 Open Access
 Total Downloads : 289
 Authors : K. Krishnam Raju, A. J Ayalaxmi, M. Chaitanya Kumar
 Paper ID : IJERTV2IS121194
 Volume & Issue : Volume 02, Issue 12 (December 2013)
 Published (First Online): 24122013
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Analylsis of Polynomial Windows for FIR Filters for Better Spectral Response
K. Krishnam raju 1 A. J ayalaxmi 2 M. Chaitanya kumar 3
1,2,3: Department of ECE, Aditya Institute of Technology and Management, Tekkali, Andhrapradesh, India.
Abstract
The analysis of timedomain functions is carried out.FIR filters are designed with windowing techniques which will improve spectral response of filter. In this paper new window functions are proposed for FIR filter design using conventional windows and polynomial windows which improve spectral response of filter than existing windowing techniques in terms of RSA(Relative side lobe attenuation).
Keywords: Polynomial window, Conventional window, RSA.
INTRODUCTION
In signal processing, a window function w(n) also known as an apodization function or tapering function is a mathematical function that is zerovalued outside of some chosen interval. When another function or waveform/datasequence is multiplied by a window function, the product is also zerovalued outside the interval. Applications of window functions include spectral analysis, filter design, and beam forming.
Polynomial window functions in the time domain, allow desired order of continuity at the boundary of the observation window. Without any loss of generality, we assume the window interval to be [1, 1]. Let us
assume that f(t) represents the signal of interest and f(t) is the windowed approximation of f(t) over the windowed interval [1, 1], i.e.,
f(t) = w(t)f(t)
Where w(t) is the window function with a compact support over [1, 1].
SectionI describes conventional window functions [3, 4] and in sectionII polynomial window functions are discussed and SectionIII describes proposed windows.
Section IV discusses the FIR filter and section discusses the spectral response of notch filter with conventional wind polynomial windows.
SECTIONI
Rectangular window:
The rectangular window (sometimes known as the boxcar or Dirichlet window) is the simplest window, equivalent to replacing all but N values of a data sequence by zeros, making it appear as though the waveform suddenly turns on and off:
For
Triangular window:
is defined by:
With
Hanning window:
The Hann window is defined by:
Blackman window: defined as:
:
The end samples are positive (equal to 2/(N + 1)). This window can be seen as the convolution of two half sized rectangular windows (for N even), giving it a main lobe width of twice the width of a regular rectangular window. The nearest lobe is 26 dB down from the main lobe.
Hamming window:
The window is optimized to minimize the maximum (nearest) side lobe, giving it a height of about onefifth that of the Hann window.
Figure1: FREQUENCY RESPONSE OF BARTLETT WINDOW:
Figure2: FREQUENCY RESPONSE OF HAMMING WINDOW:
Figure3: FREQUENCY RESPONSE OF HANNING WINDOW:
Figure4: FREQUENCY RESPONSE OF BLACKMAN WINDOW:
TABLE1: COMPARASION RELATIVE SIDE LOBE ATTENUATION OF CONVENTIONAL WINDOWS:
Window function 
Relativesidelobe attenuation in dB 
Bartlett 
26.1 
Hamming 
41.2 
Hanning 
31.5 
Blackmann 
58.2 
From TABLE 1 it is observed that relative side lobe attenuation increases from Bartlett to Blackman for the value of N=25.
SECTIONII
The time domain equation of polynomial [1] window is given as:
Where Km and Am,n are given by:
The frequency domain equation
of polynomial window is given as:
Where
Where m is called order of the polynomial window.
Figure5: FREQUENCY RESPONSE OF POLYNOMIAL WINDOW FOR m=0:
Figure6: FREQUENCY RESPONSE OF POLYNOMIAL WINDOW FOR m=1:
Figure7: FREQUENCY RESPONSE OF POLYNOMIAL WINDOW FOR m=2:
TABLE2: COMPARASION OF RELATIVE SIDE LOBE ATTENUATION OF POLYNOMIAL WINDOWS:
SECTIONIII
Proposed windowing concept:
FigureIII: convolution of conventional and polynomial windows.
Polynomial window 
Relativesidelobe attenuation in dB 
For m=0 
20.8 
For m=1 
18.7 
For m=2 
17.9 
Figure8: Response of new window1 (convolution of polynomial window for m=0 and Bartlett window):
Figure9: Response of new window2 (convolution of polynomial window for m=1 and Bartlett window):
Figure10: Response of new window3 (convolution of polynomial window for m=2 and Bartlett window):
Figure11: RESPONSE OF NEW WINDOW4 (CONVOLUTION OF POLYNOMIAL WINDOW FOR M=0 AND HAMMIING WINDOW):
igure12: RESPONSE OF NEW WINDOW5 (CONVOLUTION OF POLYNOMIAL WINDOW FOR M=1 AND HAMMIING WINDOW):
Figure13: RESPONSE OF NEW WINDOW6 (CONVOLUTION OF POLYNOMIAL WINDOW FOR M=2 AND HAMMIING WINDOW):
Figure14: RESPONSE OF NEW WINDOW7 (CONVOLUTION OF POLYNOMIAL WINDOW FOR M=0 AND HANNING WINDOW):
Figure15: RESPONSE OF NEW WINDOW8 (CONVOLUTION OF POLYNOMIAL WINDOW FOR M=1 AND HANNING WINDOW):
Figure16: RESPONSE OF NEW WINDOW9 (CONVOLUTION OF POLYNOMIAL WINDOW FOR M=2 AND HANNING WINDOW):
Figure17: RESPONSE OF NEW WINDOW10 (CONVOLUTION OF POLYNOMIAL WINDOW FOR M=0 AND BLACKMANN WINDOW):
Figure18: RESPONSE OF NEW WINDOW11 (CONVOLUTION OF POLYNOMIAL WINDOW FOR M=1 AND BLACKMANN WINDOW):
Figure19: RESPONSE OF NEW WINDOW12 (CONVOLUTION OF POLYNOMIAL WINDOW FOR M=2 AND BLACKMANN WINDOW):
CONV 
NEW 
FOR 
FOR 
FOR 

ENTIO IONAL WIND 
RSA IN dB 
WIND OW 
M=0 RSA 
M=1 RSA 
M=2 RSA 
OW 
dB 
dB 
dB 

BARTLE 

BARTLE TT 
26.1 
TT&POL YNOMI 
34.3 
35 
35.3 
AL 

HAMMI 

HAMMI NG 
41.2 
G&POL YNOMI 
53.5 
54.7 
55.2 
AL 

HANNI 

HANNI NG 
31.5 
NG&PO LYNOMI 
36 
36.4 
36.5 
AL 

BLACK 

BLACK MAN 
58.2 
MAN&P OLYNO 
72.9 
74.8 
75.7 
MIAL 
/tr>
CONV 
NEW 
FOR 
FOR 
FOR 

ENTIO IONAL WIND 
RSA IN dB 
WIND OW 
M=0 RSA 
M=1 RSA 
M=2 RSA 
OW 
dB 
dB 
dB 

BARTLE 

BARTLE TT 
26.1 
TT&POL YNOMI 
34.3 
35 
35.3 
AL 

HAMMI 

HAMMI NG 
41.2 
G&POL YNOMI 
53.5 
54.7 
55.2 
AL 

HANNI 

HANNI NG 
31.5 
NG&PO LYNOMI 
36 
36.4 
36.5 
AL 

BLACK 

BLACK MAN 
58.2 
MAN&P OLYNO 
72.9 
74.8 
75.7 
MIAL 
TABLE3: COMPARASION OF RSA OF CONVENTIONAL AND NEW WINDWOS
flow
SECTIONIV FIR FITLER DESIGN [2]:
FIGUREIVproposed filter design
SECTIONV
Spectral responses of FIR notch filter with Bartlett window:
Figure20 spectral response of notch filter
Spectral response of notch filter with poroposed polynomial windows for K=1.2:
Figure21 improved spectral response
TABLE4 comparison of RSA of Bartlett window and with polynomial windows:
Window 
RSA(dB) 
Bartlett 
28.08 
K=0 
32.71 
K=1 
31.97 
K=1.2 
32.02 
K=2.5 
31.64 
K= 0.5 
33.14 
CONCLULSION
it is clear from the table3 that for RSA of new window functions is improved by convolving conventional and polynomial windows and useful for better elimination of noise. It is clear from table4 that spectral response of FIR band reject filter is improved which is observed in terms of RSA. In this paper spectral response of notch filter is analyzed and the proposed concept.
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