Digital filtering basics

You should be familiar with analog filters, which takes an analog input signal and output analog signal. Digital filters, with the sampling device and digital-to-analog converter, can do the same job as analog filters. Analog filters are specified by its impulse response $h(t)$ and the output signal $y(t)$ for input signal $x(t)$ is given by the convolution integral

$$y(t)=\int_0^t x(t-u)h(u)\,d t$$

Similarly, digital filters are specified by its impulse $h(n)$ and the output sequence $y(n)$ for the input sequence (samples of input analog signal) $x(n)$ by the discrete convolution

$$y(t)=\sum_{k=0}^{N-0} h(k)x(n-k)$$

The filter design problem is to design the impulse response $h(n)$ so that the digital filter, together with the sample and D/A converter, performs the same processing as the analog filters. There are many possible ways to pick $h(n)$ to implement filtering.