Resonator-based Digital Filters

Implementation of digital filters has no theoretical difficulties, but, there are practical problems, because of the finite wordlength of signal processing devices. The problem arises mainly at the implementation of IIR filters of hard specification. A filter can be realized by many discrete systems, the behavior of them is not the same assuming finite wordlength implementation. By completing the resonator-based observer a very good filter structure can be made. In the following first the problems of the digital implementation are reviewed, then resonator-based filters are introduced.

Problems of Digital Implementation

Digital filters are linear, time-invariant discrete time systems, their transfer function is a rational function of z:
 

where B(z) and A(z) are polynomials. The calculation of the filter generally can be done using the following equation:

where x(n) and y(n) are the samples of the excitation and the response at time instant n, respectively, a_i and b_i are the coefficients of A(z) and B(z), respectively. In the case of FIR filters the second term of the sum is zero.

FIR and IIR filters are different, from implementation point of view, as well. While FIR filters can be calculated by the above equation, it is a rare occasion that IIR filters can be implemented this - so called direct - way. The coefficients a_i e.g. realize the denominator of H(z). The relation between the roots and the coefficients is highly nonlinear, and even a small coefficient mismatch results in a large root difference. This is why quantized coefficients of the designed filter can lead to unstable system. The greater the order of the filter, the more dangerous is the direct implementation. If the system remains stable, it will be possibly out of specification. Furthermore, the quantization error during the operation can also lead to unstable system. Such problems are summarized in Fig. 1.

Because of the problems mentioned above, the transfer function H(z) is implemented by special structures. A common solution is the decomposition of the transfer function into second order blocks, which can be implemented directly. Serial or parallel connection of them realizes the designed filter. Lattice and wave digital filters have very good properties. During the operation the rounding should lessen the absolute value of the number to be quantized. The resonator-based filter is also a special structure. It has excellent sensitivity properties, does not need scaling. Rounding towards zero guarantees that no limit cycles occur, and it produces minimal noise.

Table 1. evaluates some structures, assuming several aspects. The numbers in the table are marks used in the education. (The worst and best marks in Hungary are 1 and 5, respectively.) The base of the evaluation is that how the structure to be marked can be applied for the implementation of any transfer function H(z).

Resonator-based Filters

The block diagram of the filter can be seen in Fig. 2. The input of the systems is s_n, while y_n is the output. The resonator-based observer is completed by the coupling represented by the multipliers w_i, and by the coefficient d representing the delay-free forward path.
 

The parameters of the structure can be calculated using the original H(z) as follows:

1. The calculation of the forward coefficient is the following:

i.e. it is the ratio of the zero-order terms in the transfer function expressed as a function of z^-1.

2. By the denominator A(z) one can determine the roots of the following equation:

A(z^-1) can be simply determined by reversing the order of the coefficients of A(z). Since there are two sings in the equation, the calculation results in two root sets. One of them will be chosen as resonator set.

3. The following parameters are to be calculated for both root sets:

which are real numbers if the calculation was precise. z_i, z_j denote the potential resonator poles and p_j stands for the poles of H(z). The resonator set is to be chosen for which the following inequality is satisfied:

The coefficients g_i in the figure can be calculated as follows:

4. The coupling coefficients can be easily calculated by sampling the original transfer function at the resonator frequencies:

As an example a tenth-order Cauer-type IIR filter can be presented, the magnitude response of which can be seen in Fig. 3.

The resonator poles are the following:

The resonator poles are indicated by small blue circles in Fig. 4. Note that the resonators are concentrated in the passband.

The further parameters are the following:

The representation of these parameters is not a problem, even if only fixed-point format is used. According to the experiences, the resonator-based structure can implement IIR filters of very hard specification, even if 16 bits fixed-point format is used.

Related Publications:

G. Péceli, "Resonator based digital filters", IEEE Transactions on Circuits and Systems vol. CAS-36, pp. 156-159, Jan. 1989.	Introduction to the resonator-based filters.
T. Tölyhi, "Programozható digitális szuro tervezése", M.Sc. thesis (in Hungarian), Budapest University of Technology and Economics, Budapest, 2004, 99 p.	Filtering of tight specification is carried out by resonator-based filters. The thesis includes several measurement results.

Further publications on this topic can be found on this page.

Further information: László Sujbert