International Journal of Engineering Trends and Technology (IJETT) – Volume 4 Issue 10 - Oct 2013
Analysis Of The Fixed Window Functions In The Fractional
Fourier Domain
Mahendra Malviya1, Dr. Ashutosh Datar2, Dr. S.N. Sharma3
1
PG student, Dept. of Electronics and Communication Engg., SATI, Vidisha, M.P., India.
2
Head, Bio Medical Engg., SATI, Vidisha, M.P., India.
3
Head, Electronics and Communication Engg.,SATI, Vidisha, M.P., India.
ABSTRACT
In this paper, the equivalent noise bandwidth, coherent gain and scallop loss in the Fractional Fourier domain have been characterized
and the characterized expression for these parameters has been provided. Along with these parameters, 3db bandwidth, 6 db band
width, Highest side lobe, Main lobe width, Worst case processing loss for different fixed windows, have also been evaluated in
Fractional Fourier domain and the result is analyzed.
Keywords
( )
FrFT: Fractional Fourier Transform, ENBW: Equivalent Noise
Bandwidth, BW: bandwidth, HSL: Highest Side Lobe, MLW:
Main Lobe width, WCPL: Worst Case Processing Loss.
Where
by :
(
1. Introduction
( )
(
∫
)
( )
is known as the kernel of the transformation, is given
)
(
√
((
)
))
The window functions are highly useful in the FIR realization
and spectral analysis of any transfer function. The selection of
any transfer function is depends upon the window parameters
like ENBW, 3dB BW, 6 dB BW, HSL etc.
where is the imaginary unit,
The Fractional Fourier transform of window functions provides a
wide range of possibilities regarding spectral analysis and FIR
realization. We may use FRFT of window function at any
fractional order, for directly convolution in the Fourier domain or
for the truncation of infinite series. Apart from this, In the
presence of time variant noise the filtering is highly beneficial if
it done in fractional domain. Here also windowed filter can
perform much better in fractional domain also. For all these
purpose we need the knowledge of window function in the
fractional domain.
The FrFT realize on the basis of parameter
and can be
interpreted as an angle in the time frequency plane and is
known as the fractional order. One of the special case of the
FrFT with
, corresponds to the Fourier transform and the
FrFT corresponds to the
corresponds to the identity
operator. For
the FrFT response is the equal to the
flipping of the signal along the time axis.
2. The Fractional Fourier transform
For any given window we can define the equivalent noise
bandwidth as the broadband noise included in the peak power of
the window. [1]
The Fractional Fourier transform is the generalization of classical
Fourier transform.[2,5,6]
The ath order Fractional Fourier transform
define as
ISSN: 2231-5381
( ) may be
Here
for
(
)
(
and
) and
.
(
)
(
) respectively
.
3. The Equivalent Noise bandwidth in the Fourier and
Fractional Fourier Domain
Let ( ) is a continuous window function exist from –T/2 to T/2,
( ) is the Fourier transform of
( ) and
is the Noise
power per unit bandwidth, then the equivalent noise bandwidth
can be given as:
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International Journal of Engineering Trends and Technology (IJETT) – Volume 4 Issue 10 - Oct 2013
( )
∫
In the fractional domain the coherent gain is evaluated by
[ ( )]
Here the numerator term is the total noise energy in frequency
domain. The total signal energy is equal in both time and
frequency domain, even it is equal in all the domains. The
dominator is equal to the peak spectral power.
Now let the ( ) is the fractional domain representation of the
window function ( ) at some fractional order and ( ) is the
noise power per unit bandwidth in fractional domain corresponds
to order . Then the equivalent noise bandwidth for ( ) can
be given as:
(
) ∫
( )]
The ENBW of the window function should be smaller for good
selectivity and good cut off of the FIR characteristic
implemented through the window; on the other hand if we have
the requirement to cover out more spectral energy, we would
need the high ENBW.
definition.
5. Scalloping Loss
The scalloping loss is given by the ratio of the half bin spectral
gain to the peak spectral gain both in the Fourier and Fractional
Fourier domain [1]. This factor shows the reduction in the signal
amplitude due to the change in the signal frequency. It is given
by:
|
1.
Coherent gain can be described as the ratio of output signal to
noise ratio to the input signal to noise ratio.[1] Fourier domain
the coherent gain is given by:
3.
[ ( )]
4.
( )
Thus in Fourier domain coherent gain can be given by the
reciprocal of the ENBW. The evaluation of the CG is done by
In the fractional domain the coherent gain is given by:
( )]
( )
The term
in the numerator shows the bandwidth of the
signal in the fractional domain, if we consider the unit signal
bandwidth in the Fourier domain.
3dB and 6dB bandwidth: These are denoted by the
frequency at which the gain becomes 3dB and 6dB
respectively. The 3dB and 6dB BW should be smaller for the
sharp cutoff of the window implemented FIR filter or other
transfer function.
Highest side lobe: It is the maximum power level of the side
lobe among all the side lobes of the window. The side lobe
level is corresponding to the spectral leakage.
Half Main lobe width: It can be defined by the frequency
range at which the gain becomes equal to the side lobe level.
This parameter should be small for good frequency
selectivity.
Worst case processing loss: It can be given by the sum of the
scalloping loss and the coherent gain both in dB. It should be
small.
7. Different window Function
There are 2 types of window functions, that is fixed window
function and the variable window functions. [1] The fixed
windows are the windows for which the parameter is constant,
where as for the variable windows the shape of the window is
determine by the window’s parameter. The fixed window
functions used for the analysis are given below:
1.
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)|
6. Some other Parameters
2.
[
(
Scalloping loss should not be higher for the window otherwise it
results in the higher reduction in the processing gain (CG) due to
signal frequency.
4. Coherent gain in the Fourier and Fractional
Fourier domain
∫
The coherent gain should be larger, as per its
( )
The strategy used above for the calculation of ENBW in Fourier
domain is also used for calculation of ENBW in the Fractional
Fourier domain.
( )
.
( )
[
∫
( )
the
Bartlett-Hanning window
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International Journal of Engineering Trends and Technology (IJETT) – Volume 4 Issue 10 - Oct 2013
The equation for computing the coefficients of a Modified
Bartlett-Hanning window is:
0
a=1.0
a=0.5
a=0.2
a=0.05
-20
)
|
{
|
(
*
+)
-40
Gain in dB
(
0
a=1.0
a=0.5
a=0.2
a=0.05
-10
-20
-60
-80
-100
-120
Gain in dB
-30
-140
-40
0
0.05
0.1
0.15
normalized frequency
0.2
0.25
-50
Fig 7.3 FrFT response of Bohman window for different Fractional order
-60
4.
-70
Flat Top window
-80
Flat top windows are summations of cosines. The coefficients of
a flat top window are computed from the following equation:
-90
-100
0
0.05
0.1
0.15
normalized frequency
0.2
0.25
(
Fig 7.1 FrFT of Bartlet window for different Fractional order
)
(
)
(
)
(
)
( )
{
2.
Blackman-Harris window
where:
(
(
)
(
)
(
)
)
0
{
a=1.0
a=0.5
a=0.2
a=0.05
-20
where:
-40
Gain in dB
0
a=1.0
a=0.5
a=0.2
a=0.05
-20
-80
-40
Gain in dB
-60
-60
-100
-80
-120
0
0.05
0.1
0.15
normalized frequency
0.2
0.25
-100
Fig 7.4 FrFT response of Flat top window for different Fractional order
-120
-140
0
0.05
0.1
0.15
normalized frequency
0.2
0.25
Fig 7.2 FrFT response of Blackman Harris window for different Fractional order
3.
In the above discussion the FrFT of the different window
functions are also shown , the main lobe of the window get
shrink with the gradual decrement in the fractional order
alpha.[3]
Bohman window
(
[
)
]
[
]
[
]
{
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International Journal of Engineering Trends and Technology (IJETT) – Volume 4 Issue 10 - Oct 2013
3.
8. Results and analysis
Results for Bohman window
We implemented the methodology of the FrFT evaluation given
in [3] and used it in our work. The different parameters, for the
different fixed window functions, are evaluated in this section
and Here the result is tabulated below.
Table 8.3 parameters of Bohman window
1. Results for Bartlett-Hanning window
Table 8.1 parameters of Bartlett-Hanning window
EN
BW
(Bins)
CG
Scall
op
Loss
3-dB
BW
(Bins)
6-dB
BW
(Bins)
HSL
(dB)
MLW
(Bins)
WC
PL
(Bins)
1
1.49
0.49
1.46
1.42
1.98
35.89
3.87
3.18
0.9
3.18
0.33
1.50
1.40
1.95
35.89
3.83
6.52
0.8
6.28
0.23
1.62
1.35
1.88
35.89
3.68
9.59
0.7
9.71
0.18
1.85
1.27
1.76
35.88
3.45
11.72
0.6
13.71
0.15
2.25
1.15
1.60
35.88
3.13
13.62
0.5
18.78
0.12
2.97
1.01
1.40
35.88
2.74
15.71
0.4
25.71
0.09
4.37
0.83
1.16
35.87
2.28
18.47
0.3
36.49
0.07
7.60
0.65
0.90
35.86
1.76
23.22
0.2
56.98
0.04
19.05
0.44
0.61
0
0
36.60
α
2. Results for Blackman-Harris window
Table 8.2 parameters of Blackman-Harris window
ENB
W
(Bins)
Scall
op
Loss
3-dB
BW
(Bins)
6-dB
BW
(Bins)
HSL
(dB)
CG
1
2.04
0.35
0.79
1.93
2.71
92.10
8.06
3.90
0.9
2.79
0.30
0.81
1.91
2.68
92.10
7.96
5.27
0.8
4.57
0.23
0.88
1.84
2.58
92.10
7.66
7.48
0.7
6.82
0.18
1.00
1.72
2.42
92.10
7.18
9.34
0.6
9.54
0.15
1.22
1.56
2.20
92.10
6.52
11.01
0.5
13.02
0.12
1.59
1.37
1.92
92.10
5.70
12.74
0.4
17.84
0.09
2.32
1.14
1.60
92.10
4.75
14.83
0.3
25.38
0.07
3.91
0.88
1.23
92.10
3.68
17.96
0.2
39.74
0.04
8.64
0.60
0.84
92.11
2.53
24.64
α
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MLW
(Bins)
WC
PL
(Bins)
α
ENB
W
(Bins)
CG
Scall
op
Loss
3-dB
BW
(Bins)
6-dB
BW
(Bins)
HSL
(dB)
MLW
(Bins)
WC
PL
(Bins)
1
1.82
0.40
0.98
1.73
2.42
46.00
5.52
3.59
0.9
2.80
0.32
1.01
1.71
2.39
46.00
5.45
5.49
0.8
5.07
0.23
1.09
1.65
2.30
46.00
5.25
8.14
0.7
7.71
0.18
1.24
1.54
2.16
46.00
4.92
10.11
0.6
10.85
0.15
1.51
1.40
1.96
46.00
4.46
11.86
0.5
14.83
0.12
1.98
1.23
1.71
46.00
3.90
13.69
0.4
20.33
0.09
2.89
1.02
1.42
46.00
3.24
15.97
0.3
28.90
0.07
4.93
0.79
1.10
46.00
2.51
19.53
0.2
45.23
0.04
11.20
0.53
0.75
46.00
1.71
27.76
4.
Results for Flat-top window
Table 8.4 parameters of Flat-top window
EN
BW
(Bins)
CG
Scal
lop
Loss
3-dB
BW
(Bins)
6-dB
BW
(Bins)
HSL
(dB)
MLW
(Bins)
WC
PL
(Bins)
1
3.84
0.21
0.00
4.15
5.03
0
0
3.59
0.9
3.21
0.23
0.00
4.10
4.97
0.00
0.00
0.8
3.73
0.21
0.00
3.95
4.79
0.00
0.00
5.07
0.7
5.01
0.17
0.00
3.70
4.48
0.00
0.00
5.72
0.6
6.75
0.14
0.12
2.79
3.50
67.74
7.63
7.01
0.5
9.03
0.12
0.02
2.94
3.56
0.00
0.00
8.41
0.4
12.25
0.09
0.06
2.44
2.96
0
0
9.58
0.3
17.33
0.07
0.21
1.89
2.29
0.00
0.00
10.94
0.2
27.04
0.04
1.14
1.28
1.56
0.00
0.00
12.60
α
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International Journal of Engineering Trends and Technology (IJETT) – Volume 4 Issue 10 - Oct 2013
Here the graphical comparison of the different window
parameters, for different-different windows are given below:
0
-2
60
-4
Bartlet-Hann
Blackman Harish
Bohman
Flattop
-6
Scallop loss
50
40
-8
-10
ENBW
-12
30
-14
Bartlet Hann
-16
Blackman Harris
20
Bohman
-18
-20
10
0
1
0.9
0.8
0.7
0.6
alpha
0.5
0.4
0.3
0.2
Fig 7.5 Graph between ENBW and Fractional order for different windows
0.5
Bartlet-Hann
Blackman Harish
Bohman
Flattop
0.45
0.4
CG
0.3
0.25
0.2
0.15
0.1
0.05
1
0.9
0.8
0.7
0.6
alpha
0.5
0.4
0.3
1
0.9
0.8
0.7
0.6
alpha
0.5
0.4
0.3
0.2
Fig 7.7 Graph between Scalloping loss and Fractional order for different window
functions
The scallop loss increases with decrement in the fractional order.
The rate of decrement is higher for the Barttlet-Hann window.
The flattop window shows almost a constant value of scallop loss
up to a fractional order 0.3.
As the WCPL is the sum of CG and the Scallop loss (all in
dB).As the Scallop loss increases with the decrement in the
fractional order and CG decreases with the same that why its dB
value will increase with a negative sign, So WCPL also increases
with the decrement in the fractional order.
0.35
0
Flattop
0.2
The 3dB BW and the 6 dB BW decreases with the decrement in
the fractional order. The 3dB and 6dB bandwidths of flat top
widow are relatively higher as compare to other windows. We
also get some irregularity in its characteristic near the fractional
order 0.6
Fig 7.6 Graph between CG and Fractional order for different windows
0
From the above plot it is clear that the ENBW increases when we
gradually decrease the fractional order alpha form 1 to 0. The
Rate of growing up of the ENBW is higher for the Barttlet-Hann
window.
Barttlet Hann
Blackman Harris
Bohman
Flattop
-5
-10
WCPL
-15
The coherent gain decreases with decreasing the fractional order
alpha. The coherent gain is higher among the all four for alpha
equals to1. But with the gradual decrement in the fractional order
the CG characteristic of the different windows are almost overlap
after a value of alpha equals to 0.7.
-20
-25
-30
-35
-40
1
0.9
0.8
0.7
0.6
alpha
0.5
0.4
0.3
0.2
Fig 7.8 Graph between WCPL and Fractional order for different windows
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International Journal of Engineering Trends and Technology (IJETT) – Volume 4 Issue 10 - Oct 2013
9
4.5
Battlet Hann
Blackman Harris
Bohman
Flattop
4
3.5
Barttlet Hann
Blackman Harris
Bohman
Flattop
8
7
6
3
MLW
3dB BW
5
2.5
2
4
3
1.5
2
1
1
0.5
0
0
-1
1
0.9
0.8
0.7
0.6
alpha
0.5
0.4
0.3
0.2
Fig 7.9 Graph between 3dBBW and Fractional order for different windows
Barttlet Hann
Blackman Harris
Bohman
Flattop
4.5
4
3.5
6dB BW
0.9
0.8
0.7
0.6
alpha
0.5
0.4
0.3
0.2
Fig 7.11Graph between MLW and Fractional order for different windows
The HSL remains almost constant and MLW gradually decreases
with the fractional order. But the flat top window shows some
irregularity near the fractional order 0.6 in both HSL and MLW
characteristic. Otherwise MLW of flat top window is zero for all
other values. The Barttlet -Hann window also shows some
unevenness for the value of fractional order lower than 0.25
5.5
5
1
3
9. Conclusion
2.5
2
1.5
1
0.5
1
0.9
0.8
0.7
0.6
alpha
0.5
0.4
0.3
0.2
Fig 7.10 Graph between 6dBBW and Fractional order for different windows
10
0
With the diverse opportunities provided by the FrFT we may use
different window shapes for the normally used windowing for
Filter or Filter bank implementation [4] or for some other FIR
implementation. And the results can be optimized by adjusting
the value of alpha according to the prior knowledge of the
window parameters.
-10
-20
-30
HSL
Thus we have explained characterization of the ENBW, CG and
scallop loss with the characterized expression. Along with these
parameters we evaluated some other parameters which are
tabulated and the comparative analysis of the different window
function is done on the basis of these different parameters using
the graphical representation. Thus with the help of the given
results the behavior of the window functions can easily
understand.
-40
-50
-60
Barttlet Hann
Blackman Harris
Bohman
Flattop
-70
-80
-90
-100
1
0.9
0.8
0.7
0.6
alpha
0.5
0.4
0.3
0.2
Fig 7.11 Graph between HCL and Fractional order for different windows
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For the lower order windows we might get so many irregularities
in the characteristic of the different parameters with different
fractional order but for the higher order windows the behavior of
the different parameters is regular, on the basis of which we may
provide the following conclusions: With the reduction in the
fractional order, the 3dB and 6dB bandwidth are reducing, which
is good for the pulpous of spectral analysis. It provides the good
frequency characteristic in terms of sharp cut off. The ENBW is
increasing in the graph which is undesirable for the good
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International Journal of Engineering Trends and Technology (IJETT) – Volume 4 Issue 10 - Oct 2013
frequency selectivity and FIR cut off characteristic but it’s good
if we preferring the spectral analysis with less samples per unit
bandwidth. With the reduction in the fractional order the
coherent gain is decreasing so in terms of coherent gain we don’t
having any benefit by FrFT. The MLW is reducing in the graph
which corresponds to the good frequency selectivity. HSL is
almost constant in the characteristic shown but it has still some
of the minor reduction, so for HSL we are at little profit without
loss. The scallop loss and WCPL are increasing which is
undesirable. So the choice of the window function for the
spectral analysis or FIR implementation, should be as per the
concluded benefit and drawbacks, so as to get a good window to
get the optimum performance for some specific application. In
future we may try to minimize the drawbacks and to enhance the
benefits of the spectral analysis and implementation of the FIR
through the window function obtained through FrFT.
10. References
[1] F.J. Harris, “On the use of the window for the harmonic analysis using
Discrete Fourier transform”, IEEE, vol. 16, no.1, Jan. 1978.
[2] H. M. Ozaktas, O. Arikan, “ Digital computation of the Fractional Fourier
Transform”, IEEE trans. signal processing, vol 44, no. 9, Sep. 1996.
[3] S. N. Sharma, R. Saxena, S.C. Saxena, “Tuning of FIR filter transition
bandwidth using fractional Fourier Thransform”, Signal Processing, vol.
87, pp. 3147-3154, 2007.
[4] A. Datar, A. Jain, P.C. Sharma, “Design and performance analysis of
adjustable window functions based cosine modulated filter bank” Digital
signal processing , vol.23, pp.412-417,2013.
[5] R. Saxena, K. Singh, “Fractional Fourier Transform: A novel tool for signal
processing”, Indian Inst. Of Sci., vol. 85, pp. 11-26, 2005.
[6] V. A. Narayana, K. M. M. Prabhu “The Fractional Fourier Transform:
Theory, implimentation and erroe analysis”,
Microsystems, vol. 27, pp. 511-521, 2003.
Microprocessor and
[7] S. Chaturvedi, G. Parmar, P. Shukla, “Sharpening the response of an FIR
filter using Fractional Fourier Transform”, IJECSE, vol. 1, no. 2.
[8] P.V.Muralidhar, V. L. Nsastry D, S.K.Nayak, “Interpretation of Dirichlet,
Bartlett, Hanning and Hamming windows using Fractional Fourier
Transform”, International Journal of Scientific & Engineering Research,
vol. 4, issue 6, June-2013.
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