International Journal of Engineering Research and Development (IJERD)
ICCCNT1108-7
1. Optimized Quality Factor of Fractional Order Analog
Filters with Band-Pass and Band-Stop Characteristics
Presented by:
Anish Acharya
Department of Instrumentation and Electronics
Engineering,
Jadavpur University, Salt-Lake Campus, Kolkata, India.
ICCCNT-2012, Paper ID: ICCCNT1108-7
2. LOGO
Overview of the Presentation
Introduction
FO Band-pass Filters with Order Less Than Two
FO Band-stop Filters with Order Less Than Two
Quality Factor Optimization of Fractional BP and BS Filters with
Asymmetric Magnitude Response
Conclusion
References
3. LOGOINTRODUCTION
Given the required specifications the required order may be fractional
Have to overspecify the specifications , opt for the next higher order
With fractional filter exact requirements satisfied
Modern applications of BP and BS filters require very precise filtering,to
remove interference from other closely spaced frequency bands.
Q-Factor of the filters need to be considerably high
Q-factors have been optimized with real coded Genetic Algorithm
4. LOGO
FO BAND-PASS FILTERS WITH ORDER LESS
THAN TWO
Expression for the magnitude response of the filter
( )
2 2
cos sin
2 2
cos sin
2 2
2 cos
2
FBPF
b j
T j
j a
b
a a
β
α
β
α α
βπ βπ
ω
ω
απ απ
ω
ω
απ
ω ω
+ ÷ ÷ ÷
=
+ + ÷ ÷ ÷
=
+ + ÷
( ) ( )
( )
( )
FBPF FBPF
b jbs
T s T j
s a j a
ββ
αα
ω
ω
ω
= ⇒ =
+ +
5. LOGO
Contd…
pole and zero parameters being
associated fractional orders being
Band pass for
real-coded GA based optimization approach to obtain optimum filter parameters
symmetric band-pass characteristics if
the frequency corresponding to maximum is equal to center frequency
{ },a b
{ },α β
α β>
{ }, , ,a b α β
mω 0ω
2α β=
9. LOGO
Contd…
Gain of a filter at exactly its center frequency is equal to its quality factor
For symmetric FO filters
Q-factor is a function of the filter parameters for a user specified peak
(center) frequency.
To maintain the symmetry of the magnitude response of the filter, the independent
parameters influencing the quality factor are chosen as
Q
( ) 0
0
2 2
0 02 cos
2
FBPF
b
T j Q
a a
β
ω ω
α α
ω
ω
απ
ω ω
=
= =
+ + ÷
0mω ω=
{ }, , ,a b α β
{ }, ,a b β
13. LOGO
Contd…
Number of population members in GA chosen to be 20.
The crossover and mutation fraction are chosen to be 0.8 and 0.2 respectively
Real CodedGA based optimization attempted to maximize the Q-factor of the FO
band-pass filter within the parametric bounds and
Algorithm has been run several times for a specified center frequency
and the best results are reported .
[ ]0,2β ∈ { } [ ], 0,20a b ∈
0 1.5 / secradω =
22.6017, 0.996307, 18.2033, 0.924351Q a b β= = = =
15. LOGO
FO BAND-STOP FILTERS WITH ORDER LESS
THAN TWO
band-stop filter is just an inverse of the FO band-pass filter transfer function
The best found optimization result for
( ) ( )
( )
( )
FBSF FBSF
j as a
T s T j
bs b j
αα
ββ
ω
ω
ω
++
= ⇒ =
( ) ( )2 2
2 cos
2
FBSFT j a a bα α βαπ
ω ω ω ω
= + + ÷
( ) 0
2 2
0 0
0
2 cos
2 1
FBSF
a a
T j
b Q
α α
βω ω
απ
ω ω
ω
ω=
+ + ÷
= =
0 1.5 / secradω =
21.2739, 0.99767, 17.11228, 0.92593Q a b β= = = =
20. LOGO
QUALITY FACTOR OPTIMIZATION FOR FRACTIONAL
SECOND ORDER FILTERS
• upon optimization the second term of the denominator vanishes to produce a spike in
the frequency domain.
• GA based quality optimization has been applied for the fractional second order
structure
( )
( )
( )
( ) ( )
2
2
2
2
FBPF II
FBPF II
ds
T s
s as b
d j
T j
j a j b
α
α α
α
α α
ω
ω
ω ω
−
−
=
+ +
⇒ =
+ +
( )
( ) ( )( )
( )( )
2
4 3 2 2
2
cos sin
2 2
cos sin
2 cos sin
2 2
4 cos 4 2 cos
2
4 cos
2
FBPF II
d j
T j
j
a j b
d
a a b
ab b
α
α
α
α
α α α
α
απ απ
ω
ω
ω απ απ
απ απ
ω
ω
απ
ω ω απ ω
απ
ω
−
+ ÷ ÷ ÷
=
+
+ + + ÷ ÷ ÷
=
+ + + ÷
+ + ÷
21. LOGO
Contd…
• The optimization with high quality factor enforces to tend towards zero
• diminishing the fractional second order filter structure to produce the symmetric
fractional first order filter.
• Due to the above mentioned shortcomings of fractional second order structure its
parametric optimization for high quality is not recommended.
a
22. LOGOCONCLUSION
More work is needed in this area, and comparison with equivalent optimum digital filtering
techniques also needs to be pursued. In future, optimization of irrational filters, having
fractional power of a rational transfer function can be investigated .
23. LOGO
Reference
1. M.E. Van Valkenburg, “Analog filter design”, CBS College Pub., 1982.
2. M. Alper Uslu and Levent Sevgi, “A MATLAB-based filter-design program: from
lumped elements to microstrip lines”, IEEE Antennas and Propagation Magazine, vol.
53, no. 1, pp. 213-224, Feb. 2011.
3. Ahmed S. Elwakil, “Fractional order circuits and systems: an emergency
interdisciplinary research area”, IEEE Circuits and Systems Magazine, vol. 10, no. 4,
pp. 40-50, 2010.
4. A.G. Radwan, A.M. Soliman, and A.S. Elwakil, “First-order filters generalized to the
fractional domain”, Journal of Circuits, Systems, and Computers, vol. 17, no. 1, pp.
55-66, 2008.
5. A.G. Radwan, A.S. Elwakil, and A.M. Soliman, “On the generalization of second-order
filters to the fractional-order domain”, Journal of Circuits, Systems, and Computers,
vol. 18, no. 2, pp. 361-386, 2009.
6. A.G. Radwan, A.M. Soliman, A.S. Elwakil, and A. Sedeek, “On the stability of linear
systems with fractional-order elements”, Chaos, Solitons & Fractals, vol. 40, no. 5,
pp. 2317-2328, June 2009.
7. A.G. Radwan, A.M. Soliman, and A.S. Elwakil, “Design equations for fractional-order
sinusoidal oscillators: four practical cicuit examples”, International Journal of Circuit
Theory and Applications, vol. 36, no. 4, pp. 473-492, June 2008.
