Linear Phase Analog Filters - getting one
what is offered
An expert service for the design ( approximation and realization ) of linear phase, or equiripple group delay, analog filters. These will have applications, in particular, as anti-aliasing filters in state-of-the art digital systems.recommendation: for a first-time visitor it is suggested that the content of this page be read straight through from beginning to end, ignoring the links. Later, use the links to jump about as the occasion warrants.
- the classical and preferred solutions.
- send me your specification
- my solution
- how much will it cost?
- hardware realization
- contact me by email
- return to the home page.
conventional wisdom
It is often said that the ideal lowpass filter for anti-alias applications would have a narrow transition bandwidth (‘sharp cutoff’) whilst preserving the shape of the signal being filtered.To obtain this characteristic it is claimed that a flat group delay , or perhaps a linear phase response, would be the ideal.
To find such a response amongst the classical filters is impossible!
- elliptic (Cauer) filters have narrow transition bands, but poor phase responses.
- Bessel and Butterworth filters have good phase responses, but their amplitude responses (for filters of moderate order) leave a lot to be desired.
- Chebyschev and inverse Chebyschev filters lie somewhere in between these two extremes.
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the 'classical' solution:
The first choice to realise a filter having both a narrow transition band and a ‘good’ phase response might be an elliptic filter cascaded with an allpass equalizer.This is a less than efficient solution.
The equalizer has to contend with a large spike in the group delay (within the passband). This spike is caused by the filter pole with the high Q. It requires a high order equalizer to 'swamp' this spike, which will otherwise remain (although of reduced relative magnitude) within the passband of the filter.
Even if the inevitable delay spike can be eliminated, the equiripple delay response of the allpass equalized filter seldom extends to the passband edge.
An additional disadvantage is that the narrow transition bandwidth of the elliptic filter is achieved with relatively high Q poles. These will increase the difficulties of realization (they introduce stability problems).
In practice the ‘solution’ to the problems introduced by the elliptic filter is to capitulate and to use a high-order Bessel or Butterworth response, or perhaps a lower order with added imaginary zeros to steepen the response at the passband edge (this technique does not disturb the group delay response. The software package OpFil can do this.
There is a preferred solution.
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a preferred solution
One is inclined to ask 'would it not be better to re-locate the poles and zeros of the combined filter and equalizer so that they work together to approximate the desired amplitude and phase, or group delay, responses, rather than independently'.The answer is definitely 'yes'!
This can be achieved by using a joint optimization technique. This technique optimizes the amplitude and phase (or group delay) response of the transfer function simultaneously, and results in:
- an equiripple group delay response, extending beyond the passband edge (there is no peak near the passband edge.)
- a lower order approximation for the same (or, generally, better) set of responses
- a significant reduction of the highest pole Q
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send me your specification
If you need such a filter, let me have the specification, and I will return you a set of responses meeting that specification within a day or so.your specification
There are many ways that a filter response can be specified. Over specifying will generally increase the order unnecessarily, and possibly result in higher-than-need-be pole Qs. This complicates the realization, and probably degrades its stability.What I need to know is:
- acceptable passband ripple in peak-to-peak dB.
- transition bandwidth
- stopband attenuation. The amplitude response will be equiripple beyond this point.
- acceptable group delay ripple magnitude in seconds (peak-to-peak) at the working frequency. If the ripple magnitude is not know, then omit it. The phase response will be essentially linear.
- passband corner frequency
The above high-lighted, and other relevant terms, are defined in the glossary . In particular you might like to refer to the specification mask for further clarification.
A favour: it would be appreciated if you could indicate the sort of filter you are using at the moment. That is, its order and other properties so that I can judge the relative improvement (if any) I can offer.
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the solution
The solution will be attached (as a .PDF file) to an email, as a set of response plots, with other details including system order, and the magnitude of the highest pole Q.cost ?
There will be no charge for the solution as described above, and no obligation for you to proceed following its receipt.If the responses are acceptable to you, and you would like to receive information regarding the disposition of the poles and zeros, then it will be necessary to negotiate a charge. This could be in the form of a once-only payment, or a royalty stream.
realization
Remember that the above solution to your problem, in filter theory terminology, is an approximation. This is a set of poles and zeros of transmission, which give rise to a transfer function approaching that of the specification.An electronic circuit can be produced, from a knowledge of the locations of the poles and zeros, whose transmission characteristic matches that of the approximation.
This realization needs to be in active form, since the approximation will contain right half plane zeros. These are difficult to realize (impossible?) in passive LC (inductor, capacitor) form.
If you are unable to convert the approximation to a realization, then I can take care of that too. This service is not free.