I know the difference between ideal and practical filterI just wanted to know why it not realisable?


Most typical examples of the ideal filters are the classical brickwall frequency selective filters (lowpass, highpass, bandpass etc) which are defined as ideal because of their impossible to realize ideally selective frequency responses.

Their frequency responses include exactly flat passbands, exactly flat stopbands having absolute zero gain, and zero width transition bands. These features make it impossible to realize them using either existing physical devices or finite time computable algorithms...

Assuming an LTI system model for such filters, then their impulse responses tend to be of infinite length and typically be acausal; another reason for denying their physical realization. Yet causality alone is only a concern for real-time processing of time domain signals. It does not prevent realization of such filters in offline procesing where the whole signal data with its past and future are already available. Yet even in such an offline processing mode the fact that they require an infinite length of data is the reason for preventing their realization.

Another point of view into their unrealizability is that input/output relationship of such ideal filters cannot be described by finite order differential / difference equations. In principle any filter whose input/output relation can not be described by a finite order differential / difference equation is said to be unrealizable; i.e., there is no finite time computable algorithm to implement it...

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  • $\begingroup$ Better answer, that prompted me to update $\endgroup$ – Laurent Duval Nov 10 '19 at 13:24
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    $\begingroup$ Thanks @LaurentDuval , indeed your answer is also inspiring ! $\endgroup$ – Fat32 Nov 10 '19 at 13:35

I have meet several definitions behind "being a realizable" filter. For instance, in Wikipedia Causal filter, there is:

Systems (including filters) that are realizable (i.e. that operate in real time) must be causal


In the context of physical systems, realizability is the property of having some way of implementing a mathematically specified system with physical components.

The most standard (IMO) for linear filter is that its impulse response is stable and causal.

Causal is relatively clear (see the addition below): it should use past and present signal values only, because a physical system cannot look into the future.

Stable often means "bounded input/bounded output" or BIBO stability. Because being of finite "power", a physical system should not be able to produce infinite values.

In some contexts, I have seen a narrower definition: realizability defined as causal and using only a finite number of operations. This includes causal linear FIR or some recursive IIR filters, but non-linear causal filters as well; a backward running median filter can be realizable.

In more extreme cases, some impose that the computations should be perform exactly, for instance using integers or dyadic rationals.

[ADDITION after the excellent Fat32 answer] A common way to build (same wikipedia source as above):

a realizable filter [is] by shortening and/or time-shifting a non-causal impulse response

but this is (to me) a slight distorsion.

An ideal linear filter, containing at least a segment of constant values (like zero on an interval) is generally non realizable.

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