Mathematical Methods for Information Science and Economics
Rectangular base function CLAUDE ZIAD BAYEH1, 2 1
Faculty of Engineering II, Lebanese University 2 EGRDI transaction on mathematics (2012) LEBANON Email:
[email protected]
NIKOS E.MASTORAKIS WSEAS (Research and Development Department) http://www.worldses.org/research/index.html Agiou Ioannou Theologou 17-2315773, Zografou, Athens,GREECE
[email protected] Abstract: - The Rectangular base function is a new function introduced in the mathematical domain by the author and it has important applications in mathematics, signal processing and signal theory in which it can be used as a filter for three cases at the same time by varying only two parameters, it can be used as a low pass filter, band pass filter and high pass filter. The concept of this function is similar to traditional rectangular function with some different points, for example the Rectangular base function do not have a center as the traditional rectangular function but it has a beginning point (or edge) and an ending point and these points can be finite or infinite which is not the case of the traditional rectangular function. The output value is equal to 1 between the beginning and the ending points and it is equal to zero otherwise (outside the beginning and ending points). This difference between the traditional rectangular function and the new Rectangular base function gives the new function many advantages and an important role to play in the filtering process of signals. Key-words:- Rectangular function, Signal processing, Signal theory, Signals, Mathematics. filter, band pass filter and the high pass filter, the traditional one can only behave as the two first filters. The fourth different point is that the new function has variable edges (left and right edges are variables and they can vary from −∞ to +∞), this is not the case of the traditional one. So as conclusion the new function has many advantages ahead the traditional one especially in signal theory and signal processing when the three filters (low pass, band pass and high pass) are needed in the same time in some requirements.
1 Introduction In mathematics, in signal theory and in signal processing, the rectangular function and the boxcar function are very important tools used to filter the signals and to separate the used signals from the unused signals. They can be used in the time domain and in the frequency domain; in general they are used in the frequency domain to separate the wanted frequencies from the unwanted frequencies, in this way we can filter the signals [1-8]. But the rectangular function and the boxcar function are used as low pass filter and band pass filter; they can’t behave as a high pass filter due to their limitation, so this is the disadvantage of these functions. In this paper the author introduced a new function called Rectangular base function; this function is similar to the traditional rectangular function with some difference points. The first different point is that the new function has only two outputs values which are 0 and 1 comparing to the traditional rectangular function that has 3 outputs values (0, ½ and 1). The second different point is that the new function can have an infinite rectangular width which is not the case of the traditional one. The third different point is that the new function can behave as three different filters which are the low pass
ISBN: 978-1-61804-148-7
In the second section, a definition of the traditional rectangular function is presented. In the third section, a definition of the Rectangular base function is presented. In the fourth section, some applications of the Rectangular base function in signal theory are presented. And finally in the section 5, a conclusion is presented.
2 Definition of the Rectangular function
traditional
The rectangular function (also known as the rectangle function, rect function, Pi function, gate
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function, unit pulse, or the normalized boxcar function) is defined as:
3 Definition of the Rectangular base function
⎧ 0 𝑓𝑓𝑓𝑓𝑓𝑓 |𝑥𝑥| > 2 ⎪1 1 (1) 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟(𝑥𝑥) = 2 𝑓𝑓𝑓𝑓𝑓𝑓 |𝑥𝑥| = 2 ⎨ ⎪ 1 𝑓𝑓𝑓𝑓𝑓𝑓 |𝑥𝑥| < 1 ⎩ 2 This function can be used in signal theory and signal processing, it can be used as an ideal filter for low pass filter and band pass filter but it can’t be used as high pass filter.
1
The Rectangular base function is defined as the following: 1 𝑓𝑓𝑓𝑓𝑓𝑓 𝑥𝑥 ∈ [𝑥𝑥𝑏𝑏 ; 𝑥𝑥𝑒𝑒 ] 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑥𝑥 𝑏𝑏 ;𝑥𝑥 𝑒𝑒 (𝑥𝑥) = � (2) 0 𝑓𝑓𝑓𝑓𝑓𝑓 𝑥𝑥 ∉ [𝑥𝑥𝑏𝑏 ; 𝑥𝑥𝑒𝑒 ]
Fig. 1: represents the traditional rectangular function.
Fig. 2: represents the Rectangular base function.
In other meaning it is also equal to: 1 𝑓𝑓𝑓𝑓𝑓𝑓 𝑥𝑥𝑏𝑏 ≤ 𝑥𝑥 ≤ 𝑥𝑥𝑒𝑒 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑥𝑥 𝑏𝑏 ;𝑥𝑥 𝑒𝑒 (𝑥𝑥) = � (3) 0 𝑜𝑜𝑜𝑜ℎ𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 With 𝑥𝑥𝑏𝑏 ≤ 𝑥𝑥𝑒𝑒 , 𝑥𝑥𝑏𝑏 ∈ ℝ and 𝑥𝑥𝑒𝑒 ∈ ℝ 𝑥𝑥𝑏𝑏 is the beginning point of the function, it can take any value less or equal to 𝑥𝑥𝑒𝑒 . 𝑥𝑥𝑒𝑒 is the ending point of the function, it can take any value greater or equal to 𝑥𝑥𝑏𝑏 .
This function is similar to the boxcar function but with some difference points. The first different point is that the new function has only two outputs values which are 0 and 1 comparing to the traditional rectangular function that has 3 outputs values (0, ½ and 1). The second different point is that the new function can have an infinite rectangular width which is not the case of the traditional one. The third different point is that the new function can behave as three different filters which are the low pass filter, band pass filter and the high pass filter, the traditional one can only behave as the two first filters. The fourth different point is that the new function has variable edges (left and right edges are variables and they can vary from −∞ to +∞), this is not the case of the traditional one because the boxcar function is limited and can’t take an infinite value as it is equal to Πa,b (𝑥𝑥) = 𝑢𝑢(𝑥𝑥 − 𝑎𝑎) − 𝑢𝑢(𝑥𝑥 − 𝑏𝑏) which is limited to a rectangular form, in this case the values of 𝑎𝑎 and 𝑏𝑏 can’t be infinite, but the Rectangular base function can take an infinite value.
2.1 Boxcar function The boxcar function is the generalized function of the rectangular function and it is defined as: Πa,b (𝑥𝑥) = 𝑢𝑢(𝑥𝑥 − 𝑎𝑎) − 𝑢𝑢(𝑥𝑥 − 𝑏𝑏) = 0 𝑓𝑓𝑓𝑓𝑓𝑓 𝑥𝑥 < 𝑎𝑎 𝑎𝑎𝑎𝑎𝑎𝑎 𝑥𝑥 > 𝑏𝑏 1
�2 𝑓𝑓𝑓𝑓𝑓𝑓 𝑥𝑥 = 𝑎𝑎 𝑎𝑎𝑎𝑎𝑎𝑎 𝑥𝑥 = 𝑏𝑏
(2)
1 𝑓𝑓𝑓𝑓𝑓𝑓 𝑎𝑎 < 𝑥𝑥 < 𝑏𝑏
In this function, the values of 𝑎𝑎 and 𝑏𝑏 can’t be infinite because it is included in the Heaviside function 𝑢𝑢(𝑥𝑥 − 𝑎𝑎) and 𝑢𝑢(𝑥𝑥 − 𝑏𝑏). So it can’t be used as a high pass filter. This function can be used in signal theory and signal processing, it can be used as an ideal filter for low pass filter and band pass filter but it can’t be used as high pass filter. This is the disadvantage of this function because the value of 𝑎𝑎 and 𝑏𝑏 can’t take the value ±∞.
ISBN: 978-1-61804-148-7
This function has enormous applications in signal processing and signal theory in which it can behave as an ideal low pass filter, band pass filter and high
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pass filter by varying only two parameters. This is not the case of any other function, in fact the traditional rectangular function can behave as a low pass filter and a band pass filter but it can’t behave as a high pass filter.
4 Application of the Rectangular base function in signal theory The Rectangular base function can be used as a filter for signals in real time domain or in frequency domain. By varying only two parameters we can change from one filter to the other. So a simple circuit can be made with two variables in order to be as a filter and this filter will change its edges to behave as one of the three filters (low pass, band pass and high pass). We can change from one filter to the other by changing only two variables which can be controlled electronically using Thyristors or any other active electronic components.
Fig. 4: represents the band pass filter for the function Rectangular base function. Example: for 𝑥𝑥𝑏𝑏 = 2 and 𝑥𝑥𝑒𝑒 = 5 1 𝑓𝑓𝑓𝑓𝑓𝑓 𝑥𝑥 ∈ [2; 5] 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅2;5 (𝑥𝑥) = � 0 𝑓𝑓𝑓𝑓𝑓𝑓 𝑥𝑥 ∉ [2; 5]
4.3 High pass filter
For high pass filter we have 𝑥𝑥𝑏𝑏 is limited and 𝑥𝑥𝑒𝑒 = +∞ is unlimited
4.1 Low pass filter
For low pass filter we have 𝑥𝑥𝑏𝑏 = 0 and 𝑥𝑥𝑒𝑒 is limited Therefore, 1 𝑓𝑓𝑓𝑓𝑓𝑓 𝑥𝑥 ∈ [0; 𝑥𝑥𝑒𝑒 ] 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅0;𝑥𝑥 𝑒𝑒 (𝑥𝑥) = � 0 𝑓𝑓𝑓𝑓𝑓𝑓 𝑥𝑥 ∉ [0; 𝑥𝑥𝑒𝑒 ] Fig. 5: represents the high pass filter for the function Rectangular base function. Example: for 𝑥𝑥𝑏𝑏 = 2 and 𝑥𝑥𝑒𝑒 = +∞ 1 𝑓𝑓𝑓𝑓𝑓𝑓 𝑥𝑥 ∈ [2; +∞] 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅2;+∞ (𝑥𝑥) = � 0 𝑓𝑓𝑓𝑓𝑓𝑓 𝑥𝑥 ∉ [2; +∞]
Remark: in mathematics, usually we write (+∞[ ) or ( ] − ∞). But here, when we write (+∞] ) or ( [−∞), we mean that the edge is open and it is going to the infinite and it is impossible to be changed from 0 to 1 or from 1 to 0.
Fig. 3: represents the low pass filter for the function Rectangular base function. Example: for 𝑥𝑥𝑒𝑒 = 5 1 𝑓𝑓𝑓𝑓𝑓𝑓 𝑥𝑥 ∈ [0; 5] 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅0;5 (𝑥𝑥) = � 0 𝑓𝑓𝑓𝑓𝑓𝑓 𝑥𝑥 ∉ [0; 5]
5 Conclusion
4.2 Band pass filter
In this paper, a new function called Rectangular base function is introduced by the author. It is similar to the traditional rectangular function with some different points which are presented in this paper. As conclusion, this new function has enormous applications is signal theory and signal processing and it can be used as a filter with three different types (low pass filter, band pass filter, and high pass filter) in the same circuit. Many studies
For band pass filter we have 𝑥𝑥𝑏𝑏 and 𝑥𝑥𝑒𝑒 are limited in values 1 𝑓𝑓𝑓𝑓𝑓𝑓 𝑥𝑥 ∈ [𝑥𝑥𝑏𝑏 ; 𝑥𝑥𝑒𝑒 ] 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑥𝑥 𝑏𝑏 ;𝑥𝑥 𝑒𝑒 (𝑥𝑥) = � 0 𝑓𝑓𝑓𝑓𝑓𝑓 𝑥𝑥 ∉ [𝑥𝑥𝑏𝑏 ; 𝑥𝑥𝑒𝑒 ]
ISBN: 978-1-61804-148-7
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will follow this one in order to find more applications in mathematics, signal theory and signal processing to emphasize the good role of this function in filtering signals. References: [1] von Seggern D. , “CRC Standard Curves and Surfaces”, Boca Raton, FL: CRC Press, (1993) p. 324. [2] Bracewell R., “Rectangle Function of Unit Height and Base, ”, In “The Fourier Transform and Its Applications”. New York: McGraw-Hill, (1965), pp. 52-53. [3] Bracewell R., “The Fourier Transform and Its Applications, 3rd ed.”, New York: McGrawHill, (1999), pp. 100-101. [4] Arfken G., “Development of the Fourier Integral,” “Fourier Transforms-Inversion Theorem,” and “Fourier Transform of Derivatives.” §15.2-15.4 in “Mathematical Methods for Physicists, 3rd ed.” Orlando, FL: Academic Press, (1985), pp. 794-810. [5] Blackman R. B., Tukey J. W., “The Measurement of Power Spectra, From the Point of View of Communications Engineering.”, New York: Dover, (1959). [6] Brigham E. O., “The Fast Fourier Transform and Applications.” Englewood Cliffs, NJ: Prentice Hall, (1988). [7] Kammler D. W., “A First Course in Fourier Analysis.” Upper Saddle River, NJ: Prentice Hall, (2000). [8] Körner T. W., “Fourier Analysis.” Cambridge, England: Cambridge University Press, (1988).
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