Digital filters can be more subtly specified than analogue filters, and so are specified in a different way:
Whereas analogue filters are specified in terms of their '3dB point'and their 'rolloff', digital filters are specified in terms of desired attenuation, and permitted deviations from the desired value in their frequency response:
The passband need not necessarily extend to the 3 dB point: for example, if passband ripple is specified as 0.1 dB, then the passband only extends to a point at which attenuation has increased to 0.1 dB.
Between the passband and the stopband lies a transition band where the filter's shape may be unspecified.
Note that the stopband attenuation is formally specified as the attenuation to the top of the first sidelobe of the filter's frequency response.
Digital filters can also have an 'arbitrary response': meaning, the attenuation is specified at certain chosen frequencies, or for certain frequency bands.
Digital filters are also characterised by their response to an impulse: a signal consisting of a single value followed by zeroes:
The impulse response is an indication of how long the filter takes to settle into a steady state: it is also an indication of the filter's stability - an impulse response that continues oscillating in the long term indicates the filter may be prone to instability.
The impulse response defines the filter just as well as does the frequency response.