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Matrix notation

We denote the N × N identity matrix with I N , and diagonal matrices with

diag 0 k < N ( γ k ) = γ 0 γ N - 1 .

The N × N stride permutation matrix is defined for N = K M by the permutation

L M N : i K + j j M + i

for 0 i < K , 0 j < M . This definition shows that L M N transposes a K × M matrix stored in row-major order. Alternatively, we can write

L M N : i iM mod N - 1 , for 0 i < N - 1 , N - 1 N - 1 .

For example ( · means 0),

L 2 6 = 1 · · · · · · · 1 · · · · · · · 1 · · 1 · · · · · · · 1 · · · · · · · 1 .

L N / 2 N is sometimes called the perfect shuffle.

Further, we use matrix operators; namely the direct sum

A B = A B

and the Kronecker or tensor product

A B = [ a k , B ] k , , for A = [ a k , ] .

In particular,

I n A = A A = A A

is block-diagonal.

We may also construct a larger matrix as a matrix of matrices, e.g.,

A B B A .

If an algorithm for a transform is given as a product of sparse matrices built from the constructs above, then an algorithm for the transpose orinverse of the transform can be readily derived using mathematical properties including

( A B ) T = B T A T , ( A B ) - 1 = B - 1 A - 1 , ( A B ) T = A T B T , ( A B ) - 1 = A - 1 B - 1 , ( A B ) T = A T B T , ( A B ) - 1 = A - 1 B - 1 .

Permutation matrices are orthogonal, i.e., P T = P - 1 . The transposition or inversion of diagonal matrices is obvious.

Radix-2 fft

The DFT decomposes A = C [ s ] / ( s N - 1 ) with basis b = ( 1 , s , , s N - 1 ) as shown in [link] . We assume N = 2 M . Then

s 2 M - 1 = ( s M - 1 ) ( s M + 1 )

factors and we can apply the CRT in the following steps:

C [ s ] / ( s N - 1 ) C [ s ] / ( s M - 1 ) C [ s ] / ( s M + 1 )
0 i < M C [ s ] / ( x - W N 2 i ) 0 i < M C [ s ] / ( x - W M 2 i + 1 )
0 i < N C [ s ] / ( x - W N i ) .

As bases in the smaller algebras C [ s ] / ( s M - 1 ) and C [ s ] / ( s M + 1 ) , we choose c = d = ( 1 , s , , s M - 1 ) . The derivation of an algorithm for DFT N based on [link] - [link] is now completely mechanical by reading off the matrix for each of the threedecomposition steps. The product of these matrices is equal to the DFT N .

First, we derive the base change matrix B corresponding to [link] . To do so, we have to express the base elements s n b in the basis c d ; the coordinate vectors are the columns of B . For 0 n < M , s n is actually contained in c and d , so the first M columns of B are

B = I M * I M * ,

where the entries * are determined next. For the base elements s M + n , 0 n < M , we have

s M + n s n mod ( s M - 1 ) , s M + n - s n mod ( s M + 1 ) ,

which yields the final result

B = I M - I M I M - I M = DFT 2 I M .

Next, we consider step [link] . C [ s ] / ( s M - 1 ) is decomposed by DFT M and C [ s ] / ( s M + 1 ) by DFT -3 M in [link] .

Finally, the permutation in step [link] is the perfect shuffle L M N , which interleaves the even and odd spectral components (even and odd exponents of W N ).

The final algorithm obtained is

DFT 2 M = L M N ( DFT M DFT -3 M ) ( DFT 2 I M ) .

To obtain a better known form, we use DFT -3 M = DFT M D M , with D M = diag 0 i < M ( W N i ) , which is evident from [link] . It yields

DFT 2 M = L M N ( DFT M DFT M D M ) ( DFT 2 I M ) = L M N ( I 2 DFT M ) ( I M D M ) ( DFT 2 I M ) .

The last expression is the radix-2 decimation-in-frequency Cooley-Tukey FFT. The corresponding decimation-in-time version isobtained by transposition using [link] and the symmetry of the DFT:

DFT 2 M = ( DFT 2 I M ) ( I M D M ) ( I 2 DFT M ) L 2 N .

The entries of the diagonal matrix I M D M are commonly called twiddle factors .

The above method for deriving DFT algorithms is used extensively in [link] .

General-radix fft

To algebraically derive the general-radix FFT, we use the decomposition property of s N - 1 . Namely, if N = K M then

Questions & Answers

what is variations in raman spectra for nanomaterials
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RAW Reply
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Damian
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Professor
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LITNING
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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what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
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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 ?
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yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
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biomolecules are e building blocks of every organics and inorganic materials.
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research.net
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Introduction about quantum dots in nanotechnology
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Source:  OpenStax, Fast fourier transforms. OpenStax CNX. Nov 18, 2012 Download for free at http://cnx.org/content/col10550/1.22
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