Elliptic Fourier Transform

Transcription

1 NCUR, ART GRIGORYAN Frequency Analysis of the signals 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 Tel: () , Fax: () Faculty Advisor: Dr. Artyom M. Grigoryan Abstract Discrete Fourier transform (DFT) is widely used in data compression, pattern recognition, and image reconstruction. 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 Discrete Fourier Transform, by considering the block-wise representation of the transform in the real space, which is effective and that can be generalized to obtain new methods in spectral analysis. For that, we introduce the N-point Elliptic Discrete Fourier Transform (EDFT), with basic x 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 x, matrix whose groups of motion move a point around the ellipses. The properties of the EDFT are described and examples are provided. The EDFT preserves main properties of the DFT. Properties such as shifting and energy preservation among others. The EDFT distinguishes well from the carrying frequencies of the signal in both real and imaginary parts. It also has a simple inverse matrix. It is parameterized and includes also the DFT. Our preliminary results show that by using different parameters, the EDFT can be used effectively for solving many problems in signal and image processing field, in which includes problems such as image enhancement, filtration, encryption and many others. Keywords: -D Fourier transform, Givens transforms, elementary rotations, signal filtration, image enhancement

2 NCUR, ART GRIGORYAN I. Introduction Many fast unitary 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 the Fourier transform-based methods are effective for solving many problems in 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. The linear convolution is the main operation for linear time invariant system in the -D case, and linear spatial invarinat systems in the -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, and different fast algorithms have been introduced []-[5]. 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 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, the block-wise representation of the matrix of the discrete Fourier transform (DFT) is analyzed in the real space R N, and the Givens transformation for multiplication by twiddle coefficients are substituted by other basic two-point transformations. These transformations are defined by different N th roots of the unit matrix, whose powers, or one-parametric groups move points around ellipses. The case, when all ellipses are circles, is referred to as the known N-point DFT. When representing the N-point DFT in the real space, that is the space R N, all components of the signal, as points on the real plane, are rotated around their circles with different speed, on each stage of calculation in the DFT. Thus, the DFT represents a beautiful system rotating the data on the plane around circles. The data are referred to as the -D signals, or images in the -D case. The periodic rotation of the data around ellipses is described by the presented elliptic DFT. For the EDFT, the basic transformation is determined by the matrix that we call elliptic. This is the matrix cos(ϕ)i + sin(ϕ)r, where the matrix R such that R = I and I is identity matrix. Examples of application of the proposed N-block EDFT-II in signal and image processing are given.

3 NCUR, ART GRIGORYAN II. DFT in the real space We consider the N-point discrete Fourier transform of the signal f n, F p = R p + ii p = N n= W np f n, p = : (N ), which determines the decomposition of the signal by N roots of the unit W k = W k N = e i π N k = c k is k = cos( π N k) isin(πk), k = : (N ). N Here R p and I p denote respectively the real and imaginary parts of the transform. The roots are located on the unit circle, (W k ) N =. We consider the signal as the vector f = (f, f, f,..., f N ). The DFT has the following matrix in the complex space C N : W W W W N W [F N ] = W 4 W 6 W N W N W N 4 W N 6 W W N W N W N W. We now describe in matrix form the multiplication of the complex number x = x +ix, which is considered as the column-vector (x, x ) or row-vector (x, x ), by the twiddle coefficients W k, x = x x W k x = (c k js k )(x + jx ) = c kx + s k x c k x s k x, k = : (N ), where W k = (c k, s k ) = cos ϕ k isin ϕ k, and the angles ϕ k = πk/n. In matrix form, this multiplication can be written as T k x = cos ϕ k sin ϕ k sin ϕ k cos ϕ k x x. The matrix of the Givens rotation by the angle ϕ k = kϕ is denoted by T k. In the k = case, ϕ =, T = I, and there is no rotation of the signal-data. Thus we transfer the complex plane into the -D real space, C R, and consider each operation of multiplication by the twiddle coefficient as the elementary rotation, or the Givens transformation, W k T k,

4 NCUR, ART GRIGORYAN 4 k = : (N ). In general, the signal f k may be complex. Therefore we denote the real and imaginary parts of the signal components by r k = Ref k and i k = Imf k for k = : (N ). We next consider the inclusion of the complex space C N into the real space R N by f = (f, f,..., f N ) f = (r, i, r, i,..., r N, i N ), The vector f is composed from the original vector, or signal f, and its vector-component is denoted by f k = ( f k, f k+ ) = (r k, i k ). The N-point DFT of f is represented in the real space R N as the N-point transform F p = R N p = T kp f N k = T kp r N k = cos(kϕ p) sin(kϕ p ) r k, () I p k= k= k= sin(kϕ p ) cos(kϕ p ) where p = : (N ). i k In matrix form, the DFT in the space R N is described by the following matrix N N : I I I I I I I T T T T N X = I T T 4 T 6 T N. () I I T N T N T N T The (n, p)-th blocks of this matrix is T np, where n, p = : (N ). The matrix I = I is the identity matrix. Note the rotation matrices T k compose the one-parametric group with period N. In other words i k T k+k = T k T k, (T = T N = I), for any k, k = : (N ). According to (), when calculating the component of the DFT at the frequency-point p, the vectorcomponents, or points (r k, i k ) are rotated by the corresponding angles kϕ p. Thus the first point (r, i ) stays on its place, the second point (r, i ) is rotated around the circle of radius r + i by angle ϕ p. The next point (r, i ) is rotated around the circle of radius r + i by twice larger angle ϕ p, or twice faster, than the point (r, i ), and so on. Then the coordinates of all rotated points are added and one point is defined,

5 NCUR, ART GRIGORYAN 5 which represents the DFT at frequency-point p. For point p =, there is no rotation; the sums of coordinates of the original points define the component F. As example, consider the vector-signal of length with the following real and imaginary parts: {r k ; k = : 9} = { 5,,, 5,,, 4,, 7, 6} {i k ; k = : 9} = {,, 4,, 7,, 5, 5,, } This signals is plotted in form of ten points on the plane in Figure. These points are marked by the bullets (in black) and labeled as f p, p = : 9. Figure shows also the result of N rotations of these points, that are labeled as F p. These ten points represents the -point DFT with the following coordinates: 8 point DFT composition: F(p), p=:9. f 4 6 f 7 imaginary part y(n) of the signal z(n)=x(n)+jy(n) 4 4 f 9 f f 6 f F() F() F(8) F(9) F(6) F() F() F(7) f f 5 F(5) F(4) f f Fig.. -point signal and ten rotations of the signal in the DFT. {R p } = { } {I p } = { } where p = : 9.

6 NCUR, ART GRIGORYAN 6 III. Elliptic DFT We consider the concept of the Nth roots of the identity matrix, that was introduced by Grigoryan [7], [8]. These basic transformations are defined by transformations different from the Givens rotations. A new class of parameterized transformations which are called elliptic-type I discrete Fourier transformations are defined in the following way. Given an integer N > and angle ϕ = ϕ N = π/n, the following matrix is considered: C = C(ϕ) = cosϕ cosϕ. cos ϕ + cos ϕ This matrix can be written as C = C(ϕ) = cos ϕu + V = cos ϕ +, or C = cos ϕ I + sin ϕ R () where R = R(ϕ) = tan(ϕ/), (det R = ). cot(ϕ/) This definition leads to the equality C N (ϕ) = I for any integer N, even or odd. The matrix R satisfies the condition R (ϕ) = I. Example : For the N = 8 case we have the following: C = C π =.77.99, ϕ = π, (detc = ) The multiplication of a vector x by matrix C requires two multiplications by cos(ϕ). The matrix C defines the one-parametric group with period N. This group is defined by I, C, and other three matrices C 5 = C = , C =, C 6 = , C 4 =, C 7 =,

7 NCUR, ART GRIGORYAN 7 and C 8 = I. The 8-block C-GDT is defined by the following matrix: I I I I I I I I I C C C I C 5 C 6 C 7 I C I C 6 I C I C 6 I C C 6 C I C 7 C C 5 X(C) = I I I I I I I I I C 5 C C 7 I C C 6 C I C 6 I C I C 6 I C I C 7 C 6 C 5 I C C C (C 4 = I). (4) This matrix is orthogonal, X 8 (C) = 496I = 8 4 I, det(x(c)) = 8 8, and the inverse matrix equals X (C) = 8 X(C ). We now consider the orbit of a point x = (x, x ) with respect to the group of motion C = {C k ; k = : 7}. In other words, we consider the movement of the point in the plane, which pass the following points: x Cx C(Cx) C(C (x)) C(C (x)) C(C 4 (x)) C(C 5 (x)) C(C 6 (x)) C(C 7 (x)) = x. If instead of C we consider the matrix of rotation W by the angle ϕ = π/8, then the point x will move around a circle. Let us now consider the matrix C and the point x = (, ). Figure in part a shows the locations of eight points y k = C k (x), k = : 7, which are numbered as k, and the drawing of the full path of the point (a) (b) Fig.. The movement of the point (, ) around the (a) ellipse (by the EDFT) and (b) circle (by the DFT), for the case N = 8. x during its movement. As we see, this drawing is not a circle; the points y k move along the perimeter of the

8 NCUR, ART GRIGORYAN 8 ellipse, x + y + cos ϕ b =, b = cos ϕ =.44. (5) For comparison, the movement of the same point by the rotation group {W8 k ; k = : 7} around the unit circle is shown in part b. The following points describe the movement of the point x during the EDFT: The movement of this points during the DFT is described by The value of (ϕ) = b can be considered as a measure of difference of the elliptic and traditional discrete Fourier transformations. In the considered N = 8 case, (ϕ) = (π/8) =.44 =.44. For comparison with the DFT, we consider the data shown in Figure. Figure shows the result of N rotations of these points, that are labeled as E p, when using the EDFT over the same signal. The ten points represents the ten-block EDFT with the following coordinates: {R p } = { } {I p } = { } where p = : 9. To illustrate better the difference between the EDFT and DFT for this example, Figure 4 shows the stars of these two transforms. Each star is drown by connecting points with the point F that equals E. We next show that the EDFT can distinguish the carrying frequencies of the cosine or sine waves as the DFT does. Example : For the N = case, we consider the discrete-time signal f sampled from the cosine wave f(t) = cos(ω t), ω = π/n, in the time interval [, π]. The time-points t = t n, n = : (N ), are uniformly distributed in the interval. The N-dimensional vector-signal is composed as f = (f(),, f(t ),, f(t ),,...,, f(t N ), ). We denote by g r and g i the real and imaginary parts of the N-block EDFT of f, i.e. g r = (g(), g(), g(4), g(6),..., g(n 4), g(n )), g i = (g(), g(), g(5), g(7),..., g(n ), g(n )),

9 BRIEF NOTES IN DSP 9 point EDFT composition: component F(9). 5 imaginary part y(n) of the signal z(n)=x(n)+jy(n) 5 5 f 9 f f 6 f 4 f 7 f F() F() F(9) f f 5 F(4) F(6) F(8) F() F(5) F() F(7) f f Fig.. -point signal and ten rotations of the signal in the EDFT. where g = X(C)f. Figure 5 shows the discrete-time signal f in part a, along with the real part g r of the N-block EDFT in b. The basic transform equals C = C(π/) = The EDFT recognizes the frequency ω in the frequency-points p = and = p, as in the DFT case. Another signal f sampled from the wave cos(6ω t) is shown in c and the real part g r of the N-block EDFT of this signal in d. This transform also recognizes the carrying frequency 6ω at the frequency-points p = 6 and 8 = p. These frequencies can also be seen in the imaginary part g i of the N-block EDFT of the signal cos(6ω t), as shown in Figure 6 in part a. For comparison, the imaginary part of the DFT is given in b. One can notice that the DFT does not have maximums on the carrying frequency-points p = 6 and 6. In addition, the imaginary part of the DFT by amplitude is smaller than the EDFT.. References [] N. Ahmed and R. Rao, Orthogonal transforms for digital signal processing, Springer-Verlag, 975.

10 BRIEF NOTES IN DSP E() E() E(9) F() E(8) E() E(6) E(5) E(4) F() F(8) F(6) F(9) F() F() F(5) F(4) E() F(7) E(7) 4 4 (a) 4 4 (b) Fig. 4. (a) Star of the -point DFT and (b) star of the -bloch EDFT. Original Signal Cos(ω t) EDFT (a) Fig. 5. (a) The cosine signal with frequency ω and (b) the -block EFT of the signal. [] D.F. Elliot and K.R. Rao, Fast transforms - Algorithms, Analyzes, and Applications, New York, Academic, 98. [] P. Duhamel and M. Vetterli, Fast Fourier transforms: A tutorial review and state of the art, Signal Processing, vol. 9,

11 NCUR, ART GRIGORYAN Original Signal Cos(6ω t) EDFT (a) Fig. 6. (a) The imaginary part of the -block EDFT of the sampled cosine wave with frequency 6ω and (b) the imaginary part of the -block DFT of the wave. pp , 99. [4] I.C. Chan and K.L. Ho, Split vector-radix fast Fourier transform, IEEE Trans. on Signal Processing, vol. 4, no. 8, pp. 9-4, Aug. 99. [5] A.M. Grigoryan and S.S. Agaian, Multidimensional Discrete Unitary Transforms: Representation, Partitioning and Algorithms, Marcel Dekker Inc., New York,. [6] A.M. Grigoryan and V.S. Bhamidipati, Method of flow graph simplification for the 6-point discrete Fourier transform, IEEE Trans. on Signal Processing, vol. 5, no., pp , Jan. 5. [7] 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. 9. [8] A.M. Grigoryan and M.M. Grigoryan, Brief Notes in Advanced DSP: Fourier Analysis with MATLAB, Taylor& Francis /CRC Press, New York, 9.