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