Design and Performance Analysis of Filters Using a Proposed Window Function

International Journal of Research and Scientific Innovation (IJRSI) | Volume IV, Issue VII, July 2017 | ISSN 2321–2705
www.rsisinternational.org Page 66
Design and Performance Analysis of Filters Using a
Proposed Window Function
Amit Kumar Patil
H.O.D., EC Department, TSEC Burhanpur, Madhya Pradesh, India
Abstract: A new window function is presented which like the well
known Hamming window offers a preferred property for use in
signal spectrum analysis: the sum of window coefficients with its
shifted version by half of the order is constant for the overlapped
region in the time domain. In high orders, the new window has
main-lobe width equal to Hamming window. For low orders, the
window parameters are modified to have smaller main-lobe
width compared to Hamming window, while maintaining smaller
maximum side-lobe peak. The results indicate performance
improvement of the proposed window compared to Kaiser and
Gaussian windows. The FIR filters designed by windowing
method show the efficiency of the new window.
Keywords: FIR, Gaussian, Kaiser, Proposed, Window.
I. INTRODUCTION
indow functions are widely used in digital signal
processing for the applications in signal analysis and
estimation, digital filter design and signal processing. FFT
windows reduce the effects of leakage but can not only
eliminate leakage entirely. In effect, they only change the
shape of the leakage. In addition each type of window affects
the spectrum in a slightly different way. Many different
windows have been proposed over time each with its own
advantage and disadvantage relative to the others. Some are
effective for specific types of signal types such as random or
sinusoidal. Some improve the frequency resolution, that is,
they make it easier to detect the exact frequency of a peak in
the spectrum. Some improve the amplitude accuracy that is
they most accurately indicate the level of the peak. The best
type of window should be chosen for each specific
application.
Two main applications of the windows in digital signal
processing are: data analysis based on Fast Fourier Transform
(FFT) and design of Finite Impulse Response filters from
Infinite Impulse Response filters. For FFT analysis, windows
are employed to suppress the so called “leakage effect”, and
for FIR filter design according to the “windowing method”,
Gibbs oscillations are attenuated. Desirable characteristics for
a window in the frequency domain are small main-lobe width
and side-lobe peak (high attenuation). However, these two
requirements are contradictory, since for a given length, a
window with a narrow main-lobe has a poor attenuation and
vice versa. Also a third preferred property of window when
applied in data spectrum analysis is that, in the time domain,
the sum of window function (w[n]) with its shifted version by
M/2 samples (M is the window order) would be constant:
w [n] + w [n- M/2] = constant, M/2 ≤ n ≤ M (1)
In this paper, a new window function is presented which can
be considered as a special case of the important class of
windows, named raised cosine windows. The proposed
window has 2~4 dB more side-lobe attenuation than that of
Hamming window, while offering approximately the same
main-lobe width and still satisfying the property in equation
(1). The window parameters are modified to avoid the
performance degradation for lower window lengths, which
happens for Hamming window.
II. ANALYSIS OF WINDOWING TECHNIQUE
2.1 TYPES OF WINDOWS
There are different types of windows starting from simple
type (rectangular window) to more complex type (Kaiser
window). The main goal of these windows is to truncate the
impulse response of the filter in order to generate fixed length
filter.
2.1.1 Kaiser window
Syntax: w = kaiser(L,beta)
Description: It returns an L-point Kaiser window in the
column vector w. Beta is the Kaiser window β parameter that
affects the sidelobe attenuation of the Fourier transform of the
window. The default value for beta is 0.5. Kaiser window is
defined by:
(2)
2.1.2 Hamming window
Syntax: w = hamming(L)
Description: It returns an L-point symmetric Hamming
window in the column vector w. L should be a positive
integer. The coefficients of a Hamming window are computed
W

from the following equation. Hamming window is optimized
to minimize the maximum side lobe. It is defined by:
(3)
2.1.3 Gaussian window
Syntax: w = gausswin(N), w = gausswin(N,Alpha)
Description: It returns an N-point Gaussian window in the
column vector w. L is a positive integer. The shape of this
window is similar in the frequency domain because the
Fourier transform of a Gaussian is also a Gaussian.
2.1.4 Taylor window
Syntax: w = taylorwin(n)
Description: A Taylor window allows to make tradeoffs
between the mainlobe width and sidelobe level. The Taylor
distribution avoids edge discontinuities.Taylor windows are
typically used in radar applications, such as weighting
synthetic aperature radar images and antenna design. It returns
an n-point Taylor window in a column vector w. The values in
this vector are the window weights or coefficients. n must be a
positive integer.
2.1.5 Blackman window
Syntax: w = blackman(L)
Description: It returns the L-point symmetric Blackman
window in the column vector w, where L is a positive integer.
Blackman windows have slightly wider central lobes and less
sideband leakage than equivalent length Hamming and Hann
windows. Blackman window is defined by:
(4)
2.1.6 Rectangular window
Syntax: w = rectwin(L)
Description: It returns a rectangular window of length L in the
column vector w. This function is provided for completeness;
a rectangular window is equivalent to no window at all. It is
defined by:
(5)
2.1.7 Hanning window
Syntax: w = hann(L)
Description: It returns an L-point symmetric Hann window in
the column vector w. L must be a positive integer. The
coefficients of a Hann window are computed from the
following equation.
(6)
2.2 WINDOWING TECHNIQUE
The windowing method involves multiplying the ideal
impulse response with a window function to generate a
corresponding filter, which tapers the ideal impulse response.
Like the frequency sampling method, the windowing method
produces a filter whose frequency response approximates a
desired frequency response. The windowing method,
however, tends to produce better results than the frequency
sampling method.
The impulse response of ideal filters is of infinite duration. It
is not possible to evaluate the corresponding frequency
response and implement the filter by hardware or software.
Thus the impulse response must be truncated at both ends
with respect to the central. Even the impulse response can be
truncated when it is small enough but such a sudden cut off
will cause some undesired effects. The window method will
reduce them.
In the time domain windowing means to multiply the infinite
impulse response hD(n) by a finite duration window function
w(n) to get a truncation. The resulted impulse response h(n) is
their product and is given as follows:
H(n) = hD(n)*w(n) , 0 ≤ n ≤ M (7)
In the window design method we first evaluate the desired
filter response hD(n) from the given desired frequency
response HD( ) and then apply an appropriate window. Thus
the method should be called Fourier window method. In this
method, use is the made of the fact that the frequency
response of the filter, HD( ) and the corresponding impulse
response, hd(n) are related by inverse Fourier transform. The
subscript D is used to distinguish between ideal and practical
impulse response.
HD(n) = )(


DH
(8)
The basic idea behind the Window method of filter design is
that the ideal frequency response of the desired filter is equal
to 1 for all the pass band frequencies, and equal to 0 for all the
stop band frequencies and then the filter impulse response is

obtained by taking the Discrete Fourier Transform (DFT) of
the ideal frequency response. Unfortunately, the filter
response would be infinitely long since it has to reproduce the
infinitely steep discontinuities in the ideal frequency response
at the band edges. To create a Finite Impulse Response (FIR)
filter, the time domain filter coefficients must be restricted in
number by multiplying by a window function of a finite
width. Many windows are used for truncating the signal and
the simplest window function is the rectangular window
which corresponds to truncating the sequence after a certain
number of terms.
Fig.1 Windowing Technique
Fig. 2 Spectral Response

III. PROPOSED WINDOW
The goal is to modify Hamming window to lower its
maximum side-lobe peak, while holding the main-lobe width
unchanged, and still satisfying property of equation (4.3). In
the following, the derivation of the new window is explained.
All windows described later, have zero valued coefficients
outside the interval 0 ≤ n ≤ m Hamming window has the
shape of
WH[n]= 0.54-0.46 cos(2πn/M), 0 ≤ n ≤ M (9)
This window satisfies the property mentioned in eq. (3), i.e.
WH[ n]+ WH[ n- M/2]= 0.54+0.54, M/2≤n ≤M (10)
On the other hand, Blackman window is composed of three
terms as:
WH[n]=0.42-0.5*cos(2πn/M)+0.08*cos(4πn/M), 0 ≤n≤M (11)
that is, it has a DC term, a cosine function with frequency
2/M, and its second harmonic. For this window, due to the
second harmonic, the property in eq.(3) is not satisfied
wH[ n] +wH[ n-M/2]=0.84+0.16*cos(4πn/M ), M/2≤n≤M (12)
However, it can be found that if the third harmonic is added to
the Hamming window function, then the property of eq. (3)
will be satisfied. Therefore, the main idea in obtaining the
new window is to insert a third harmonic of cosine function
into eq. (1). Thus, an extra degree of freedom is obtained in
tuning the window coefficients. In this way, the proposed
window will be as
W[ n]= a0-a1*cos(4πn/M)-a3cos(6πn/M), 0≤n ≤M (13)
Where for normalization, i.e. W[M/2]=1, one have:
a0+a1+a3 = 1 (14)
The new window is also symmetric about point M/2; thus it
has a generalized linear phase, like the other common
windows. Checking for the property in eq. (3), one find that
W[n]+W[n-M/2]= 2a0 (15)
Another point of view states that eq. (5) is a four-term raised
cosine window, with restriction that the third term is zero:
W[n]= ∑ ai cos(2iπn/M) 0 ≤ n ≤ M, k = 3, a2 = 0 (16)
The new window can be analyzed in the frequency domain. Its
Fourier transform is:
W(ὣ)={a0 D(ω) + a1/2[D(ω - 2π/M) + D(ω + 2π/M)] +
a3/2[D(ω -6π/M)+D(ω +6π/M)]}* exp(-jMω/2) (17)
where D(ω) is Dirichlet kernel.
D(ω) = sin((M+1) ω/2)/sin(ω/2) (18)
Noting the above condition and the normalizing condition in
eq. (8), one can apply a simple optimization algorithm to find
the optimal values of the window parameters. For sufficiently
large orders, the derived window is of the form
W[n] = 0.536-0.46cos*(2πn/M)-0.003*cos(6πn/M), 0≤n≤M
(19)
Just like Hamming window, the frequency response of the
new window is degraded for low orders; therefore depending
on the window order, the above parameters are modified to
maintain the efficiency. It shows the dependence of a0 and a1
on M. It can be easily verified that the coefficients are
composed of a monotonic function and a DC term. Some
simple formulas are tried to present the dependence of these
parameters on M, the following formulas approximately fit the
data obtained from the optimization:
a0= 0.537 - 0.3/(M+15); a1 = 0.46 + 0.25/(M+15); a3 = 1 - a0 -
a1 (20)
IV. RESULTS
The various methodologies are adopted in developing the
programs for proposed window, its comparison with other
windows, performance evaluation, filter designing and
plotting their magnitude response, phase response, pole-zero
plot, impulse response, step response, phase delay, group
delay and filter information. It discusses the simulation results
for window based FIR filters design.
0 10 20 30 40 50 60 70 80 90 100
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
time
amplitude
comparision of different windows
hamm
hann
black
kaiser
gauss
taylor
bart
Fig. 3 Time-Magnitude response of different window
0 50 100
0
0.5
1
hamm
0 50 100
0
0.5
1
hann
0 50 100
0
0.5
1
black
0 50 100
0.9
0.95
1
kaiser
0 50 100
0
0.5
1
gauss
0 50 100
0
1
2
taylor
0 50 100
0
0.5
1
bart
0 50 100
0
1
2
rect
0 50 100
0
0.5
1
cheby
Fig. 4 Plot of different windows in matrix form

2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
samples
amplitude
proposed
kaiser
Fig. 5 Time-Magnitude response of Kaiser window & Proposed window
0 0.5 1 1.5 2 2.5 3 3.5
-150
-100
-50
0
50
magnitude
proposed
0 0.5 1 1.5 2 2.5 3 3.5
-100
-50
0
50
ksiser
magnitude
frequency
Fig. 6 Frequency-Magnitude response of Kaiser window & Proposed window
0 0.5 1 1.5 2 2.5 3 3.5
-4
-2
0
2
4
phase
proposed
0 0.5 1 1.5 2 2.5 3 3.5
-4
-2
0
2
kaiser
frequency
phase
Fig. 7 Frequency-Phase response of Kaiser window & Proposed window
2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Samples
Amplitude
Proposed
Blackman
Fig. 8 Time-Magnitude response of Blackman window & Proposed window
2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Samples
Amplitude
Proposed
Gaussian
Fig. 9 Time-Magnitude response of Gaussian window & Proposed window
2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Samples
Amplitude
Proposed
Taylor
Fig. 10 Time-Magnitude response of Taylor window & Proposed window
V. CONCLUSION
A proposed window is derived for FIR filter designing and
spectral analysis which gives better response in comparison
with other windows. Performance evaluation is done in
comparison with Hamming, Kaiser and Gaussian window.
Window based FIR filter is designed. Windowing gives good
results for proposed window and it is more convenient. It
designs a FIR filter with finite impulse response. It reduces
the ripples in the pass-band and the stop-band due to
Gibbs’phenomenon. By choosing the window carefully,
various trade-offs can be managed so as to maximize
the filter-design quality in a given application. The window
method for digital filter design is fast, convenient, and robust.
A novel efficient window function has been designed which
minimizes the sidelobes. The new window has the main lobe
width less than or equal to that of the Hamming window,
while offering less maximum side-lobe peak. The
performance comparison of the proposed window with the
other windows showed the better performance of the proposed
window. The average reduction in the side lobe peak of the
new window compared to that of the Hamming, Kaiser, and
Gaussian windows is 3 dB, 3.3 dB and 8 dB respectively. The
FIR filter designed with the proposed window achieves less
ripple ratio, than those obtained using the windows like
Kaiser, Hamming, Blackman, Gaussian, Hanning, Taylor etc.
for all window lengths. The obtained results indicate that the

implemented filter has good performance and stability, and
during the implementation of various cases, it is possible to
reach an optimal design used for final design.
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