Transform of Walsh

In Mathematical, and more precisely in analyzes harmonic the transformed of Walsh is the analog of the discrete Transformée of Fourier.

It operates on a Corps finished of the modular Arithmétique in the place of the complex numbers.

It is used in Information theory at the same time for the linear codes and the Cryptographie.

Definition

That is to say G a abelian Group finished of order G and exhibitor a power n^ième of a Prime number p , F _pⁿ the Body finished of cardinal p ⁿ, χ a character with value in F _pⁿ and F a function of G in F _pⁿ.

* the transformed of Walsh is a function, often noted $\ widehat f$ of the whole of the characters of G in the body F _pⁿ defined by:

\ widehat F (\ chi) \ = \ frac 1g \ sum_ {S \ in G} F (S) \ chi^ {- 1} (S)

Analyzes harmonic on a finished abelian group

See also: harmonic Analysis on an abelian group finished

The context is identical to that of the traditional harmonic analysis of a finished abelian group. The bilinear form associated with the algebra of the group is then the following one:

\ forall F, H \ in \ mathbb F_ {p^n} ^G = \ frac 1g \ sum_ {S \ in G} F (S) ^ {- 1}. \, H (S) \;

The whole of the results of the theory of the harmonic analysis applies, one has thus of the equality of Parseval, the Théorème of Plancherel, a Produit convolution, Dualité of Pontryagin or of the Formule sommatoire of Poisson.

Case of a finished vector space

See also: harmonic Analysis on a vector space finished

There exists a particular case, that or G groups it is the additive group of a finished vector space. A particular case is that or G is a body.

The discrete transformation of Fourier is given by

$f_j= \ sum_ {k=0} ^ {n-1} x_k \ left (e^ {- \ frac {2 \ pi I} {N}} \ right) ^ {jk} \ quad \ quad j=0, \ dowries, n-1$

The theoretical transformation of number operates on a continuation of N numbers, modulo a Prime number p of the form $p = \ xi N + 1 \,$ , where $\ xi \,$ can be any positive integer.

The number $e^ {- \ frac {2 \ pi I} {N}} \,$ is replaced by a number $\ omega^ {\ xi} \,$ where $\ Omega \,$ is a “primitive Racine” of p , a number where more the positive whole small number $\ alpha \,$ where $\ omega^ {\ alpha} = 1 \,$ is $\ alpha = p - 1 \,$ . There should be a quantity d' $\ Omega \,$ which sticks to this condition. Let us note that the two numbers $e^ {- \ frac {2 \ pi I} {N}} \,$ and $\ omega^ {\ xi} \,$ high with the power N are equal to 1 (MOD p), all the lower powers different from 1.

The theoretical transformation of number is given by

$f (X) _j= \ sum_ {k=0} ^ {n-1} x_k (\ omega^ \ xi) ^ {jk} \ MOD p \ quad \ quad j=0, \ dowries, n-1$

Context

The theoretical transformation of opposite number is given by

$f^ {- 1} (X) _h=n^ {p-2} \ sum_ {j=0} ^ {n-1} x_j (\ omega^ {p-1- \ xi}) ^ {hj} \ MOD p \ quad \ quad h=0, \ dowries, n-1$

$\ omega^ {(p-1- \ xi)} = \ omega^ {- \ xi} \,$ , the reverse of $\ omega^ {\ xi} \,$ , and $n^ {p-2} = n^ {- 1} \,$ , the reverse of N . (MOD p)

The reverse is true, because $\ sum_ {k=0} ^ {n-1} z^k$ is N for z=1 and 0 for all the others Z where $z^n = 1 \,$ . A demonstration of this (should go for all Algèbre of division) is

$z \ left (\ sum_ {k=0} ^ {n-1} z^k \ right) +1= \ sum_ {k=0} ^nz^k$

$z \ sum_ {k=0} ^ {n-1} z^k= \ sum_ {k=0} ^ {n-1} z^k$ (withdrawing $z^n = 1 \,$ )

$z=1 \,$ if $\ sum_ {k=0} ^ {n-1} z^k \ 0$ (dividing the two with dimensions ones)

If Z =1 then we could see in a commonplace way that $\ sum_ {k=0} ^ {n-1} z^k= \ sum_ {k=0} ^ {n-1} 1=n$ . If $z \ 1 \,$ then the right-sided must be false to avoid a contradiction.

We can now supplement the demonstration. We take the reverse transformation of the transformation.

$f^ {- 1} (F (X))_h=n^ {p-2} \ sum_ {j=0} ^ {n-1} \ left (\ sum_ {k=0} ^ {n-1} x_k \ left (\ omega^ \ xi \ right) ^ {jk} \ right) (\ omega^ {p-1- \ xi}) ^ {hj} \ MOD p$

$f^ {- 1} (F (X))_h=n^ {p-2} \ sum_ {j=0} ^ {n-1} \ sum_ {k=0} ^ {n-1} x_k (\ omega^ \ xi) ^ {jk-hj} \ MOD p$

$f^ {- 1} (F (X))_h=n^ {p-2} \ sum_ {k=0} ^ {n-1} x_k \ sum_ {j=0} ^ {n-1} (\ omega^ {\ xi (KH)}) ^j \ MOD p$

f^ {- 1} (F (X))_h=n^ {p-2} \ sum_ {k=0} ^ {n-1} x_k \ left \ {\ begin {matrix} N, &k=h \ \ 0, &k \ H \ end {matrix} \ right \} \ MOD p

(since

\ omega^ {\ xi} = 1 \,

)

$f^ {- 1} (F (X))_h=n^ {p-2} x_hn \ MOD p$

$f^ {- 1} (F (X))_h=x_h \ MOD p$

See too

Convolution
Algorithm of multiplication

External bond

Site which (fortunately) reports the same thing as this article (in English)

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