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 H 0 (z) F 0 (z) H 1 (z) F 1 (z)

4 Tay-Kingsbury idea PR condition Take Define or

5 Tay-Kingsbury transformation 1-D halfband filter factorized 2-D transformation G(z 1,z 2 ) 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 where 0 0 elementary Toeplitz Gram matrix

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 H 0

15 Frequency response F 0

16 Improvement of synthesis filter Since usually degF 0 >degH 0, 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 F 0