# Dft definition and properties

 Page 1 / 1
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

$X(k)=\sum_{n=0}^{N-1} x(n)e^{-(i\frac{2\pi nk}{N})}$
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,

$x(n)=\frac{1}{N}\sum_{k=0}^{N-1} X(k)e^{i\frac{2\pi nk}{N}}$
and can thus also be computed efficiently using FFTs .

## Periodicity

Due to the $N$ -sample periodicity of the complex exponential basis functions $e^{i\frac{2\pi nk}{N}}$ 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.

$(x((n-m)\mod N), X(k)e^{-(i\frac{2\pi km}{N})})$ The modulus operator $p\mod N$ means the remainder of $p$ when divided by $N$ . For example, $9\mod 5=4$ and $-1\mod 5=4$

## Time reversal

$(x(-n\mod N)=x((N-n)\mod N), X((N-k)\mod N)=X(-k\mod N))$ Note: time-reversal maps $(0, 0)$ , $(1, N-1)$ , $(2, N-2)$ , etc. as illustrated in the figure below.

## Complex conjugate

$(\overline{x(n)}, \overline{X(-k\mod N)})$

## Circular convolution property

Circular convolution is defined as $\doteq ((x(n), h(n)), \sum_{m=0}^{N-1} x(m)x((n-m)\mod N))$

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. $(x(n)h(n), \frac{1}{N}(X(k), H(k)))$

## 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. $\sum_{n=0}^{N-1} \left|x(n)\right|^{2}=\frac{1}{N}\sum_{k=0}^{N-1} \left|X(k)\right|^{2}$

## 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  $X(k)=\overline{X((N-k)\mod N)}$ (This implies $X(0)$ , $X(\frac{N}{2})$ are real-valued.)

#### Questions & Answers

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

### Read also:

#### Get the best Algebra and trigonometry course in your pocket!

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
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'The dft, fft, and practical spectral analysis' conversation and receive update notifications?

 By By Mistry Bhavesh