On the Optimization of the Tay-Kingsbury 2-D Filterbank

1. On the Optimization of the Tay-Kingsbury 2-D Filterbank Bogdan Dumitrescu Tampere Int. Center for Signal Processing Tampere University of Technology, Finland

2. Summary • Problem: design of 2-channel 2-D FIR filterbank • Idea: 1-D to 2-D McClellan transformation • Math tool for optimization: • sum-of-squares polynomials (on the unit circle) • Optimization tool: semidefinite programming

3. 2-Channel 2-D Filterbank • Quincunx sampling • Non-separable filters • FIR H0(z) F0(z) H1(z) F1(z)

4. Tay-Kingsbury idea • Take • Define • PR condition or

5. Tay-Kingsbury transformation • 1-D halfband filter factorized • 2-D transformation G(z1,z2) with • PR filterbank:

6. Transformation properties • Ideal frequency response • Denote • Property:

7. Optimization of the transformation • Minimize stopband energy where is the vector of coefficients and is a positive definite matrix • Constraint: !!!

8. Stopband shape

9. Sum-of-squares polynomials • A symmetric polynomial is sum-of-squares on the unit circle if • A sos polynomial is nonnegative on the unit circle

10. Positive polynomials • Basic result: all polynomials positive on the unit circle can be expressed as sum-of-squares • However, theoretically it is possible that

11. Parameterization of sos polynomials • A symmetric polynomial is sos if and only if there exists a positive semidefinite matrix Q such that Gram matrix where 0 elementary Toeplitz 0

12. Resulting optimization problem • Semidefinite programming (SDP) problem • Unique solution, reliable algorithms

13. Example of design • 1-D halfband prototype • Transformation degree: 3 (symmetric polynomial) • Execution time: 1.2 seconds

14. Frequency response H0

15. Frequency response F0

16. Improvement of synthesis filter • Since usually degF0>degH0, a new synthesis filter of same degree can be obtained via lifting • Optimization of a quadratic with linear constraints A is obtained by solving a linear system.

17. Frequency response of improved F0