research papers mathematics and applied mathematics science journal 4658

The General Science Journal Claude Ziad BAYEH

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Hyperbolic Tangent Filter in Signal Processing

CLAUDE ZIAD BAYEH1, 2

1Faculty of Engineering II, Lebanese University 2EGRDI transaction on mathematics (2012)

LEBANON Email: [email protected]

Abstract: - The Hyperbolic tangent Filter is an original method introduced by the author in signal processing in 2012; it is a filter type in the Amplitude-oriented design of filters. It is more similar to the Butterworth filter but it uses the function Hyperbolic Tangent for frequencies instead of the polynomial “w” frequency. It has many advantages over the traditional filters such as -It has no ripples, similar to the Butterworth filter. -It attenuates faster than other existing filters for the same order (compared to Butterworth, Chebyshev, Elliptic Filter…). -It is more flexible than other filters by varying some parameters. -It is an excellent attenuator for unneeded frequencies and excellent conservator for the needed frequencies. Briefly, a comparative study is presented in this paper to compare the traditional filters such as Butterworth and Chebyshev with the new proposed filter Hyperbolic Tangent Filter. Key-words:- Filters, Analog filters, Digital filters, Signal processing, Mathematics, Amplitude oriented design.

1. Introduction In signal processing, a filter is a device or process that removes from a signal some unwanted component or feature [1-3]. Filtering is a class of signal processing, the defining feature of filters being the complete or partial suppression of some aspect of the signal [6-7]. Most often, this means removing some frequencies and not others in order to suppress interfering signals and reduce background noise. However, filters do not exclusively act in the frequency domain; especially in the field of image processing many other targets for filtering exist [10-12]. The drawback of filtering is the loss of information associated with it. Signal combination in Fourier space is an alternative approach for removal of certain frequencies from the recorded signal. There are many different bases of classifying filters and these overlap in many different ways; there is no simple hierarchical classification. Filters may be: • analog or digital • discrete-time (sampled) or continuous-time • linear or non-linear • Time-invariant or time-variant, also known as shift invariance. If the filter operates in a spatial

domain then the characterization is space invariance. • passive or active type of continuous-time filter • Infinite impulse response (IIR) or finite impulse response (FIR) type of discrete-time or digital

filter.

mailto:[email protected]


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In fact, many scientists introduced many filters in order to filter the signal, keep the needed frequencies and attenuate the unneeded frequencies. The more important thing is the implementation of these mathematical expressions in electronics and the design of analog and digital circuits. The important filters are Butterworth, Chebyshev 1, Chebyshev 2, and Elliptic filter. In this paper, the author introduced a new filter design (amplitude oriented design) in signal processing in order to increase the efficiency of filtering signals. The name of this filter is “Hyperbolic Tangent Filter”. This filter has many advantages over previous ones, and these advantages are discussed in this paper. In the second section, a brief review of ideal filter is presented. In section 3, Amplitude oriented design is discussed with some examples of filters such as Butterworth and Chebyshev filters. In section 4, the author proposed a new filter based on Hyperbolic Tangent function. And finally in the section 5, a conclusion is presented. 2. Brief review of the Ideal Filter In this section, we are going to see a brief introduction to the Ideal Filter [1]. Let’s consider a system, as shown in figure 1 whose input is 𝑓𝑓(𝑡𝑡) and output is 𝑔𝑔(𝑡𝑡). With the original notation of the Fourier transform we have 𝑓𝑓(𝑡𝑡) ↔ 𝐹𝐹(𝑗𝑗𝑗𝑗) (1) And 𝑔𝑔(𝑡𝑡) ↔ 𝐺𝐺(𝑗𝑗𝑗𝑗) (2) The transfer function of the system is: 𝐻𝐻(𝑗𝑗𝑗𝑗) = 𝐺𝐺(𝑗𝑗𝑗𝑗 )

𝐹𝐹(𝑗𝑗𝑗𝑗 )= |𝐻𝐻(𝑗𝑗𝑗𝑗)| ∙ 𝑒𝑒𝑗𝑗𝑗𝑗 (𝑗𝑗) (3)

Where |𝐻𝐻(𝑗𝑗𝑗𝑗)| is the amplitude response and 𝑗𝑗(𝑗𝑗) is the phase response. The system is called an ideal filter if its amplitude is constant within certain frequency bands, and exactly zero outside these bands.

Fig. 1: A linear system with input 𝑓𝑓(𝑡𝑡) and output 𝑔𝑔(𝑡𝑡)

In addition, in the bands where the amplitude is constant, the phase is a linear function of 𝑗𝑗. The amplitude response of the ideal filter is shown in figure 2 for the four main types of low-pass, high-pass, band-pass and band-stop filters.


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Fig. 2: The ideal filter amplitude characteristics

The ideal phase response for the low-pass case is shown in figure 3 and is given by 𝑗𝑗(𝑗𝑗) = −𝑘𝑘𝑗𝑗 with |𝑗𝑗| ≤ 𝑗𝑗0 Where 𝑘𝑘 is a constant.

Fig. 3: The ideal low-pass filter phase characteristics

To appreciate why these are the desirable ideal characteristics, let’s consider the low-pass case described by the transfer function

𝐻𝐻(𝑗𝑗𝑗𝑗) = �𝑒𝑒(−𝑗𝑗𝑘𝑘𝑗𝑗 ) 𝑓𝑓𝑓𝑓𝑓𝑓 0 ≤ |𝑗𝑗| ≤ 𝑗𝑗00 𝑓𝑓𝑓𝑓𝑓𝑓 |𝑗𝑗| > 𝑗𝑗0

� (4)

Then 𝐺𝐺(𝑗𝑗𝑗𝑗) = 𝐻𝐻(𝑗𝑗𝑗𝑗) ∙ 𝐹𝐹(𝑗𝑗𝑗𝑗) So we have for 0 ≤ |𝑗𝑗| ≤ 𝑗𝑗0 𝐺𝐺(𝑗𝑗𝑗𝑗) = 𝑒𝑒(−𝑗𝑗𝑘𝑘𝑗𝑗 ) ∙ 𝐹𝐹(𝑗𝑗𝑗𝑗) (5) The inverse Fourier transform of the equation (5) is 𝑔𝑔(𝑡𝑡) = 𝑓𝑓(𝑡𝑡 − 𝑘𝑘) (6) It means that the output is an exact replica of the input, but delayed by a constant time value 𝑘𝑘. It means that any input signal with spectrum lying within the pass-band of the ideal filter will be


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transmitted without attenuation and without distortion in its phase spectrum; the signal is simply delayed by a constant time value. If we take the inverse Fourier transform of 𝐻𝐻(𝑗𝑗𝑗𝑗), we obtain the impulse response of the ideal filter as ℎ(𝑡𝑡) = 𝐹𝐹−1{𝐻𝐻(𝑗𝑗𝑗𝑗)} = sin(𝑗𝑗0(𝑡𝑡−𝑘𝑘))

𝜋𝜋(𝑡𝑡−𝑘𝑘) (7)

which is a cardinal sine function (𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠) and clearly exists for negative values of time, so that the required system is non-causal, which means that it is unrealizable by physical components. So in conclusion, the ideal filter can’t be realizable for the instance. Therefore, many scientists have introduced their own methods to resolve this problem, and the following section will introduce briefly the main methods used to implement realizable filters. 3. Amplitude-oriented design We now discuss the amplitude approximation problem. This consists in finding a realizable magnitude function |𝐻𝐻(𝑗𝑗𝑗𝑗)| or |𝐻𝐻(𝑗𝑗𝑗𝑗)|2 which is capable of meeting arbitrary specifications on the amplitude response of the filter. The discussion will be on the design of low-pass filters because the others are deduced from the low-pass filters. The low-pass approximation problem is to determine |𝐻𝐻(𝑗𝑗𝑗𝑗)|2 such that the typical specifications shown in figure 4 are met.

Fig. 4: Tolerance schemes for amplitude-oriented filter design, magnitude-squared (in the left) and

attenuation (in the right) The frequency 𝑗𝑗0 is called the pass-band edge or cut-off frequency while 𝑗𝑗𝑠𝑠 is referred to as the stopband edge. The amplitude-squared function of the filter may be written as |𝐻𝐻(𝑗𝑗𝑗𝑗)|2 = ∑ 𝑎𝑎𝑓𝑓𝑗𝑗2𝑓𝑓𝑚𝑚

𝑓𝑓=01+∑ 𝑏𝑏𝑓𝑓𝑗𝑗2𝑓𝑓𝑠𝑠

𝑓𝑓=1 (8)

The problem may be posed as one of determining the coefficients 𝑎𝑎𝑓𝑓 and 𝑏𝑏𝑓𝑓 such that the above function is capable of meeting an arbitrary set of specifications. In the following subsections we are going to see the existing filters such as Butterworth filter and Chebyshev filter. 3.1. Maximally Flat Response (Filter) It is also called Butterworth response.


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Fig. 5: General appearance of the maximally flat amplitude response for the Butterworth Filter

This is obtained by forcing the maximum possible number of derivatives of |𝐻𝐻(𝑗𝑗𝑗𝑗)|2, with respect to 𝑗𝑗, to vanish at 𝑗𝑗 = 0 and 𝑗𝑗 = ∞. So the Butterworth filter takes the following form: |𝐻𝐻(𝑗𝑗𝑗𝑗)|2 = 1

1+𝑗𝑗2𝑠𝑠 (9) With 𝑠𝑠 is an integer and 𝑠𝑠 ∈ ℕ, it is called the degree of the filter and the 3dB occurs at 𝑗𝑗 = 1 for all 𝑠𝑠 3.2. Chebyshev Response For the same degree 𝑠𝑠 as the Butterworth Filter, a considerable improvement in the rate of cut-off, over the Butterworth response, results if we require |𝐻𝐻(𝑗𝑗𝑗𝑗)|2 to be equiripple in the passband while retaining the maximally flat response in the stopband. The function takes the following form: |𝐻𝐻(𝑗𝑗𝑗𝑗)|2 = 1

1+𝜀𝜀2𝑇𝑇𝑠𝑠2(𝑗𝑗) (10)

Where 𝑇𝑇𝑠𝑠(𝑗𝑗) is chosen to be an odd or even polynomial which oscillates between -1 and +1 the maximum number of times in the passband |𝑗𝑗| ≤ 1 and is monotonically increasing outside the interval. The size of the oscillations or ripple can be controlled by a suitable choice of the parameter 𝜀𝜀. The polynomial 𝑇𝑇𝑠𝑠(𝑗𝑗) which leads to these desired properties is the Chebyshev polynomial of the first kind defined by:

𝑇𝑇𝑠𝑠(𝑗𝑗) = �cos(𝑠𝑠 ∙ 𝑠𝑠𝑓𝑓𝑠𝑠−1(𝑗𝑗)) 𝑓𝑓𝑓𝑓𝑓𝑓 0 ≤ 𝑗𝑗 ≤ 1cosh(𝑠𝑠 ∙ 𝑠𝑠𝑓𝑓𝑠𝑠ℎ−1(𝑗𝑗)) 𝑓𝑓𝑓𝑓𝑓𝑓 |𝑗𝑗| > 1

� (11)

Using the recurrence formula, we obtain, 𝑇𝑇𝑠𝑠+1(𝑗𝑗) = 2𝑗𝑗𝑇𝑇𝑠𝑠(𝑗𝑗) − 𝑇𝑇𝑠𝑠−1(𝑗𝑗) (12) With 𝑇𝑇0(𝑗𝑗) = 1, 𝑇𝑇1(𝑗𝑗) = 𝑗𝑗


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Fig. 6: Chebyshev response for 𝑠𝑠 = 3 and 𝜀𝜀 = 0.5

The Chebyshev approximation is known to be the optimum solution to the problem of determining an |𝐻𝐻(𝑗𝑗𝑗𝑗)|2 which is constrained to lie in a band for 0 ≤ |𝑗𝑗| ≤ 1 and attain the maximum value for all 𝑗𝑗 in the range 1 < 𝑗𝑗 < ∞ for a given degree 𝑠𝑠. 4. Proposed Filter by the Author (Hyperbolic Tangent Filter) This proposed filter by the author works mainly as low-pass filter. The high-pass filter, band-pass filter and band-stop filter can be deduced from the low-pass filter by varying some parameters as shown in section 4.2. Its advantages are presented in the section 4.3. 4.1. Hyperbolic Tangent Filter-Low pass filter The expression of the filter for a causal system is: 𝑇𝑇𝑎𝑎𝑠𝑠ℎ𝐹𝐹(𝑗𝑗) = �1 − tanh(𝑓𝑓 ∙ 𝑗𝑗2𝑠𝑠)2𝑚𝑚 (13) With - 𝑓𝑓 > 0 and 𝑓𝑓 ∈ ℝ+, for unitary filter we take it equal to 1. - 𝑠𝑠 ∈ ℕ∗ is the order of the filter - 𝑚𝑚 ∈ ℕ∗ is the sharpness of the filter In the following figure 7, different orders and different shapes are formed. To form a low-pass filter, we shall take the following parameters: - 𝑓𝑓 = 1 - 𝑠𝑠 ∈ ℕ is the order of the filter - 𝑚𝑚 = 1


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a) 𝑠𝑠 = 2

b) 𝑠𝑠 = 4

Fig. 7: Different forms formed by changing the order of the filter (𝑠𝑠). 4.2. Hyperbolic Tangent Filter, other types of filters -For the high-pass filter we change the value of 𝑗𝑗 to be equal to 𝑏𝑏

𝑗𝑗.

-For the band-stop filter we change the value of 𝑗𝑗 to be equal to 1

𝛽𝛽�𝑗𝑗𝑏𝑏− 𝑏𝑏𝑗𝑗�.

-For the band-pass filter we change the value of 𝑗𝑗 to be equal to 𝛽𝛽 �𝑗𝑗

𝑏𝑏− 𝑏𝑏

𝑗𝑗�.

With 𝛽𝛽 and 𝑏𝑏 > 0 4.3. Advantages of the Hyperbolic Tangent Filter -It has no ripples, similar to the Butterworth filter. -It attenuates faster than other existing filters for the same order (compared to Butterworth, Chebyshev, Elliptic Filter…). Refer to figures 8 through 10. -It is more flexible than other filters by varying some parameters. -It is an excellent attenuator for unneeded frequencies and excellent conservator for the needed frequencies.

Fig. 8: Comparison between the proposed filter Hyperbolic Tangent Filter (Black color), Chebyshev

Filter (Blue color) and Butterworth (Red color) for the same Order (𝑠𝑠 = 2).


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Fig. 9: Comparison between the proposed filter Hyperbolic Tangent Filter (Black color), Chebyshev

Filter (Blue color) and Butterworth (Red color) for the same Order (𝑠𝑠 = 3).

Fig. 10: Comparison between the proposed filter Hyperbolic Tangent Filter with 𝑚𝑚 = 3 (Black color),

Chebyshev Filter (Blue color) and Butterworth (Red color) for the same Order (𝑠𝑠 = 3). In conclusion, we can remark the importance of the Hyperbolic Tangent Filter and compare it with other types of filters. The attenuation of Hyperbolic Tangent Filter is much more important than other types of filters such as Butterworth, Chebyshev, Elliptic filter… 5. Conclusion In this paper, the author introduced a new method in signal processing. A new filter is defined named “Hyperbolic Tangent Filter”. This filter has many advantages over the traditional filters as discussed in the previous sections. In the section 2, a brief introduction about the ideal filter is discussed; the ideal filter is not applicable as it is a non-causal system. In the section 3, some important filters are presented such as Butterworth and Chebyshev filters, these types of filters can be realized using electronic components such as resistors, capacitances, inductors and semiconductors. In the section 4, the author proposed a new filter based on Amplitude oriented design. Some advantages are discussed and finally some pictures are presented to compare three filters with the same order which are: Butterworth, Chebyshev and the proposed filter Hyperbolic Tangent Filter. References: [1] Hussein Baher, “Signal processing and integrated circuits”, Published by John Wiley & Sons Ltd., ISBN:

9780470710265, 2012.


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[2] John G. Proakis, Dimitris G. Manolakis, “Digital Signal Processing, Principles, Algorithms, and Applications” Fourth edition, Pearson International Edition, ISBN: 0-13-228731-5.

[3] Niklaus Wirth, “Digital Circuit Design”, Springer, ISBN: 3-540-58577-X. [4] N. Senthil Kumar, M. Saravanan, S. Jeevananthan, “Microprocessors and Microcontrollers”, Oxford

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on Mathematics, Issue 9, Volume 8, September 2009, pp. 551-560. [17] N. Moubayed, Claude 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] Eric W. Weisstein, Trigonometric Addition Formulas, Wolfram MathWorld, 1999-2009. [20] Paul A. Foerster, Algebra and Trigonometry: Functions and Applications, Addison-Wesley publishing

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Pearson Education Canada, 2006. [23] E. Demiralp, Applications of High Dimensional Model Representations to Computer Vision, WSEAS

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Transactions on Computers Journal, 2(4), pp. 1039 – 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, 3rd

edition, New York: McGraw-Hill, pp. 61-65, 2000.

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printing, New York: Dover, 1972.

[30] Vitit Kantabutra, On hardware for computing exponential and trigonometric functions, IEEE Transactions on Computers, Vol. 45, issue 3, pp. 328–339, 1996.

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[33] Cyril W. Lander, Power electronics, third edition, McGraw-Hill Education, 1993. [34] Claude Bayeh, “Introduction to the Rectangular Trigonometry in Euclidian 2D-Space”, WSEAS

Transactions on Mathematics, ISSN: 1109-2769, Issue 3, Volume 10, March 2011, pp. 105-114. [35] Claude Ziad Bayeh, “Introduction to the Angular Functions in Euclidian 2D-space”, WSEAS

TRANSACTIONS on MATHEMATICS, ISSN: 1109-2769, E-ISSN: 2224-2880, Issue 2, Volume 11, February 2012, pp.146-157

[36] Claude Ziad Bayeh, “Introduction to the General Trigonometry in Euclidian 2D-Space”, WSEAS TRANSACTIONS on MATHEMATICS, ISSN: 1109-2769, E-ISSN: 2224-2880, Issue 2, Volume 11, February 2012, pp.158-172.

[37] Claude Bayeh, “Application of the Elliptical Trigonometry in industrial electronic systems with analyzing, modeling and simulating two functions Elliptic Mar and Elliptic Jes-x”, WSEAS Transactions on Circuits and Systems, ISSN: 1109-2734, Issue 11, Volume 8, November 2009, pp. 843-852.

[38] Claude Bayeh, “A 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-x”, in the book “ Latest Trends on Circuits, Systems and Signals”, publisher WSEAS Press, ISBN: 978-960-474-208-0, ISSN: 1792-4324, July 2010, pp.96-108.

About the author Claude Ziad Bayeh (or El-Bayeh), (born in 11 September 1982-) is a Lebanese Scientist and Researcher, he holds the following degrees: - PhD in Electronics and Signal Processing from (Princely International University)-USA-2012. - PhD in Physics and Relativity from (Princely International University)-USA-2013. - Currently PhD in Electrical Engineering from (Universidad Empresarial de Costa Rica)-Costa Rica. - Master in Electrical and Electronic Engineering from (Lebanese University Faculty of Engineering II)-Lebanon-2008. - Master in Organizational Management from (Quebec University UQAC)-Canada-2012. He has published numerous international papers in Mathematics, Engineering, Physics, Management and Chemistry (more than 100 papers). The most published papers are considered as revolutionary papers in their fields. You can find a brief history using the link: http://www.linkedin.com/pub/claude-ziad-bayeh/34/9b9/ab6 For any additional information, any question or suggestions, please don’t hesitate to contact the author.

http://www.linkedin.com/pub/claude-ziad-bayeh/34/9b9/ab6

research papers mathematics and applied mathematics science journal 4658

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