Proceedings of the National Conference On Undergraduate Research (NCUR) 2012 Weber State University, Ogden Utah March 29-31, 2012

Frequency Analysis on Ellipses: Elliptic Fourier Transform Joao Donacien N. Nsingui Department of Electrical and Computer Engineering The University of Texas at San Antonio One UTSA Circle, San Antonio, TX 78249-0669 Faculty Advisor: Dr. Artyom Grigoryan Abstract The discrete Fourier transform (DFT) is widely used in data compression, pattern recognition, and image reconstruction. The study of properties of the DFT shows that this transform can be generalized and used mere effectively for solving many problems in such areas as signal and image processing. The traditional N-point DFT is defined as the decomposition of the signal by N roots of the unit, which are on the unit circle. On different stages of the DFT, the data of the signal are multiplied by the exponential factors, or rotated around the circles. This paper presents a novel concept of the DFT by considering the block-wise representation of the transform in the real space, which can be generalized to obtain new methods in spectral analysis. For that, the new concept of the N-point elliptic discrete Fourier transform (EDFT) is introduced, whose basic 2×2 transformations that are not Given transformations, but rather rotations around ellipses. The elliptic transformation is therefore defined by different Nth roots of the identity matrix 2×2, matrix whose groups of motion move a point around the ellipses. The main properties of the EDFT are described and examples are provided. The EDFT preserves main properties of the DFT, such as shifting and energy preservation among others. The EDFT distinguishes the carrying frequencies of the signal in both real and imaginary parts better than the DFT. It also has a simple inverse matrix. It is parameterized and includes the DFT, as a partial case. The preliminary results show that by using different parameters, the EDFT can be used effectively for solving many problems in signal and image processing field, which includes image enhancement, filtration, encryption and others. Keywords: 1-D Fourier transforms, elementary rotations, image enhancement

1. Introduction Many fast transforms such as the Fourier, Hadamard, Haar, and Cosine transforms, are widely used in different areas of engineering and science. The theory of the Fourier transform is well developed and its basic methods are effective for solving several problems in fields such as signal and image processing, communication, and biomedical imaging. The most important property of this transform is the representation of the linear convolution as the simple operation of multiplication in the frequency domain. It is also the main operation for the linear-time invariant system in the 1D case, and linear-spatial invariant systems in the 2-D case, particularly in image processing. Since the introduction of the fast algorithm of the discrete Fourier transform (DFT), the Fourier analysis has become one of the most frequently used tools in signal and image processing and communication systems [1]. In the discrete case of signals and systems, the transformation of the data from the time domain, or image plane to the frequency domain, can be extended by introducing a more general concept than the Fourier transform. This paper describes the one dimensional N-point elliptic discrete Fourier transforms (EDFT) by introducing the concept of the T-generated block discrete transformation (T-GDT) in the real space. For that purpose, the block-wise representation of the matrix of the discrete Fourier transform (DFT) is analyzed in the real space R2N ,and the Givens transformation for multiplication by twiddle coefficients are substituted by other basic two-point transformations.

These transformations are defined by the different Nth roots of the unit matrix 2×2 whose powers or one-parametric groups move points around the ellipses. However, the case when all the ellipses are circles is referred to as the Npoint DFT [5]. When representing the N-point DFT in the real space, that is the space R2N, all components of the signal as points on the real plane are rotated around their circles with different speeds on each stage of calculation in the DFT. Thus, the DFT represents a stunning system rotating the data on the plane around circles. The data are referred to as the 1D signals or images in the 2-D case. The elliptic Discrete Fourier Transform describes the periodic rotation of these data around the ellipses. The matrix on the EDFT, determines the basic transformation that is called elliptic. It is the matrix cos ! ! + sin!(!)!!, where R is the matrix such that R2 = -I and I is a 2×2 identity matrix.

2. DFT In The Real Space Consider the N-point discrete Fourier transforms of the signal !! , !!!

! !" !! !!!!!!!!!!!!!!!!!!!! = 0: ! − 1 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!![1]

!! = !! + !"! = !!!

that determines the decomposition of the signal by N roots of the unit !!

! ! = !!! = ! ! ! = !! − !"! = cos

2! 2! ! − ! sin ! , !!!!! = 0: ! − 1 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!![2] ! !

Here !! and !! denote respectively the real and the imaginary parts of the transform. The roots are located on the !

unit circle, (! ! )! = 1. Now consider the signal as the vector ! = !! , !!, !! , … !!!! . The DFT has the following matrix in the complex ! !

!!

1 1 1 = 1 1 1

1 !! !! ⋮ ! !!! ! !!!

1 !! !! ⋮ ! !!! ! !!!

1 !! !! ⋮ ! !!! ! !!!

1 ⋮ ⋮ ⋮ ⋮ ⋮

1 ! !!! ! !!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!![3] ⋮ ! ! !!

The matrix form is described as the multiplication of the complex number ! = !! − !"! , which is considered as the column vector (!! − !"! )! , or simply row vector (!! , !! ), by the twiddle coefficients ! ! , !! !! + !! !! !! ! = ! → ! ! ! = (!! − !"! ) (!! + !"! = ,!!!!! = 0: ! − 1 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!![4] ! !"! − !! !! Here ! ! = (!! , −!! ) = ! cos !! + sin!(!! ), and the angle !! = 2!" !. In matrix form, this multiplication can be written as

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!!! =

cos !! !!!sin !! −sin !! !!!cos !!

!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!![5] !!

The matrix of the givens rotation by the angle !! = !!! !is denoted by ! ! . In the k=0 case, !! = 0, ! = ! and there is no rotation of the signal-data. Thus the complex plan is transferred into the 2-D real space, ! → ! ! , and considering each operation of multiplication by the twiddle coefficient as the elementary rotation, or the givens transformation, ! ! → ! ! , ! = 0: ! − 1 .! In general, the signal !! may be complex. Therefore, the real and imaginary parts of the signal are denoted by components !! = !"!! and !! = !"!! k = 0: (N-1). Considering the inclusion of the complex space ! ! into the real space ! !! by

! = !! , !!, !! , … , !!!!

!

→ ! = !! , !! , !! , !! , … , !!!! , !!!! ! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!![!]

The vector ! is composed from the original vector, or signal !, and its vector component is denoted by !! = !!! , !!!!! ′ = (!! , !! )′. The N-point DFT of ! is represented in the real space ! !! as the 2N-point transform

!! !!! = = !!

!!!

!!! !"

! !! =

! !!!

!!!

!"

!! !! =

!!!

!!!

cos !!! !!!!!!!sin !!! −sin !!! ! !!!cos !!!

!! !! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!![7]

where p = 0 : (N-1). In matrix form, the DFT in the real space ! !! !is described by the following matrix 2N×2N: 1 1 1 !!!!!!!!!!!!!!!!!!!!!!!!!!! = 1 1 1

1 !! !! ⋮ ! !!! ! !!!

1 !! !! ⋮ ! !!! ! !!!

1 !! !! ⋮ ! !!! ! !!!

1 ⋮ ⋮ ⋮ ⋮ ⋮

1 ! !!! ! !!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!![8] ⋮ !! !!

The (n, p)-th block 2×2 of this matrix is ! !" , where n, p = 0 : (N-1). The matrix ! = !! is the identity matrix 2×2. Note that the rotational matrices ! ! compose the one-parametric group with period N. In other words ! !!!!! = !!! !!! , ! ! = ! ! = 1 ,!!!for!any!k = 0: N − 1 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!![9] According to the equation 1, when calculating the components of the DFT at the frequency-point p, the vector components or points !! , !! are rotated by the corresponding angle !!! . Thus the first point !! , !! stays on its place, the second point !! , !! is rotated around the circle of radius !!!

!!!

!!! + !!! by angle !! . The next point !! , !!

is

+ by twice larger angle 2!! , or twice faster than the point !! , !! , and so rotated around the circle of radius on. Then the coordinates of all rotated points are added and one is defined, which represents the DFT at frequencypoint p. Note that for point p = 0, there is no rotation. The sums of coordinates of the original points define the first component !! .

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Example 1: Consider the vector-signal of length 10 with the following real and imaginary parts: !! ; ! = 0: 9 ! = ! !!!!5 −3!! 2! −5 −1 3 −4 −2 !! ; ! = 0: 9 = −2!!! !!1! 4 !−3 !!!7 !2 −5!!!! 5!

!!7 −6 −2 !3 !

This signal is plotted in form of ten points on the plan in Figure 1. These points are marked by bullets in black and labeled as !! , ! = 0: 9. The figure shows also the result of N rotations of these points. These rotations are labeled as !! .

Figure$1.$10/point$signal$and$ten$rotations$of$the$signal$by$the$DFT. These ten points represent the 10-point DFT with the following coordinates:

1.0

−0.4 −0.0268 0.7142 −2.0394 4.1488 2.2 −0.160 0.0978 −0.3030 0.7684 −0.6464! −0.2558 !0.5791 −0.1716 −0.6 −0.552 −1.5027! −0.0206 !0.1700

where ! = 0: 9, and the first row is the real part !! and the second is the imaginary part !! respectively.

3. Elliptic Discrete Fourier Transform (EDFT) Lets consider the concept of the Nth roots of the identity matrix 2×2 that was introduced by Dr. Grigoryan [6]. These basic transformations are defined by different transformations from the Givens rotations. A new class of parameterized transformations, which are called, the elliptic-type I discrete Fourier transformations, or simply EDFT-I are defined in the following way [6,7] Given an integer ! > 1!and!angle!! = !! = 2! ! , the following matrix is considered:

!=! ! =

$

cos ! cos ! + 1

cos ! − 1 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!![10] cos !

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The matrix above can also be written as 1 1

! = ! ! = cos ! ! + ! = cos !

!=! ! =

0 cot!(! 2)

1 0 + 1 1

−1 !!!!!!!!!"!!!!!!!!! = cos ! ! + sin ! !! 0

−tan!(! 2) !,!!!!!!!!!!(!"#$ = 1) 0

[11]

This definition leads to the equality ! ! ! = ! for any integer N, even or odd. The matrix R satisfies the condition ! ! ! = −!. Example 2: Consider the case when N = 8. The angle ! = 2! 8 = ! 4. The matrix C will be computed as

!=! ! 4 =

cos ! 4 cos ! 4 + 1

cos ! 4 − 1 −0.7071 = cos ! 4 1.7071

−0.2929 !, 0.7071

(det!(!) = 1)

The multiplication of a vector x by the matrix C requires two multiplications by cos!(!). The matrix C defines the one-parametric group with period N. This group is defined by matrix I, C, and other three matrices as show below

!! =

!! =

0 2.4142

0.7071 −1.7071

−0.4142 −0.7071 , !! = 0 1.7071

0.2929 0 , !! ! = −0.7071 −2.4142

−0.2929 −1 , !! ! = −0.7071 0

0.4142 0.7071 , !! ! = 0 −1.7071

0 −1 0.2929 0.7071

where ! ! = !. The following matrix defines the 8-block C-Generalized discrete transform (C-GDT) ! ! ! ! ! = ! ! ! ! !

! !! !! !! −! !! !! !!

! !! −! !! ! !! −! !!

! !! !! !! −! !! !! !!

! −! ! −! ! −! ! −!

! !! !! !! −! !! !! !!

! !! −! !! ! !! −! !!

! !! !! ! ! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!![12] −! !! !! !! !

This matrix is orthogonal, ! ! ! = 4096! = 8! , and the inverse matrix equals ! !! ! = !(! !! ). !

We now

!

consider the orbit of the point ! = (!! , !! )!with respect to the group of motion ! = {! ; ! = 0: 7}. In other word, consider the movement of the point ! = (!! , !! )!in the plan, which passes the following points: ! → !! → ! !! → ! ! ! !

→ ! !! !

→ ! !! !

→ ! !! !

→ ! !! !

→ ! !! !

=!

If instead of C, we use the matrix rotation W by the angle ! = 2! 8, the point x will move around a circle.

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Example 3: Lets consider the matrix C and the point ! = (1,1)′. Figure 2 shows the locations of eight points !! = ! ! ! , ! = 0: 7, that are numbered as k, and the drawing of the full path of the point x during its movement. 3

3

2

2 1

1

2

0

1

0

0

−1

7

−1

3

4

−2

1

0

3

7 4

5

6

−2

5 −3 −3

2

−2

−1

6 0

1

2

3

−3

−2

(a)

0

2

(b)

Figure$2.$The$movement$of$point$(1,1)$around$the$(a)$ellipse$(by$the$EDFT)$and$(b)$circle$(by$the$DFT)$for$the$N&=&8.

As it can be seen in part a, this drawing is not a circle. The points !! move along the perimeter of the ellipse, !! !! + !! = !1!!!!!!!!!where!!!!!!! = 1!!!!!!and!!!!!!! = !! !!

1 + cos!(!) !!! = !!2.4142!!!!!!!!!!!!!!!!!!!!![13] 1 − cos!(!)

For comparison, the movement of the same point by the rotational group {!!! !; !! = 0: 7} around the unit circle is shown in part b. The following points describe the movement of the point x during the EDFT 1 0.4142 −0.4142 −1 −1 −0.4142 0.4142 1 1 → → → → → → → → 1 2.4142 2.4142 1 −1 −2.4142 −2.4142 −1 1 The movement of these points during the DFT is described by 1 0 1 0 −1 −1.4142 −1 0 1 → → → → → → → → 0 1 1.4142 −1 −1.4142 −1 1 1.4142 1 The value of ! ! = ! − 1 can be considered as a measure of difference of the elliptic and the traditional discrete Fourier transformations. In the N = 8 case, we have Δ ! = ! − 1 = Δ 2! 8 = 2.4142 − 1 = 1.4142. For comparison with the DFT, consider the data shown in Figure 1. Figure 3 shows the result of N rotations of these points. These points are labeled as !! , when using the EDFT over the same signal. The 10 points represent the 10block EDFT with the following coordinates: !!!!!!! −0.4 −0.2416 0.3708 −1.318 2.6944 2.2 1.2944 −0.6236 0.0403 0.500 1.0 −1.4944 −0.5000 2.7416 0.2236 −0.6 −0.9472 −3.6652 0.2236 1.0180 Here ! = 0: 7, and the first row represent the real !! part and the second the imaginary part !! of the signal respectively.

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$ Figure$3.$10/point$signal$and$ten$rotations$of$the$signal$by$the$EDFT. To illustrate better the difference between the EDFT and DFT in this example, Figure 4 shows the stars of these two transforms. Note that each star on this figure is drawn by connecting each point with the point !! that equals !! .

Figure 4. (a) Star of the 10 point DFT and (b) star of the 10 block EDFT. The EDFT can distinguish the carrying frequencies of the cosine or sine waves, as the DFT does. Example 3: For the case N = 32, consider the discrete-time signal f sampled from the cosine wave ! = cos 2!! ! ,!!!!! = 2! ! , in the time interval 0,2! . The time-points ! = !! !, ! = 0: (! − 1) are uniformly distributed in the interval 0,2! . The 2N-dimensional vector signal is composed as !! = ! 0 , 0, ! !! , ! !! , 0, … . ,0, ! !!!! , 0 ! . The real and imaginary part of the N-block EDFT of the signal f is denoted as !! , !"#!!! i.e. !! = (! 0 , ! 2 , ! 4 , ! 6 , . . , ! 2! − 4 , !(2! − 2))′ !! = (! 1 , ! 1 , ! 5 , ! 7 , . . , ! 2! − 3 , !(2! − 1))′

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where g = X(C)f . Figure 5 shows the discrete-time signal f in part a, along with the real part !! of the N-block EDFT in b. Cosine signal with frequency 2wt 2

(a)

0

−2

0

5

10

15

20

25

30

32−block EDFT of the signal with frequency 2wt 20 (b)

10

0 0

5

10

15

20

25

30

Figure$4.$$(a)$The$cosine$signal$with$frequency$2!! $$and$(b)$the$$real$32/block$EDFT$of$the$signal.

The basic transform equals !! = ! 2! 32 =

0.9808 1.9808

−0.0192 . !!0.9808

The EDFT recognizes the frequency 2!! in the frequency-points p = 2 and 30 = 32-p, as in the DFT case. Another signal sampled from the wave cos!(6!! ) is shown in Figure 6 part c, and the real part !! of the N-block EDFT of the signal is given in d. This transform also recognizes the carrying frequency 6!! at the frequency-points p = 6 and 28 = 32 – p.

Signal f= cos(6wt) 2

(c)

0

−2

0

5

10

15

20

25

30

32−block EDFT of the signal with frequency 6wt 20 (d)

10

0 0

5

10

15

20

25

30

Figure$5.$$(a)$The$cosine$signal$with$the$frequency$6!! $and$(b)$the$$real$part$of$32/block$EDFT.

Figure$6.$(a)$The$imaginary$part$of$32/block$EDFT$of$cosine$with$frequency$!!!! $and$(b)$imaginary$part$of$DFT.

These frequencies can also be seen in the imaginary part of the DFT of the cosine signal cos(6!! ), as shown in Figure 7 in part a. For comparison, the imaginary part of the DFT is also given in b.

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5. Conclusion The block wise representation of the N-point EDFT allows for constructing new parameterized integer transformations with structures that are similar to the N-point DFT. The EDFT also conserves the same proprieties as the DFT does. Even though the EDFT possesses the same basic properties of the traditional DFT, parts of these proprieties are more sensitive to frequencies on EDFT than on DFT. The EDFT distinguishes better frequencies than the traditional transforms do. On Figure 7, the DFT does not have maximums on the carrying frequency-points p=6 and 26. In addition to that, the imaginary part of the DFT by amplitude is smaller than on the EDFT. The preliminary results show that by using different parameters, the EDFT together with DFT can be used effectively for solving many problems related to signal and image enhancement, filtration, encryption and many others.

6. References 1. Ahmed and R. Rao, Orthogonal transforms for digital signal processing, Springer-Verlag, 1975. 2. D.F. Elliot and K.R. Rao, Fast transforms-Algorithms, Analysis, and Applications, New York, Academic, 1982. 3. I.C. Chan and K.L. Ho, “Split vector-radix fast Fourier transform,” IEEE Trans. on Signal Processing, vol. 40, no. 8, pp. 2029-2040, Aug. 1992. 4. A.M. Grigoryan and S.S Again, Multidimensional Discrete Unitary Transforms: Representation, Partitioning and Algorithms, Marcel Dekker Inc., New York, 2003. 5. A.M. Grigoryan and V.S. Bhamidipati, “Method of flow graph simplification for the 16-point discrete Fourier transform,” IEEE Trans. on Signal Processing, vol. 53, no. 1, pp. 384-389, Jan. 2005. 6. A.M. Grigoryan and M.M. Grigoryan, “Discrete integer Fourier transform in real space: Elliptic Fourier transform,” SPIE Conf.: Electronic Imaging, San Diego, CA, Jan.2009. 7. A.M. Grigoryan and M.M. Grigoryan, Brief Notes in Advanced DSP: Fourier Analysis with MATLAB, Taylor & Francis/CRC Press, New York 2009.

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