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Comprehensive Analysis of π Base Exponential Functions as a Window by ijcsiseditor


									                                                          (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
     , 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

                                                                                                       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

                                                                                                          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

   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.

                                                                                                                    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

                                                                                                                  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
                  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

                                                                                                                       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
   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. Kitney, ―Real-time heart rate variability
                                                                                                 extraction using the Kaiser window,‖ IEEE Trans. On Biomedical
                                                                                                 Engineering, vol. 44, no. 10, pp. 990–1005, 1997.
                               VI.           CONCLUSION
                                                                                           [3]   R. M. Rangayyan, Biomedical Signal Analysis: A Case-Study Approach,
    In this paper an improved class of window family based on                                    Wiley-IEEE Press, New York, NY, USA, 2002.
exponential function with π base is proposed. The proposed                                 [4]   S. He and J.-Y. Lu, ―Sidelobe reduction of limited diffraction beams
window has been derived in the same way of the derivation of                                     with Chebyshev aperture apodization,‖ Journal of the Acoustical Society
Kaiser Window, but it has the advantage of having no power                                       of America, vol. 107, no. 6, pp. 3556–3559, 2000.
series expansion in its time domain function. The spectrum                                 [5]   E. Torbet, M. J. Devlin,W. B. Dorwart, et al., ―Ameasurement of the
                                                                                                 angular power spectrum of the microwave background made from the
comparisons with the Kaiser window for the same window

                                                                                                                            ISSN 1947-5500
                                                                  (IJCSIS) International Journal of Computer Science and Information Security,
                                                                  Vol. 10, No. 9, September 2012

      high Chilean Andes,‖ The Astrophysical Journal, vol. 521, pp. L79–L82,             Circuits and Systems (ISCAS ’89), vol. 1, pp. 359– 362, Portland, Ore,
      1999.                                                                              USA, May 1989.
[6]   B. Picard, E. Anterrieu, G. Caudal, and P. Waldteufel, ―Improved              [10] F. J. Harris, ―On the use of windows for harmonic analysis with the
      windowing functions for Y-shaped synthetic aperture imaging                        discrete                    Fourier transform,‖ Proceedings of the IEEE,
      radiometers,‖ in Proc. IEEE International Geoscience and Remote                    vol. 66, no. 1, pp. 51–83, 1978.
      Sensing Symposium (IGARSS ’02), vol.5, pp. 2756–2758, Toronto, Ont,           [11] R. L. Streit, ―A two-parameter family of weights for nonrecursive digital
      Canada, June 2002.                                                                 filters and antennas,‖ IEEE Trans. Acoustics, speech, and Signal
[7]   P. Lynch, ―The Dolph-Chebyshev window: a simple optimal filter,‖                   Processing, vol. 32, no. 1, pp. 108–118, 1984.
      Monthly Weather Review, vol. 125, pp. 655–660, 1997.                          [12] A. G. Deczky, ―Unispherical windows,‖ in Proc. IEEE Int. Symp.
[8]   J. F. Kaiser, ―Nonrecursive digital filter design using I0-sinh window             Circuits and Systems (ISCAS ’01), vol. 2, pp. 85–88, Sydney, NSW,
      function.,‖ in Proc. IEEE Int. Symp. Circuits and Systems (ISCAS ’74),             Australia, May 2001.
      pp. 20–23, San Francisco, Calif, USA, April 1974.
[9]   T. Saram¨aki, ―A class of window functions with nearly minimum
      sidelobe energy for designing FIR filters,‖ in Proc. IEEE Int. Symp.

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