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The discrete Fourier transform (DFT) maps a finite number of discrete time-domain samples to the same number of discrete Fourier-domain samples. Being practical to compute, it is the primary transform applied to real-world sampled data in digital signal processing.The DFT has special relationships with the discrete-time Fourier transform and the continuous-time Fourier transform that let it be used as a practical approximation of them through truncation and windowing of an infinite-length signal. Different window functions make various tradeoffs in the spectral distortions andartifacts introduced by DFT-based spectrum analysis.

Discrete-time fourier transform

The Discrete-Time Fourier Transform (DTFT) is the primary theoretical tool for understanding the frequency content of a discrete-time (sampled) signal.The DTFT is defined as

X ω n x n ω n
The inverse DTFT (IDTFT) is defined by an integral formula, because it operates on a continuous-frequency DTFT spectrum:
x n 1 2 ω X k ω n

The DTFT is very useful for theory and analysis, but is not practical for numerically computing a spectrum digitally, because

  • infinite time samples means
    • infinite computation
    • infinite delay
  • The transform is continuous in the discrete-time frequency, ω

For practical computation of the frequency content of real-world signals, the Discrete Fourier Transform (DFT) is used.

Discrete fourier transform

The DFT transforms N samples of a discrete-time signal to the same number of discrete frequency samples, and is defined as

X k n 0 N 1 x n 2 π n k N
The DFT is invertible by the inverse discrete Fourier transform (IDFT):
x n 1 N k N 1 0 X k 2 n k N
The DFT and IDFT are a self-contained, one-to-one transform pair for alength- N discrete-time signal. (That is, the DFT is not merely an approximation to the DTFT as discussed next.) However, the DFT is very often used as a practical approximation to the DTFT .

Relationships between dft and dtft

Dft and discrete fourier series

The DFT gives the discrete-time Fourierseries coefficients of a periodic sequence ( x n x n N ) of period N samples, or

X ω 2 N X k δ ω 2 k N
as can easily be confirmed by computing the inverse DTFT of the corresponding line spectrum:

x n 1 2 ω 2 N X k δ ω 2 k N ω n 1 N k N 1 0 X k 2 n k N IDFT X k x n

The DFT can thus be used to exactly compute the relative values of the N line spectral components of the DTFT of any periodic discrete-time sequence with an integer-length period.

Dft and dtft of finite-length data

When a discrete-time sequence happens to equal zero for all samples except for those between 0 and N 1 , the infinite sum in the DTFT equation becomes the same as the finite sum from 0 to N 1 in the DFT equation. By matching the arguments in the exponential terms, we observe that theDFT values exactly equal the DTFT for specific DTFT frequencies ω k 2 k N . That is, the DFT computes exact samples of the DTFT at N equally spaced frequencies ω k 2 k N , or

X ω k 2 k N n x n ω k n n 0 N 1 x n 2 π n k N X k

Dft as a dtft approximation

In most cases, the signal is neither exactly periodic nor truly of finite length; in such cases, the DFT of a finite block of N consecutive discrete-time samples does not exactly equal samples of the DTFT at specific frequencies. Instead, the DFT gives frequency samples of a windowed (truncated) DTFT X ^ ω k 2 k N n N 1 0 x n ω k n n x n w n ω k n X k where w n 1 0 n N 0 else Once again, X k exactly equals X ω k a DTFT frequency sample only when n n 0 N 1 x n 0

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..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
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.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
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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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