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(IJCSIS) International Journal of Computer Science and Information Security, Vol. 10, No. 9, September 2012 Comprehensive Analysis of π Base Exponential Functions as a Window Mahdi Nouri1, Sepideh Lahooti2, Sepideh Vatanpour3, and Negar Besharatmehr4 Dept. of Electrical Engineering Iran University of Science & Technology1, A.B.A Institute of Higher Education2,3,4 Tehran1, Abeyek2,3,4, Iran1,2,3,4 1 mnuri@elec.iust.ac.ir , s.lahooti@gmail.com2, s.vatanpoor@gmail.com3, n.besharatmehr@gmail.com4 Abstract—A new simple form window with the application of FIR window is to choose a proper ideal frequency-selective filter filter design based on the π Base exponential function is proposed which always have a noncausal, infinite-duration impulse in this article. An improved window having a closed simple response and then truncate (or window) its impulse response formula which is symmetric ameliorates ripple ratio in comparison with Kaiser and cosine hyperbolic windows. The [n] to obtain a linear-phase and causal FIR filter [3]. proposed window has been derived in the same way as Kaiser Window, but its advantages have no power series expansion in its ( ) h[n]= [n] w[n] ; w[n]= , - (1) time domain representation. Simulation results show that proposed window provides better ripple ratio characteristics which are so important for some applications. A comparison with Where ( ) is function of n , (M+1 ) is the length, h[n] Kaiser window shows that the proposed window reduces ripple represented as the product of the desired response [n] and a ratio in about 6.4dB which is more than Kaiser’s in the same mainlobe width. Moreover in comparison to cosine hyperbolic finite-duration ―window‖, w[n]. So the Fourier transform of window, the proposed window decreases ripple ratio in about h[n], H( ), is the periodic convolution of the desired 6.5dB which is more than cosine hyperbolic’s. The proposed frequency response, ( ), with Fourier transform of the window can realize different criteria of optimization and has window, W( ). Thus, H( ) will be a spread version of lower cost of computation than its competitors. ( ). Fourier transforms of windows can be expressed as sum Keywords-component; Window functions; Kaiser Window; FIR of frequency-shifted Fourier transforms of the rectangular filter design; Cosine hyperbolic window windows. Two desirable specifications for a window function are smaller main lobe width and good side lobe rejection I. INTRODUCTION (smaller ripple ratio). However these two requirements are FIR filters are particularly useful for applications where incongruous, since for a given length, a window with a narrow exact linear phase response is required. The FIR filter is main lobe has a poor side lobe rejection and contrariwise. The generally implemented in a non-recursive way which rectangular window has the narrowest mainlobe, it yields the guarantees a stable filter. FIR filter design essentially consists sharpest transition of H( ) at a discontinuity of ( ). of two parts, approximation problem and realization problem. So by tapering the window effortlessly to zero, side lobes are The approximation stage takes the specification and gives a transfer function through four steps [1,2]. They are as follows: greatly reduced in amplitude [2]. By increasing M, W( ) becomes narrower, and the smoothing provided by W( ) is 1) A desired or ideal response is chosen, usually in the reduced. The large sidelobes of W( ) result in some frequency domain. undesirable ringing effects in the FIR frequency response 2) An allowed class of filters is chosen (e.g. the length N H( ), and also in relatively larger sidelobes in H( ). So for a FIR filters). using windows that don’t contain abrupt discontinuities in 3) A measure of the quality of approximation is chosen. their time-domain characteristics, and have correspondingly low sidelobes in their frequency-domain characteristics is 4) A method or algorithm is selected to find the best required [3]. filter transfer function. There are different kind of windows and the best one is The realization part deals with choosing the structure to depending on the required application, Windows can be implement the transfer function which may be in the form of categorized as fixed or adjustable [9]. Fixed windows have circuit diagram or in the form of a program. The essentially only one independent parameter, namely, the window length three well-known methods for FIR filter design are the which controls the main-lobe width. Adjustable windows have window method, the frequency sampling technique and two or more independent parameters, namely, the window Optimal filter design methods [2]. The basic idea behind the 79 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 10, No. 9, September 2012 length, as in fixed windows, and one or more additional parameters that can control other window’s characteristics. The Kaiser window is a kind of two parameter windows, that have maximum energy concentration in the mainlobe, it control the mainlobe width and ripple ratio [4,8,9]. In this paper an improved two parameter window based on the exponential function is proposed, that performs better ripple ratio and lower sildelobe ( 6.42 db ) compared to the Kaiser and Cosine hyperbolic windows, while having equal mainlobe width. Also its computation reduced because of having no power series. Windows are time-domain weighting functions that are resulted from the truncation of a Fourier series. They are utilized in a variety of additional signal processing applications Figure 1. A typical window’s normalized amplitude spectrum including power spectral estimation, beamforming, signal analysis and estimation, digital filter design and speech II. CHARACTERIZATION OF WINDOW processing. In spite of their maturity, windows functions (or First, confirm that you have the correct template for your windows for short) maintain to find new roles in the paper size. This template has been tailored for output on the applications of today. The best window depends on the US-letter paper size. If you are using A4-sized paper, please applications. Very recently, windows have been used to smooth close this file and download the file for ―MSW A4 format‖. the progress of the revealing of irregular and abnormal heartbeat patterns in patients in electrocardiograms [1]. A window, w(nT), with a length of N is a time domain Medical imaging systems, such as the ultrasound, have also function which is defined by: illustrated enhanced performance when windows are used to improve the contrast resolution of the system [2]. Windows have also been utilized to aid in the classification of cosmic | | ( ) ( ) , (2) data [3, 4] and to improve the consistency of weather prediction models [5]. Windows have independent parameter, Windows are generally compared and classified in terms of namely, the window length which controls the main-lobe width their spectral characteristics. The frequency spectrum of w(nT) and one or more additional parameters that can control other can be introduced as [7]: window characteristics [6, 7, 8, 9, 12]. The Dolph-chebyshev window [10] has two parameters and produces the minimum main-lobe width for a specified maximum side-lobe level. The ( ) Kaiser window [8- 9] has two parameters and achieves close W( )= ( ) (3) approximations to distinct prelate functions that have maximum energy concentration in the mainlobe. The Kaiser and Dolph-Chebyshev windows can direct the amplitude of the Where W( ) is called the amplitude function, N is the sidelobes relative to that of the mainlobe. Kaiser window is a window length, and T is the space of time between samples. well-known flexible window and extensively used for FIR Two parameters of windows in general are the null-to null filter design and spectrum analysis applications [2], since it width BN and the main-lobe width BR. These quantities are achieves close approximation to the distinct prelate spheroid defined as BN = 2ωN and BR = 2ωR, where ωN and ωR are the functions that have maximum energy focus in the mainlobe half null-to-null and half mainlobe widths, respectively, as with adjusting its two independent parameters. Windows can shown in Fig. 1, an important window parameter is the ripple be classified as fixed or adaptable. Kaiser window is an ratio r which is defined as adaptable window that has a better sidelobe roll-off characteristic than the other well-known adjustable windows. The paper is organized as follows: Section II presents the r= ( ) characterization of window to distinguish the windows performance, and introduces Cosh and Kaiser Windows. Section III introduces the proposed window and presents Having small proportion less than unity permit to work with numerical simulations and discusses the final results. Section the bilateral of r in dB, which is IV shows the time required to compute the window coefficients for the Cosh, Kaiser and proposed windows. Section V is given a numerical comparison example for the filters using the Proposed and Kaiser windows. Finally, conclusion is given in R = 20 log ( ) (5) section VI after which the paper is equipped with related references. 80 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 10, No. 9, September 2012 R clarifies as the minimum side-lobe attenuation relative to the main lobe and −R is the ripple ratio in dB. S is the side-lobe ( √ ( ) roll-off ratio, which is defined as ( ) { | | ( ) ( ) ( ) This window provides better sidelobe roll-off ratio, but worse ripple ratio for the same window length and mainlobe is the largest side lobe and is the lower one which is width compared with Kaiser Window. It has the advantage of furthest from the main lobe. If S is the side-lobe roll-off ratio having no power series expansion in its time domain function so the Cosine hyperbolic window has less computation in dB, then s is given by compared with Kaiser one. ( ) These spectral characteristics are important performance III. PROPOSED WINDOW measures for windows. In some applications of FIR filters it is necessary to reduce the level of sidelobes below -45 dB. The goal of this work is to find a window with simple closed form formula, having equal A. Kaiser window main lobe width and smaller side lobe peak compared to the Kaiser window is one of the most useful and optimum other windows. The FIR filter is designed with the new windows. It is optimum in the sense of providing a large window to evaluate its efficiency which is given by Eq. 13,14. mainlobe width for a given stopband attenuation, which implies the sharpest transition width [1]. The trade-off between the √ ( ) ( ) mainlobe width and sidelobe area is quantified by seeking the window function that is maximally concentrated around w=0 in the frequency domain [2]. ( ) ( ) ( ) Where the is the adjustable shape parameter. The is ( ) ∑ * ( ) + ( ) defined optimum to gain better ripple ratio for proposed window. In discrete time domain, Kaiser Window is defined by [5]: ( ( ) ( )) ( ) ( √ ( ) From Fig.2, it can be easily seen that as in the case for ( ) { | | ( ) proposed window, when increases the mainlobe width ( ) increases and ripple ratio decreases. Fig.3 shows the relationship between the shape parameter and ripple ratio for Where α is the shape parameter, N is the length of window the proposed window. From this figure, the ripple ratio remains and I0(x) is the modified Bessel function of the first kind of almost constant for a change in the window length. For some order zero. applications such as the spectrum analysis, the design equations which define the window parameters in terms of the spectrum parameters are required. From Fig.3, an approximate B. Cosh Window relationship for the adjustable shape parameter can be found in terms of the ripple ratio (R) by using the curve fitting The hyperbolic cosine of x is expressed as: method as ( ) ( ) {( ) ( ) ∑ ( ) ( ) ( ) ( ) ( ) ( ) Fig. 2, shows that the functions Cosine hyperbolic(x) and I0(x) have the same Fourier series characteristics [7]. Cosine- The approximation model for the adjustable shape parameter hyperbolic window is proposed as: given by Eq.16 is plotted in Fig.4 It can be seen that the model provides a good approximation with an error plotted in Fig.4 The largest deviation in alpha is 0.1 which corresponds to an error of 0.4dB in actual ripple ratio. 81 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 10, No. 9, September 2012 plotted in Fig.e. The percentage error in the model changes between 0.2 and −0.25. Figure 2. Proposed window spectrum in dB for = 0, 2 and N=50 Figure 4. Error curve of approximated given for N = 51 Figure 3. The relation between and R for the proposed window Figure 5. Relation between ripple ratio and Dw for cosine hyperbolic and proposed window in N =50 As for the Kaiser model given in, the largest deviation in alpha is 0.07, but this corresponds to an error of 0.44dB in This error range satisfies the error criterion in which states actual ripple ratio. More accurate results can be obtained by that the predicted error in the normalized width must be smaller restricting the range, but the proposed model is adequate for than 1%. An integer value of the window length (N) can be predicted from most applications like Kaiser model. The second design equation is the relation between the ( ) window length and the ripple ratio. To predict the window length (N) for a given quantities of the ripple ratio (R) and half To find a suitable window which satisfies the given mainlobe width ( ), the normalized width D = 2 (N − 1) is prescribed filter specification, it is necessary to obtain the used. The relation between the normalized width and the ripple relation between the window parameters and filter parameters. ratio for the proposed window with N = 51 is given in Fig.5. By Fig.6 shows the relation between the window adjustable using the curve fitting method for Fig.5, an approximate parameter ( ) and the minimum stop band attenuation ( ) for Design relationship between the normalized width (D) and the N = 50. It is seen that as the window parameter increases the ripple ratio(R) can be established as: minimum stop band attenuation also increases. By using the curve fitting method, an approximate expression as a first filter design equation can be found as: { ( ) ( ) ( ) The approximation model for the normalized width given by {( ) ( ) ( ) ( ) Eq.17 is plotted in Fig.5. The relative error of approximated ( ) ( ) ( ) normalized width in percent versus the ripple ratio for N = 50 is 82 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 10, No. 9, September 2012 Figure 6. Relation between and the minimum stop band attenuation for the Figure 8. Comparison between the proposed and Kaiser windows with N=50 proposed window with N =50 ∑ ( ) + ( ) As known from the fixed windows such as the rectangular and Hamming windows, while the window length increases the mainlobe width decreases ,but the ripple ratio remains almost constant .As for the adjustable parameter , a larger value of results in a wider mainlobe width and a smaller ripple ratio. Fig.7 shows the Comparison of the proposed and Kaiser Windows for N =51 and =6. It can be observed that the both of window have same mainlobe but the proposed widow have smaller and narrower mainlobe width. By decreasing =5.6 both of window have same mainlobe but proposed window have smaller ripple ratio. Figure 7. Error curve of approximated given by Eq.(19) versus As for N =50 The approximation model for the adjustable shape parameter given by Eq.19 is plotted in Fig.6. It is seen that the model provides a good approximation with an error plotted in Fig.7. IV. PERFORMANCE ANALYSIS A. Kaiser window In discrete time domain, the Kaiser window is defined by . √ ( ) / ( ) { ( ) ( ) Figure 9. Comparison between the proposed and Cosine hyperbolic Where is the adjustable parameter, and (x) is the windows with N=50 and modified Bessel function of the first kind of order zero which can be described by the power series expansion as 83 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 10, No. 9, September 2012 A. Cosine Hyperbolic Window length and normalized width show that the proposed window provides a better ripple ratio than Kaiser window and larger The cosine hyperbolic window is defined by sidelobe roll-off ratio which maybe useful for some ( √ ( ) ) applications. The last spectrum comparison is performed with ( ) { ( ) window, and two specific examples show signs of that for ( ) narrower mainlobe width and smaller ripple ratio. Moreover, the paper presents the application of the proposed window in From Fig.8 the functions cosh(x) and (x) have the same the area of FIR filter design. The filter design equations for the shape characteristic. Fig.9 shows the Comparison of the proposed window to meet the given lowpass filter specification proposed Window with the cosh window with N=50. It can be are established and the comparison with Kaiser window is observed that the both of window have often same mainlobe discussed. The simulation results show that the filters designed but the proposed widow have smaller and narrower ripple ratio. by the proposed window provide better minimum stopband attenuation also they perform significantly better maximum stopband attenuation than the filters designed by other windows. V. APPLICATION TO FIR FILTER DESIGN FIR filter design is almost entirely restricted to discrete time ACKNOWLEDGMENT (HEADING 5) implementations. The design techniques for FIR filters are The preferred spelling of the word ―acknowledgment‖ in based on directly approximating the desired frequency response America is without an ―e‖ after the ―g‖. Avoid the stilted of the discrete time system [2]. In order to show the efficiency expression, ―One of us (R. B. G.) thanks . . .‖ Instead, try ―R. B. G. thanks‖. Put sponsor acknowledgments in the unnum- of the proposed window and compare the results with the other bered footnote on the first page. windows, an example of designing an FIR low pass filter by windowing of an ideal IIR low pass filter is considered. Having REFERENCES a cut-off frequency of ωC, the impulse response of an ideal low The template will number citations consecutively within pass filter is: brackets [1]. The sentence punctuation follows the bracket [2]. Refer simply to the reference number, as in [3]—do not use ( ) ( ) ( ) ( ) ―Ref. [3]‖ or ―reference [3]‖ except at the beginning of a sentence: ―Reference [3] was the first . . .‖ ( ) ( ) ( ) Number footnotes separately in superscripts. Place the actual footnote at the bottom of the column in which it was cited. Do not put footnotes in the reference list. Use letters for table footnotes. Unless there are six authors or more give all authors' names; do not use ―et al.‖. Papers that have not been published, even if they have been submitted for publication, should be cited as ―unpublished‖ [4]. Papers that have been accepted for publication should be cited as ―in press‖ [5]. Capitalize only the first word in a paper title, except for proper nouns and element symbols. For papers published in translation journals, please give the English citation first, followed by the original foreign-language citation [6]. [1] Shukla, P.; Soni, V.; Kumar, M ―Nonrecursive Digital FIR Filter Design by 3-Parameter Hyperbolic Cosine Window: A High Quality Low Order Filter Design‖ International Conference on Digital Object Identifier, communications and signal processing (ICCSP), 2011, Figure 10. The filters designed by the Proposed and Kaiser Windows pp.331-335 for wct = 0.4 and w = 0.2 rad /sample with N = 50 [2] S. R. Seydnejad and R. I. 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