KK Linear Phase Filter
A kind visitor fellow provided me with a huge table of "Equiripple Delay Time Linear Phase Low Pass Filter".
He also granted for disclosure of the table on this site.
I named it as KK Linear Phase Filter.
I discussed about Group Delay (GD hereafter) relative to higher order of "Symmetrical Equiripple Linear Phase Filter" in the page of ==>this (sorry, it's in Japanese). The comparison was made against same percentage of GD ripple. This fellow pointed out that the comparison should be made based on same GD time ripple.
I agreed with him. The comparison was unfair, e.g. high order filters had advantage.
The purpose of linear phase filter usage is to minimize pulse shape distortion. To that end, we must keep same delay time as higher frequency as possible. Not %. If we employ percentage (instead of time), the ripple amount will become larger when filter order goes higher.
The pictures below demonstrate comparisons among the 2nd order to the 8th order of KK Linear Phase Filters with 5 ms (note: time) ripple.
We should use this graph to select order number for filter design.
Previous comparison is shown just below of it. The ratios of upper(8th) and lower(2nd) limiting frequency are significantly different between 1% ripple and 5ms ripple..
The line of 7th in the upper picture is aligned at about 0.5sec GD and 5ms ripple, which is equivalent to 1%. So, the graph on the left is apple-to-apple comparison.
The excerpt data out of the provided table follows.
(Copied here by courtesy of "kmn")
- Note that the unit of frequency is Hz.
- This filter has Equiripple GD but not Symmetrical GD ripple.
[Note]
The reference book 1 uses ripple error in % for comparison.
The reference book 2 further refers 3, and shows table of "degree" error of phase (not GD).
The article of this page is the first one in the world that uses GD error of time.
The reference books 2 and 3 employs "degree", that may be confusing as the unit of GD is second. The picture on the right will clarify the confusion. "Linear Phase Filter" aims ideal linear line. The actual error is the difference from this line in degree. The period of ripple become shoter as frequency goes high. So, the GD ripple which is a derivative of phase, becomes slightly larger as it goes right.
［Reference Books］
1. 「伝送回路網およびフィルタ」、 電子通信学会／コロナ社、 矢崎銀作 武部幹、 No ISBN
2. Electronic Filter Design Handbook, Forth Edition, Arthur Williams and Fred Taylor,
McGrow-Hill, ISBN 007-147171-5
3. Handbook of Filter Synthesis, Anatol. I. Zverev, John Wily and Sons, 1967
(revised 2004, ISBN-13: 978-0471749424)