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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

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.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
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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