Frequency-Domain Versus Time Domain Thinking

Thinking about analog filters, most engineers are
comfortable in the time domain. For example, the operation
of an RC lowpass filter can easily be envisioned as a
capacitor charging and discharging through a resistor.
Likewise, it is easy to envision how a negative-feedback
active filter uses phase shift as a function of frequency,
which is a time domain operation.
A digital filter is better conceptualized in the frequency
domain. The filter implementation simply performs a
convolution of the time domain impulse response and the
sampled signal. A filter is designed with a frequency domain
impulse response which is as close to the desired ideal
response as can be generated given the constraints of the
implementation. The frequency domain impulse response is
then transformed into a time domain impulse response
which is converted to the coefficients of the filter.