1. Joint Optimization of Multiple Behavioral and Implementation Properties of Digital IIR Filter Designs
Magesh Valliappan, Brian L. Evans, and Mohamed Gzara
The University of Texas at Austin
Miroslav D. Lutovac
Telecom & Electronics Institute University of Belgrade
Dejan V. Tošić
Dept. of Electrical Engineering
University of Belgrade
2000 IEEE Int. Sym. on Circuits and Systems

2. Introduction Problem Goal Solution
Simultaneously optimize multiple characteristics of an existing digital IIR lowpass filter design
Goal
Develop an extensible automated framework
Solution
Solve constrained nonlinear optimization problem by using sequential quadratic programming (SQP)
Program Mathematica to derive formulas and generate Matlab programs to perform optimization
SQP is more robust and converges faster than other gradient based methods, especially when the search space is complicated due to nonlinear constraints.
We use Matlab to leverage its powerful optimization toolbox, which includes a built in SQP routine. Matlab also provides common filter design tools.

3. Modeling Free parameters Properties Compute
Set of n conjugate pole pairs ak exp(± j bk)
Set of m conjugate zero pairs ci exp(± j di)
Properties
Behavioral: magnitude response, phase response
Implementation: quality factors
Compute
Cost function as a weighted mean of distance measures
Constraints to enforce numerical stability
Closed-form symbolic gradients for robustness
The poles and zeros are represented in a polar format. This simplifies the
mathematical expressions.
We optimize the filter to improve the magnitude and phase response and reduce the quality factor. Implementations of IIR digital filters can suffer from instability and quantization noise. The quality factor provides an architecture independent measure of the filter’s numerical requirements for implementation.
We compute an objective function as a weighted mean of distance measures. We add constraints on the desired properties and also to ensure numerical stability. We symbolically compute expressions for gradients because the optimization is very sensitive to errors in gradient estimation by finite differences.

4. Objective Measures Magnitude response Unwrapped phase response
Scale to be unity at DC
Unwrapped phase response
Constrain zeros to be outside passband
Quality factor Q
For each pole pair ak exp(± j bk): 0.5  Qk  
Effective quality factor is geometric mean of Qk factors
This slide shows the three objective measures.
The phase response over the passband is obtained in unwrapped form only when the zeros are located outside the passband and all poles are within the unit circle. This does not adversely affect the search space of possible lowpass filters and guarantees a smooth differential expression for phase.
(This actually depends on the exact equations used for the phase)
The quality factor is as defined in the book.

5. Distance Measures Non-negative differentiable measures
A value of zero means the ideal case
Differentiability necessary for SQP formulation
Deviation in magnitude response
L2 norm of deviation from ideal in passband, stopband and transition bands
Deviation of quality factors
Minimum effective quality factor Qeff is 0.5
Deviation measured as sq = Qeff - 0.5
This slide talks about the distance measures used to quantify the objective measures. All the measures represent a ‘norm’.
For magnitude we use the L2 norm, but this can be easily changed to any norm that is symbolically differentiable.
Quality factor has a minimum of 0.5 and so Qeff-0.5 can be used as a norm to measure deviation from ideal.

6. Distance Measures Deviation from linear phase in passband
L2 norm of deviation from perfect linear phase at optimal slope within (0, l) where l  p
Approximate optimal slope to obtain analytic form
Weighted mean of two first-order estimates
Accurate up to four Taylor series terms
 = ;  = ; r1= ; r2 =
The optimization is setup to minimize the deviation from linear phase over the passband or a portion of the passband. To obtain an analytical expression for the deviation we need to symbolically determine the optimal straight line model for the phase for the filter.
This is done by a simple linear combination of two first order estimates. This is identically equal to the slope of the optimal best line fit for any phase expression accurate up to 4 terms of the Taylor series about the origin.

7. Overall Cost Function Weighted sum of distance measures
Passband, transition band, stopband, and phase response are normalized by bandwidth
Quality factor
User-defined weights
Wp,Wt,Ws, and Wphase for passband, transition band, stopband, and phase response, respectively
Wq for quality factor

8. Constraints Zero locations outside of the passband
Numerical stability of phase response
Quality factor of each pole pair less than Qmax
Qmax determined by technology
Pole locations inside unit circle
Magnitude constraints
Passband 1 - p < | H(e j) | < 1 + p
Stopband | H(e j) | < s p
This slide describes the constraints being used.
The zeros are forced out of the passband to ensure unwrapped phase in the passband.
The value of Qmax is determined by the implementation technology.
The magnitude response constraints are similar to other design procedures.
Even though it is shown as a continuous constraint it is implemented at a few discrete points. The number of points has to be reasonably larger than the order of the filter to make sure we do not sample only at peaks and valleys.
| H(e j) | s 

9. Implementation Mathematica Matlab
Compute cost function, constraints, and gradients
Generate efficient Matlab code for SQP solution to optimization problem
Matlab
Set all user-definable parameters
Initial filter design
User-supplied or default computed elliptic filter
Preferably overdesigned
This slide is an overview of the operation of the framework.
Mathematica computes all the functions symbollically and then generates Matlab programs. The Matlab code generation process and the Matlab code itself can be optimized.
* Code generation -
The code generation can be sped up significantly by creating Matlab functions for commonly occurring expressions and their derivatives.
All the expressions are equivalent with respect to pole and zero pairs. They can be generated by computing with respect to a pole zero combination and then swapping variable names to generate all other combinations.
* Matlab Code execution
The Mathematica to Matlab code generation package performs common subexpression elimination to reduce the execution time of the Matlab code.
The user can set the tolerances and weights in the generated Matlab script. Matlab is used to generate an initial filter. If this filter barely meets the requirements then there is no room for further optimization. So we prefer to start with a filter of order higher than necessary.

10. Design Example
Optimization of phase response with constraints on quality factors and magnitude response
Phase optimized over the entire passband
Magnitude response constraints
Passband: (0 - 1) rad, ripple 0.05
Stopband: (1.8 - ) rad, ripple 0.01
Initial filter
Fourth-order elliptic filter generated by Matlab
Fails quality factor constraints (SQP relaxation)
We optimize a filter satisfying given magnitude constraints to make the phase more linear and reduce quality factor.
The initial filter satisfies magnitude constraints but does not meet constraint of
maximum quality factor = 2.
When the initial filter does not meet the constraints the program tries to obtain a feasible solution by minimizing the constraint violation.

11. Design Example Optimized filter
Phase response closer to linear (shown on next slide)
Lower quality factors
Satisfies magnitude response constraints
The new filter is better in terms of both phase response and quality factor but at the same time meets magnitude constraints.
Q factor reduced from 3.62 to 2.00

12. Design Example Magnitude Response Phase Response Initial filter
The optimization has traded off excess bandwidth in the transition band to improve the quality factor and reduce the nonlinearity of phase.
Magnitude Response
Phase Response
Initial filter
Optimized filter

13. Conclusions
Extensible framework for automated digital IIR filter design optimization
Symbolic computation eliminates algebraic errors
Error-free generation of source code
Robust due to symbolic computation of gradients
Easy to change objective functions, measures and constraints
Software available at