The basic filter characteristics like low-pass, high-pass and band-pass have beein introduced earlier. In addition to these characteristics, different filter designs further specifiy the properties of the filter, respectively the transfer function. Different designs have their benifits and drawbacks and may be preferred in specific applications.
There is a general trade-off between the strength of ripples in the pass-band or stop-band and the steepness of the slope. The more effective a filter is, the more ripples it will introduce. This is similar for the phase. A filter with a gentle roll-off (the slope between pass-band and stop-band) will have less effect on the phase of the signal.
The following filter types originally stem from the analog domain. They are now commonly used to design digital filters.
Butterworth¶
The Butterworth design is popular and frequently used for several reasons:
- flat frequency response in the passband (maximally flat filter technique)
- decent roll-off
- smooth phase response
- simple to design (using bilinear transformation or impulse invariance)
A Butterworth filter with the cutoff frequency $\omega_c$ nad filter order $N$ has the following transfer function,:
$$ |H(jΩ)| = \frac{1} {1 + \sqrt{( \omega / \omega_c)^{2N}}} $$
Roll-Off¶
Butterworth filters have linear slope in the passband when visualized in the Bode plot. The steepness of the slope - the roll-off - depends on the order of the filter. Per order it is $-6 \mathrm{dB}$ per octave, respectively $-20 \mathrm{dB}$ per decade, resulting in:
- 1st order: -6dB
- 2nd order: -12dB
- 3rd order -18dB
- ...
Bessel¶
Bessel filters have one specific advantage over the other standard filters in this section:
- maximum linear phase response (maximum flat group delay)
Thie means they do not change the sinal's phase but only the amplitude spectrum. As a downside, the Bessel filter is the least effective filter type per order.
The plot below shows a Bessel low-pass filter with different orders.
Chebyshev¶
Chebyshev filters can have a steeper slope in the transition than the Butterworth design.
As a trade-off, this design shows ripples in the magnitude response:
- Chebyshev Type I: ripples in the pass-band
- Chebyshev Type II: ripples in the stop-band
The height of these ripples is a design parameter.
Elliptic (Cauer)¶
Elliptic filters have ripples in both pass-band and stop-band, leading to more distortions. They also cause significant phase distortions.
However, they can achieve the sharpest cutoff from the standard filter types introduced here.
When designing Elliptic filters, both the level of pass-band and stop-band ripples can be used as paramters.
