Elliptic discrete fourier transforms of type II

Abstract

This paper presents a novel concept of the iV-point elliptic DFT of type II (EDFT-II), by considering and generalizing the iV-point DFT in the real space R^2N. In the definition of such Fourier transformation, the block-wise representation of the matrix of the DFT is reserved and the Givens transformations for multiplication by the twiddle coefficients are substituted by other basic transformations. The elliptic transformations are defined by different iVth roots of the identity matrix 2 × 2, whose groups of motion move the point (1, 0) around ellipses. The elliptic DFTs of type II are parameterized by two vector-parameters, exist for any order N, and differ from the class of elliptic DFT of type I whose basic transformations are defined by the elliptic matrix cos(¿)I + sin(¿)R, where R is such a matrix that R² = -I and I is the identity matrix 2 × 2. Examples of application of the proposed iV-block EDFT-II in signal and image processing are given.

Full text

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Elliptic Discrete Fourier
Transforms of Type II
Artyom M. Grigoryan
Department of Electrical and Computer Engineering
The University of Texas at San Antonio
One UTSA Circle, San Antonio, TX 78249-0669, USA
Tel: (210) 458-7518
Fax: (210) 458-5589
amgrigoryan@utsa.edu
1Art Grigoryan, UTSA 2009

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Outline
 Introduction
 Discrete Fourier transform (DFT) in the real space
 N-block T-transform generated discrete transform
 N-block elliptic DFT of type I
 N-block elliptic DFT of type II
 Properties of elliptic DFT of type II
 Experimental results
 Examples
 Summary
2Art Grigoryan, UTSA 2009

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Introduction
 Fourier analysis is one of the most frequently used tools
in signal/image processing and communication systems.
 The N-point discrete Fourier transformation can be
defined in the real space R2Nby the block-wise matrix.
Each 2x2 block of this matrix represents the Givens
rotation, or multiplication by the twiddle factors.
These coefficients are roots of the unit and represent the
Givens rotations by angles φk=(2π/N)k, k=0:(N-1).
 We introduce a concept of the elliptic DFTs in the real
space, which are defined by 2-point transformations
different from the Givens rotations.
Matrices of these transformations describes the
movement of points around ellipses.
3Art Grigoryan, UTSA 2009

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Transformation: CNto R2N
Consider the transformation of the signal
),...,,,(
1210
N
fffff
and components of the vector as
,(
22
kkk
fff
The N-point DFT as 2N-point in real space R2N
has the following
matrix:
4Art Grigoryan, UTSA 2009
), ,...,,,,,,(
11221100
NN
iriririrf
. )
k
Im ,(Re ),()
1
kkk
ffir
.
...
...
T
.........
...
...
...
][
121
242
121
TTI
I
TTTI
TTTI
IIII
F
NN
N
N
bN
).(
,, ,
0
21
2121
ITT
kkTTT
N
kkkk
f
),1 ( :0 ,
cossin
sin cos
1
0
1
0
NpfT
i
r
I
R
F
N
k
k
kp
N
k
k
k
p
kk
kk
p
p
p

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Elliptic DFT of type I
T-generated N-block discrete transformation, or the N-block
T-GFT is defined by the block-wise DFT type matrix
(
mn
THH
T is a matrix 2x2, det T=1, and it defines a one-parametric
group with period N.
Case T=W: The N-block W-GFT (N-block DFT)
Example: Given the angle consider the matrix
coscos
)(
5Art Grigoryan, UTSA 2009
) 1( : 0,),(
)
Nmn
nm
RITT
sincos
cos1cos
1
,/2
N
). , 1det( ,
0) 2/cot(
) 2/tan(0
)(
2
-IRRRR
.)(
ITN
. 1det ,
6235 . 06235. 1
3765 . 0 6235 . 0
7
2
TTT