Filter Optimization Packages for Matlab and Mathematica
Niranjan Damera-Venkata and
Brian L. Evans
Embedded Signal Processing
Department of Electrical and Computer Engineering
The University of Texas at Austin, Austin, TX 78712-1084
- Software accompanying the book
Design for Signal Processing (2001)
for Filter Optimization provide a Web-based interface to the
filter optimization packages
- Version 1.1, June 23, 1998:
simultaneously optimizing analog filter designs for any combination of
magnitude, phase, and step responses, and quality factors,
based on pole-zero locations
- Version 1.0, November 21, 1997:
simultaneously optimizing all-pole analog filter designs for any
combination of magnitude, phase, and step responses, and quality factors,
based on pole locations
The traditional approach to infinite impulse response (IIR) filter
implementation is to first design a conventional IIR filter to
double-precision accuracy in the pole and zero locations and then
iteratively adjust the poles and zeros in the design for the
target implementation technology (breadboard components, switched
capacitor, custom integrated circuit, digital signal processor,
heterogeneous, and so forth). Due to the implementation technology,
it is possible for a stable filter to become unstable due to the
truncation of the pole and zero locations.
A key technology-independent measure of a filter's robustness under
implementation is a quality factor. A quality factor is associated
with each conjugate pair of poles. For analog filters, the quality
factor is the radius of the pole divided by -2 times its real part.
The quality factor varies from 0.5 (purely real pole; 100% damping)
and infinity (imaginary pole; 100% oscillator). The approximate
technology limits are 10 (low-end breadboard components and 8-bit
microcontrollers), 25 (high-end breadboard components and 16-bit
digital signal processor), and 40 (switched capacitor).
A recurring problem in filter implementation is that the conventional
filter design formulas yield filters with quality factors in the
40-80 range, even for low filter orders. For example, using the
elliptic filter design formulas, Matlab returns a 4th-order elliptic
filter with quality factors 1.7 and 61 for a given magnitude
specification. This filter can only be realized with expensive
custom integrated circuitry or 32-bit floating-point processors
(using sampling and quantization). Using the Filter Optimization
Packages, we have developed Matlab code that can take this filter as the
initial guess, and in less than a minute, return a 4th-order filter
that meets the original specifications but with quality factors of
0.68 and 10. The optimized design can now be easily implemented in
a wide variety of technologies.
We have developed Mathematica 2.2/3.0 packages to provide a framework for
generating Matlab 4/5 programs to optimize multiple characteristics of
existing filter designs subject to constraints on filter characteristics.
The current version simultaneously optimizes analog filter designs for
magnitude response, deviation from linear phase, peak overshoot, and
quality factors while satisfying constraints on the same criteria.
New characteristics can be easily added.
The release includes the generated Matlab code in the release for fourth-,
sixth-, and eight-order filters.
The generated Matlab code relies on the Optimization Toolbox.
We formulate filter optimization as a Sequential Quadratic
Programming (SQP) problem.
SQP requires real-valued, continuous, twice differentiable cost
function and constraints.
Provided that the solution space is convex, SQP in theory
always converges to the global optimum.
In practice, one must supply closed-form expressions for the
gradients of the objective function and constraints.
We use Mathematica to derive analytic objective functions and
constraints, compute the gradients of the objective function
and constraints, and convert the algebraic formulation into
correct Matlab code.
These packages were initially developed by Brian L. Evans during the
1993-1994 academic year while he was a post-doctoral researcher at
the University of California at Berkeley with the
The packages are copyright © 1993-1998 by the
Regents of the University of California.
The packages are released under a very liberal
freely distributable copyright and licensing agreement.
The packages are explained by the following papers:
- N. Damera-Venkata and
B. L. Evans,
"An Automated Framework for Multi-criteria Optimization of
Analog Filter Designs",
IEEE Transactions on Circuits and Systems II:
Analog and Digital Signal Processing,
vol. 46, no. 8, pp. 981-990, Aug. 1999.
- N. Damera-Venkata,
B. L. Evans,
M. D. Lutovac, and
D. V. Tosic,
"Joint Optimization of Multiple Behavioral and Implementation
Properties of Analog Filter Designs",
Proc. IEEE Int. Sym. on Circuits and Systems,
vol. 6, pp. 286-289, Monterey, CA, May 31 - Jun. 3, 1998.
- B. L. Evans,
D. R. Firth,
K. D. White, and
E. A. Lee,
Generation of Programs That Jointly Optimize Characteristics
of Analog Filter Designs",
Proc. of European Conference on Circuit Theory and Design,
pp. 1047-1050, Istanbul, Turkey, August 27-31, 1995.