<< Chapter < Page Chapter >> Page >

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 .

FFT

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

How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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.
Bharti
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
Daniel
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
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
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
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
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.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Sampling rate conversion. OpenStax CNX. Sep 05, 2013 Download for free at http://legacy.cnx.org/content/col11529/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Sampling rate conversion' conversation and receive update notifications?

Ask