Adjustable Spectral Characteristics to Design FIR Filter

░ ABSTRACT: This paper present a new dynamic window function having variable or adjustable spectral characteristics. A new dynamic window function proposed in this paper is a combination of hamming, Blackman-Harris, chebwin, and Kaiser Window function. Blackman-Harris, chebwin, and Kaiser Window functions have been used to compare with the suggested proposed window function i.e. a combination of hamming, Blackman-Harris, chebwin, and Kaiser Window function. Compare observation has been done with the help of MATLAB simulation. The simulation results show that the suggested proposed window function provides better ripple-ratio, main lobe width and side lobe roll-off ratio as compared to Blackman-Harris, chebwin, and Kaiser-window function. While comparing with Blackman-Harris, chebwin, and Kaiser Window functions, proposed window function provide quite good spectral performances with a very small compromisation of one of the spectral characteristics.


░ 1. INTRODUCTION
For an input sequence x(n) with discrete-time, discrete-time filter generated an output sequence y(n) with discrete-time. As know that filter has a device which gives us desirable outcomes in term of frequency and these discrete-time filters have been shown a lot of application like signal processing, suppression of noise and also an enhancement in images [1][2].
A Filter required may be required to have a given frequency response, or a specific response to an impulse, step, or ramp, or simulate an analogue system. Digital filters are categories either FIR or IIR, depending upon the form of the unit pulse response of the system. A digital FIR filter has a finite duration of the impulse response, it means the finite number of non-zero terms. Assume that a system has an equation (difference) with input and output sequence x(n)and y(n) respectively define as follows; M-length FIR filter is described as follows by the difference equation Where { } is the set of filter coefficient. So it is very clear from the above equation, the output response of the FIR filter depends upon only on the present and past input sample sequence.
The FIR filters exhibit exact linear phase and they are always stable, design methods are also generally linear, the filter start up transients have a finite duration and also realization in hardware so efficiently. Because of these advantages FIR filters are highly preferable if no phase distortion is required. There was the number of approaches presented by the Windowing is a process of truncation of the impulse response from infinite impulse response filter. Truncation of the impulse response from infinite impulse response filter is equal to the product of and w[n]. Because of this digital FIR filter design using window approach is a better approach to design a finite impulse response filter. Generally, window function helps us to provide zero value outside the certain interval so can modify the value of adjustable window by modified the generally window function help us to provide zero value outside the certain interval so can modify the value of adjustable window by modifying the one or more variable parameters [2] [6]. This paper presents several windows for quite better spectral responses than commonly used windows.
The basic desirable need or requirement for implementing any window function, smaller main lobe width, high ripple ration (in a negative sense)and high side-lobe roll-off ratio [8]. But for some instances, these characteristics oppose each other and also shows some limitation like a window having higher side lobe has smaller main lobe width and vice versa [3][4][5].
There are three desired specifications for a window function which are defined as: Main-lobe width (WB) Width of the main-lobe x 2π Ripple-Ratio (R) Maximum side-lobe amplitude (in dB) -Main-lobe amplitude (in dB) = S1 Side-lobe roll-off ratio (S) Maximum side-lobe amplitude (in dB) -Minimum side-lobe amplitude (in dB) = S1-SL A new dynamic window function proposed in this paper is a combination of hamming, Blackman-Harris, chebwin, and Kaiser Window function. Blackman-Harris, chebwin, and Kaiser Window functions have been used to compare with the suggested proposed window function i.e. a combination of hamming, Blackman-Harris, chebwin, and Kaiser Window function. Compare observation has been done with the help of MATLAB simulation.
This section of paper discusses the introductory part of FIR filter and their advantage and also discusses the window design approach, rest of the paper describe as follows, section II discusses the suggested proposed window function to design FIR filter and comparison discusses in the section 3 of this paper and last but not the least conclusion discusses in section 4.

░ 2. SUGGESTED WINDOW FUNCTION
A new dynamic window function proposed in this paper is a combination of Blackman-Harris, hamming, Kaiser and chebwin-window function provided in equation respectively; The function of the Blackman-Harris window given as; The function of the hamming window given as; The function of Kaiser Window given as; Where N is the length of FIR filter require and is the zerothorder modified Bessel function of the first kind and α is inverse of the standard deviation of Kaiser Window function.

The function of Chebwin Window given as;
A new dynamic window function proposed is a combination of hamming, Blackman-Harris, chebwin, and Kaiser Window function. The suggested window functions are given by the equation below; Where 'a' setup the gain of the window and 'r' is the spectral control parameter.

░ 3. RESULTS & DISCUSSION
A new dynamic window function proposed in this paper is a combination of Blackman-Harris, hamming, Kaiser and chebwin Window function provided in the equation as follows; Website: http://www.ijeer.forexjournal.co.in Adjustable Spectral Characteristics to Design FIR Filter  As above depicted figure 2 shows that the frequency domain characteristics of the suggested window for the different specific value of 'a' and 'r'. The figure shows the frequency domain characteristics of proposed window for 'a' = 200, 300, 600 and value of 'r' = 0.6, 0.7, 0.8 respectively, from the frequency domain characteristics graph shows that the proposed window achieved better performance for 'a' = 600 and 'r' = 0.8.
░  This section of the paper presents a comparative analysis of suggested proposed window with black-man Harris, Kaiser and Chebwin window function.

Black-man Harris Window:
The Black-man Harris window function is defined as the equation below;

Kaiser Window Function
Kaiser window function is defined as: Where the zero-order is modified Bessel function of the first kind.   As above depicted table 3 shows that suggested window (a=600, r=0.8) gives smaller ripple-ratio and higher side-lobe roll-off ratio compared to Kaiser Window as desired.

Chebwin Window Function
Chebwin window function is defined as:  As above depicted table 4 shows that suggested window (a=600, r=0.8) gives smaller main-lobe width and higher sidelobe roll-off ratio compared to Chebwin window as desired. But there is a compromisation of 25 dB in RR.

░ 4. CONCLUSION
The proposed window function can adjust the spectral characteristics i.e. ripple-ratio main-lobe width, and side-lobe roll-off by using two adjustable parameters 'a' and 'r'. It shows better performance than Kaiser Window in term of SLRR and RR. The proposed window has better main-lobe width and SLRR than Blackman Harris Window but with a compromisation of ripple ratio (approx. 18 dB). While comparing with Chebwin window we have concluded that the proposed window gives better main-lobe width and SLRR and but with a compromisation of approx. 25 dB in RR.