2.6. Stability – Digital Filters Design for Signal and Image Processing

2.6. Stability

The fact that the transfer function is a rational fraction naturally leads us to the issue of stability, which can be studied from considering the z-transform of the impulse response.

2.6.1. Bounded input, bounded output (BIBO) stability

A linear time invariant system is BIBO stable if its impulse response verifies the following relation (see also Chapter 10):

The transfer function is the z-transform of the impulse response; from there, we have, for all of z belonging to the ROC:

Now, on the unity circle in the complex plan z, we have:

From this the following result is obtained:

Many stability criteria have been developed to study the stability of filters. Among these, we first will look at the test of pole positions of the transfer function, then at Routh's and Jury's criteria.

2.6.2. Regions of convergence

In causal systems, a necessary and sufficient condition of stability is that all the poles of the transfer function must be inside the unity circle in the z-plane.

The decomposition of the basic elements of the transfer function of a discrete causal system Hz(z) introduces two types of terms.

admits the pole pi = a, and admits the complex conjugates as poles of the module equal to |pi| = c.

Here we see that the z-transform of the sequence x1(k) = aku(k) converges if, and only if, |a|<1 and equals . In addition, according to Table 2.1, x2(k) = αk sin(ω0kTs).u(k) and x2(k) = αk cos(ω0kTs).u(k) admit, respectively, the following z-transfonns:

on condition that |α| < 1. A specific linear combination of x2(k) and of x3(k) gives us a z-transform in the form of with |c| < 1.

For an anti-causal system, a necessary and sufficient condition of stability is that all the poles of the transfer function must be strictly outside the unity circle.

EXAMPLE 2.10.– let the following formula be the transfer function of a discrete causal system:

It admits for zero z1 = 2 and for poles and . The stability is verified because |p1| <1 and |p2| <1

Figure 2.9. Diagram of poles and zeros of

2.6.2.1. Routh's criterion

The first approach we will consider for looking at stability uses Routh's criterion. In general, Routh's criterion is used to study the stability of continuous systems, usually with looped systems. It helps us learn the number of zeros of the real part of a polynomial by examining its coefficients.

Routh's criterion has been adapted to discrete systems by changing variables with the follow ing transform:

We then continue by analyzing the denominator of H(λ) that is expressed as:

We formulate the following table:

Table 2.2. Table for application of Routh's criterion

Routh's theorem states that the number of zeros of Hz(λ) of the strictly positive real part is equal to the number of sign changes. We can verify this by looking at the first column of Table 2.2 from top to bottom.

EXAMPLE 2.11.-

Let us look again at the example where:

First, we carry out the change of the variable indicated in equation (2.38). We get:

From this, the following table is constructed from the coefficients of D(λ):

7 7
6  
3  

Table 2.3. Application of Routh's criterion

There is no change of sign in the first column; this means there will be no zeros in the strictly positive real part of D(λ). We conclude from this that there will be stability.

2.6.2.2. Jury's criterion

Let be the transfer function. Jury's criterion is an algebraic criterion that allows us to determine if the polynomial roots A(z) are inside the circle of radius unity in the z-plane.

So we get:

where the coefficients ak are real and a0 > 0.

We construct a table of 2(M−l)-3 rows. The first two lines of this table are filled, respectively, by polynomial coefficients according to the increasing, then decreasing, powers in z.

The following lines are respectively deduced by using the determinant of specific coefficients of the two proceeding lines, as follows:

This gives us the following table:

Table 2.4. Table for establishing and verifying Jury's criterion

According to Jury's criterion, the polynomial roots are inside the circle of radius unity in the z-plane if the following M conditions are met:

A(1)> 0 and A(−1) > 0 if M-1 is even or A(−1) < 0 if M-1 is odd.

– |aM−1| < a0.

– |βM−2| > |β0| |γM−3| > |γ0|, … and |q2| > |q0|.

EXAMPLE 2.12.– looking again at the example of with A(z)=4 − 2z−1 + z−2. The corresponding Jury table is as follows:

In addition, since A(1) =3 > 0, A(−1) = 7 > 0, the poles of the transfer function are inside the unity circle. In Chapter 10, we will discuss stability in more depth.