*non-causal*filter. For example, see the following paper:

M. Unser, A. Aldroubi, and M. Eden,

B-spline Signal Processing: Part II---Efficient Design and Applications,

*Signal Processing, IEEE Transactions on*, vol.41, no.2, pp.834-848, Feb 1993

A non-causal filter as above is no problem in image processing. However, in a real time system, in particular, in a feedback loop, non-causality should cause a problem of implementation. The simplest way to avoid this is

*truncation*of the non-causal part of the filter impulse response, say, {

*f*(

*n*):

*n*<0}. This absolutely degrades the filter performance, and the second simplest is to truncate {

*f*(

*n*):

*n*<-

*N*} (often with a window) and shift the residual by

*N*.

Recently, much more sophisticated methods are proposed, via constraint least squares, which is related to H

^{2}optimization:

M. Unser and M. Eden,

FIR approximations of inverse filters and perfect reconstruction filter banks,

*Signal Processing*, Vol. 36, No. 2, pp. 163-174, 1994.

and via H

^{∞}optimization:

M. Nagahara and Y. Yamamoto,

H∞ optimal approximation for causal spline interpolation,

*Signal Processing*, Vol. 91, No. 2, pp. 176-184, 2011.

As I mentioned in a previous entry that H

^{∞}optimization leads to robustness against signal uncertainty. If you like robustness (or if you don't like risk-taking behavior), then you must choose H

^{∞}! You can use MATLAB codes for the H

^{∞}design here.

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