
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