Why is the transition band of a least-square linear-phase FIR filter seems always monotonic

Asked Dec 21 '13 at 07:34

Active Dec 23 '13 at 10:14

Viewed 119 times

Given desired magnitude response and linear-phase constraint in predefined pass band and stop band, we can get the desired frequency response in both bands. By sampling the frequency in both bands, we can solve the optimal FIR filter coefficients in least-square sense. In the above discussion, we didn't give any constraints on the frequency response in transition band, but we always get a monotonic decreasing curve in transition band.

I think least-square FIR filter design is somewhat like trigonometric approximation. But in trigonometric approximation, if we have two groups of data points, the fitting curve will not always be monotonic in the region between two groups, as shown in figure below.

Example of trigonometric approximation

So why is it seems always true in filter design? Is there a mathematical proof about this?

edited Dec 23 '13 at 10:14

Martin Sleziak

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asked Dec 21 '13 at 07:34

Jeffrey

The ([tag:filters]) tag is for a different type of filters, see the tag-wiki. But I do not know what would be correct tags for this question. – Martin Sleziak Dec 22 '13 at 09:52
Thank you for your suggestion. – Jeffrey Dec 23 '13 at 05:27

Why is the transition band of a least-square linear-phase FIR filter seems always monotonic

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