Introduction

The traditional way to transform a signal U i.e., to change amplitude-frequency characteristic(AFC) of U in some desired manner, is:

use Fourier transform to obtain the AFC of the signal;

apply some mechanism to recalculate the row coefficients; and

use inverse Fourier transform to obtain the transformed signal.

Introduction

Also, the traditional way to calculate values of a function U + i*V, analytical inside a unit circle, from given values on the unit circle, so that new values resemble values of that function on some other concentric circle, is:

use Fourier transform to obtain row coefficients;

recalculate the coefficients to reflect that the new values are on the circle with radius ‘r’ and starting argument ‘’; and

use reverse Fourier transform to obtain the desired values.

The Traditional Way to Transform a Signal Also Has a Well-known Obstacle.

The obstacle is the increasing relative cost of calculations that occurs with an increasing number of sampling points in cases where the growing numbers are not powers of 2, so that dividers of those numbers have to be found. A worst-case scenario is the absence of small prime dividers, which results in costs proportional to n^2.

I intend to prove here that one circulant matrix operator can do both transforms: 1)transform the AFC, and2)transform from analytical function values on the unit circle to the values on the concentric circle with a different radius and starting point argument.In both transforms, only U is used as an input vector, while the result of the transform is a complex vector.

1. Transform AFC of Signals by Circulant Matrix

AFC of a digital signal U with N number of points

Row coefficients for a function, analytical inside a unit circle

1

0

))2cos((*2 N

ttk N

kt

N Ua

1

0

))2sin((*2 N

ttk N

kt

N Ub

)2exp())2(exp(*1

0

1Njt

N

tNt

NjiiWc

Corollary I

The matrix

))2(exp((*),,(2/)1(

1

2,

kir Njlk

N

kkNtjl rM

The transformTrigonometric form

Complex form

K

kNkj

kNkj

kj newbnewaUM1

))2sin(_)2cos(_()*(

K

kNkj

kNkj

k newanewbi1

))2sin(_)2cos(_(*

))sin()cos((_ kk barnewa kk

k

kk

))sin()cos((_ kk abrnewb kk

k

kk

)2exp(_)*(1

1N

jtN

ttj inewcUM

)exp(_ jicrnewc jj

jj

Proof: Two base parts of operator:

M1 is similar to the way in which a Szego kernel is constructed, second part is harmonically conjugated.

TNj

Nj

kNl

Nl kkkkM ))2(sin()),2(cos(*))2(sin()),2(cos(1

TNj

Nj

kNl

Nl kkkkM ))2(sin()),2(cos(*))2(cos()),2(sin(2

2/)1(

1jl, )))2(sin())2(sin())2(cos())2((cos(M1

N

kNj

Nl

Nj

Nl

kkkkk

2/)1(

1jl, )))2(sin())2(cos())2(cos())2((sin(M2

N

kNj

Nl

Nj

Nl

kkkkk

2/)1(

1jl, ))]2()2(sin())2()2([cos(M2)*i(M1

N

kNj

Nl

Nj

Nl

kkkikk

Demo of the transformed AFC on Maplesoft platformM(, 1, 0)K = (Dimension(signal)-1)/2k = 1..K

k = exp(-(0.3*K-k)^2/K^2*4)

Evaluating Analytical Functions. set k1 and use the formula for a sum of geometrical progression, then the operator will become usable to evaluate analytical functions:

Further set values r=1 and =0 and extract the imaginary part, then the operator produces values of the harmonically conjectured function:

1)2exp(

1)2/)1)(2exp((2,

2/)1(

*)2exp(),(

ii

Nii

Njl

NjlN

jlN

jlN

riirrMA

))(sin(

)1(*))(cos())(cos(

,

)2,mod()2,mod(1

jlN

jljl

jl N

jlNNNMH

Connection between harmonic equations in rectangular and polar coordinate systems

. It is also well known that arg(z) is harmonically conjugated to ln(|z|). Corollary II in the paper states the more generalized fact:

])))0()(([(~)( WWnormLn

]))(([(~)()( '2 WnormLn

Proof:Row representation of the analytical, one-to-one function W(z) is Function W1

has the following row representation:

W1(0) = W’(0) 0If W1() = 0, then *W1()

= 0, but *W1() is one listed & 0*W1(0) = 0.

Ln(W1) is analytical and its real and imaginary parts are harmonically conjugated.

0

)(n

nn zCzW

zWzWW )0()(1

01)(1

n

nn zCzW

The logarithm of a derivative expressed in polar coordinates is an analytical function, so its real and imaginary parts are harmonically conjugated.

Conformal Mapping Connections between harmonic equations in rectangular and polar coordinate systems are powerful tools for solving Riemann’s task of finding conformal mapping from unit disk to simple-connected area surrounded by Jordan’s curve

For j [0; N-1] ns(j) = exp [~((a_(j)-(π/2+j*2π/N)) \ -(a(j)-j*2π/N))]* \ sqrt[norm(w’(j))/norm(w (j)-w0)];ns *= (N/sqrt(norm (ns)),where a(j) =arg(w(j)-w0), a_(j)-j*2π/N = arg(w’(j)).

))))()(((exp(~))(( 2' Wnorm

Conformal mapping Demos Ellips

Cat face

Symmetrical

Big and ugly

why it works:

Let -*Sin(kt+) be a deviation component of frequency k ideal solution is:

for the deviated distribution the ratio is

))(exp(~' f

'/))(exp(~ 11 f

Reinstating Wave Function

From the norm known on the circle, flat wave function is:

where Ψ(ζ) is the wave function, and denotes its norm.

])))(([~exp(*)()(22 Lni

( ) ( ) 2

Harmonic Covariation

For two oscillative function U(t) and U(t), having mean(U) = 0 and mean(V)=0, integrated on interval [0; 2], the Harmonic Covariation is:

))](~*[]*[(1

~2

0

2

0

dtVUidtVUVU

The properties of the tilde operator are: 1) (α + i*β) ~ u = α*u + β*(~u),2) λ ~ u = u ~ conj(λ),3) (λ + μ)~u = λ ~ u + μ ~ u,4) u ~ v = conj(v ~ u), and5) u~(λ~x + μ~y) = conj(λ)*(u~x) + conj(μ)*(u~y),

Property number 2 is not obvious; an illustrative example is as follows:[ sin(a) ~ cos(a) ] ~ sin(a) = i ~ sin(a) & sin(a) ~ [cos(a) ~ sin(a)] = sin(a) ~ (-i) i~sin(a) = sin(a)~(-i)

Harmonic Correlation

))()*((~),( VnormUnormsqrt

VUVUHC

1))0exp(Real(W()HC(Real(W( ii ))exp(Real(W()HC(Real(W( 2

1))exp(Real(W()HC(Real(W( i

ii ))exp(Real(W()HC(Real(W( 23

Application to financial data:

method of harmonic correlation was used to sort market data to find companies with share price behavior that is most similar to some particular company’s share price behavior.market data was sorted by using the standard correlation coefficient and harmonic correlation coefficient to find the stocks whose price history most closely tracked that of International Business Machines Corporation (IBM).

Top 3 by correlation coefficient

Top 3 by Harmonic Correlation coefficient