The specification of filter is usually given by the tolerance scheme.
The above example uses a discrete-time filter to process a continuous-time signal after periodic sampling. In practice, many applications may not use this approach.
The art of continuous time IIR design is highly advanced. Many useful continuous-time IIR designs have relatively simple closed-form formulas. The standard approximation methods working well for continuous time IIR do not lead to simple closed-form design formulas when they are applied to discrete-time IIRs.
The distortion in the frequency axis manifests itself as a warping of the phase response of the filter.
The most widely used classes of frequency- selective continuous-time filters are Butterworth, Chebyshev and elliptic filter designs. We expect the discrete Butterworth, Chebyshev and elliptic filters can retain the monotonicity and ripple characteristics of the corresponding continuous-time filters.
Butterworth lowpass filters are defined by the property that the magnitude response is the maximally flat in the passband and that the magnitude response is monotonic in the passband and stopband.