NONLINEARIZAD OF FAST FOURIER TRANSFORM Текст научной статьи по специальности «Математика»

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УДК 517.443 DOI 10.29141/2782-4934-2023-2-2-1 EDN TDVEMG

Valeriy Labunets1, Victor Chasovskikh1, Evgeniy Starikov1

!Ural State University of Economics, Ekaterinburg, Russia

Nonlinearizad of Fast Fourier Transform

Abstract. A unified mathematical form of reversible nonlinear transformations based on a nonlinear tensor product is presented in the form of fast algorithms. The main goal of this article is to show that almost all Fourier transforms (FFTs) can be both generalized and non-linear. Nonlinearity and generalization of the FFT are based on two recursive rules, which generate nonlinear transformations using a fast algorithm. For each rule, simple relations indicate the number of elementary nonlinear operations required by the fast algorithm. The resulting scheme is formed in three stages. The first step involves the so-called 2x2 Basic Non-Linear Transforms (BNLT). The second step is based on sparse nonlinear transformations (SNLTs), which are direct sums of BNLTs. The third step is Fast Nonlinear Transform (FNLT) as an SNLTS overlay product.

Key words: nonlinear transforms; nonlinear Fourier transforms; fast algorithms. Acknowledgments: This work was supported by the RFBR grant 19-29-09022\1.9. Paper submitted: April 30, 2023

For citation: Labunets V., Chasovskikh V., Starikov E. Nonlinearizad of fast Fourier transform. Digital models and solutions. 2023. Vol. 2, no. 2. DOI: 10.29141/2782-4934-20232-2-1. EDN: TDVEMG.

В. Г. Лабунец1, В. П. Часовских1, Е. Н. Стариков1

1Уральский государственный экономический университет, г. Екатеринбург, Российская Федерация

Нелинеаризация быстрого преобразования Фурье

Аннотация. В статье представлена единая математическая форма обратимых нелинейных преобразований в виде быстрых алгоритмов, основанная на нелинейном тензорном произведении. Основная цель данной статьи заключается в том, чтобы показать, что почти все быстрые преобразования Фурье (БПФ) могут быть как обобщенными, так и нелинейными. Нелинейность и обобщение БПФ основаны на двух рекурсивных правилах, которые генерируют нелинейные преобразования с использованием быстрого алгоритма. Для каждого правила простые соотношения указы-

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вают количество элементарных нелинейных операций, требуемых быстрым алгоритмом. Полученная схема формируется в три этапа. Первый шаг включает в себя базовые нелинейные преобразования 2x2 (BNLT). Второй шаг основан на разреженных нелинейных преобразованиях (SNLT), которые являются прямыми суммами BNLT. Третий шаг - это быстрое нелинейное преобразование (FNLT) в качестве продукта наложения SNLTS.

Ключевые слова: нелинейные преобразования; нелинейные преобразования Фурье; быстрый алгоритм.

Финансирование: Работа выполнена при поддержке гранта РФФИ 19-29-09022\1.9. Дата поступления статьи: 30 апреля 2023 г.

Для цитирования: Labunets V., Chasovskikh V., Starikov E. Nonlinearizad of fast Fourier transform. Digital models and solutions. 2023. Vol. 2, no. 2. DOI: 10.29141/27824934-2023-2-2-1. EDN: TDVEMG.

Introduction

Recently, nonlinear processing technology has become the main signal processing tool. Initially, modeling and nonlinear filtering were based on the generalized nonlinear Volterra convolution integral or Wiener's G-functional [1-5]. Further, for image processing [6-10], the technology of aggregation operators was used. We offer a unified view of Fast Nonlinear Transforms (FNLT).

Problem Formulation

For non-linear signal processing, 2 new waveforms are proposed, based on sequential decomposition of point DFT decomposition. The proposed forms are based on the fact that they are created by a number of fast recursive unitary transformations of arbitrary order [14]. We generalize fast linear transformations to some non-linear cases defined by parallel/sequential combining of elementary non-linear 2x2 transforms (non-linear butterfly mappings). Applying this structure, we define a large family of FUTs and obtain a number of known and new results about FFT algorithms, existing transformations and define structural properties between transformations.

The main element of the 2"-point FFT defines operations with complex data - the butterfly block (Fig. 1):

~Уо~ fo (-*Ч) ' -*ï)

Системный анализ System analysis

Fig. 1. Non in-place operations

The butterfly has two entrances and two exits. The values at the inputs are called a/1 and bi2, the values at the outputs are called Cj1 and j «^.Complex number W(twiddle factor) is the weight factor, and is different for each butterfly. By combining the butterfly operations in a suitable manner, a 2N point FFT is created (Fig. 2). The proposed butterfly FFT algorithm shows data flow and data operations in graphical form.

Fig. 2. 8-point Fast Fourier transform (with interconnections between butterflies)

The Invertible Nonlinear "Butterfly" Transformations

In this section we introduce the simplest version of the explicitly invertible transformation: it consists of a change of variables, involving 2 arbitrary functions g0 (•,•), g1(v)from two quantities x1, x2 to two quantities y1, y2 and vice versa:

Уо~ goCon) Xq g0 (x0, X1 )

Уг. Mo'-i). A

If £о(-ом) = Soo(•<>) +Soi('i)> gi(vi) = Sio('o)-&i(-i) then

V

У1-

£oo(-o) + £oi(-i) &o("o)~£n(-i)

Xn

Системный анализ System analysis

8oo('o) ! goi('i) Sio('o) ! £ll('l)

о

Soo C*o ) Soi^X)

In particular, we are going to use the following transforms

where /(•„)= &о(*о)> /Oi) = S>i(*i)> ш('о)> =ЯиСч)- For these transforms we have

= k(A)

Уо

У\.

Уо У\.

Уо У\.

Уо

У1.

= Х/>

= Г{fу

А(НУ

l('o) 1 Oi-^O

K-o) \ l(-i)_ l<-*vl _

l( o) ! fi'i ) 0^*0

\ !(',)_ 0 1<-XI

'/(• 0) ! К-,)'

l('o) ! 0 k- 1

/г(х0)+х,

¡Ю,)' 1(' о ) IЩ

л x0 + f(x,)

X,

/(x0)+x,

- " X, "

The most remarkable aspect of these transformations is its explicitly invertible character:

16 basis transforms

Using these transforms we can construct 16 basis transforms BT2 (h,f) (Table), depending on two nonlinear functions:

BT,(Af/)e{^) r(/2), Af,),^) k(/4).....

...,M^)oM/4)}={[BT2(/!,/),2BT2(/f,/),...,l5BT2(/f,AlfBT2(/1,/)}.

For example,

Системный анализ System analysis

Note that both the direct and inverse transforms BT2(h,f) involve only the two arbitrary functions f,h and not their inverses.

The Nonlinear Tensor Superposition Product

Let us introduce two elementary nonlinear (2x2)-transforms, called nonlinear butterfly transforms: BTi:V2 ^ V2, BT2 :V2 ^ V2. Each of them is described by two arbitrary functions BT2 [go, g1 ], BT2 [go, g1]. We will denote them in matrix-like way:

¿"o

y i.

Уо

л

ВТ¡[gl,g2]

(X- So (' 0 > ' 1)" 0 x0

. 1<- g\ (* I) ' ' A

0<- 01) (' 0 ' 0

_ ]<-8\ ('«' 1). A

0<-g0(*0>*l) (x0>x\)

Sometimes we will use weights woo = -wh = cos©1, woi = w\o = sin©1; and woo = -w íi = = cos02, woi = wio = sin02, woi = wio = sin02, for variable xo, xi and "amplitudes" ao = e^o1 ), ai = e^i1, a0 = e^o, ai = e^i2 for non-linear functions:

Уо

л

CK-

Y ло

0 —

where k = 1,2. In this case BT2 [^0, 01; go, gi] and BT2 [^0, ^f, 02; go, gi] depend on three scalar parameters 0 = (^0, ^1,0) £ [0, 2n]3 and two nonlinear functions g = (go, gi).

Definition 1. Tensor product of two nonlinear (2x2)-transforms BT2 [go, gf] and BT22 [g(, gf] is a nonlinear (4x4)-transform that is defined as

Системный анализ System analysis

l<-£|("0' 'l)

®

0<-g)(*0> *|) 1i-S ('О' "|)

where

S\ (-0' 'l)

g)('l)> 'l) S\ ("о» 'l)

и

оо> ÜI 10' II

00' Ol 10' II

= gi

<?o("№ *0|) S\ ('oo' 'oi)

gl ( go > go J gi ( Si >g\ J

go('io' 'il)

Sl2(-

10' II'

gj g0 g! •g2,

is the rule for the composition of input data addresses and

(X-So

o<- go

0<-6(l 1 ^g] J

/

к

S1

0<-Ö0

J

o<-So

-il J

00<

0I<

- go ( go ' go )

So(Si >g\ )

go > g0

1Ск-

1I<-Sl (SI'S*

00<- go

oi<-So 8 if

l(k- gl *So

Ik- g\

is the rule for the composition of output data addresses. Here by definition gl0 • gl g^ ( gl, g^ j, gi • g? go ( g? >g? ) ' g\ • go - gl ( go ' g02 ) ' g\ • gi - g\ ( g}, g? ) and • is the symbol of left superposition of nonlinear functions. So,

ВТ] [g\,g\]® ВТ2 [gl,g;] = BT4 [(g[,g\)i (g2, g,2)] :

= BT4[

J r „2

go * go ' So * g\ ' g\ * So ' g\ * g

Системный анализ System analysis

Definition 2. Tensor product of two nonlinear transforms BT'.. g^.,, and

is i ; i y. , y .....y is a nonlinear (N1 * N2)-transform that is defined as

(5)

Definition 3. We define w-fold tensor product of n nonlinear transforms

RT'1

N„

8o ->8t ■>•■■■>S^-Î

' as the following nonlinear (N x N)-transform

)p= I

where N = N1 N2...Nn.

Example 1. Let ВТ! [gj, g? ] be the identical transform

ВТ

2 [&0>£|]

<Х-£о(*0» "l) ("0> '] )

0<-

Add( Add

,(l(-o)>0(-i))" (О(-о)Д(-,))

then

If ВТ,2 , gf ] is the identical transform, then

Системный анализ System analysis

ВТî[glo,g\

&o('0> 'l) (*0> "l )

0<-

Add^K-oXO^))"

m_£i(Adde(K-n>.<K-U».Add4|(4;H,X0(11»)'

00<-

01+

£o('oi»" 11 )

UX-Ä1 ( 00' "ю) lk-Ä| ("op' ll )

It easy to check that

(Bljy.g}]®!«)-^®!«?

So »gl

)=Bt[glsi]®vt[glgî],

where BT2 gj] <§> I2x2 and I2x2® BT22[g^,g12 are nonlinear radix-2 sparse nonlinear transform (SNLT).

Theorem 1. Each «-fold tensor product of n nonlinear transforms j BT£ |

is the ° - product of sparse nonlinear transforms SNLT

FNLT

№ * Я J(|=o, i,=0,.... /„=0

ЩV,„.JV„ jt<-l

- ВТ

r r

n«-l

- SNLT."

g;,...,g;.] °... о snlt^ [gov.^-i

-SNLT'

(6)

where SNLT;, [gl,g\,...,gTK_^ ® ВТ; [g^gf,...,^]® that are fast nonlinear

transforms (FNLTs), where N0 :-1, Nr_t ... Nr_ p r = 1,2,n and Ii * i = 1. In particular,

FNLT,

H*- I 1

- n°(w ® BT2 [*Uir] ® I w = n°SNLT; [g[,g[\ =

r=] r=l

Л<—1 _ »4-1

= U°(l2 ^ ® BT2 [gro>g[]® Г-) = П° SNLT2 [si =

r=1

= SNLT2"[g0"JgI"]oSNLT22[g02ïg12]0.„oSNLT^[g;îg1I]

Г=1

(7)

Системный анализ System analysis

is the radix-2 FNLT.We define the "bar" operation on the NLT for obtain "nonlinear transpose" transform

1. The Nonlinear Superposition Product

Let Vn = Span{eo,ei,...,eN-i} be a N-D space spanned on basis {eo,e\,...,eN-\}. We suppose that N = 2n. We divide the space VN on M = 2n-1 2-D spaces

Further, we introduce the following basis (2x2)-transforms: acting as

A

family F

Jo >J\

So 'Si

ip ip 4) >ч

} of S = M - 1 = 2й 1 -1 an arbitrary basis transforms )p=a

generates in Vn some nonlinear sparse nonlinear transform (SNLT)

Each sparse transform contains S basis transforms. Let

вт2„ ?

Jo » J i

ВТ

2"

/¿\^]г=12/,©вт2@12Я

-2(p+l)

Jo >Jl ВТ I ¡0 ,i\

So 'Si

2(^+1)

Then sparse matrices can be presented as superposition o - product

s s

Y

,,=0

ei

SNLT2„ =ПоВТ' =m*2p@[jsj?\w>%hp~

р=0 р=0

2"-2(р+\)

Системный анализ System analysis

if; il, Z,1;...;^, if

Two sequences -j °0' '0' ' "s !> form two addressing schemes of basis transforms

s [7o ->J\ >-">7o iJ\ J

| HT. ■ for the so-called non in-place radix-2 fast NLT.

Definition 4. Let j|rBTf = [ 'jf rgf J | be a set of L families of an arbitrary

basis transforms. We define generalized tensor product of these families in the form of the superposition o - product of sparse nonlinear transforms:

(8)

It is called fast radix-2 nonlinear transforms of L stages (L = log2 N = n are typically used for linear unitary transformations). In details this fast transform has the following form:

(9)

The following address schemes are used in linear digital signal processing [4-8]

where pr = 2r [p/2r 1] + (p mod 2r 1). For these schemes we have the following classical archi-

tecture for fast nonlinear transforms

n S

FNLT2,=nn°M

r-I p=0

2"-2 (p+1)

(2)

_p„pr+T-i\'VY\pr,pr+r

1 FNLTr = П П° ОК © \Pr. Pr + 2' "11 " BT2

r= 1 p=О

where П=П or П=П ■ Figure 3 shows the data flow for the radix-2 (1)FNLTs

(3)1

п д<—1 n n

FNLT2^nno(i2pe

r=l p=0

n S

ei

2"-2(p+l)

r=l r-l I Г-I

Системный анализ System analysis

1-5-3 3

Fig. 3. Яас//х-2<"FNLTS =П°П0(1;,® [rffI?]® Wo)

A wide family of invertible basis transforms work [7]. Using these BTf we obtain invertible fast non

Л - h

li 1 /, can find in our previous inear transforms FNLT2W.

Some Concluding Remarks

A unified approach to nonlinear transformations using a fast algorithm has been developed. Using recursive rules to describe nonlinear transformations allows you to systematically view unknown fast nonlinear transformations [7]. These transformations are the product of a superposition of "sparse" nonlinear transformations and describe a fast algorithm for NLWT. We believe that the proposed nonlinear wavelet-like transformations can be useful in wired and wireless communication applications.

Источники

1. Rugh W. J. Nonlinear system theory: the Volterra/Wiener approach. Baltimore: The Johns Hopkins University Press, 1981. 325 p. ISBN 0-8018-2549-0.

2. Schetzen M. The Volterra und Winer theorems of nonlinear systems. New York: Wiley, 1980. 531 p. ISBN 0-471-04455-5.

3. Kim K. I., Powers E. J. A digital method of modeling quadratically nonlinear systems with a general random input // IEEE Transactions on Acoustics, Speech, and Signal Processing. 1988. Vol. 36, iss. 11. P. 1758-1769. DOI: https://doi.org/10.1109/29.9013

4. Schetzen M. Non-linear system modeling based on the Wiener theory // Proceedings of the IEEE. 1981. Vol. 69, iss. 12. P. 1557-1573. DOI: https://doi.org/10.1109/PR0C.1981.12201

5. Pages-Zamora A., Lagunas M.A., Najar M., Perez-Neira A. The K-filter: a new architecture to model and design nonlinear systems from Kolmogorov's theorem // Signal Processing. 1995. Vol. 44, iss. 3. P. 249-267. DOI: https://doi.org/10.1016/0165-1684(95)00028-C.

6. Labunets V., Ostheimer E. Systematic approach to nonlinear filtering associated with aggregation operators. Part 1. SISO-filters // Procedia Engineering. 2017. Vol. 201. P. 385-397. DOI: https://doi.org/10.1016/j.proeng.2017.09.652.

ЦИФРОВЫЕ МОДЕЛИ И РЕШЕНИЯ

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Системный анализ System analysis

7. Labunets V., Ostheimer E. Systematic approach to nonlinear filtering associated with aggregation operators. Part 2. Frechet MIMO-filters // Procedia Engineering. 2017. Vol. 201. P. 397-411. DOI: https://doi.org/10.1016/j.proeng.2017.09.655.

8. Ostheimer E., Labunets V., Komarov D., Fedorova T. Fréchet filters for color and hyper-spectral images filtering // Communications in Computer and Information Science. 2015. Vol. 542. P. 57-70. DOI: https://doi.org/10.1007/978-3-319-26123-2_6.

9. Ostheimer E., Labunets V., Myasnikov F. Families of Heron digital filters for image filtering // CEUR Workshop Proceedings. 2015. Vol. 1452. P. 56-63.

10. Ostheimer E., Labunets V., Kurganski A., Komarov D., Artemov I. New bi-, tri-, and four-lateral filters for color and hyperspectral images filtering // Communications in Computer and Information Science. 2015. Vol. 542. Р. 102-113. DOI: https://doi.org/10.1007/978-3-319-26123-2_10.

11. Колмогоров А. Н. О представлении непрерывных функций нескольких переменных в виде суперпозиций непрерывных функций одного переменного и сложения // Доклады Академии наук СССР. 1957. (114. С. 369-373.

12. Арнольд В. И. О представлении непрерывных функций трех переменных суперпозициями непрерывных функций двух переменных // Доклады Академии наук СССР. 1957. № 114. С. 679-681.

13. Гашков С. Б. Сложность реализации булевых функций схемами из функциональных элементов и формулами в базисах, элементы которых реализуют непрерывные функции // Проблемы кибернетики. 1980. № 37. С. 57-118.

14. Wyner A. D. The wire-tap channel // The Bell System Technical Journal. 1975. Vol. 54, iss. 8. P. 1355-1387. DOI: https://doi.org/10.1002/j.1538-7305.1975.tb02040.x

Information about the authors

Valery G. Labunets, Doctor of Technical Sciences, Professor of Chess Art and Computer Mathematics Dept. Ural State University of Economics, 620144, Russia, Ekaterinburg, 8 Marta/Narodnaya Volya St., 62/45. E-mail: vlabunets05@yahoo.com

Viktor P. Chasovskikh, Doctor of Technical Sciences, Professor of Chess Art and Computer Mathematics Dept. Ural State University of Economics, 620144, Russia, Ekaterinburg, 8 Marta/Narodnaya Volya St., 62/45. E-mail: u2007u@yandex.ru

Evgeny N. Starikov, Candidate of Economic Sciences, Associate Professor, Acting Head of Chess Art and Computer Mathematics Dept. Ural State University of Economics, 620144, Russia, Ekaterinburg, 8 Marta/Narodnaya Volya St., 62/45. E-mail: starikov_en@usue.ru.

Информация об авторах

Лабунец Валерий Григорьевич, доктор технических наук, профессор кафедры шахматного искусства и компьютерной математики. Уральский государственный эконо-

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мический университет, 620144, Россия, г. Екатеринбург, ул. 8 Марта/Народной Воли, 62/45. E-mail: vlabunets05@yahoo.com

Часовских Виктор Петрович, доктор технических наук, профессор кафедры шахматного искусства и компьютерной математики. Уральский государственный экономический университет, 620144, Россия, г. Екатеринбург, ул. 8 Марта/Народной Воли, 62/45. E-mail: u2007u@yandex.ru

Стариков Евгений Николаевич, кандидат экономических наук, доцент, и. о. заведующего кафедрой шахматного искусства и компьютерной математики. Уральский государственный экономический университет, 620144, Россия, г. Екатеринбург, ул. 8 Марта/ Народной Воли, 62/45. E-mail: starikov_en@usue.ru