Performance Of Modified Cosh Window Function

In this paper, we have analyzed the performance of modified Cosh window function.A new parameter has been proposed to the previous 2-parameter Cosh window to improve its spectral characteristics in terms of ripple-ratio and transition width.It is observed by several iterations that by increasing the values of new parameter so proposed there is a significant variation in ripple ratio and transition width of the window spectrum.By the suitable combination of two parameters of the proposed window function, an optimum modified Cosh window is derived for N=51 and N=101.Optimum window provides better spectral characteristics. This modified window has applications in digital FIR designing, image processing as it provides better minimum stop band attenuation and contrast ratio respectively.An example has been taken to show the better performance of the proposed window for FIR filter designing for specified requirements.


INTRODUCTION
Electronic filters are electronic circuits which perform signal processing functions, specifically to remove unwanted frequency components from the signal. Digital filters are classified as finite impulse response (FIR) and infinite impulse response (IIR) filters based on the duration of their impulse response. Window functions are widely used to truncate the infinite response of a filter to make the filter realizable. Various window functions have been proposed in literature.
Due to the flexible spectral properties of windows various windows like Dolph-Chebyshev, Kaiser, Barlett etc. have been proposed for signal processing applications. Since, the windows are sub-optimal solutions; every window is used according to the specifications required [13]. Cosh window when modified by introducing a new parameter, performs better in terms of ripple ratio. It provides better MSA as compared to Kaiser window, Cosh window, Exponential window in FIR filter designing [14].

WINDOW DESIGN METHOD
The desired frequency response of any digital filter is periodic in frequency and can be expanded in Fourier series [2,9], i.e.
Window functions (or simply windows) are widely used in digital signal processing for the applications in signal analysis and estimation, digital filter design and speech processing. There are many windows that have been proposed and all are suboptimal solutions, the best window depends on the specific application.

WINDOW FUNCTIONS
Window functions (or simply windows) are widely used in digital signal processing for the applications in signal analysis and estimation, digital filter design and speech processing. There are many windows that have been proposed and all are suboptimal solutions, the best window depends on the specific application. The Key parameters of a window function are [2,3]:

COSH WINDOW
The abbreviation COSH stands for Hyperbolic Cosine. A window based on this function is called as COSH window. The COSH Function and COSH window are given by following mathematical equations [10,12]: Here, N is the length of window in time domain, α is the adjustable window parameter.

MODIFIED COSH WINDOW
Modified Cosh window is simply a 3-parameter Cosh window. The basis function for this window is again Hyperbolic Cosine function. Mathematically, the modified Cosh window is given as [10,13,14]: Where, N = length of window function α, ρ = Shape changing parameters for window Fig. 1 [10] shows that Cosh window, Kaiser window, Exponential window and Modified Cosh window all have same shape characteristics. J u l y 2 0 , 2 0 1 3

Fig 1Similar Basis Functions of Several Windows
The procedure to calculate the window's spectral parameters is show by fig. 2. A typical window is taken and its frequency spectrum is plotted. The dotted lines mark the values of spectral parameters in db and rad/sec respectively.
Further the values are shown, how to observe and calculate them accurately.

Fig. 2 Amplitude Spectrum for a general window
Form the above figure [10,14], Ripple Ratio = Maximum side lobe amplitude in dB -Main lobe amplitude in db = S1 db.
Side lobe Roll off Ratio = Maximum side lobe amplitude in dB-Minimum side lobe amplitude in dB =S1 -SL db.
Normalized Main lobe width i.e. D = 2 wr (N-1) rad/sec [10].  From the above table, it is observed that as we keep on increasing the values of ρ, the ripple ratio and transition width increases and sidelobe roll-off ratio decreases as well respectively. So the parameter so proposed has a significant effect on all the three spectral properties of a window function.

FIR FILTER DESIGNING
Modified Cosh window provides better MSA for the LPF designing. It provides better FSA for same window length.

RIPPLE RATIO VS. NORMALIZED WIDTH FOR DIFFERENT VALUES OF α AND ρ FOR N=51
Ripple ratio vs. normalized width of different modified Cosh windows and basic Cosh window have been plotted. By keeping α constant and varying ρ, various values of ripple ratio and normalized width of the proposed window have been observed and plotted as shown in fig. 5.
At larger values of α, Modified Cosh window behaves like a Cosh window. Modified Cosh window will perform better for some particular values of its parameters. It does not have power series expansion in its time domain function [2,12,14]. It performs better ripple ratio than 3-parameter ultra spherical window for wider main lobe width and larger side lobe roll-off ratio. Filters designed by Modified Cosh window provide better far end stop band attenuation than the filters designed by Kaiser Window. J u l y 2 0 , 2 0 1 3

RELATION BETWEEN FOR α AND ρ N=51 & N=101
The relation between α and ρ of modified Cosh window can be developed by keeping window size constant, then varying one of them and producing a table between these parameters and Ripple ratio. R ranging from -13.26 ≤ R ≤ -60 db [12].

OPTIMUM MODIFIED COSH WINDOW
Now, from the above table, an optimum modified Cosh window is generated for N=51 & N=101.Several readings are taken to plot the relationship between two windows 1 st window, N=51 2 nd Window, N=101

Fig. 8 R vs. D For optimum modified Cosh window
From the plot, we found Optimum Modified Cosh window for different window sizes, N and fixed R and wr. It can be seen that on increasing the window size i.e. N the ripple ratio increases for the same transition width and same parameters. It is found that Optimum proposed window occurs at the parameters where their combinations provide equal amplitude of side-lobes including the first one adjacent to the main lobe. This also shows that on increasing the window size ripple ratio also increases.   The following plots have been recovered by using the above requirements for designing of a LPF and specifications for the window functions.