Why is it hard to build an ideal digital filter, e.g. an ideal lowpass band filter

From the mathematical point of view such a filter has an infinite spectrum in the time domain, i.e. it is not casual. What does it mean for software or hardware implementation? Consider two close frequencies one below cutoff frequency f and one above it, f-d and f+d respectively, where d is an infinitesimal and the filter must remove the second from its output. If a digital filter performs sampling of an input signal then for such close frequencies it would see the same values for an infinite time as a difference is below its resolution threshold defined by its internal ALU implementation, the filter will see one frequency

and it will take an infinite time as d approaches zero before both signals split up far enough so the filter recognises the difference, i.e. you must have an

From the mathematical point of view such a filter has an infinite spectrum in the time domain, i.e. it is not casual. What does it mean for software or hardware implementation? Consider two close frequencies one below cutoff frequency f and one above it, f-d and f+d respectively, where d is an infinitesimal and the filter must remove the second from its output. If a digital filter performs sampling of an input signal then for such close frequencies it would see the same values for an infinite time as a difference is below its resolution threshold defined by its internal ALU implementation, the filter will see one frequency

**infinite delay**in a filter.
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