Design of Filters

Filters are essential parts for almost all technical applications. There is hardly any electric device which does not contain at least one filter.

Filterdesign in the Frequency Domain

Filters are useful to suppress unwanted parts of a signal. A lowpass filter only lets frequencies up to a certain frequency limit pass. A highpass filter suppresses all frequencies up to a certain limit and a band pass filter lets pass all frequencies within a band. A band reject (notch) filter suppresses all frequencies within a band. All these filters can be obtained from a lowpass prototype filter, which is therefore the filter which is usually designed and from which the other types easily follow. In practice it is only possible to approximate the behavior of an ideal lowpass filter. One accepts a certain non-ideal behavior. Deviations from the ideal behaviour are usually measured in decibel (dB), which is a logarithmic measure which is defined in terms of power ratios. A decibel is ten times the logarithm to base ten of the respective power ratios. As we usually consider voltage- (or current-)ratios this then leads to 20 times the logarithm of the voltages involved. For example an amplifier which amplifies an input voltage by the factor 1000 has an amplification of 60 dB (=> 20 lg 1000=60). Or a filter which suppresses unwanted frequencies by a factor of 100 attenuates them by 40 dB (=> 20 lg 0.01=-40).

Conversion from a ratio V to dB is done here: VdB

The complexity of a filter depends on the deviation to the ideal lowpass filter which an application can tolerate. This is usually measured with three parameters. One uses normalised (angle) frequencies. The pass band goes from zero to 1. In the pass band one accepts a maximal attenuation of αp dB. The stop band goes from ωs to infinity. In the stop band one demands at least an attenuation of α s dB. The specification can be fulfilled by certain filter classes described below. The complexity is given by the filter degree. The filter degree n is an indication of the hardware complexity to build the filter. In the simplest case a degree n=1 filter consist of a resistor and a capacitor.

The Lowpass Prototype Filter

The degrees of filters with common approximation methods can be determined here: Compute Degrees.

Hint: To understand the following some basic knowledge of Laplace transforms is necessary.

Allpole Filters

Allpole filters have a transfer function H(s) which only has poles and no zeros in the left half plane, i.e. H(s)=const/P(s) where P(s)=(s-p1)(s-p2)...(s-pn).

Butterworth Filters

Butterworth filters have a magnitude M(ω2)=H(j ω) H(-j ω) which is maximally flat. We have M(ω2)=1/(1+ε2 ω2n). The poles are located on a circle in the left half plane. For the computation of the poles of Butterworth filters click here.

Tschebyscheff Filters

Among allpole filters the Tschebyscheff filters are the filters which achieve the greatest steepness at unity (i.e. at the end of the passband). This follows from an extremal property of Tschebyscheff polynomials Tn(x) which are used. We have M(ω2)=1/(1+ε2 Tn(ω)2). The poles are located on an ellipse in the left half plane. For the computation of the poles of Tschebyscheff filters click here.

Novel Filters which perfectly match specifications

These filters cover the whole range between Butterworth and Tschebyscheff filters. If nB is the degree of a Butterworth filter and nT the degree of a Tschebyscheff filter which fulfills a certain specification the novel filters deliver allpole filters of degree n where nT ≤ n ≤ nB-1 which exactly meet the specification. As Tschebyscheff filters and Butterworth filters usually do not meet the specifications exactly, the novel filters in almost all cases lead to improved filters. Details can be found in K. Huber, 'All-Pole Filters Matched to Specifications', Journal of the Audio Engineering Society, Vol.61, No.12, pp.1022-1025, December 2013 (For the website of the Audio Engineering Society see here: AES). Some tables are given here: Novel Allpole Filters

Class-L Filters

These filters achieve the highest attenuation at the end of the passband under the constraint that the amplitude is monotonic. Due to this constraint the selectivity is less than the selectivity of Tschebyscheff filters but much better than the selectivity of Butterworth filters (this holds for n≥3, for n=2 the class-L filter is identical to the Butterworth filter). The phase linearity is better than the linearity of Tschebyscheff filters but not as good as the linearity of Butterworth filters. We have M(ω2)=1/(1+ε2 Ln(ω2)). For the parameters and some tables of poles of such filters see here.

Filters with Zeros in the Transfer Function

Inverse Tschebyscheff Filters ...(see at some later time)

Cauer Filters

Determination of the parameters of Cauer filters can be done here: Parameters of Cauer filters.
For the computation of the zeros and poles of Cauer filters click here.

Approximation Methods for a linear Phase

Allpole Filters

Bessel Filters

Bessel filters (also called Thomson filters) are allpole filters which are designed to approximate a linear phase. The poles of Bessel filters for degrees n=1,2,...,10 are given here.

Filters with Zeros in the Transfer Function

Novel Filters ...(see at some later time)

Low-degree Designs

Filters of degree n are often built by using low degree sections in series. If n is even one usually uses n/2 second order sections. For n odd one can use a first order section and (n-1)/2 second order sections or a third order section followed by (n-3)/2 second order sections (for odd n greater than three).

First and Second Order Circuits

A first order section which realises a real pole on the left half plane is essentially an RC term, (i.e. consisting of a resistor and a capacitor).
A very popular second order allpole circuit which realises a complex conjugate pole pair is the Sallen-Key circuit here. For the realisation of e.g. Cauer filters one also needs circuits which realise zeros of the transfer function on the imaginary axis. One such circuit is the Bainter circuit.

Third Order Allpole Circuits

Analog designs (active or passive) of degree 3 with explicit design formulas are given here: thirdorder (or at the external researchgate page: Scheme for improved realisation of third order filters).
To compute the circuit values from the poles click here.

Filterdesign in the Time Domain

For data transmission applications time domain approximations are used.

Allpole Filters

... (see at some later time)

Filters with Zeros in the Transfer Function

... (see at some later time)