INTRODUCTION
Computations of variance is required in the statistical analysis of quantization
effects in recursive digital filters. In the case of 1-D filters, a method using
Laurent series[1] expansion is presented to evaluate the Complex
integral or variance. In this method, one has to first take it for granted that
the 1-D transfer function H(z) is BIBO stable and then proceed with the decomposition
of H(z)H(z-1) of a particular type and finally arrive at a value
for 
,
where hn=Z-1 H (z) is the impulse response of the 1-D
filter.
In the case of 2-D digital filter transfer function H (z1, z2),
though the same technique of decomposition of H (z1, z2)
H (z1-1, z2-1) is not in general
applicable, a modified method is presented by Hwang[2] where 2-D
complex integral
is evaluated, by evaluating one integral at a time. This method always works
and gives positive value for
where hmn = Z-1 H (z1, z2), if H (z1, z2) is BIBO stable. But we find that this method does not give a positive value for the integral when the given H (z1, z2) is not BIBO stable. We bank on this and give a method for testing the BIBO stability of 2-D recursive filters.
We give some concepts on stability and present a theorem on stability for ready reference. In section III, we present briefly the method of Hwang[2] to calculate the variance in the case of 2-D recursive digital filters and we discuss the way by which one can judge whether the given transfer function is BIBO stable or not depending on the nature of the value the 2-D Complex integral yields when we apply the method of Hwang[2]. In section V we make use of recently reported results on a simple method of testing a 2-D filters for BIBO stability and apply it to some already available filters having the NSSK and come to the same conclusion regarding their l1-stability and their inverses.
Stability aspects: We presented two concepts of stability and discuss them as they apply to 2-D recursive filters. We assume that the 2-D Z-transform is defined with negative powers of the variable z.
Theorem 1: The 2-D transfer function H (z1, z2)
is given by
is BIBO stable iff
,
where hmn is the inverse Z-transform of H (z1, z2).
Theorem 1 is well known and can be found at many places in the literature[3].
The BIBO stability is also known as l1-stability. A 2-D recursive
filter transfer function H (z1, z2) is l2-stable
if
A 2-D recursive filter may be l2-stable but l1-unstable and not the other way. For example a 2-D transfer function is said to have non-essential singularities of the second kind[4] when both A (z1, z2) and B (z1, z2) are zero for some z1 and z2, |z1|=1, |z2|=1. In this case the filter may be l2 stable but not l1 stable. Whenever a filter having non-essential singularities of the second kind is l1 stable, it is also l2 stable.
Condition for stability: In the Hwang’s method of calculating variance from the 2-D transfer function H (z1, z2), one has to decompose the function H (z1, z2) H (z1-1, z2-1) as
by equating the like coefficients of the equation (1) on
either side by expressing the numerator A (z1, z2) and
denominator B (z1, z2) as polynomials in variable z2,
the coefficients being the polynomials in z1. We get the matrix equation
of the type Qq=a with qi and pi being the entries of the
column matrix q. We have to then solve the matrix equation and get q0
(z1-1). Then 
is obtained as equal to the 1-D integral
When we talk about the variance calculation using the method suggested by Hwang[2], there is a possibility that we may end up with a negative value for variance. This happens when we evaluate the 2-D Complex integral without testing the 2-D transfer function for BIBO stability. So we can use the method used to evaluate the variance to test for BIBO stability. Let us now workout some examples.
We have found that H (z1, z2) clearly has no non-essential
singularities of the second kind. Also we found by some other method that the
filter is BIBO stable (l1 stable). This is the same example which
Hwang[2] has given and it has a positive value for variance.
Example 2:
Let
This filter is found to be BIBO unstable. Let us now check what value we get
for variance by using the method suggested by Hwang[2].
The matrix equation we get is[2]
Solving the above matrix equation for q0 (z1-1)
we have,
Since
is negative, We can say that the given filter is unstable since variance cannot
be negative.
Example 3: Consider the transfer function[5]
The transfer function has non-essential singularities of the second kind since the numerator and denominator are equal to zero at z1=1 and z2=1. The matrix equation[2] corresponding to this H(z1, z2) is
we get after solving the above matrix equation
and the corresponding integral[2] value
So in this case the filter is l2 stable.
As mentioned earlier a 2-D filter of this type may be l2 stable
but need not be l1 stable. If we use the same variance method, the
determinant of the co-efficient matrix in equation (2) is
given by 
The Δ is equal to zero for z1=0 since it has a factor (z1-1)2 in the numerator. So when we try to decompose the function H(z1,z2)H(z1-1,z2-1) as in[2], we get values for q0(z1-1), q1(z1-1) and p1(z1) which may be infinity for z1=1. But we find that all q0(z1-1), q1(z1-1) and p1(z1) have factors (z1-1)2 in the numerators cancelling with the factor (z1-1)2 in the numerator of Δ. So H(z1-z2) may be BIBO stable.
Example 4: Consider the following example[5]
This function H(z1-z2) also has non-essential singularities of the second kind at z1=1 and z2=1. The matrix equation[2] is
The determinant of the coefficient matrix in (4) is
This Δ does not have zeros on the unit circle at z1=1. So we
have a finite solution for equation (4) when z1=1
and z2=1. In this case the filter will be l1 stable as
well as l2 stable as proved in[5].
BIBO stability of filters having the NSSK: We deal exclusively with the 2-D filter transfer functions having the NSSK.
We continue to assume that the 2-D transform is defined with negative powers of the complex variable z. We know the following theorem very well regarding the BIBO stability of 1-D recursive filters, given the denominator polynomial B(z) of a transfer function.
Theorem 1: A 1-D digital filter is BIBO stable iff
In a recent study[6] we have proved that to test the stability of a 2-D filter transfer function whose denominator is B(z1,z2) we need to test for stability only the stability of 1-D polynomials B(z1,z2N+1, z) and/or B(z,z2N+1) where N is the degree of both the variables in B(z1,z2). We make use of this fact and test some well known transfer functions which have non-essential singularities of the second kind for BIBO stability (l1- stability).
Example 5: Let
be required to be tested for BIBO stability. We note that this is the same
H(z1,z2) as in example 3. We have
The filter is of order N=1.
Now look at the denominator polynomial factor (2z2+z+1). It has
two complex conjugate poles at 
having
magnitudes of 
.
Since the poles lie outside the unit circle, the filter is BIBO unstable. With
this we can say that the 2-D transfer function H(z1,z2)
and hence the 2-D filter is BIBO unstable. Unfortunately in this method we are
not able to comment anything on the l2 stability of the filter. But
on seeing (5) we can comment that the inverse transfer function
is also l1 unstable since in (5), three zeros are on |z|=1, which is in conformity with the conjecture[7].
Example 6: Consider the following transfer function[8].
we have,
It is easy to see that the 1-D function in (6) has all its zeros and poles inside the unit circle. Thus the function is BIBO stable. Further on seeing the numerator of (6) we can say that the inverse of H(z1,z2) is also stable. So the corresponding 2-D transfer function H(z1,z2) and its inverse are BIBO stable as proved by Swamy and Roytman[8].
In this study, the method to find variance for 2-D recursive digital filters
is applied to test whether the given 2-D transfer function H(z1,z2)
is BIBO stable or not. We also applied the same technique to 2-D transfer functions
having non-essential singularities of the second kind and could conclude successfully
whether these transfer functions are l1 stable (BIBO stable) or l2
stable or both. In summary if the recursive 2-D transfer function has no non-essential
singularities of the second kind (NSSK), it will be l1 unstable if
the value of
is negative.
Unfortunately we have to restrict ourselves to lower order transfer functions since the complexity involved in evaluating the determinant of the matrix to obtain q0 (z1-1) and the consequent 1-D integral evaluation is too much in Hwang’s method. We are trying to propose altogether a different simple approach to evaluate the variance in the case of 2-D recursive filters.
We also used the recently proved results[6] on the stability testing of 2-D recursive filters and shown with two examples, what was conjectured in[9] and proved in[8,10] are also true. This method is also very simple and can comment on the l1 stability of the 2-D transfer function and its inverse when it has the NSSK.
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