## 7.3. Structure of IIR filters

### 7.3.1. *Direct structures*

In this section, we will refer to the direct canonic structure that is characterized by the following transfer function:

First, we introduce an intermediary output *x*_{1}(*k*), so that:

The transfer function in equation (7.1) can then be decomposed into a product of two transfer functions:

All this occurs as if the filter *H _{z}*(

*z*) has been obtained by cascading an FIR filter

*H*

_{1}(

*z*) and an IIR filter

*H*

_{2}(

*z*).

We end up with the most spontaneous realization, called the direct form structure 1.

We see that this structure requires *N*+*M*−2 delay cells. We can then ask the following question: can we share some cells and only use *M*-l delay cells, knowing that *M*>*N*? The answer is yes, leading us to the direct form structure 2.

An alternative approach consists of decomposing the transfer function shown in equation (7.1) into an IIR, then an FIR filter. The intermediate output *x*_{1}(*k*) then satisfies:

Here we have

So by using equations (7.5) and (7.6), we can establish the structure shown in Figure 7.5.

We can nevertheless distribute some delay cells. We then have the structure shown in Figure 7.6.

When *M* = *N* = 2, we end up with a second-order cell (see Figure 7.7).

Taking into account the role played by the second-order cell, we need to understand what happens when we consider a linear system guided by a pole near the unity disk. To generate a resonance at the frequency/0, we consider a pole inside the unity disk with *R* < 1. is also a pole of the transfer function of the second-order cell. From here, the transfer function can be written in the two equivalent forms that follow:

and

The frequency response of the filter corresponds to the Fourier transform of the impulse response, which can be obtained by taking the transfer function *Hz*(*z*) as so that:

According to the values of *R*, the resonance is more or less strong and tends towards infinity when *R* tends towards 1. So, Figure 7.8 shows the position of the poles and the corresponding frequency response in different situations. The normalized frequencies linked to the poles are equal to and the values of *R* are equal successively to 0.7, 0.9 and 0.99.

We then normalize *H*(*f*) so that |*H*(*f*_{0})| = 1, which conditions the gain value G as follows:

By definition, the cut-off frequencies. *f _{c}* of this filter at −3 dB are:

Equation (7.13) is reduced to:

both by using equation (7.11) and from the fact that |*H*(*f*_{0})| = 1:

This equation admits two solutions *f*_{1} and *f*_{2} which characterize the bandwidth Δ*f* = *f*_{2} − *f*_{1}. To avoid complex algebraic calculations, using a geometric approach helps us to easily find the result. Indeed, let *Z _{A}*,

*Z*, and

_{B}*z*be the complex numbers associated with points A, B, and Q in the complex z-plane and at frequencies

_{Q}*f*

_{1},

*f*

_{2}, and

*f*

_{0}(as in Figure 7.9). Moreover

*p*is associated with point

_{i}*P*and

*p** to P*.

_{i}We thus have:

By combining equations (7.16) and (7.17), we obtain:

Now, when *R* tends towards 1, the poles approach the unity circle while the points P, Q, and A become very close and their distance in relation to O, the source in the complex plane, is approximately the same, i.e. . Therefore, equation (7.18) becomes:

Since distances between points Q, A, and P* are approximately the same, this means that:

Taking into account the equalities in equations (7.14) and (7.20), equation (7.18) becomes:

The angle represented by segments AP and PQ is therefore approximately equal to , which means that the angle represented by segments PA and AQ is as well. The triangle PAQ is almost an isosceles.

This means that:

When *R* tends towards 1, we assimilate the arc AB to the tangent vector to the unity circle in Q. We deduce that:

from which we get:

We can demonstrate in this way that the bandwidth is proportional to (1−*R*). The more the poles are close to the unity circle in the z-plane, the more the filter is selective around the frequencies associated with the poles.

Now we place a pair of zeros and close to the poles inside the disk and associated at the same frequencies. Equation (7.9) becomes:

or

The frequency response of the filter then satisfies:

According to the values of *r*, we will observe an accentuation effect of the resonance at normalized frequencies when *r* < *R* and weakening if *r* > *R*. The accentuation or weakening level is controlled by the proximity of *r* to *R*. The spike width is always related to the proximity of the poles in relation to the unity circle (see Figures 7.10 to 7.13).

If we choose the value of *r* equal to 1, the expression of the transfer function given in equations (7.25) and (7.26) becomes:

and also

Whether it has two poles or two poles and two zeros, we learn several facts from studying this kind of filter:

– we learn that the resonance acuity is regulated by the proximity degree of the pole in relation to the unity circle;

– we learn the association of zeros in the transfer function allows us to modify the response curve of the filter. There will be positions of zeros that will lessen the curve, and there will be others that instead favor resonance, as shown in the diagrams in Figure 7.15.

This technique can be generalized to synthesize a filter having a finite number of slots. It is enough to generate a numerator *N*(*z*) whose zeros are situated on the unity circle and correspond to the desired slots. We then construct the denominator which must equal *N*(*R*^{-1}*z*) to deduce the transfer function.

We thus obtain a generalized expression of the following formula:

APPLICATION.- with a signal sampled at 500 Hz, we want to eliminate a periodic signal whose fundamental frequency equals 50 Hz.

The normalized frequencies to be eliminated are represented as:

The zeros of *H _{z}*(

*z*) thus correspond to:

and correspond to the tenth-degree root of the unity. Thus, we have:

and we take, for example:

Slot filters allow us to remove harmonies, while comb filters can increase periodicities; that is, they increase the signals containing harmonic frequencies (multiples of the fundamental). These filters are used in audio applications for devices that create reverberations. The filters act as reflectors of sound waves and favor certain periodic signals.

### 7.3.2. *The cascade structure*

The cascade structure decomposes the filter of the transfer function *H _{z}*(

*z*) with a succession of first and second-order cells

*H*(

_{i}*z*). However, we can see that a first order cell is a specific example of a second-order cell by taking the values associated with

*z*

^{−2}that are equal to zero:

or the following formula:

or:

### 7.3.3. *Parallel structures*

A parallel structure decomposes the filter of the transfer function *H _{z}*(

*Z*) with a parallel interconnection of filters the transfer function of which is the transfer function

*H*(

_{i}*z*). We have, as a result, the following formula for the transfer function:

In the following section, we will look more closely at direct and cascade structures.