6.1. Introduction to infinite impulse response filters – Digital Filters Design for Signal and Image Processing

6.1. Introduction to infinite impulse response filters

Infinite impulse response (IIR) filters are recursive mode filters that are characterized by the following difference equation:

where at least one of the coefficients {bi}1≤i≤N−1 is non-null. We can easily reduce this to a relation where a0 = 1. From here, we will assume that this hypothesis is satisfied.

Equation (6.1) is verified for all the values of k. We thus have:

By reinjecting equation (6.2) of y(k − 1) in the difference equation in (6.1), we see that y(k) depends on the preceding values of the output y(k − 2),…, y(kM) and of N + 1 values of the input signal x(k),…, x(kN). By repeating this step to infinity, we express the output y(k) as a linear combination of an infinity of terms of the input signal x(k). The filter is therefore an IIR filter.

The z-transform on equation (6.2) helps us obtain the transfer function of the filter, which is a rational fraction in z:

The division following the increasing powers of the numerator by the denominator then leads to an infinite sum of terms.

If the order of a finite impulse response (FIR) filter is between 25 and 400, that of an equivalent IIR filter is generally lower and typically varies between 5 and 20 (or sometimes more).

This chapter is organized as follows: after giving several examples of IIR filters, we will discuss their synthesis. Synthesis can be carried out in several ways. One of the most commonly used methods consists of using continuous-time synthesis techniques (as discussed in Chapter 4), then going from the continuous domain to the digital. For that, it is necessary to follow the rules of changing from the continuous to the discrete domain. To this effect, we will use the following two approaches: that of the invariance of the impulse response and that of the bilinear transformation.

6.1.1. Examples of IIR filters

Let us look at two types of discrete filters, which we assume are causal.

The first, expressed as y = T1[x] is regulated by the following difference equation:

From here, the transfer function of the system is written as:

Now we look at a second transformation y = T2[x] so that:

We obtain the following transfer function H2(z):

Now we consider the stability of the two transformations. For that, we look at the position of the poles of the two transfer functions H1(z) and H2(z). In both cases,

We can then obtain the impulse responses h1(n) and h2(n), respectively from H1(z) and H2(z). Since H1(z) = , i.e. that the transfer function is the z-transform of the impulse response of the system, we carry out the polynomial division of N(z) by D(z).

We now directly calculate the impulse response of the transformations T1 and T2 using difference equations. In order to begin calculating the output sequence of the causal system, we see that the input and output sequences are causal (except in the special case when we have information about the system's initialization). So, let x(n) = δ(n).

We can easily conclude that:

and for n ≥ 1, which we can rewrite in the following form:

We proceed in the same way to calculate h2(n), the impulse response linked to the second transformation T2.

The frequency responses linked to the transformations T1 and T2 are shown in Figures 6.1 and 6.2.

Figure 6.1. Amplitude diagram of the T1 transformation (low-pass)

Figure 6.2. Amplitude diagram of the transformation T2 (high-pass)

Now we will look at the following system links: T1]T2[.]], T2]T1[.]] and T2]T1[.]].

We calculate their respective impulse responses by using the laws of association of invariant linear systems.

First, the impulse responses of the links T1]T2[.]] and T2]T1[.]] are identical. So the transfer functions and the corresponding impulse responses are, respectively:

and h1(n)*h2(n) = h2 (n)*h1(n).

More precisely, we obtain:

The difference equation related to the links T1[T2[.]] and T2[T1[.]] then equals:

Figure 6.3. Amplitude diagram of the transformation T1[T2[.]]

Now we consider the link T2[.] + T1[.]. We see that the transfer function satisfies the identity:

The corresponding impulse response is h1+2 (n) = δ(n).

6.1.2. Zero-loss and all-pass filters

A linear filter is a zero-loss filter if the energy of the signal is conserved during filtering for all input signals. This means we have:

Therefore, the energies of the input signals x(k) and the output signals y(k) can be calculated in the frequential domain using Parseval's theorem. We thus have:

and

According to equations (6.5) and (6.6), a linear filter is zero-loss if we have:

We assume the filter is stable. So that equation (6.7) is verified for all input signals x(k), we must have, with the possible exception of a finite number of responses, for all frequency values:

A zero-loss filter that is stable and of finite order is thus an all-pass filter of unity gain.

The simplest all-pass filters are FIR filters such as H(z) = ±l or H(z) = ±zm. For IIR type filters, the most general expression of the transfer function of a causal all-pass filter is:

where a*k designates the conjugate of ak.

EXAMPLE 6.1.– we assume that the coefficients ak are real.

Since the filter is stable, the poles are inside the unity circle in the complex z-plane. The zeros are then necessarily outside the unity circle.

Figure 6.4. Diagram of poles and zeros for an all-pass filter

6.1.3. Minimum-phase filters

6.1.3.1. Problem

Here we look at a filter represented by its transfer function H(z). We use the input x(k) and try to characterize the output y(k).

Figure 6.5. Filtering of input x(k)

However, now we assume that we use the output y(k) and that H(z) is known. We can then determine the z-transform of x(k) as follows:

Figure 6.6. Inversion or deconvolution

The operation of the change from , called deconvolution, is only possible if is stable; that is, if all the zeros of H(z) that have become poles of inside the unity disk.

At this point, the following question arises: can we bring the zeros that are outside the unity circle to the inside of the circle? The answer is discussed in the following section.

6.1.3.2 Stabilizing inverse filters

We say that a minimal phase, causal filter is stable whose transfer function zeros are inside or on the unity circle. This kind of filter presents a minimum group delay in the ensemble of filters having the same transfer function module, no matter what z is in the z-plane.

Let a filter be characterized by its transfer function H(z). Let Hint(z) be the transfer function constructed from the poles and the zeros of H(z) situated in the unity disk and Hext(z) the transfer function constructed from the zeros of H(z) situated outside the unity disk. We then obtain the following decomposition of H(z):

This decomposition can develop in the same way by introducing , the inverse of the conjugate of zi

The zeros of H1(z) all being inside the unity circle since is at minimum phase. In addition, . Indeed, we have:

Consequently, we can always decompose a filter into a minimum phase and all-pass filter. We thus have:

Since we have:

Then from equation (6.14), we get:

and by deriving equation (6.16) In relation to the frequency on the interval we get:

Now, we can demonstrate by using equation (6.12) that:

We then have:

In Chapter 4, we introduced the concept of group delay, represented as follows:

Using equations (4.73) and (6.19), we conclude that:

Consequently, a minimum phase filter is a filter whose group delay is the lowest among causal and stable filters with the same transfer function module throughout f.