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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.


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.


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


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
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


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

find the 15th term of the geometric sequince whose first is 18 and last term of 387
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The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
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kinnecy Reply
can someone help me with some logarithmic and exponential equations.
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I don't understand what the A with approx sign and the boxed x mean
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I'm not sure why it wrote it the other way
I got X =-6
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oops. ignore that.
so you not have an equal sign anywhere in the original equation?
is it a question of log
Commplementary angles
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a perfect square v²+2v+_
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algebra 2 Inequalities:If equation 2 = 0 it is an open set?
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or infinite solutions?
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
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Differences Between Laspeyres and Paasche Indices
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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
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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?
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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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