ELLIPTIC JES WINDOW FORMS IN SIGNAL PROCESSING
-
Upload
independent -
Category
Documents
-
view
6 -
download
0
Transcript of ELLIPTIC JES WINDOW FORMS IN SIGNAL PROCESSING
Elliptic Jes Window Forms in Signal Processing
Claude Ziad Bayeh
EEE GROUP R&D department
LEBANON Email: [email protected]
ABSTRACT The Elliptic Jes window forms are original studies introduced by the author in Mathematics and Signal Processing in 2012. They are based on an Elliptical Trigonometry function βEjesβ in which it can produce a large number of different signals and shapes by varying only one parameter. In this paper, the developed study is the application of the Elliptical Trigonometry in signal processing in which some formulae are introduced using the function βEjesβ, these formulae has many advantages ahead the traditional window functions such as improving the convergence of the Fourier series at the discontinuity more rapidly compared to the traditional window functions, the proposed window functions are used to truncate the Fourier series with variable window shapes that keep the necessary information about the signal even after truncation. The proposed window functions are variable in form; they can take a huge number of different forms by varying only a few numbers of parameters. The proposed window functions can be used in both analog and digital design of filters. In fact, the General trigonometry and its sub-topics such as Elliptical Trigonometry can have also other applications in any scientific field that uses the trigonometry and it can improve all previous studies by replacing the traditional trigonometric functions such as cosine and sine by General trigonometric functions such as Gjes and Gmar or other functions. KEYWORDS Elliptical Trigonometry, Window functions, Signal processing, Fourier series, Truncated series.
1 INTRODUCTION In mathematics and in signal processing, the definition of a window function is that it has a zero-valued outside of some chosen interval [1-3]. A typical example is a rectangular window in which any curve inside the window is conserved and any curve outside the window is set to be equal to zero [6-15]. When another function or a signal (data) is multiplied by a window function, the product is also zero-valued outside the interval: all that is left is the part where they overlap. Applications of window functions include filter design, spectral analysis, beamforming⦠[4-5], [28] and [33]. In typical applications, the window functions used are non-negative smooth "bell-shaped" curves, though rectangle, triangle, and other functions are sometimes used [1-2]. Thy are used mainly to converge the Fourier series at the discontinuity [1-2]. In this paper, the author introduced four new window functions using an Elliptical Trigonometric function such as Elliptic Jes function [16-17]. The Elliptical trigonometry is also introduced by the author and it can be considered as the basis of the new generation of Signal Processing, Electronics and Electrical systems based on variable signals [17]. The new window functions based on the Elliptical Trigonometry has huge advantages over the traditional window functions based on the traditional trigonometry. This will be discussed in this paper.
International Journal of Digital Information and Wireless Communications (IJDIWC) 3(4): 77-95The Society of Digital Information and Wireless Communications, 2013 (ISSN: 2225-658X)
77
2 BRIEF INTRODUCTION TO THE βELLIPTIC JESβ FUNCTION The Elliptic Jes function (πΈπΈπΈπΈπΈπΈπΈπΈππ(π₯π₯)) is a function of the Elliptical Trigonometry which is defined in the papers [16-17]. In this paper we will use it only to create new window functions based on the Elliptical Trigonometry. β’ The function πΈπΈπΈπΈπΈπΈπΈπΈππ(π₯π₯) can describes an infinite number of forms but in this section we will see only some important forms as depicted in the following figures. Figures 1.a to 1.f represent multi form signals obtained by varying one parameter ( ππ ) of the function πΈπΈπΈπΈπΈπΈπΈπΈππ(π₯π₯).
a) ππ = 0.001
b) ππ = 0.2
c) ππ = β3/3
d) ππ = 1
e) ππ = 3
International Journal of Digital Information and Wireless Communications (IJDIWC) 3(4): 77-95The Society of Digital Information and Wireless Communications, 2013 (ISSN: 2225-658X)
78
f) ππ = 90
Figure 1, multi form signals of the function πΈπΈπΈπΈπΈπΈπΈπΈππ(π₯π₯) and for different values of ππ > 0. Important signals obtained using this function: Impulse train with positive and negative part, elliptic deflated, quasi-triangular, sinusoidal, elliptical swollen, square signal, rectangular signalβ¦[40]. These types of signals are widely used in power electronics, electrical generator, signal processing and in transmission of analog signals [16-17], [35-45]. 3 βELLIPTIC JESβ WINDOW FORM 3 FUNCTION The Elliptic Jes window form 3 function is the application of the Elliptic Jes function in signal processing. It takes the following forms: π€π€ = 25
46β 21
46πΈπΈπΈπΈπΈπΈπΈπΈππ οΏ½
2ππππππβ1
οΏ½ (1) With 0 β€ ππ β€ ππ β 1 and ππ β β And π€π€ππ = 25
46+ 21
46πΈπΈπΈπΈπΈπΈπΈπΈππ οΏ½
πππππποΏ½ (2)
With βππ β€ ππ β€ ππ and ππ β β
So the truncated Fourier series using the Elliptic Jes window form 3takes the following form:
ππππ(ππ) = οΏ½ π€π€πππππππΈπΈπΈπΈππππππ
ππ=βππ
= οΏ½ οΏ½2546
+2146
πΈπΈπΈπΈπΈπΈπΈπΈππ οΏ½ππππππ οΏ½οΏ½
πππππΈπΈπΈπΈππππππ
ππ=βππ
(3) 3.1 Variable shapes of window formed by Elliptic Jes window form 3 The formed shapes of this function can be drawn using MATLAB. In the figures 2.a to 2.f, different shapes of the window function are formed by varying only one parameter which is ππ.
a) ππ = 0.001
b) ππ = 0.2
International Journal of Digital Information and Wireless Communications (IJDIWC) 3(4): 77-95The Society of Digital Information and Wireless Communications, 2013 (ISSN: 2225-658X)
79
c) ππ = β3/3
d) ππ = 1
e) ππ = 3
f) ππ = 90
Figure 2, multi form signals of the function Elliptic Jes window form 3 and for different values of ππ > 0. In fact, this window is very important as it has variable amplitude that can be changed as we wish over a period or a half period. Applications of window functions include spectral analysis, filter design, beamforming and telecommunications. A more general definition of window functions does not require them to be identically zero outside an interval, as long as the product of the window multiplied by its argument is square integrable, that is, that the function goes sufficiently rapidly toward zero. 3.2 Programming the function Elliptic Jes window form 3 using MATLAB %----------------------------------------------------------- %Elliptic Jes Window form 3 %Introduced by Claude Ziad Bayeh in 2012-06-21 clc close all M=2; a=1; x=0:0.0001:M-1; fprintf('---Elliptic Jes Window form 3 Introduced by Claude Ziad Bayeh in 2012-06-21---\n'); fprintf('---------------------------------------------\n'); repeat='y'; while repeat=='y' b=input('determine the form of the Elliptic trigonometry: b='); fprintf('b is a variable can be changed to obtain different signals \n'); %b is the intersection of the Ellipse and the axe y'oy in the positive part.
International Journal of Digital Information and Wireless Communications (IJDIWC) 3(4): 77-95The Society of Digital Information and Wireless Communications, 2013 (ISSN: 2225-658X)
80
if b<0, b error('ATTENTION: ERROR b must be greater than Zero'); end; Ejes=(1./(sqrt(1.+((a/b).*tan(x)).^2))).*angx(x); % the Elliptic Jes "Ejes" Emar=(1./(sqrt(1.+((a/b).*tan(x)).^2))).*angx(x).*tan(x).*a/b; % the Elliptic Mar "Emar" % Elliptic Jes Window form 3 EjesW3=25./46-21./46.*((1./(sqrt(1.+((a/b).*tan(2.*pi.*x)).^2))).*angx(2.*pi.*x)); plot(x,EjesW3); %xlabel('X''OX axis'); ylabel('f(x)=AEjesx(x)'); title('Absolute Elliptic Jes-x: AEjesx(x)'); axis([0 M-1 0 1.1]); grid on;%grid Minor (for more details)/ grid on (for less details) fprintf('Do you want to repeat ?\nPress y for ''Yes'' or any key for ''No''\n'); repeat=input('Y/N=','s'); %string input clc close all end; %End while %----------------------------------------------------------- 3.3 Advantages of the function Elliptic Jes window form 3 over the traditional window functions Similar to other windows used in signal processing such as: Hamming, Hanning, Blackman, Kaiser, Lanczos, Tukey and many other windows, the main goal of introducing the Elliptic Jes window form 3 is to improve the convergence of the Fourier Series at the discontinuity. The advantages of the new window function over the traditional windows are: -The proposed window function is variable in form; it can take more than 6 different forms by varying only one parameter. -It can help the Fourier series to converge more rapidly compared to the traditional ones. βIt can be used in both analog design of filters and digital design of filters.
βIt is used to truncate the Fourier series with a variable window shape that keep the necessary information about the signal even after truncation. 4 βELLIPTIC JESβ WINDOW FORM 4 FUNCTION The Elliptic Jes window form 4 function is the application of the Elliptic Jes function in signal processing. It takes the following forms:
π€π€ = οΏ½12οΏ½1 β πΈπΈπΈπΈπΈπΈπΈπΈππ οΏ½
2ππππππβ1
οΏ½οΏ½π π
(4)
With 0 β€ ππ β€ ππ β 1 and ππ β β With π π is the radical of the function, it is an integer with π π β β And
π€π€ππ = οΏ½12οΏ½1 + πΈπΈπΈπΈπΈπΈπΈπΈππ οΏ½
πππππποΏ½οΏ½
π π (5)
With βππ β€ ππ β€ ππ and ππ β β So the truncated Fourier series using the Elliptic Jes window form 4 takes the following form:
ππππ(ππ) = οΏ½ π€π€πππππππΈπΈπΈπΈππππππ
ππ=βππ
= οΏ½ οΏ½12οΏ½
1 + πΈπΈπΈπΈπΈπΈπΈπΈππ οΏ½ππππππ οΏ½οΏ½
π π πππππΈπΈπΈπΈππππ
ππ
ππ=βππ
(6) 4.1 Variable shapes of window formed by Elliptic Jes window form 4 The formed shapes of this function can be drawn using MATLAB. In the figures 3.a to 3.f, different shapes of the window function are formed by varying only one parameter which is ππ . We consider in this case that the value of π π = 1.
International Journal of Digital Information and Wireless Communications (IJDIWC) 3(4): 77-95The Society of Digital Information and Wireless Communications, 2013 (ISSN: 2225-658X)
81
a) ππ = 0.001 and π π = 1
b) ππ = 0.2 and π π = 1
c) ππ = β3/3 and π π = 1
d) ππ = 1 and π π = 1
e) ππ = 3 and π π = 1
f) ππ = 90 and π π = 1
Figure 3, multi form signals of the function Elliptic Jes window form 4 and for different values of ππ > 0 and π π = 1. If we change the value of π π , we can increase the performance of the window at the discontinuity
International Journal of Digital Information and Wireless Communications (IJDIWC) 3(4): 77-95The Society of Digital Information and Wireless Communications, 2013 (ISSN: 2225-658X)
82
and at each point within the window. The formed shapes of this function can be drawn using MATLAB. In the figures 4.a to 4.f, different shapes of the window function are formed by varying only one parameter which is ππ . We consider in this case that the value of π π = 4.
a) ππ = 0.001 and π π = 4
b) ππ = 0.2 and π π = 4
c) ππ = β3/3 and π π = 4
d) ππ = 1 and π π = 4
e) ππ = 3 and π π = 4
f) ππ = 90 and π π = 4
Figure 4, multi form signals of the function Elliptic Jes window form 4 and for different values of ππ > 0 and π π = 4.
International Journal of Digital Information and Wireless Communications (IJDIWC) 3(4): 77-95The Society of Digital Information and Wireless Communications, 2013 (ISSN: 2225-658X)
83
In fact, this window is very important as it has variable amplitude that can be changed as we wish over a period or a half period. Applications of window functions include spectral analysis, filter design, beamforming and telecommunications. A more general definition of window functions does not require them to be identically zero outside an interval, as long as the product of the window multiplied by its argument is square integrable, that is, that the function goes sufficiently rapidly toward zero. This window function is considered as the most powerful window function of all existing ones. Because we can form an infinite number of forms and shapes that help the Fourier series to converge more or less rapidly than ever. 4.2 Programming the function Elliptic Jes window form 4 using MATLAB %-------------------------------------------------------------------- %Elliptic Jes Window form 4 %Introduced by Claude Ziad Bayeh in 2012-06-21 clc close all M=2; a=1; x=0:0.0001:M-1; fprintf('---Elliptic Jes Window form 4 Introduced by Claude Ziad Bayeh in 2012-06-21---\n'); fprintf('-----------------------------------------------------------\n'); repeat='y'; while repeat=='y' b=input('determine the form of the Elliptic trigonometry: b='); fprintf('b is a variable can be changed to obtain different signals \n'); %b is the intersection of the Ellipse and the axe y'oy in the positive part. if b<0, b error('ATTENTION: ERROR b must be greater than Zero'); end; R=input('determine the root of the window: R='); fprintf('R is a variable can be changed to obtain variable amplitude of the window function\n'); if R<=0, R error('ATTENTION: ERROR "R" must be greater than Zero'); end;
Ejes=(1./(sqrt(1.+((a/b).*tan(x)).^2))).*angx(x); % the Elliptic Jes "Ejes" Emar=(1./(sqrt(1.+((a/b).*tan(x)).^2))).*angx(x).*tan(x).*a/b; % the Elliptic Mar "Emar" % Elliptic Jes Window form 4 X=(2.*pi.*x); EjesW4=(1./2.*(1-(1./(sqrt(1.+((a/b).*tan(X)).^2))).*angx(X))).^(1./R); plot(x,EjesW4); %xlabel('X''OX axis'); ylabel('f(x)=EjesW4(x)'); title('Elliptic Jes Window form 4'); axis([0 M-1 0 1.1]); grid on;%grid Minor (for more details)/ grid on (for less details) fprintf('Do you want to repeat ?\nPress y for ''Yes'' or any key for ''No''\n'); repeat=input('Y/N=','s'); %string input clc close all end; %End while %-------------------------------------------------------------------- 4.3 Advantages of the function Elliptic Jes window form 4 over the traditional window functions Similar to other windows used in signal processing such as: Hamming, Hanning, Blackman, Kaiser, Lanczos, Tukey and many other windows, the main goal of introducing the Elliptic Jes window form 4 is to improve the convergence of the Fourier Series at the discontinuity. The advantages of the new window function over the traditional windows are: -The proposed window function is variable in form; it can take an infinite number of different forms by varying only two parameters. -It can help the Fourier series to converge more rapidly compared to the traditional ones. βIt can be used in both analog design of filters and digital design of filters. βIt is used to truncate the Fourier series with a variable window shape that keep the necessary information about the signal even after truncation.
International Journal of Digital Information and Wireless Communications (IJDIWC) 3(4): 77-95The Society of Digital Information and Wireless Communications, 2013 (ISSN: 2225-658X)
84
5 ELLIPTIC JES WINDOW FORM 5 FUNCTION The Elliptic Jes window form 5 function is the application of the Elliptic Jes function in signal processing. It takes the following forms:
π€π€ = 12οΏ½1 + πΈπΈπΈπΈπΈπΈπΈπΈππ οΏ½
12
(ππ+2)(ππ+1)
οΏ½2ππππππβ1
β πποΏ½οΏ½οΏ½ (7)
With 0 β€ ππ β€ ππ β 1 and ππ β β And
π€π€ππ = 12οΏ½1 + πΈπΈπΈπΈπΈπΈπΈπΈππ οΏ½
12
(ππ+2)(ππ+1)
οΏ½πππππποΏ½οΏ½οΏ½ (8)
With βππ β€ ππ β€ ππ and ππ β β So the truncated Fourier series using the Elliptic Jes window form 5takes the following form:
ππππ(ππ) = οΏ½ π€π€πππππππΈπΈπΈπΈππππππ
ππ=βππ
= οΏ½12οΏ½
1 + πΈπΈπΈπΈπΈπΈπΈπΈππ οΏ½12
(ππ + 2)(ππ + 1) οΏ½
ππππππ οΏ½οΏ½οΏ½
πππππΈπΈπΈπΈππππππ
ππ=βππ
(9) 5.1 Variable shapes of window formed by Elliptic Jes window form 5 The formed shapes of this function can be drawn using MATLAB. In the figures 5.a to 5.g, different shapes of the window function are formed by varying only one parameter which is ππ.
a) ππ = 0
b) ππ = 0.01
c) ππ = 0.2
d) ππ = β3/3
International Journal of Digital Information and Wireless Communications (IJDIWC) 3(4): 77-95The Society of Digital Information and Wireless Communications, 2013 (ISSN: 2225-658X)
85
e) ππ = 1
f) ππ = 3
g) ππ = 90
Figure 5, multi form signals of the function Elliptic Jes window form 5 and for different values of ππ > 0. In fact, this window is very important as it has variable amplitude that can be changed as we wish
over a period or a half period. Applications of window functions include spectral analysis, filter design, beamforming and telecommunications. A more general definition of window functions does not require them to be identically zero outside an interval, as long as the product of the window multiplied by its argument is square integrable, that is, that the function goes sufficiently rapidly toward zero. 5.2 Programming the function Elliptic Jes window form 5 using MATLAB %-------------------------------------------------------------- %Elliptic Jes Window form 5 %Introduced by Claude Ziad Bayeh in 2012-06-21 clc close all M=2; a=1; x=0:0.0001:M-1; fprintf('---Elliptic Jes Window form 5 Introduced by Claude Ziad Bayeh in 2012-09-21---\n'); fprintf('--------------------------------------------------------------------------------\n'); repeat='y'; while repeat=='y' b=input('determine the form of the Elliptic trigonometry: b='); fprintf('b is a variable can be changed to obtain different signals \n'); %b is the intersection of the Ellipse and the axe y'oy in the positive part. if b<0, b error('ATTENTION: ERROR b must be greater than Zero'); end; Ejes=(1./(sqrt(1.+((a/b).*tan(x)).^2))).*angx(x); % the Elliptic Jes "Ejes" Emar=(1./(sqrt(1.+((a/b).*tan(x)).^2))).*angx(x).*tan(x).*a/b; % the Elliptic Mar "Emar" % Elliptic Jes Window form 5 X=1/2.*(b+2)./(b+1).*(2*pi*x-pi); EjesW5=1/2.*(1+(1./(sqrt(1.+((a/b).*tan(X)).^2))).*angx(X)); plot(x,EjesW5); %xlabel('X''OX axis'); ylabel('f(x)=EjesW1(x)'); title('Elliptic Jes Window form 5');
International Journal of Digital Information and Wireless Communications (IJDIWC) 3(4): 77-95The Society of Digital Information and Wireless Communications, 2013 (ISSN: 2225-658X)
86
axis([0 M-1 0 1.1]); grid on;%grid Minor (for more details)/ grid on (for less details) fprintf('Do you want to repeat ?\nPress y for ''Yes'' or any key for ''No''\n'); repeat=input('Y/N=','s'); %string input clc close all end; %End while %-------------------------------------------------------------- 5.3 Advantages of the function Elliptic Jes window form 5 over the traditional window functions Similar to other windows used in signal processing such as: Hamming, Hanning, Blackman, Kaiser, Lanczos, Tukey and many other windows, the main goal of introducing the Elliptic Jes window form 5is to improve the convergence of the Fourier Series at the discontinuity. The advantages of the new window function over the traditional windows are: -The proposed window function is variable in form; it can take more than 6 different forms by varying only one parameter. -It can help the Fourier series to converge more rapidly compared to the traditional ones. βIt can be used in both analog design of filters and digital design of filters. βIt is used to truncate the Fourier series with a variable window shape that keep the necessary information about the signal even after truncation. 6 ELLIPTIC JES WINDOW FORM 6 FUNCTION The Elliptic Jes window form 6 function is the application of the Elliptic Jes function in signal processing. It takes the following forms:
π€π€ = οΏ½12οΏ½1 + πΈπΈπΈπΈπΈπΈπΈπΈππ οΏ½
1πποΏ½2ππππππβ1
β πποΏ½οΏ½οΏ½π π
(10)
With 0 β€ ππ β€ ππ β 1 and ππ β β β’π π is an integer greater than zero with π π β β, it is a variable parameter and it is used to amplify the window when the value is smaller than 1.
β’ππ is a real number greater than zero with ππ β β In general we take 1 β€ ππ β€ 2, it is also a variable parameter which represents the frequency of the window function and it is used to justify the attenuation to zero in the extremity of the window. And
π€π€ππ = οΏ½12οΏ½1 + πΈπΈπΈπΈπΈπΈπΈπΈππ οΏ½
1πποΏ½πππππποΏ½οΏ½οΏ½
π π (11)
With βππ β€ ππ β€ ππ and ππ β β So the truncated Fourier series using the Elliptic Jes window form 6takes the following form:
ππππ(ππ) = οΏ½ π€π€πππππππΈπΈπΈπΈππππππ
ππ=βππ
= οΏ½ οΏ½12οΏ½
1 + πΈπΈπΈπΈπΈπΈπΈπΈππ οΏ½1ππ οΏ½ππππππ οΏ½οΏ½οΏ½
π π
πππππΈπΈπΈπΈππππππ
ππ=βππ
(12) 6.1 Variable shapes of window formed by Elliptic Jes window form 6 The formed shapes of this function can be drawn using MATLAB. In the set of figures 5 to 7, different shapes of the window function are formed by varying one of the three parameters which are ππ, ππ,ππππππ π π .
a) ππ = 0.01, ππ = 1,π π = 1
International Journal of Digital Information and Wireless Communications (IJDIWC) 3(4): 77-95The Society of Digital Information and Wireless Communications, 2013 (ISSN: 2225-658X)
87
b) ππ = 0.2, ππ = 1,π π = 1
c) ππ = β3/3, ππ = 1,π π = 1
d) ππ = 1, ππ = 1,π π = 1
e) ππ = 3, ππ = 1,π π = 1
f) ππ = 90, ππ = 1,π π = 1
Figure 5, multi form signals of the function Elliptic Jes window form 6 and for different values of ππ > 0 and ππ = 1,π π = 1.
a) ππ = 0.01, ππ = 1,π π = 4
International Journal of Digital Information and Wireless Communications (IJDIWC) 3(4): 77-95The Society of Digital Information and Wireless Communications, 2013 (ISSN: 2225-658X)
88
b) ππ = 0.2, ππ = 1,π π = 4
c) ππ = β3/3, ππ = 1,π π = 4
d) ππ = 1, ππ = 1,π π = 4
e) ππ = 3, ππ = 1,π π = 4
f) ππ = 90, ππ = 1,π π = 4
Figure 6, multi form signals of the function Elliptic Jes window form 6 and for different values of ππ > 0 and ππ = 1,π π = 4.
a) ππ = 0.01, ππ = 1.5,π π = 1
International Journal of Digital Information and Wireless Communications (IJDIWC) 3(4): 77-95The Society of Digital Information and Wireless Communications, 2013 (ISSN: 2225-658X)
89
b) ππ = 0.2, ππ = 1.5,π π = 1
c) ππ = β3/3, ππ = 1.5,π π = 1
d) ππ = 1, ππ = 1.5,π π = 1
e) ππ = 3, ππ = 1.5,π π = 1
f) ππ = 90, ππ = 1.5,π π = 1
Figure 7, multi form signals of the function Elliptic Jes window form 6 and for different values of ππ > 0 and ππ = 1.5,π π = 1. In fact, this window is very important as it has variable amplitude that can be changed as we wish over a period or a half period. It is considered as the most important Window function ever written because of its wide variation forms and its simplicity in the design of circuits. Applications of window functions include spectral analysis, filter design, beamforming and telecommunications. A more general definition of window functions does not require them to be identically zero outside an interval, as long as the product of the window multiplied by its argument is square integrable, that is, that the function goes sufficiently rapidly toward zero.
International Journal of Digital Information and Wireless Communications (IJDIWC) 3(4): 77-95The Society of Digital Information and Wireless Communications, 2013 (ISSN: 2225-658X)
90
6.2 Programming the function Elliptic Jes window form 6 using MATLAB %-------------------------------------------------------------- %Elliptic Jes Window form 6 %Introduced by Claude Ziad Bayeh in 2012-06-21 clc close all M=2; a=1; x=0:0.0001:M-1; fprintf('---Elliptic Jes Window form 6 Introduced by Claude Ziad Bayeh in 2012-09-21---\n'); fprintf('--------------------------------------------------------------------------------\n'); repeat='y'; while repeat=='y' b=input('determine the form of the Elliptic trigonometry: b='); fprintf('b is a variable can be changed to obtain different signals \n'); %b is the intersection of the Ellipse and the axe y'oy in the positive part. if b<0, b error('ATTENTION: ERROR b must be greater than Zero'); end; R=input('determine the root of the window: R='); fprintf('R is a variable can be changed to obtain variable amplitude of the window function\n'); if R<=0, R error('ATTENTION: ERROR "R" must be greater than Zero'); end; c=input('determine the frequency of the window: c='); fprintf('"c" is a variable can take the values between 1 and 2\n'); if c<=0, c error('ATTENTION: ERROR "c" must be greater than Zero'); end; Ejes=(1./(sqrt(1.+((a/b).*tan(x)).^2))).*angx(x); % the Elliptic Jes "Ejes" Emar=(1./(sqrt(1.+((a/b).*tan(x)).^2))).*angx(x).*tan(x).*a/b; % the Elliptic Mar "Emar" % Elliptic Jes Window form 5 X=1./c.*(2*pi*x-pi); EjesW6=(1/2.*(1+(1./(sqrt(1.+((a/b).*tan(X)).^2))).*angx(X))).^(1./R);
plot(x,EjesW6); %xlabel('X''OX axis'); ylabel('f(x)=EjesW6(x)'); title('Elliptic Jes Window form 6'); axis([0 M-1 0 1.1]); grid on;%grid Minor (for more details)/ grid on (for less details) fprintf('Do you want to repeat ?\nPress y for ''Yes'' or any key for ''No''\n'); repeat=input('Y/N=','s'); %string input clc close all end; %End while %-------------------------------------------------------------- 6.3 Advantages of the function Elliptic Jes window form 6 over the traditional window functions Similar to other windows used in signal processing such as: Hamming, Hanning, Blackman, Kaiser, Lanczos, Tukey and many other windows, the main goal of introducing the Elliptic Jes window form 6 is to improve the convergence of the Fourier Series at the discontinuity. The advantages of the new window function over the traditional windows are: -The proposed window function is variable in form; it can take infinite different forms by varying three parameters. -It can help the Fourier series to converge more rapidly compared to the traditional ones. βIt can be used in both analog design of filters and digital design of filters. βIt is used to truncate the Fourier series with a variable window shape that keeps the necessary information about the signal even after truncation. 7 EXISTING WINDOW FUNCTIONS In this section, the author present other types of window functions used to converge a Fourier series at the limit [1-2] and [39]. Such as: 7.1 Rectangular window The rectangular window (sometimes known as the boxcar or Dirichlet window) is the simplest window, equivalent to replacing all but N values of
International Journal of Digital Information and Wireless Communications (IJDIWC) 3(4): 77-95The Society of Digital Information and Wireless Communications, 2013 (ISSN: 2225-658X)
91
a data sequence by zeros, making it appear as though the waveform suddenly turns on and off: π€π€(ππ) = οΏ½1 ππππππ 0 β€ ππ β€ ππ β 1
0 πΈπΈπππΈπΈπΈπΈπ€π€βπΈπΈπππΈπΈ οΏ½ (13)
Figure 8, Rectangular window function (from Wikipedia [46]). Other windows are designed to moderate these sudden changes because discontinuities have undesirable effects on the discrete-time Fourier transform (DTFT) and/or the algorithms that produce samples of the DTFT. 7.2 Triangular window The triangular window is defined by:
π€π€(ππ) = 1 β οΏ½ππβππβ1
2ππ+1
2οΏ½ (14)
Figure 9, Triangular window function (from Wikipedia [46]). The end samples are positive (equal to 2/(N + 1)). This window can be seen as the convolution of two half-sized rectangular windows (for N even), giving it a main lobe width of twice the width of a regular rectangular window. The nearest lobe is β26 dB down from the main lobe. 7.3 Welch window The Welch window consists of a single parabolic section:
π€π€(ππ) = 1 β οΏ½ππβππβ1
2ππ+1
2οΏ½
2
(15)
Figure 10, Welch window function (from Wikipedia [46]).
International Journal of Digital Information and Wireless Communications (IJDIWC) 3(4): 77-95The Society of Digital Information and Wireless Communications, 2013 (ISSN: 2225-658X)
92
The defining quadratic polynomial reaches a value of zero at the samples just outside the span of the window. 7.4 Hann (Hanning) window The Hann window also known as the Hanning is defined by: π€π€(ππ) = 1
2οΏ½1 β cos οΏ½2ππππ
ππβ1οΏ½οΏ½ (16)
Figure 11, Hann window function (from Wikipedia [46]). 7.5 Tukey window The Tukey window, also known as the tapered cosine window, can be regarded as a cosine lobe of width Ξ±N/2 that is convolved with a rectangular window of width (1 β Ξ±/2)N π€π€(ππ)
=
β©βͺβͺβ¨
βͺβͺβ§ 1
2οΏ½1 + cosοΏ½ππ οΏ½
2πππΌπΌ(ππ β 1) β 1οΏ½οΏ½οΏ½ ππππππ 0 β€ ππ β€
πΌπΌ(ππ β 1)2
1 ππππππ πΌπΌ(ππ β 1)
2β€ (ππ β 1) οΏ½1 β
πΌπΌ2οΏ½
12οΏ½1 + cosοΏ½ππ οΏ½
2πππΌπΌ(ππ β 1) β
2πΌπΌ
+ 1οΏ½οΏ½οΏ½ ππππππ (ππ β 1) οΏ½1 βπΌπΌ2οΏ½ β€ ππ β€ (ππ β 1)
οΏ½
(17)
Figure 12, Tukey window function (from Wikipedia [46]). And so onβ¦ The main purpose of developing these windows is to obtain the smoother form that helps the attenuation of the desired signal in the extremity of the window function at the same time obtaining the minimum amplitude of the side-lobes and maximum width of the main lobe (refer to figure 10). This is not possible with the existing window functions. So there is a compromise to do. The disadvantage of these window functions is that their frequency response doesnβt converge to zero outside the interval, and moreover, their amplitudes are not negligible.
International Journal of Digital Information and Wireless Communications (IJDIWC) 3(4): 77-95The Society of Digital Information and Wireless Communications, 2013 (ISSN: 2225-658X)
93
Figure 13, window function in the frequency domain (from Wikipedia [46]). This problem is resolved with the proposed window function by the author based on the Elliptical Trigonometry in which we can regulate the shape of the window to obtain a very smooth form within the window function and obtain at the same time a wider main lobe and very small side lobes in magnitude. 8 CONCLUSION In this paper, the author introduced new window functions based on the Elliptical Trigonometry. These new window functions have many advantages as cited in the previous sections. The main goal of introducing these new window functions is to improve the convergence of the Fourier Series at the discontinuity. As we have seen, the shapes of the window functions using the Elliptical Trigonometry have variable shapes and we can regulate the shape in a way to improve the convergence and moreover to control the frequency of the signals that we want and that we donβt want, the Elliptical Trigonometry has other applications such as [16-17] and [35-45]. These new window functions have enormous applications in mathematics and in signal processing and precisely in the design of analog and digital filters. 6 REFERENCES 1. H. Baher, βSignal processing and integrated circuitsβ,
Published by John Wiley & Sons Ltd., ISBN: 9780470710265, (2012).
2. J. G. Proakis, Dimitris G. Manolakis, βDigital Signal Processing, Principles, Algorithms, and Applicationsβ Fourth edition, Pearson International Edition, ISBN: 0-13-228731-5.
3. N. Wirth, βDigital Circuit Designβ, Springer, ISBN: 3-540-58577-X.
4. N. Senthil Kumar, M. Saravanan, S. Jeevananthan, βMicroprocessors and Microcontrollersβ, Oxford University Press, ISBN-13: 978-0-19-806647-7.
5. G. Wade, βSignal Coding and Processingβ Second Edition, Cambridge University Press, ISBN: 0-521-42336-8.
6. S. J. Orfanidis, βIntroduction to Signal Processingβ, Rutgers University, ISBN 0-13-209172-0, 2010.
7. B. Gold and C. M. Rader, βDigital Processing of Signalsβ, McGraw-Hill, New York, (1969).
8. A. V. Oppenheim and R. W. Schafer, βDiscrete-Time Signal Processingβ, Prentice Hall, Englewood Cliffs, NJ, (1989).
9. A. V. Oppenheim and R. W. Schafer, βDigital Signal Processingβ, Prentice Hall, Englewood Cliffs, NJ, (1975).
10. L. R. Rabiner and B. Gold, βTheory and Application of Digital Signal Processingβ, Prentice Hall, Englewood Cliffs, NJ, (1975).
11. S. K. Mitra and J. F. Kaiser, eds., βHandbook of Digital Signal Processingβ, Wiley, New York, (1993).
12. T. W. Parks and C. S. Burrus, βDigital Filter Designβ, Wiley, New York, (1987).
13. A. Antoniou, βDigital Filters: Analysis and Designβ, 2nd ed., McGraw-Hill, New York, (1993).
14. D. F. Elliott, βHandbook of Digital Signal Processingβ, Academic Press, New York, (1987).
15. L. R. Rabiner and C. M. Rader, eds., βDigital Signal Processingβ, IEEE Press, New York, (1972).
16. C. Bayeh, M. Bernard, N. Moubayed, Introduction to the elliptical trigonometry, WSEAS Transactions on Mathematics, Issue 9, Volume 8, (September 2009), pp. 551-560.
17. N. Moubayed, C. Bayeh, M. Bernard, A survey on modeling and simulation of a signal source with controlled waveforms for industrial electronic applications, WSEAS Transactions on Circuits and Systems, Issue 11, Volume 8, (November 2009), pp. 843-852.
18. M. Christopher, From Eudoxus to Einstein: A History of Mathematical Astronomy, Cambridge University Press, (2004).
19. E. W. Weisstein, Trigonometric Addition Formulas, Wolfram MathWorld, (1999-2009).
20. P. A. Foerster, Algebra and Trigonometry: Functions and Applications, Addison-Wesley publishing company, (1998).
21. F. Ayres, Trigonométrie cours et problèmes, McGraw-Hill, (1991).
22. R. C. Fisher and Allen D.Ziebur, Integrated Algebra and Trigonometry with Analytic Geometry, Pearson Education Canada, (2006).
23. E. Demiralp, Applications of High Dimensional Model Representations to Computer Vision, WSEAS Transactions on Mathematics, Issue 4, Volume 8, (April 2009).
24. A. I. Grebennikov, Fast algorithm for solution of Dirichlet problem for Laplace equation, WSEAS
International Journal of Digital Information and Wireless Communications (IJDIWC) 3(4): 77-95The Society of Digital Information and Wireless Communications, 2013 (ISSN: 2225-658X)
94
Transactions on Computers Journal, 2(4), pp. 1039 β 1043, (2003).
25. I. Mitran, F.D. Popescu, M.S. Nan, S.S. Soba, Possibilities for Increasing the Use of Machineries Using Computer Assisted Statistical Methods, WSEAS Transactions on Mathematics, Issue 2, Volume 8, (February 2009).
26. Q. Liu, Some Preconditioning Techniques for Linear Systems, WSEAS Transactions on Mathematics, Issue 9, Volume 7, (September 2008).
27. A. I. Grebennikov, The study of the approximation quality of GR-method for solution of the Dirichlet problem for Laplace equation. WSEAS Transactions on Mathematics Journal, 2(4), pp. 312-317, (2003).
28. R. Bracewell, Heaviside's Unit Step Function. The Fourrier Transform and its Applications, 3
rd edition,
New York: McGraw-Hill, pp. 61-65, (2000). 29. M. Abramowitz and Irene A. Stegun, eds, Handbook of
mathematical functions with formulas, graphs and mathematical tables, 9th
printing, New York: Dover,
(1972). 30. V. Kantabutra, On hardware for computing exponential
and trigonometric functions, IEEE Transactions on Computers, Vol. 45, issue 3, pp. 328β339, (1996).
31. H. P. Thielman, A generalization of trigonometry, National mathematics magazine, Vol. 11, No. 8, (1937), pp. 349-351.
32. N. J. Wildberger, Divine proportions: Rational Trigonometry to Universal Geometry, Wild Egg, Sydney, (2005).
33. C. W. Lander, Power electronics, third edition, McGraw-Hill Education, (1993).
34. C. Bayeh, βIntroduction to the Rectangular Trigonometry in Euclidian 2D-Spaceβ, WSEAS Transactions on Mathematics, ISSN: 1109-2769, Issue 3, Volume 10, (March 2011), pp. 105-114.
35. C. Z. Bayeh, βIntroduction to the Angular Functions in Euclidian 2D-spaceβ, WSEAS Transactions on Mathematics, ISSN: 1109-2769, E-ISSN: 2224-2880, Issue 2, Volume 11, (February 2012), pp.146-157.
36. C. Z. Bayeh, βIntroduction to the General Trigonometry in Euclidian 2D-Spaceβ, WSEAS Transactions on Mathematics, ISSN: 1109-2769, E-ISSN: 2224-2880, Issue 2, Volume 11, (February 2012), pp.158-172.
37. C. Bayeh, βApplication of the Elliptical Trigonometry in industrial electronic systems with analyzing, modeling and simulating two functions Elliptic Mar and Elliptic Jes-xβ, WSEAS Transactions on Circuits and Systems, ISSN: 1109-2734, Issue 11, Volume 8, (November 2009), pp. 843-852.
38. C. Bayeh, βA survey on the application of the Elliptical Trigonometry in industrial electronic systems using controlled waveforms with modeling and simulating of two functions Elliptic Mar and Elliptic Jes-xβ, in the book β Latest Trends on Circuits, Systems and Signalsβ,
publisher WSEAS Press, ISBN: 978-960-474-208-0, ISSN: 1792-4324, (July 2010), pp.96-108.
39. C. Z. Bayeh, βElliptic Jes window form 2 in Signal Processingβ, International Journal of Digital Information and Wireless Communications (IJDIWC) 3(3). The Society of Digital Information and Wireless Communications, 2013 (ISSN: 2225-658X), pp.1-9.
40. C. Z. Bayeh, βIntroduction to the Elliptical Trigonometry in Euclidian 2D-space with simulation of four elliptical trigonometric functions Jes, Jes-x, Mar and Ritβ, WSEAS Transactions on Mathematics, ISSN: 1109-2769, E-ISSN: 2224-2880, Issue 9, Volume 11, September 2012, pp.784-795.
41. C. Z. Bayeh, βIntroduction to the Rhombus Trigonometry in Euclidian 2D-space with simulation of four Rhombus trigonometric functions RhJes, RhJes-x, RhMar and RhRitβ, WSEAS Transactions on Mathematics, ISSN: 1109-2769, E-ISSN: 2224-2880, Issue 10, Volume 11, October 2012, pp.876-888.
42. C. Z. Bayeh, βIntroduction to the Elliptical Trigonometry Series Using two Functions Absolute Elliptic Jes (AEjes) and Absolute Elliptic Mar (AEmar) of the First Formβ, WSEAS Transactions on Mathematics, E-ISSN: 2224-2880, Issue 4, Volume 12, April 2013, pp. 436-448.
43. C. Z. Bayeh, Nikos E. Mastorakis, βRectangular Base Functionβ, in the book βMathematical Methods for Science and Economicsβ, Proceedings of the 17th WSEAS International Conference on Applied Mathematics (AMATH '12), Montreux, Switzerland, Published by WSEAS Press, December 29-31, 2012, ISBN: 978-1-61804-148-7, pp. 105-108.
44. C. Z. Bayeh, Nikos E. Mastorakis, βElliptic Jes Window Form 1β, in the book βMathematical Methods for information Science and Economicsβ, Proceedings of the 17th WSEAS International Conference on Applied Mathematics (AMATH '12), Montreux, Switzerland, Published by WSEAS Press, December 29-31, 2012, ISBN: 978-1-61804-148-7, pp. 115-120.
45. C. Z. Bayeh, Nikos E.Mastorakis , βApplication of the Rectangular Trigonometry in industrial electronic systems with analyzing, modeling and simulating the function Rectangular Ritβ, in the book βRecent Researches in Circuits, Communications and Signal Processingβ, Proceedings of the 7th WSEAS International Conference on Circuits, Systems, Signal and Telecommunications (CSST '13), Milan, Italy, publisher WSEAS Press, ISBN: 978-1-61804-151-7, January 2013, pp.97-107.
46. Wikipedia, βWindow functionβ.
International Journal of Digital Information and Wireless Communications (IJDIWC) 3(4): 77-95The Society of Digital Information and Wireless Communications, 2013 (ISSN: 2225-658X)
95