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Discrete fourier transform

The Discrete Fourier Transform, from now on DFT, of a finite length sequence ( x 0 , ... , x K - 1 ) is defined as

k = n = 0 K - 1 x n e - 2 π j k n K ( k = 0 , ... , K - 1 )

To motivate this transform think of x n as equally spaced samples of a T -periodic signal x ( t ) over a period, e.g., x n = x ( n T / K ) . Then, using the Riemann Sum as an approximation of an integral, i.e.,

n = 0 K - 1 f ( n T K ) T K 0 T f ( t ) d t

we find

x ˆ k = n = 0 K - 1 x ( n T K ) e - 2 π j n T K k / T K T 0 T x ( t ) e - 2 π j t k / T d t = K X k

Note that the approximation is better, the larger the sample size K is.

Remark on why the factor K in [link] : recall that X k is an average while x ^ k is a sum. Take for instance k = 0 : X 0 is the average of the signal while x ^ 0 is the sum of the samples.

From the above we may hope that a development similar to the Fourier series [link] should also exist in the discrete case. To this end, we note first that the DFT is a linear transform and can berepresented by a matrix multiplication (the “exponent” T means transpose):

( x ˆ 0 , ... , x ˆ K - 1 ) T = D F T K · ( x 0 , ... , x K - 1 ) T .

The matrix D F T K possesses K lines and K rows; the entry in line k row n is e - 2 π j k n / K . A few examples are

D F T 1 = 1 D F T 2 = 1 1 1 - 1 D F T 4 = 1 1 1 1 1 - j - 1 j 1 - 1 1 - 1 1 j - 1 - j

The rows are orthogonal The scalar product for complex vectors x = ( x 1 , x 2 , ... , x K ) and y = ( y 1 , y 2 , ... , y K ) is computed as

x · y = x 1 y 1 * + x 2 y 2 * + ... + x K y K * ,
where a * is the conjugate complex of a . Orthogonal means x · y = 0 .
to each other. Also, all rows have length Length is computed as | | x | | = x · x = x 1 x 1 * + x 2 x 2 * + ... + x K x K * = | x 1 | 2 + | x 2 | 2 + ... + | x K | 2 . K . Finally, the matrices are symmetric (exchanging lines for rows does not change the matrix). So, the multiplying DFT with its conjugate complex matrix ( D F T K ) * we get K times the unit matrix (diagonal matrix with all diagonal elements equal to K ).

Inverse DFT

From all this we conclude that the inverse matrix of D F T K is I D F T K = ( 1 / K ) · ( D F T K ) * . Since ( e - α ) * = e α we find

x n = 1 K k = 0 K - 1 x ˆ k e 2 π j k n K ( n = 0 , ... , K - 1 )

Spectral interpretation, symmetries, periodicity

Combining [link] and [link] we may now interpret x ˆ k as the coefficient of the complex harmonic with frequency k / T in a decomposition of the discrete signal x n ; its absolute value provides the amplitude of the harmonic and its argument the phase difference.

If x is even, x ˆ k is real for all k and all harmonics are in phase.

Using the periodicity of e 2 π j t we obtain x n = x n + K when evaluating [link] for arbitrary n . Short, we can consider x n as equally-spaced samples of the T -periodic signal x ( t ) over any interval of length T :

x ˆ k = n = - K / 2 K / 2 - 1 x n e - 2 π j k n K .

Similarly, x ˆ k is periodic: x ˆ k = x ˆ k + K . Thus, it makes sense to evaluate x ˆ k for any k . For instance, we can rewrite [link] as

x n = 1 K n = - K / 2 K / 2 - 1 x ˆ k e 2 π j k n K

Since x ˆ k corresponds to the frequency k / T , the period K of x ˆ k corresponds to a period of K / T in actual frequency. This is exactly the sampling frequency (or sampling rate) of the original signal( K samples per T time units). Compare to the spectral repetitions.

However, the period T of the original signal x is nowhere present in the formulas of the DFT (cpre. [link] and [link] ). Thus, if nothing is known about T , it is assumed that the sampling rate is 1 (1 sample per time unit), meaning that K = T .


The Fast Fourier Transform (FFT) is a clever algorithm which implements the DFT in only K log ( K ) operations. Note that the matrix multiplication would require K 2 operations.

Matlab implements the FFT with the command fft(x) where x is the input vector. Note that in Matlab the indices start always with 1! This means that the first entry ofthe Matlab vector x , i.e. x ( 1 ) is the sample point x 0 = x ( 0 ) . Similar, the last entry of the Matlab vector x is, i.e. x ( K ) is the sample point x K - 1 = x ( ( K - 1 ) T / K ) = x ( T - T / K ) .

Questions & Answers

are nano particles real
Missy Reply
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Lale Reply
no can't
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Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
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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
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Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
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Jyoti Reply
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Crow Reply
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RAW Reply
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I think
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Brian Reply
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industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
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scanning tunneling microscope
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
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Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
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Bob Reply
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The nanotechnology is as new science, to scale nanometric
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Damian Reply
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Source:  OpenStax, Sampling rate conversion. OpenStax CNX. Sep 05, 2013 Download for free at http://legacy.cnx.org/content/col11529/1.2
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