This is the fifth in a series of blogs on the basic concepts of digital data acquisition. In the first blog we talked about how different data acquisition strategies give us different, but equally right (or wrong) results. In the second, we discussed sampling basics and \aliasing and the irrecoverable errors they cause, and in the third we got started on the discussion of what filtering and sample-rate strategies are necessary to reduce aliasing errors to “acceptable” levels. The fourth, covered the pros and cons of the filter types for that we might use for alias protection. Here, and in the next one, we will discuss the hardware concepts needed to accomplish these objectives.
We are going to discuss the strategies used to design useful analog low-pass filters from basic building blocks. We are not going to stray very far into the unknown so we won’t have to worry about stability and such. I hope that won’t offend our more-mathematically-inclined readers.
As soon as Shannon and his co-researchers discovered the basic concepts and problems associated with digital data measurements, solutions to the aliasing problem appeared. Analog low pass filters were developed that attenuated the energy above the Nyquist Frequency (FN) and consequently reduced aliasing. Later, when computing power increased, digital filters joined the fray.
So, let’s get started with analog filters… We will do the hybrid/digital (SD) flavor next time.
Figure 1 shows the operational-amplifier circuit diagram and transfer function of a one-pole filter. Its features/characteristics are:
This is not sharp enough to be useful as an anti-alias filter, Maybe we could put a bunch of them in series to make a multi-pole filter. The result of stringing 4 of them in series (which results in the multiplying of the transfer functions) is shown in Figure 2.
At high frequency, it rolls of 4 times a fast (good) but its attenuation (amplitude distortion) in the pass band (below FC) is pretty bad. We can do better.
Let's take a break here to ask you to join the discussion.
With any luck, I have provoked some questions and/or disagreements in these blogs.
For instance, the first blog generated a comment/question about my statement that I dislike Bessel filters. I think that answered that in the previous entry. Do you agree?
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Fortunately there is a better tool. Figure 3 shows the basic circuit and transfer functions of several two-pole filters with different damping.
The critical features are:
If we put several of these circuits in series we can build a multi-pole filter. For example if we were to combine the four 2-pole circuits in Figure 3 to make an 8-pole filter, we would produce the transfer function in Figure 4. Its attenuation rate at high frequency is -8 x 6 = -48dB/Octave and it is pretty flat in the passband (<FC). Not a bad low-pass filter but we can do better.
That’s the basic idea. Adding more (appropriately configured) poles (and zeroes, as we will see shortly) will produce a sharper filter. Figure 5 shows the effect of adding more poles.
Each pole increases the roll-off slope by 6dB/octave and provides more high-frequency attenuation.
It’s as simple as that, or is it?
How do we design a filter that meets our needs? Fortunately, smart people have been here ahead of us.
In 1930 Stephen Butterworth designed a family of filter sets named for him and in 1949 W. E. Thompson applied Bessel functions to manipulate the filter coefficients to produce a "Bessel" filter set with different desirable characteristics. These have become standards in the industry.
The designs of 8-pole Bessel and Butterworth filters are shown in Figure 6.
The left frame shows the individual transfer functions of the four two-pole stages. The frequencies and Qs of the stages are shown in the lower left corner. The right frame shows the end result: the product of the individual stages that is produced by putting the four circuits in series. The inset in the lower left shows an expanded view of the filter pass band.
Comparison of the two strategies shows:
These filters represent the useable state of the art until about 1975.
In the late 70’s a new kind of filter surfaced that included “Zeroes”. A zero may be created by inverting the zero-damping, two-pole, response to produce the transfer function shown in Figure 7.
Its fundamental characteristics are:
Here we need to handle some strange nomenclature. It takes a two-pole circuit to make this filter shape. So it counted as two zeroes!
To make a filter, zeroes need to be combined with pole pairs to reduce the gain at high frequency. A useful filter must have and equal, or greater, number of poles than zeroes.
Filters that have zeroes are called Elliptical.
Figure 9 shows the characteristics of an 8 pole/8 zero elliptical filter that is optimized for alias rejection characteristics. It was offered by Tustin Electronics (R.I.P.) in the mid 1980’s.
Its fundamental characteristics are:
Note that I have not listed the attenuation rate of the filter because it does not mean much. The critical feature is the frequency ratio at which the attenuation is adequate.
The Tustin filter has remarkable characteristics that are not likely to be found in modern systems. The critical feature is the Pole Pair with a Q>10. This is very hard to achieve for practical reasons. These systems were built by elves with tiny screwdrivers.
This filter is an example of using zeroes to provide “optimum” alias-rejection performance. The use of zeroes and poles also offers options to optimize other characteristics. Examples can be found here.
An in-depth analysis of the performance and usefulness of all of these filters can be found in the previous blog.
The next blog will cover the next generation of alias-protection systems: Hybrid Oversampling and Sigma Delta (SD) filters.
The following web pointers will give you more information and, in some cases, a different view of the low-pass filtering operation.
https://en.wikipedia.org/wiki/Bessel_filter
https://en.wikipedia.org/wiki/Butterworth_filter
https://en.wikipedia.org/wiki/Elliptic_filter
http://www.analog.com/media/en/training-seminars/design-handbooks/Basic-Linear-Design/Chapter8.pdf
http://www.electronics-tutorials.ws/filter/second-order-filters.html
http://www.pfinc.com/paper_briefs/anti_alias_brief.pdf
http://www.freqdev.com/guide/analog.html
https://www.maximintegrated.com/en/app-notes/index.mvp/id/733
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