Stabilization and Stability Testing of Multidimensional Recursive Digital Filters

Lecture by Prof. Brian L. Evans
Slides by Niranjan Damera-Venkata
Embedded Signal Processing Laboratory
Dept. of Electrical and Computer Engineering
The University of Texas at Austin
Austin, TX 78712-1084
http://signal.ece.utexas.edu

Introduction

• Stability of general IIR filters
• Stability tests
  • Graphical root locus techniques
  • FFT based cepstral methods

• Stabilization of unstable filters based on
  • Double Planar Least Squares Inversion (DPLSI)
  • Discrete Hilbert Transform (DHT)

The Transfer Function

• The z-transform will not converge for all values of
• If it converges for , , then the Discrete-Time Fourier Transform exists and the system is stable.

  and Unit Bicircle

Stability of 2-D LSI Systems

• Bound-Input, Bounded-Output (BIBO) criterion

  

• Spatial domain necessary and sufficient condition for BIBO stability

  

  Implies is analytic on the unit bicircle

• For a rational transfer function

  

  of a causal system, recall that the stability condition is that all roots of B(z) should be inside unit circle

Effect of Numerator Polynomial on Stability: [Goodman]

• No effect in 1-D case: Use factorization theorem

  

• Situation is not so simple in 2-D

  

  

• What happens when
  • Note the indeterminate forms
  • Goodman established that is unstable while is stable
  • Such singularities are called non-essential singularities of the second kind.
  • There is no known method to test for stability in the presence of such singularities

Necessary and Sufficient Conditions for Stability

• Let
• (Shanks and Justice) Let be the denominator polynomial of a first quadrant multidimensional recursive digital filter. The filter is stable if and only if whenever
  simultaneously.
  Disadvantage: Whole exterior of the unit bicircle must be searched for points of singularity.
• (DeCarlo-Strintzis) is stable if and only if
  1. for where and
  2. 
     This second condition is equivalent to

     

• The DeCarlo-Strintzis Theorem suggests a stability test that consists of N 1-D stability tests plus a search for roots of B(z) over the N-dimensional surface .

The O'Connor-Huang Mapping Theorem

• How do we test the stability of an NSHP filter?
  Consider two sectors and
  , with .
  The following is an injective linear map from into

  

  with defined as:

  

• Let b(m,n) be a recursive array with angle of support . Then b(m,n) is stable if and only if with , the recursive array g(m,n) = b(m',n') is stable,
• We need to ensure that we have support in a sector and the two rays (line segments) that define the sector are not colinear.
• Example:

  

  So the mapped polynomial to be tested is

  

Root-Locus Techniques

• Consider:

  

• We can hold constant

  

• Roots of with respect to are functions of
• Plot roots in plane. Rootlets must lie completely inside the unit hyperdisk for the filter to be stable.

Cepstrum/2-D cepstral stability tests

• 2-D complex cepstrum of b(m,n)

  

• is real for a real sequence
• It is called ``complex'' due to the use of the complex logarithm.

  

• A general recursive digital filter is stable if and only if its 2-D complex-cepstrum exists and has the same minimum angle support as the original sequence.

Stabilization of unstable recursive digital filters

• In the 1-D case this is very simple
• , a reflection of the root did not change the magnitude spectrum.
• Factor denominator polynomial and reflect the roots inside the unit circle.
• Fundamental curse of multidimensional digital signal processing: no polynomial factorization algorithm
• Proposed methods
  • Double Planar Least Squares Inversion [Shanks, Treitel and Reddy]
  • Discrete Hilbert Transform [Read, Treitel, Reddy]

Double Planar Least Squares Inversion

• PLSI of a coefficient array C is an array P such that
  1. , U is the unit pulse array (of all ones)
  2. C * P = G such that U-G is minimized in least squares sense.
• Shank's conjecture: Given an arbitrary real, finite array C, any PLSI of C is minimum phase, and the applying PLSI twice to C yields minimum phase with the same magnitude spectrum as C.
• Proof of ``modified'' Shank's conjecture [Reddy]

Stabilization and Stability Testing Unified: The Multidimensional DHT

• Continuous Hilbert Transform theory involves theory of singular integrals and m-D extensions are very complicated [ Besikovitch, Calderon and Zygmund]
• DHT is the relation between the real and imaginary parts of the Fourier Transform of a causal sequence.

  

• If we assume that the complex cepstrum is causal,

  

• Expression for very complicated [ Damera-Venkata, Venkataraman, Hrishikesh and Reddy] and reduces to in the 1-D case.

Stabilization via DHT

• To stabilize
  1. Find , the minimum phase response
  2. Evaluate
  3. Take multidimensional inverse FFT
  4. Truncate coefficients to same support as
  5. Use a large size FFT for higher coefficient accuracy

Stabilization via DHT: Example Example: Consider given by:

Useful Theorems: \ [Damera-Venkata, Venkataraman, Hrishikesh and Reddy]

• Multidimensional minimum phase if it exists is unique
• If the given m-D polynomial is factorizable, then the transformed polynomial is also factorizable, and the factors of the transformed polynomial are transformed versions of the factors of the given m-D polynomial.
• The m-D polynomial of any causal m-D polynomial , not having zeros on the unit hypercircle is stable.
• Minimum phase polynomials are fixed points of the multidimensional DHT

Stability Testing using the DHT

• It is required to ascertain whether array B is stable or not.
  1. Apply the DHT to obtain array A.
  2. Compare arrays B and A.
     • If ,then B is a stable array.
     • If , then B is unstable.

Stability Testing Example

is unstable, while is stable.

• About this document ...